Bruhat–Tits Theory: A New Approach 1108831966, 9781108831963

Table of contents :
Contents
Tables
Introduction
Goals
A Brief Overview of the Theory
Our Approach
A Summary of Each Chapter
On the Logical Structure of the Exposition
Different Paths through the Book
Acknowledgements
PART ONE: BACKGROUND AND REVIEW
1. Affine Root Systems and Abstract Buildings
1.1 Metric Spaces
1.2 Affine Spaces
1.3 Affine Root Systems
1.4 Tits Systems
1.5 Abstract Buildings
1.6 The Monoid ~R
2. Algebraic Groups
2.1 Henselian Fields
2.2 Bounded Subgroups of Reductive Groups
2.3 Fields of Dimension ≼ 1
2.4 Affine Group Schemes over Perfect Fields
(a) Homomorphisms and Kernels
(b) Unipotent Groups
(c) The Reductive Quotient and Conjugacy Results
2.5 Tori
(a) Induced Tori
(b) The Valuation Homomorphism
(c) The Maximal Bounded Subgroup
(d) The Iwahori Subgroup
(e) The Lie Algebra
2.6 Reductive Groups
(a) Basic Notation
(b) The Absolute and Relative Root Datum
(c) The Relative Root System of a Tits Index
(d) The Valuation Homomorphism and the Subgroup G(k)¹
(e) The Subgroups G(k)♯ and G(k)⁰
2.7 The Group SU₃
2.8 Separable Quadratic Extensions of Local Fields
2.9 Chevalley Systems
(a) Pinnings
(b) The Split Case
(c) The Quasi-split Case
(d) Commutation Relations
(e) Simply Connected Cover
2.10 Integral Models
2.11 Group Scheme Actions and the Dynamic Method
PART TWO: BRUHAT–TITS THEORY
3. Examples: Quasi-split Simple Groups of Rank 1
3.1 The Example of SL₂
(a) The Standard Apartment
(b) The Affine Roots
(c) The affineWeyl group
(d) The Building
(e) The Moy–Prasad Filtration Subgroups
(f) The Apartment as an Affine Space
3.2 The Example of SU₃
(a) The Filtration of the Maximal Torus
(b) The Filtrations of the Root Subgroups
(c) The Standard Apartment and the Affine Roots
(d) The AffineWeyl Group
(e) Unshifted Filtrations
(f) The Building
4. Overview and Summary of Bruhat–Tits Theory
4.1 Axiomatization of Bruhat–Tits Theory
4.2 Metric
4.3 The Enlarged Building
4.4 Uniqueness of the Apartment and the Building
5. Bruhat, Cartan, and Iwasawa Decompositions
5.1 The (affine) Bruhat Decomposition
5.2 The Cartan Decomposition
5.3 The Iwasawa Decomposition
5.4 The Intersection of Cartan and Iwasawa Double Cosets
6. The Apartment
6.1 The Apartment of a Quasi-split Reductive Group
(a) Split Semi-simple Groups
(b) Quasi-split Semi-simple Groups
(c) Quasi-split Reductive Groups
6.2 Affine Reflections and Uniqueness of Valuations
6.3 Affine Roots and Affine Root Groups
6.4 The Affine Root System of a Quasi-split Group
(a) Split Groups
(b) Quasi-split Groups
6.5 Change of Valuation
6.6 The AffineWeyl Group
6.7 Projection to a Levi Subgroup
7. The Bruhat–Tits Building for a Valuation of the Root Datum
7.1 Commutator Computations
7.2 A Filtration of Z(k)
7.3 Concave Functions
7.4 Parahoric Subgroups
7.5 The Iwahori–Tits System
7.6 The (Reduced) Building
7.7 Disconnected Parahoric Subgroups
7.8 The Iwahori–Weyl Group
7.9 Change of Base Field and Automorphisms
(a) Change of Base Field
(b) Automorphisms
7.10 Passage to Completion
7.11 Absolutely Special Points
8. Integral Models
8.1 Preliminaries
8.2 General Properties of Smooth Models of G
8.3 Parahoric Integral Models
8.4 The Structure of the Special Fiber of G⁰_Ω
8.5 Integral Models Associated to Concave Functions
8.6 Passage to Completion
9. Unramified Descent
9.1 Preliminaries
9.2 Statement of the Main Result
9.3 The Building and its Apartments
9.4 The Affine Root System
9.5 Completion of the Proof of the Main Result
9.6 Valuation of Root Datum
9.7 Levi Subgroups
9.8 Concave Function Groups
9.9 Special, Superspecial, and Hyperspecial Points
9.10 Residually Split and Residually Quasi-split Groups
9.11 Restriction of Scalars
PART THREE: ADDITIONAL DEVELOPMENTS
10 Residue Field f of Dimension ≼ 1
10.1 Conjugacy of Special Tori
10.2 Superspecial Points
10.3 Anisotropic Groups
10.4 Fixed Points of Large Subgroups of Tori
10.5 Existence of Anisotropic Tori
10.6 Cohomological Results
10.7 Classification of Connected Reductive k-Groups
(a) (¹A_{n−1}, ¹A_{n−1})
(b) (¹A_{n−1}, ²A_{n−1})
(c) (²Aₙ, ²Aₙ)
(d) (Bₙ, Bₙ), n ≽ 3
(e) (Cₙ, Cₙ), n ≽ 2
(f) (¹Dₙ, ¹Dₙ), n > 4
(g) (¹Dₙ, ²Dₙ), n > 4
(h) (²Dₙ, ²Dₙ), n > 4
(i) D₄
(j) (¹E₆, ¹E₆)
(k) (¹E₆, ²E₆)
(l) E₇
11. Component Groups of Integral Models
11.1 The Kottwitz Homomorphism for Tori
11.2 The Component Groups of T^{ft} and T^{lft}
11.3 The Algebraic Fundamental Group
11.4 z-Extensions
11.5 The Kottwitz Homomorphism for Reductive Groups
11.6 The Component Groups of Parahoric Integral Models
11.7 The Case of dim(f) ≼ 1
12. Finite Group Actions and Tamely Ramified Descent
12.1 Preliminaries
12.2 Certain Group Schemes Associated to H and G
12.3 A Reduction
12.4 Apartments of B
12.5 The Polyhedral Structure on B
12.6 Identification of Parahoric Subgroups
12.7 The Main Theorem
12.8 The Case of a Finite Cyclic Group
12.9 Tamely Ramified Descent
13. Moy–Prasad Filtrations
13.1 Filtrations of Tori
13.2 Filtrations of Parahoric Subgroups
13.3 Filtrations of the Lie Algebra and its Dual
13.4 Optimal Points
13.5 The Moy–Prasad Isomorphism
13.6 Semi-stability
13.7 G-Domains in the Lie Algebra g
13.8 Vanishing of Cohomology
14. Functorial Properties
14.1 Quotient Maps
14.2 Embeddings: Isometric Properties
14.3 Embeddings: Factorization through a Levi Subgroup
14.4 Embeddings of Apartments
14.5 Adapted Points: Definition and Properties
14.6 Embeddings of Buildings via Adapted Points
14.7 The Space of Embeddings and Galois Descent
14.8 Existence of Adapted Points
14.9 Uniqueness of Admissible Embeddings
15. The Buildings of Classical Groups via Lattice Chains
15.1 The Special and General Linear Groups
15.2 Symplectic, Orthogonal, and Unitary groups
PART FOUR: APPLICATIONS
16. Classification of Maximal Unramified Tori (d’après DeBacker)
17. Classification of Tamely Ramified Maximal Tori
18. The Volume Formula
18.1 Remarks on Arithmetic Subgroups
18.2 Notations, Conventions and Preliminaries
18.3 Tamagawa Forms on Quasi-split Groups
18.4 Volumes of Parahoric Subgroups
18.5 Covolumes of Principal S-Arithmetic Subgroups
18.6 Euler–Poincaré characteristic of S-arithmetic subgroups.
18.7 Bounds for the Class Number of Simply Connected Groups
18.8 The Discriminant Quotient Formula for Global Fields
PART FIVE: APPENDICES
A. Operations on Integral Models
A.1 Base Change
A.2 Schematic Closure
A.3 Weil Restriction of Scalars
A.4 The Greenberg Functor
(a) Review of Witt Vectors
(b) Some Module Schemes and Ring Schemes
(c) Definition and Properties of the Greenberg Functor
(d) Beyond the Affine Case
(e) Applications
A.5 Dilatation
A.6 Smoothening
A.7 Schematic Subgroups
A.8 Reductive Models
B. Integral Models of Tori
B.1 Preliminaries
B.2 Split Tori
B.3 Induced Tori
B.4 The Standard Model
B.5 The Standard Filtration
B.6 Weakly induced tori
B.7 The ft-Néron Model
B.8 The Néron Mapping Properties and the lft-Néron Model
B.9 The pro-unipotent radical
B.10 The Minimal Congruent Filtration
C. Integral Models of Root Groups
C.1 Introduction
C.2 Integral Models for Filtration Subgroups of Ga
C.3 Integral Models for Filtration Subgroups of R⁰_{L/K} Ga
C.4 Integral Models for Filtration Subgroups of U_{L/K}
C.5 Summary
References
AB
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HJKLM
NOP
RS
TVWYZ
Index of Symbols
General Index
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Citation preview

Bruhat–Tits Theory Bruhat–Tits theory is an important topic in number theory, representation theory, harmonic analysis, and algebraic geometry. This book gives the first comprehensive treatment of this theory over discretely valued Henselian fields. It can serve both as a reference for researchers in the field and as a thorough introduction for graduate students and early career mathematicians. Part I of the book gives a review of the relevant background material, touching upon Lie theory, metric geometry, algebraic groups, and integral models. Part II gives a complete, detailed, and motivated treatment of the core theory. For more experienced readers looking to learn the essentials for use in their own work, there is also an axiomatic summary of Bruhat–Tits theory that suffices for the main applications. Part III treats modern topics that have become important in current research. Part IV provides a few sample applications of the theory. The appendices contain further details on the topic of integral models, including a detailed study of the integral models of tori. Ta s h o K a l e t h a is Professor of Mathematics at the University of Michigan. He is an expert on the Langlands program, and has studied arithmetic and representationtheoretic aspects of the local Langlands correspondence for p-adic groups. G o pa l P r a s a d is Raoul Bott Professor Emeritus of Mathematics at the University of Michigan. He is a leading expert on real and p-adic Lie groups and algebraic groups. Together with Ofer Gabber and Brian Conrad, he published the complete classification and structure theory of pseudo-reductive groups in the books Pseudo-reductive Groups (2010, 2015) and Classification of Pseudo-reductive Groups (2015).

N E W M AT H E M AT I C A L M O N O G R A P H S Editorial Board Jean Bertoin, B´ela Bollob´as, William Fulton, Bryna Kra, Ieke Moerdijk, Cheryl Praeger, Peter Sarnak, Barry Simon, Burt Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit www.cambridge.org/mathematics. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

S. Berhanu, P. D. Cordaro and J. Hounie An Introduction to Involutive Structures A. Shlapentokh Hilbert’s Tenth Problem G. Michler Theory of Finite Simple Groups I A. Baker and G. W¨ustholz Logarithmic Forms and Diophantine Geometry P. Kronheimer and T. Mrowka Monopoles and Three-Manifolds B. Bekka, P. de la Harpe and A. Valette Kazhdan’s Property (T) J. Neisendorfer Algebraic Methods in Unstable Homotopy Theory M. Grandis Directed Algebraic Topology G. Michler Theory of Finite Simple Groups II R. Schertz Complex Multiplication S. Bloch Lectures on Algebraic Cycles (2nd Edition) B. Conrad, O. Gabber and G. Prasad Pseudo-reductive Groups T. Downarowicz Entropy in Dynamical Systems C. Simpson Homotopy Theory of Higher Categories E. Fricain and J. Mashreghi The Theory of H(b) Spaces I E. Fricain and J. Mashreghi The Theory of H(b) Spaces II J. Goubault-Larrecq Non-Hausdorff Topology and Domain Theory ´ J. Sniatycki Differential Geometry of Singular Spaces and Reduction of Symmetry E. Riehl Categorical Homotopy Theory B. A. Munson and I. Voli´c Cubical Homotopy Theory B. Conrad, O. Gabber and G. Prasad Pseudo-reductive Groups (2nd Edition) J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson Sobolev Spaces on Metric Measure Spaces Y.-G. Oh Symplectic Topology and Floer Homology I Y.-G. Oh Symplectic Topology and Floer Homology II A. Bobrowski Convergence of One-Parameter Operator Semigroups K. Costello and O. Gwilliam Factorization Algebras in Quantum Field Theory I J.-H. Evertse and K. Gy˜ory Discriminant Equations in Diophantine Number Theory G. Friedman Singular Intersection Homology S. Schwede Global Homotopy Theory M. Dickmann, N. Schwartz and M. Tressl Spectral Spaces A. Baernstein II Symmetrization in Analysis A. Defant, D. Garc´ıa, M. Maestre and P. Sevilla-Peris Dirichlet Series and Holomorphic Functions in High Dimensions N. Th. Varopoulos Potential Theory and Geometry on Lie Groups D. Arnal and B. Currey Representations of Solvable Lie Groups M. A. Hill, M. J. Hopkins and D. C. Ravenel Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem K. Costello and O. Gwilliam Factorization Algebras in Quantum Field Theory II S. Kumar Conformal Blocks, Generalized Theta Functions and the Verlinde Formula P. F. X. M¨uller Hardy Martingales T. Kaletha and G. Prasad Bruhat–Tits Theory J. Schwermer Reduction Theory and Arithmetic Groups

Bruhat–Tits Theory A New Approach

TA S H O K A L E T H A University of Michigan, Ann Arbor G O PA L P R A S A D University of Michigan, Ann Arbor

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 103 Penang Road, #05–06/07, Visioncrest Commercial, Singapore 238467 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108831963 DOI: 10.1017/9781108933049 © Tasho Kaletha and Gopal Prasad 2023 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2023 A catalogue record for this publication is available from the British Library. ISBN 978-1-108-83196-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

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Contents

List of illustrations List of tables Introduction

PART ONE

page xii xiii xv

BACKGROUND AND REVIEW

1

Affine Root Systems and Abstract Buildings 1.1 Metric Spaces 1.2 Affine Spaces 1.3 Affine Root Systems 1.4 Tits Systems 1.5 Abstract Buildings  1.6 The Monoid R

2

Algebraic Groups 2.1 Henselian Fields 2.2 Bounded Subgroups of Reductive Groups 2.3 Fields of Dimension  1 2.4 Affine Group Schemes over Perfect Fields 2.5 Tori 2.6 Reductive Groups 2.7 The Group SU3 2.8 Separable Quadratic Extensions of Local Fields 2.9 Chevalley Systems 2.10 Integral Models 2.11 Group Scheme Actions and the Dynamic Method

vii

1 3 3 8 14 47 54 66 68 68 70 78 81 86 92 108 110 112 121 130

viii

Contents PART TWO

BRUHAT–TITS THEORY

139

3

Examples: Quasi-split Simple Groups of Rank 1 3.1 The Example of SL2 3.2 The Example of SU3

141 141 147

4

Overview and Summary of Bruhat–Tits Theory 4.1 Axiomatization of Bruhat–Tits Theory 4.2 Metric 4.3 The Enlarged Building 4.4 Uniqueness of the Apartment and the Building

157 157 168 178 180

5

Bruhat, Cartan, and Iwasawa Decompositions 5.1 The (affine) Bruhat Decomposition 5.2 The Cartan Decomposition 5.3 The Iwasawa Decomposition 5.4 The Intersection of Cartan and Iwasawa Double Cosets

186 186 187 188 191

6

The Apartment 6.1 The Apartment of a Quasi-split Reductive Group 6.2 Affine Reflections and Uniqueness of Valuations 6.3 Affine Roots and Affine Root Groups 6.4 The Affine Root System of a Quasi-split Group 6.5 Change of Valuation 6.6 The Affine Weyl Group 6.7 Projection to a Levi Subgroup

195 196 212 217 223 229 230 233

7

The Bruhat–Tits Building for a Valuation of the Root Datum 7.1 Commutator Computations 7.2 A Filtration of Z(k) 7.3 Concave Functions 7.4 Parahoric Subgroups 7.5 The Iwahori–Tits System 7.6 The (Reduced) Building 7.7 Disconnected Parahoric Subgroups 7.8 The Iwahori–Weyl Group 7.9 Change of Base Field and Automorphisms 7.10 Passage to Completion 7.11 Absolutely Special Points

235 236 245 246 255 259 263 267 272 273 276 278

8

Integral Models 8.1 Preliminaries 8.2 General Properties of Smooth Models of G 8.3 Parahoric Integral Models

283 283 286 295

Contents 8.4 8.5 8.6 9

The Structure of the Special Fiber of GΩ0 Integral Models Associated to Concave Functions Passage to Completion

ix 302 310 321

Unramified Descent 9.1 Preliminaries 9.2 Statement of the Main Result 9.3 The Building and its Apartments 9.4 The Affine Root System 9.5 Completion of the Proof of the Main Result 9.6 Valuation of Root Datum 9.7 Levi Subgroups 9.8 Concave Function Groups 9.9 Special, Superspecial, and Hyperspecial Points 9.10 Residually Split and Residually Quasi-split Groups 9.11 Restriction of Scalars

323 324 325 330 345 363 364 368 374 379 381 385

PART THREE

387

ADDITIONAL DEVELOPMENTS

10

Residue Field f of Dimension  1 10.1 Conjugacy of Special Tori 10.2 Superspecial Points 10.3 Anisotropic Groups 10.4 Fixed Points of Large Subgroups of Tori 10.5 Existence of Anisotropic Tori 10.6 Cohomological Results 10.7 Classification of Connected Reductive k-Groups

389 389 391 391 393 395 397 401

11

Component Groups of Integral Models 11.1 The Kottwitz Homomorphism for Tori 11.2 The Component Groups of T ft and T lft 11.3 The Algebraic Fundamental Group 11.4 z-Extensions 11.5 The Kottwitz Homomorphism for Reductive Groups 11.6 The Component Groups of Parahoric Integral Models 11.7 The Case of dim(f)  1

420 421 426 427 428 429 432 433

12

Finite Group Actions and Tamely Ramified Descent 12.1 Preliminaries 12.2 Certain Group Schemes Associated to H and G 12.3 A Reduction 12.4 Apartments of B

436 437 439 443 445

x

Contents 12.5 12.6 12.7 12.8 12.9

The Polyhedral Structure on B Identification of Parahoric Subgroups The Main Theorem The Case of a Finite Cyclic Group Tamely Ramified Descent

449 456 457 459 463

13

Moy–Prasad Filtrations 13.1 Filtrations of Tori 13.2 Filtrations of Parahoric Subgroups 13.3 Filtrations of the Lie Algebra and its Dual 13.4 Optimal Points 13.5 The Moy–Prasad Isomorphism 13.6 Semi-stability 13.7 G-Domains in the Lie Algebra g 13.8 Vanishing of Cohomology

466 466 468 470 471 474 478 481 485

14

Functorial Properties 14.1 Quotient Maps 14.2 Embeddings: Isometric Properties 14.3 Embeddings: Factorization through a Levi Subgroup 14.4 Embeddings of Apartments 14.5 Adapted Points: Definition and Properties 14.6 Embeddings of Buildings via Adapted Points 14.7 The Space of Embeddings and Galois Descent 14.8 Existence of Adapted Points 14.9 Uniqueness of Admissible Embeddings

490 493 494 496 497 500 502 504 505 509

15

The Buildings of Classical Groups via Lattice Chains 15.1 The Special and General Linear Groups 15.2 Symplectic, Orthogonal, and Unitary groups

513 513 524

PART FOUR

545

APPLICATIONS

16

Classification of Maximal Unramified Tori (d’après DeBacker)

547

17

Classification of Tamely Ramified Maximal Tori

553

18

The Volume Formula 18.1 Remarks on Arithmetic Subgroups 18.2 Notations, Conventions and Preliminaries 18.3 Tamagawa Forms on Quasi-split Groups 18.4 Volumes of Parahoric Subgroups 18.5 Covolumes of Principal S-Arithmetic Subgroups

558 559 564 566 573 578

Contents 18.6 18.7 18.8

Euler–Poincaré characteristic of S-arithmetic subgroups. Bounds for the Class Number of Simply Connected Groups The Discriminant Quotient Formula for Global Fields

PART FIVE

APPENDICES

xi 584 585 587

591

A

Operations on Integral Models A.1 Base Change A.2 Schematic Closure A.3 Weil Restriction of Scalars A.4 The Greenberg Functor A.5 Dilatation A.6 Smoothening A.7 Schematic Subgroups A.8 Reductive Models

593 593 593 595 603 632 645 647 649

B

Integral Models of Tori B.1 Preliminaries B.2 Split Tori B.3 Induced Tori B.4 The Standard Model B.5 The Standard Filtration B.6 Weakly induced tori B.7 The ft-Néron Model B.8 The Néron Mapping Properties and the lft-Néron Model B.9 The pro-unipotent radical B.10 The Minimal Congruent Filtration

653 653 654 656 659 667 668 675 680 684 686

C

Integral Models of Root Groups C.1 Introduction C.2 Integral Models for Filtration Subgroups of Ga C.3 Integral Models for Filtration Subgroups of R0L/K Ga C.4 Integral Models for Filtration Subgroups of UL/K C.5 Summary References Index of Symbols General Index

696 696 697 698 699 703 708 715 717

Illustrations

1.3.1 The hyperplane arrangements of the irreducible reduced affine root systems A2 (top), C2 (middle), and G2 (bottom). 1.3.2 The hyperplane arrangement of the irreducible reduced affine root system BC2 . 1.5.1 The 3-regular tree. 1.5.2 The retraction of a 3-regular tree. 7.11.1 The apartment of SU3 (k) as a subset of the apartment of SL3 ().

xii

20 21 59 64 282

Tables

1.3.3 The affine Dynkin diagrams of the non-reduced irreducible affine root systems 1.3.4 The affine Dynkin diagrams of the reduced irreducible affine root systems 1.3.5 The integers (1.3.1) for the reduced irreducible affine root systems 2.6.1 The coefficients for the highest root in the irreducible root systems 6.4.1 The affine root systems Ψ and Ψ  for an absolutely simple quasisplit group G

xiii

41 42 43 102 227

Introduction

Goals The purpose of this book is to describe the affine building associated to a connected reductive group defined over a discretely valued Henselian field and the associated structures such as the affine root system and the parahoric group schemes. Much of this theory was developed in the 1960s, 1970s, and 1980s, by Iwahori–Matsumoto, Hijikata, and in its ultimate form by Bruhat–Tits. In the ensuing years the theory has had profound applications to various branches of mathematics, most prominently in the representation theory of, and harmonic analysis on, reductive groups over non-archimedean local fields, but also in the study of arithmetic groups, and even to questions of algebraic geometry, such as the classification of fake projective planes. Our exposition is intended to be approachable by the non-expert. With this in mind, we have decided to restrict generality and consider only discrete valuations and demand that the residue field is perfect in a large number of places. In this case the main ideas come forward most clearly, and the combinatorics of chambers and facets, which underlies many arguments in the applications of the theory, takes a prominent role.

A Brief Overview of the Theory Let us give a brief summary of the outcome of the theory. Given a connected reductive group G defined over a discretely valued Henselian field k with perfect residue field, the theory produces a contractible topological space B(G) called the (reduced) Bruhat–Tits building of G. This topological space has the structure of a polysimplicial complex (cf. Definition 1.5.1) and the topological group G(k) acts on B(G) by automorphisms that preserve the polysimplicial xv

xvi

Introduction

structure. The polysimplices are customarily called facets, while those of maximal dimension are called chambers. The chambers form a single G(k)-orbit. The topological space B(G) has a metric invariant under G(k) with respect to which it is complete. There is a unique geodesic curve connecting any two points. The metric has a non-positive curvature property which implies that any bounded convex subset has a unique center of mass, which is then preserved by any group of isometries that leaves invariant the bounded convex subset; this is called the Bruhat–Tits fixed point theorem. The space B(G) comes equipped with a family of subspaces, called apartments, which is in G(k)-equivariant bijective correspondence with the family of maximal k-split tori in G. Each apartment is a polysimplicial subcomplex of B(G) that is isometric to the affine space Rn , where n is the k-rank of the derived subgroup of G. In particular, as any two maximal k-split tori in G are conjugate under G(k), any two apartments are conjugate under G(k). Any two chambers are contained in an apartment. The dimension of a chamber is equal to n. The polysimplicial structure on each apartment A is induced by a family of hyperplanes, in the sense that the chambers of A are the connected components of the complement of the union of these hyperplanes, and the lower-dimensional facets are obtained analogously. The hyperplanes in turn are defined as the vanishing loci of a set of affine functionals on A that form an affine root system Ψ, whose derivative root system Φ is the root system of the reductive group G relative to the maximal k-split torus S. The main purpose of the building is to organize the various open bounded (i.e. open compact when k is locally compact) subgroups of G(k) that play a role in its structure and representation theory. To each point x ∈ B(G) are associated various subgroups of G(k). The most obvious of them is the stabilizer G(k)x , an open subgroup of G(k) that is bounded modulo ZG (k). When ZG (k) is bounded, which is the case precisely when the maximal k-split central torus AG of G is trivial, then G(k)x is bounded. In general only G(k)x /AG (k) is bounded and there are two, essentially equivalent, ways to obtain from x a bounded subgroup of G(k)x . One is to form G(k)1x = G(k)1 ∩ G(k)x , where G(k)1 = {g ∈ G(k) | ω( χ(g)) = 0 for all χ ∈ X∗ (G)}. Here ω : k → R ∪ {∞} denotes the non-archimedean valuation on k, as well as its extension to any algebraic field extension of k. It is easy to see that G(k)1 contains every bounded subgroup of G(k). An important part of Bruhat–Tits theory is that G(k)1x , the stabilizer of x for the action of G(k)1 on B(G), is a bounded subgroup. One can show (cf. Proposition 4.2.13) that a subgroup G of G(k)1 is bounded if and only if its action on B(G) is bounded, in the sense that the orbit under G of every point of B(G) is bounded with respect to the

Introduction

xvii

metric on B(G). Therefore, the Bruhat–Tits fixed point theorem implies that any bounded subgroup of G(k) fixes a point in B(G). In this way, one shows that every maximal bounded subgroup of G(k) is of the form G(k)1x for some x ∈ B(G). Note that the converse is not true – the points x ∈ B(G) for which the bounded subgroup G(k)1x is maximal are rather special. The second way to obtain from x a bounded subgroup of G(k) is to introduce a slight modification of B(G), called the enlarged building, defined by  B(G) = B(G) × (R ⊗Z X∗ (AG )).   The group G(k) acts on B(G) and the stabilizer of each point  x ∈ B(G) is a bounded and open subgroup G(k)x of G(k). In fact, the group G(k) acts by translations on the vector space X∗ (AG ) ⊗Z R, the kernel of this action x under the is G(k)1 , and G(k)x = G(k)1x , where x ∈ B(G) is the image of  first projection map. In other words, working with the extended building is equivalent to working with the reduced building and the subgroup G(k)1 . One can also endow the extended building with a metric by taking the product of the metric on B(G) with the Euclidean metric on X∗ (AG ) ⊗Z R. It is worth noting some parallels between the Bruhat–Tits building of a reductive group over k and the symmetric space of a reductive group over R. If H is a connected reductive group defined over R, then the Lie group H(R) has a maximal compact subgroup K. Then X = H(R)/K is called the symmetric space of H. It is a complete Riemannian manifold with non-positive curvature. As such, between any two points there is a unique geodesic line. Each maximal R-split torus in H corresponds to a maximal flat submanifold, analogous to the apartment of the Bruhat–Tits building. The analog of the Bruhat–Tits fixed point theorem is the Cartan fixed point theorem, stating that a compact group acting isometrically on a non-positively curved complete simply connected Riemannian manifold has a fixed point, cf. [Ebe96, Theorem 1.4.6]. This implies that the maximal compact subgroups of H(R) are precisely the stabilizers of the points of X. In particular, they are all conjugate to K, with x ∈ H(R)/K corresponding to xK x −1 . Note here that the analogy with the non-archimedean case already breaks down, since the set of points x for which the subgroup G(k)1x is a maximal bounded subgroup is a rather small subset of the set of points of B(G), and even in this small subset there are usually multiple (although always finitely many) G(k)-orbits. Let us now return to the non-archimedean case, where the structure of the field k – the ring of integers and its maximal ideal – give further structure that does not exist in the real case. To each point x ∈ B(G), there is a descending chain of open bounded subgroups of G(k)1x . Of fundamental importance among them is the parahoric subgroup G(k)0x = G(k)x,0 . This was originally defined

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by Bruhat–Tits as the group of integral points of a certain natural integral model that will be discussed below. It was later shown by Haines–Rapoport (cf. Appendix to [PR08]) that the parahoric subgroup G(k)0x can also be described in a way similar to G(k)1x , namely as G(k)x ∩ G(k)0 , where G(k)0 is a natural open and closed characteristic subgroup of G(k). It can be described most succinctly as the kernel of the Kottwitz homomorphism defined in Chapter 11. Thus, the parahoric subgroups are precisely the stabilizers of points of B(G) for the action of G(k)0 . For every r > 0, Moy and Prasad have defined a bounded open subgroup G(k)x,r , normal and of finite index in G(k)x,0 . These groups form a descending filtration whose intersection is the trivial group. The jumps of the filtration, that  is, those r for which G(k)x,r+  G(k)x,r , where G(k)x,r+ = s>r G(k)x, s , form a discrete subset of R. It is worth pointing out that G(k)0x depends only on the polysimplex of B(G) containing x. This is not true for the subgroup G(k)1x , nor for the subgroup G(k)x,r for most values of r. All constructions reviewed so far are compatible with passage to unramified extensions of k. More precisely, let K be a maximal unramified extension of k. Then B(G K ) is equipped with an action of Gal(K/k) and B(G) is the set of fixed points for that action. For each x ∈ B(G) and r  0 the group G(K)x,r is invariant under Gal(K/k) and the subgroup of fixed points is G(k)x,r . A fundamental part of the theory is the construction of integral models of G. Let o and O be the valuation rings of k and K, respectively. To every x ∈ B(G) Bruhat and Tits construct a smooth affine o-group scheme Gx1 whose generic fiber is G and such that Gx1 (O) = G(K)1x . These two conditions characterize Gx1 uniquely. The special fiber of Gx1 may be disconnected. The relative identity component Gx0 of Gx1 is by definition the union of the identity component of the special fiber of Gx1 and the (by assumption connected) generic fiber G. Then Gx0 is an open subgroup scheme of Gx1 , called the parahoric group scheme. It has the property that Gx0 (O) = G(K)0x , and this was the original definition of the parahoric subgroup due to Bruhat–Tits. More generally, for each r  0 there is a smooth affine group scheme Gx,r with generic fiber G and connected special fiber, such that Gx,r (O) = G(K)x,r . The special fiber of Gx0 has an interesting structure. Let f be the residue field of k. The residue field of K is then an algebraic closure f of f. The special fiber of Gx0 is a smooth connected algebraic group defined over f. Its unipotent radical is usually non-trivial. The kernel of the reduction map Gx0 (O) → Gx0 (f) is equal to G(K)x,1 , while the preimage of the unipotent radical of the special fiber of Gx0 is equal to Gx,0+ (O) = G(K)x,0+ . The reductive quotient of the special fiber of Gx0 has f-points equal to G(K)x,0 /G(K)x,0+ , and f-points equal to G(k)x,0 /G(k)x,0+ . For r > 0, the quotient G(k)x,r /G(k)x,r+ is the set

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of f-points of a finite-dimensional vector space defined over f. This vector space receives an action of G(k)x,0 /G(k)x,0+ that makes it into a rational representation of the reductive quotient of the special fiber of Gx0 (under certain assumptions on r or G, cf. §13.5 for more details). The structure of the reductive quotient of the special fiber of Gx0 can be read off easily from the local structure of B(G) at x. Let F be the facet of B(G) containing x. Since G(K)0x depends only on F, the same is true for Gx0 , so we can denote it by GF0 . If F  is a facet whose closure contains F there is a natural morphism GF0  → GF0 . The image of the special fiber of this morphism is a parabolic subgroup of the special fiber of GF0 , and every parabolic subgroup arises this way. In other words, the spherical building (cf. Example 1.5.11) of the reductive quotient of the special fiber of GF0 is the link of F (i.e. the set of all facets whose closure contains F, excluding F itself) in the affine building B(G). One can also describe the root system (in fact, even the root datum, cf. Theorem 8.4.10) of the reductive quotient of the special fiber of Gx0 . Consider an apartment A ⊂ B(G) containing x and let S be the associated maximal k-split torus of G. The subset Ψx of Ψ consisting of those affine roots that vanish at x is a finite root system. The torus S has a canonical integral model S and there is a natural closed immersion S → Gx0 whose image is a maximal o-split torus and its special fiber S is a maximal f-split torus of the special fiber G x0 of Gx0 . The root system of the maximal reductive quotient of G x0 with respect to S is Φx .

Our Approach Let us now describe our approach to Bruhat–Tits theory, and how it differs from the approach presented in [BT72] and [BT84a]. These references work in the general setting of a Henselian valued field, and allow valuations with dense image, as well as non-perfect residue fields. [BT72] develops the theory of affine buildings in an axiomatic way. A key concept is that of a valuation of a root datum. Such a datum is taken as an input in [BT72], from which the building, its apartments, its metric, and various open bounded subgroups are constructed. A central theorem, proved at the end of [BT72], gives conditions under which such a valuation of a root datum descends from one root datum to a “smaller one.” This abstract theory is then applied in [BT84a] to the setting of reductive groups. For a split reductive group one can construct a valuation of the root datum explicitly. The descent theorem is then applied twice: first to pass from

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split groups to quasi-split groups, and then to pass from quasi-split groups to more general groups. These two steps are called quasi-split descent and étale descent. One can think of étale descent as the step from K to k, because the group G becomes quasi-split over K by a theorem of Steinberg under the assumption that f is perfect, cf. Corollary 2.3.8. The theory of integral models of a quasi-split group G associated to points of B(G) is developed in [BT84a] and is used in étale descent. To construct integral models, [BT84a] develops a very general procedure based on a concept called a schematic root datum. In contrast to this development, we have tried to minimize the axiomatic approach in our exposition. We work with reductive groups from the beginning. But the largest difference may be that we completely avoid using the descent theorem of Bruhat–Tits, as well as schematic root data, and we define parahoric subgroups without referring to integral models. Instead, our approach is the following. We first assume that G is quasi-split over k. We construct the valuation of the root datum of G directly. We then introduce the parahoric subgroups, in particular the Iwahori subgroups. These are generated by the bounded subgroups of the root groups coming from the valuation of the root datum, and the Iwahori subgroup of the maximal torus. The latter can be defined in an elementary way using the norm map, without reference to integral models. We furthermore define the subgroup G(k)0 of G(k), again in an elementary way. Simple group-theoretic considerations show that the Iwahori subgroup is a Tits group inside of G(k)0 and one obtains a Tits system (also called a BN-pair), which we may call the Iwahori–Tits system. The general theory of Tits systems, which we review in §1.5, associates to any Tits system a Tits building. We also define a minor modification of this construction, which we call a restricted building. The restricted building of the Iwahori–Tits system is the Bruhat–Tits building B(G) of G. This is again a slight expository difference to the articles of Bruhat–Tits, where the affine building B(G) is constructed explicitly by means of an equivalence relation (which we also give, cf. Proposition 4.4.4). We then show that the parahoric subgroups we have constructed by hand are precisely the stabilizers of points of B(G) for the action of G(k)0 , thereby connecting with the idea of Haines–Rapoport. At this stage we have constructed the affine building of G and the parahoric subgroup associated to each x ∈ B(G), provided G is quasi-split. Next we turn to the construction of integral models of the parahoric subgroups of the group G K . In fact, we construct more generally integral models associated to concave functions. Instead of using the approach of Bruhat–Tits via schematic root data, we employ a simpler and more direct method due to Jiu-Kang Yu [Yu15], based on the systematic use of Néron dilatations. Despite the different method, it is clear that we recover the same integral models, since a smooth

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integral model is determined by its group of O-points, cf. Corollary 2.10.11. Our discussion avoids the use of the more technically complicated notions of quasi-concave functions and their optimizations. This is possible since we only consider discrete valuations. In more general situations the original method of Bruhat–Tits via schematic root data is still useful; we refer to [Lou20] for a modern interpretation of this method. The method of Jiu-Kang Yu is in one regard more general than that of Bruhat– Tits: it allows for positive depth in the toral component of the concave function. In this way it produces group schemes that are not produced by the method of Bruhat–Tits. Examples of these include integral models associated to the Moy– Prasad filtration subgroups that have now become an indispensable tool in the representation theory of p-adic groups. This necessitates a careful discussion of integral models of tori and their filtration subgroups. We introduce and study in this book first the simpler notions of the standard model and the standard filtration, which are sufficient in most cases (in technical terms, when the torus becomes induced after a tamely ramified base change). For general tori we introduce the finite type Néron model and the minimal congruence filtration, adapting arguments of [BLR90] and [Yu15]. We then turn to étale descent and follow the approach developed in [Pra20b]. In this approach, we use systematically the Bruhat–Tits theory of the quasi-split group G ×k K to derive the main results of the theory over k. In particular, we show that B(G) := B(G K )Gal(K/k) satisfies the axioms of a building and obtain descent data from O to o for the parahoric integral models Gx0 associated to x ∈ B(G). Using this, we define natural filtrations of the root groups Ua (k) and extract from them a valuation of the root datum of G relative to a maximal k-split torus S, that would have been the product of the Bruhat–Tits descent theorem. As one application of the theory that is available once étale descent is complete, we give the famous uniform proof, due to Bruhat–Tits, of Kneser’s theorem that H1 (k, G) vanishes for any simply connected semi-simple group G defined over a non-archimedean local field k with perfect residue field f with dim(f)  1, cf. Theorem 10.6.4. This leads to a computation of the set H1 (k, G) for general connected reductive k-groups G, cf. Theorem 11.7.7, as well as the classification of all connected reductive k-groups by means of affine Dynkin diagrams, cf. §10.7. An important theorem about Bruhat–Tits buildings is their compatibility with tamely ramified descent. This result was originally proved by G. Rousseau in his thesis, and later generalized by Prasad and Yu in [PY02] to the following statement. Let H be a connected reductive k-group endowed with a k-rational action of a finite group Θ whose order is not divisible by the characteristic p

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of f. Then G := (H Θ )0 is reductive and B(H)Θ is the Bruhat–Tits building of G. In this book we prove this statement following [Pra20a].

A Summary of Each Chapter We now give a brief summary of the contents and purpose of each chapter. Chapters 1 and 2 collect much of the background material that is used throughout the book. Chapter 1 contains reviews of affine spaces and affine root systems, Tits systems and abstract buildings. Chapter 2 contains reviews of Henselian fields, fields of dimension  1, tori and reductive groups, bounded open subgroups of reductive groups, Chevalley systems and pinnings. Some important definitions, such as the restricted building of a Tits system and of the subgroup G(k)0 of G(k), are given in these two chapters. Chapter 3 presents the two fundamental examples of quasi-split simply connected groups of rank 1, namely SL2 and SU3 . For these groups the theory can be described very simply and with minimal notation, serving as an entryway into the general theory. Chapter 4 contains a summary of Bruhat–Tits theory and closely related material. More precisely, in §4.1 we state most major results of the theory in the form of Axioms. This section can serve as a convenient reference for the expert as well as an initial overview for the beginner before diving into the specifics of the theory. Building on these axioms, in §4.2 we define and study the metric on the Bruhat–Tits building, in §4.3 we define the enlarged building, and in §4.4 we formulate various uniqueness properties of the building. Chapter 5 provides the first major application of Bruhat–Tits theory, namely the various decompositions of the topological group G(k) known under the names of Bruhat, Cartan, and Iwasawa. These decompositions are an essential tool in the study of representation theory and harmonic analysis on G(k). The development of Bruhat–Tits theory begins with Chapter 6. The main purpose of this chapter is to define the apartment associated to a maximal k-split torus. The fundamental object used in the construction of the apartment is the notion of a valuation of the root datum due to Bruhat–Tits and the equivalence relation on such valuations called equipollence. In Chapter 6, an apartment is by definition an equipollence class of valuations of root data. When the group G is quasi-split we show that there is a canonical such equipollence class for each maximal k-split torus of G. It is constructed using a Chevalley basis, or, more generally, a Chevalley–Steinberg system. When G is not quasi-split we take the existence of an equipollence class as an abstract input, to be specified later. In

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addition to the apartment, we discuss in this chapter the affine root system and the affine Weyl group, as well as compatibility with Levi subgroups. Chapter 7 constructs the Bruhat–Tits building of G and the parahoric subgroups of G(k), using as an input the apartments constructed in Chapter 6. The key technical device here is that of a concave function. This is a function  → R, where Φ is the root system of G relative to a maximal k-split torus f:Φ  = Φ∪{0}, satisfying the concavity property f (a+b)  f (a)+ f (b) S ⊂ G and Φ  such that a + b ∈ Φ.  If A is the apartment associated to S in for all a, b ∈ Φ Chapter 6 and x ∈ A, the discussion of Chapter 6 allows one to construct an open bounded subgroup G(k)x, f of G(k). The construction of G(k)x, f can be summarized briefly as follows. For each element a of the relative root system Φ of G with respect to S, the valuation of the root datum x gives a filtration Ua, x,r of the relative root group Ua (k), and the value f (a) of the concave function at a determines which filtration subgroup of Ua (k) will contribute to G(k)x, f . There is also a filtration on the centralizer ZG (S)(k) and the value f (0) determines the contribution of that filtration to G(k)x, f . These subgroups are generalizations of the subgroups constructed by Bruhat–Tits (which we denote G(k)x, f in this book), and include as a special case the Moy–Prasad filtration subgroups. The group-theoretic properties of G(k)x, f are studied in §§7.1–7.3. The special case where f is the constant function with value 0 is particularly important, as it produces the parahoric subgroup G(k)x,0 associated to x. Using the grouptheoretic properties of the groups G(k)x, f it is shown in §7.5 that parahoric subgroups lead to a Tits system, and hence to a (restricted) Tits building. This is the Bruhat–Tits building B(G) of G, defined in §7.6. In §7.7 further open bounded subgroups associated to points of B(G), or more generally subsets of B(G), are studied, such as the stabilizer of x ∈ B(G) in G(k), or in G(k)1 . In §7.11 the notions of hyperspecial, absolutely special, and superspecial, points of B(G) are defined and studied. Chapter 8 constructs the integral models of G associated to points of B(G), or more generally pairs of a point and a concave function. These models are smooth affine o-group schemes G equipped with an isomorphism G ×o k → G. While §8.2 studies general properties of such models, the construction and their study is performed in §§8.3–8.5. In those three sections it is assumed that k is strictly Henselian, that is, k = K. However, this assumption is immaterial, since a simple application of étale descent endows these O-models with an o-structure. Associated to non-empty bounded subset Ω ⊂ B(G K ) contained in an apartment, we construct a number of integral models: the “parahoric” smooth affine group scheme GΩ0 whose special fiber is connected, the smooth affine group scheme GΩ1 whose group of O-points is the pointwise stabilizer of

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Ω in G(K)1 , the smooth affine group scheme GΩ† whose group of O-points is the subgroup of G(K)1 mapping Ω to itself, and the smooth but possibly non-affine group scheme GΩ whose group of O-points is the subgroup of G(K) mapping Ω to itself. Chapter 9 builds on the fact that for any connected reductive k-group G, the base change G K is quasi-split, and hence the Bruhat–Tits building and corresponding integral models for G K have already been constructed. From these, the arguments of Chapter 9 produce the Bruhat–Tits building and integral models for G. While the case of integral models is essentially trivial, as mentioned in the preceding paragraph, the case of the building itself and its properties is far from trivial, and is the main focus of the chapter. The formal definition of B(G), including its apartments and facets, is laid out in §9.2 and its properties are studied in §9.3. The affine root system is constructed and studied in §9.4. Since there are now two affine root systems, one over K and one over k, we refer to the former as the absolute affine root system and to the latter as the relative affine root system. In analogy with the setting of finite root systems, the absolute affine root system is always reduced. This fact might seem familiar at first, but on second look it is revealed to be somewhat mysterious, given that the finite root system of G K need not be reduced. The results of §§9.3 and 9.4 are collected in §9.5 to show that all the axioms formulated in Chapter 4 are valid for G. At this point, the basic properties of the building for G have been established. The remainder of Chapter 9 establishes further properties, not stated in Chapter 4, that are nonetheless very useful. In §9.6 the apartment of B(G) associated to a maximal k-split torus S of G is interpreted as an equipollence class of valuations of the root datum of G relative to S, unifying the point of view of Chapter 9 with that of Chapter 6. In §9.7, the building of G and that of a Levi subgroup of G are compared. In §9.8 the concave function groups for G(k) are related to those for G(K). The important properties of G being residually quasi-split or residually split are studied in §9.10. Finally, §9.11 studies the relationship between the building and the parahoric groups for a group and its Weil restriction of scalars. Chapter 10 is concerned with the additional results that hold when one assumes that the residue field f of k has dimension at most 1. This notion is reviewed in §2.3. For example, finite fields and algebraically closed fields have this property, so the results of Chapter 10 apply to local fields, their maximal unramified extensions, or the field C((x)) of Laurent series with complex coefficients. Among the main results of this chapter are the following: (1) superspecial points of B(G) exist only when G is quasi-split;

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(2) an anisotropic group G is automatically of type An ; if k is in addition assumed to be a local field, then G is of inner type An ; (3) anisotropic tori exist when G is semi-simple and the arithmetic of f is sufficiently rich; (4) H1 (k, G) can be described in terms of the building of G; it vanishes when G is simply connected. Building on the last result, we give in §10.7 a complete and explicit description of all inner forms of a given quasi-split connected reductive k-group G using the combinatorics of the Bruhat–Tits building. Since quasi-split groups themselves are described, over any field, in a very combinatorial way via the notion of a based root datum, this gives a combinatorial description of all connected reductive k-groups. At the heart of this description is the notion of folding of an affine Dynkin diagram. This is the affine notion of a concept for a finite Dynkin diagram explored in §2.6(c). Both are used in the classification. Chapter 11 introduces the Kottwitz homomorphism and uses it to describe the component group of the special fiber of a given parahoric integral model. In particular, we describe the component group of the special fiber of the ft-Néron and lft-Néron models of a k-torus. Chapter 12 considers a connected reductive K-group H equipped with an action of a finite group Θ and studies the relationship between the buildings of H and G = (H Θ )0 , under the assumption that the order of Θ is prime to the residue field characteristic p. It is shown that B(G) can be identified with the set of fixed points B(H)Θ , and that the facets of B(G) and the associated parahoric groups can be related to those for H. A special case of this set-up occurs when we take a connected reductive K-group G and set H = R L/K (G L ), where L is a finite tamely ramified Galois extension of K. The results then relate Bruhat–Tits theory for G L and G. This is called tamely ramified descent, and the resulting isomorphism B(G L )Gal(L/K) = B(G) was originally obtained in the thesis of Guy Rousseau by an entirely different method. Chapter 13 introduces the Moy–Prasad filtrations of the group G(k), its Lie algebra g(k), and its dual g∗ (k). These filtrations were originally defined in [MP94] for simply connected groups and in [MP96] for arbitrary connected reductive groups “by hand.” In this book, our treatment of groups associated to concave functions is more general than that of [BT72, BT84a] in that we use not only filtrations of root groups, but also filtrations of the centralizer of a maximal k-split torus. This additional generality makes the definition of Moy–Prasad filtration groups a simple application of the theory of concave function groups developed in §7.3. Moreover, the results of §8.5 provide associated group schemes, whose Lie algebras turn out to be the required filtration lattices of

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g(k). Despite this different approach of definition, we do recover the definition given in [MP94]. With regards to [MP96], the situation is a bit more subtle. As was discovered by Jiu-Kang Yu, the definition given in [MP96], which is just the translation of the definition of [MP94] to the case of arbitrary connected reductive groups, may have a pathological behavior in the presence of wild ramification. In particular, the Moy–Prasad isomorphism, one of the essential results of the theory, may fail. This problem occurs already when the connected reductive group G is a torus. We follow Yu’s suggestion given in [Yu15] and equip the category of tori with Yu’s minimal congruent filtration, discussed in Appendix §B.10. This leads to Moy–Prasad filtrations that may differ from those defined in [MP96] when the group is not simply connected or adjoint and does not split over a tamely ramified field extension. Nonetheless, the main results of [MP96] do still hold for this modified filtration, including those which turn out to fail for the original filtration. Chapter 14 discusses the functorial property of the Bruhat–Tits building. More precisely, given a homomorphism G → H of connected reductive kgroups, this chapter studies the set of G(k)-equivariant maps B(G) → B(H),   as well as B(G) → B(H). Results about this problem were originally obtained by Landvogt in [Lan00]. Chapter 15 gives explicit descriptions of the buildings of classical groups – general and special linear groups, symplectic, orthogonal, and unitary groups – in terms of lattice chains. This description is also given by Bruhat–Tits in [BT84b, BT87b]. In this book we have restricted generality by considering only quasi-split classical groups and avoiding the case p = 2. While the omitted cases are certainly important, we did not see how to improve the exposition of Bruhat–Tits while maintaining full generality. On the other hand, in the restricted generality of Chapter 15, the exposition becomes simpler, because we can use the more intuitive notion of self-dual norms, rather than that of maxi-minorant norms. The next three chapters represent a sample of applications of the theory developed in this book. Chapter 16 presents the work [DeB06] of DeBacker on parameterization of G(k)-conjugacy classes of maximal unramified tori in G using Bruhat–Tits theory. These results, which give a complete and explicit classification, have become an essential tool in representation theory. Chapter 17 gives a similar, albeit less complete, parameterization of the G(k)-conjugacy classes of tamely ramified maximal tori. Chapter 18 presents Prasad’s formula [Pra89] for the covolume of S-arithmetic subgroups of an absolutely simple simply connected group over a global

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field. The derivation of this formula involves a considerable amount of Bruhat– Tits theory, even if S consists only of archimedean places. The three appendices to this book contain technical discussions that are used in Chapter 8. In Appendix A we review the various operations that can be performed with integral models. While most material in the appendix is well known, some parts of it have not been covered in the literature in a way that we have found suitable for our purposes. Appendix B contains a careful discussion of integral models of tori. These include the standard model discussed in [BT84a, §4.4.6] under the name “le schéma canonique de fibre générique T” and in [VKM02] and [Pop01] under the name “standard model,” the ft-Néron model briefly mentioned in [CY01], the lft-Néron model constructed in [BLR90, §10], as well as various congruence models related to the standard filtration and the minimal congruent filtration. Appendix C discusses the integral models of root groups and their Lie algebras. The reader will notice that there are a few topics that we have not covered. These include the compactification of the Bruhat–Tits building discussed in [Lan96] (a reference that also includes a very useful review of Bruhat–Tits theory which we highly recommend).

On the Logical Structure of the Exposition The core of Bruhat–Tits theory is developed in Chapters 6–9. Chapters 6 and 7 are written for arbitrary connected reductive groups. However, many results of these chapters are unconditional only for quasi-split connected reductive groups. For general connected reductive groups, the results of Chapters 6 and 7 are conditional on the existence of objects, which is only proved in Chapter 9. At the same time, Chapter 9 is conditional on the results of Chapters 6 and 7 for quasi-split groups. Thus, while there is no circular logic, the situation is a bit subtle, and we want to explain it here. Chapter 6 studies the properties of equipollence classes of valuations of the root datum for a connected reductive group G and a maximal k-split torus S. The apartment associated to S is such an equipollence class. When G is quasisplit, a canonical such class is constructed in §6.1. When G is not quasi-split, the existence of valuations of the root datum is not proved in Chapter 6; rather, it is taken for granted. Chapter 7 constructs the building for G taking as input an equipollence class of valuations of the root datum for a maximal k-split torus S. Given the construction of §6.1, the building of G is thus constructed unconditionally

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Introduction

provided G is quasi-split, while for general G it is conditional on the existence (as well as the specification) of an equipollence class of valuations of the root datum for a maximal k-split torus S. The discussion of concave function groups in Chapter 7 requires as further input a filtration of the group ZG (S)(k) for a maximal k-split torus S. In the case when G is quasi-split, so that ZG (S) is a maximal torus, appropriate filtrations are constructed and studied in Appendix B independently. Thus, again, when G is quasi-split, these results are unconditional. For the construction of the building of G, only the filtration subgroup ZG (S)(k)0 is needed. This subgroup is already defined in Chapter 2 unconditionally. Therefore, the construction of the building for general G in Chapter 7 is conditional only on the existence of valuations of root data. The filtration subgroups ZG (S)(k)r for r > 0 are needed for the study of more general concave function groups, including the Moy–Prasad filtration. Most results in Chapter 8, in particular the construction of various group schemes and the structure of their special fibers, require the base field to be strictly Henselian. Chapter 9 uses as input the results of Chapters 6–8 for quasi-split groups, which are unconditional, to derive some, but not all, of the analogous results for arbitrary connected reductive groups. In particular, the Bruhat–Tits building for such groups is constructed in that chapter. One of the outcomes of this chapter, formulated in §9.6, is the existence of a canonical (up to equipollence) valuation of the root datum for a maximal k-split torus. This valuation can now be used as input to Chapters 6 and 7 to obtain another interpretation of the Bruhat–Tits building. Another outcome of this chapter is the construction of a natural filtration on ZG (S)(k). This filtration can then be used as an input to Chapter 7 for the construction and properties of general concave function groups.

Different Paths through the Book This book is written to serve a variety of different audiences. We present here different possible paths through the book, depending on the goal of the reader. Chapters 1 and 2 contain a rather long collection of background material used in the book, and are not meant to be read linearly. For example, readers interested in the base field Q p need not read through the generalities of §2.1. We recommend that the reader skim through the various sections so that they are roughly aware of what is there and can refer back when needed. The index of notation and terminology should provide help with that.

Introduction

xxix

Most readers will likely be interested in understanding the theory in broad strokes, so that they can apply it to particular problems in other areas. Such readers might find it most useful to study in detail Chapter 4, and especially §4.1, where the main results of Bruhat–Tits theory are summarized in an axiomatic way. Chapters 5 and 15 might be a natural follow-up. A reader who would like to learn the Bruhat–Tits theory more deeply should first study the examples in Chapter 3 carefully, and then begin with Chapter 6, skipping Chapters 4 and 5. If such a reader wishes to stay away from scheme theory and only stick to Lie theory, they can read Chapters 6 and 7 with general connected reductive groups in mind, taking on faith the existence of valuations of root data and the filtration of ZG (S)(k) when G is not quasi-split. The material of Chapter 8, will of course have to be skipped, but the summary of the main results given in §4.1 could serve as a substitute. On the other hand, a reader willing to use scheme theory and looking for the quickest route under this assumption should read Chapters 6 and 7 assuming that the base field is strictly Henselian. In that case G is quasi-split and many arguments simplify. For example, abstract arguments involving the axioms of a valuation of root datum can almost always be replaced by concrete computations with SL2 or SU3 , many of which are given in Chapter 3, or as alternative proofs in Chapters 6 and 7. The reader should then continue to Chapter 8 and then to Chapter 9.

Acknowledgements This book has profited from the help of many fellow mathematicians. First and foremost, the contributions of Brian Conrad cannot be overstated. He supplied numerous useful suggestions, corrections, and wrote some parts of the arguments used in Appendix A, including the entire section on the Greenberg functor. He also reread the three appendices and inserted corrections in the LaTeX-files himself. We thank Richard Taylor and the participants of the Conrad–Taylor learning seminar on Bruhat–Tits theory at Stanford, and Jessica Fintzen and the students in her course on Bruhat–Tits theory at the University of Bonn, who worked through an early version of the book and gave many useful comments and corrections. We thank Ulrich Görtz, George McNinch and Nguyen Quoc Thang for their careful reading and numerous comments, and Pierre Deligne, Jessica Fintzen, Thomas Haines, João Lourenço, Michael Rapoport, Timo Richarz, and Loren Spice for helpful comments. We thank Guy Rousseau and Anne Parreau for supplying diagrams for the 2-dimensional

xxx

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apartments, and Stella Gastineau, Ben McKay and Amit Prasad for helping with LaTeX. The material of this book reflects the insights of many mathematicians who have improved and refined the original work of Bruhat and Tits. Among others, we have included ideas due to Thomas Haines, Robert Kottwitz, Michael Rapoport, and Jiu-Kang Yu. G.P. thanks his teacher M. S. Raghunathan. He also thanks his surgeons Dr. Jonathan Eliason and Dr. Himanshu Patel, his PCP Dr. Jasmine Parvaz, the physicians and nurses of the University of Michigan hospital and New YorkPresbyterian for their excellent care during his recent health crises, and George Lusztig for his friendship. G.P. thanks his beloved wife Indu for her unwavering support and encouragement and for bearing the burden of numerous responsibilities and struggles. She has been an inspiration to so many in the Prasad family. Finally, he expresses his deep appreciation to the families of his son Anoop and daughter Ila Fiete, for their love and support. T.K. thanks his teacher Robert Kottwitz. He also thanks his wonderful wife Corinna, his son Ilian, and his parents Gallina and Holger for their love and support. T.K. acknowledges support from the National Science Foundation with grant DMS–1801687. We both thank Diana Gillooly, Tom Harris, Clare Dennison, David Tranah and the copy-editor Siriol Jones of the Cambridge University Press for all their help during the publication of this book.

P A R T ONE BACKGROUND AND REVIEW

1 Affine Root Systems and Abstract Buildings

In this chapter we collect some of the background material used throughout the book. We encourage the reader to skim this chapter, rather than read it linearly, and refer back to it as needed. While some important objects, like the subgroups G(k)0 and G(k)1 , are defined in this chapter and used throughout the book, the index of notation should help the reader locate the appropriate places in this chapter as needed.

1.1 Metric Spaces Let (X, d) be a metric space. Recall the notion of the ball B(x, r) = {y ∈ X | d(x, y) < r } of radius r with center x. Definition 1.1.1 (1) A subset A ⊂ X is called bounded if there exist x ∈ X and r > 0 such that A ⊂ B(x, r). (2) For any two non-empty subsets A, B ⊂ X their joint diameter is diam(A, B) = sup{d(a, b) | a ∈ A, b ∈ B}. (3) The diameter of A ⊂ X is diam(A) = diam(A, A). Note that diam(A) is finite precisely when A is bounded. A curve is a continuous map c : [0, 1] → X. The length of the curve is defined as (c) = sup

n−1 

d(c(ti ), c(ti+1 )),

(1.1.1)

i=0

where the supremum is taken over the set of finite sequences 0 = t0 < t1 < 3

4

Affine Root Systems and Abstract Buildings

· · · < tn = 1. The curve is said to join the points x, y ∈ X if c(0) = x and c(1) = y. Definition 1.1.2

A curve c is called

(1) rectifiable, if (c) < ∞, (2) geodesic, if c is rectifiable and for all 0 < t1 < t2 < 1 the equalities hold (c|[t1 ,t2 ] ) = d(c(t1 ), c(t2 )) = d(c(0), c(1))|t1 − t2 |, where c|[t1 ,t2 ] : [0, 1] → X is defined by s → c(t1 + s(t2 − t1 )). Remark 1.1.3 If c is rectifiable, then so is c|[t1 ,t2 ] . Hence the definition of geodesic makes sense. It is clear from the definition that if c1, c2 : [0, 1] → X both join x, y ∈ X and c1 is geodesic, then d(x, y) = (c1 )  (c2 ). The second equality in the definition of geodesic expresses the fact that c is parameterized by arc length. Definition 1.1.4 The space (X, d) is called geodesic, if any two points x, y ∈ X are connected by a geodesic. It is called uniquely geodesic if that geodesic is unique. The length metric d on X is defined as d (x, y) = inf (c), where the infimum is taken over all rectifiable curves c joining x and y, assuming that at least one such curve exists (as is the case with a geodesic space). Since d (x, y)  d(x, y) one sees easily that d is a metric. Definition 1.1.5

The space (X, d) is called a length space if d = d .

Fact 1.1.6 Every geodesic space is a length space. Example 1.1.7 Consider the circle S1 as a metric space, where the metric d is the restriction of the Euclidean metric on R2 . For x, y ∈ S1 we have d(x, y) = ||x − y||, where || − || is the Euclidean norm on R2 , while d (x, y) = arccos( x, y ) > d(x, y). Thus (S1, d) is not a length space. Tautologically, (S1, d ) is a length space, and in fact a geodesic space. However, it is not uniquely geodesic, because two antipodal points are joined by two distinct geodesics. Definition 1.1.8 The space (X, d) is called non-positively curved, if for every x, y ∈ X there exists m ∈ X such that for all z ∈ X, d(x, z)2 + d(y, z)2  2d(m, z)2 + (1/2)d(x, y)2 .

1.1 Metric Spaces

5

Remark 1.1.9 In a Euclidean space, if we let m be the midpoint between x, y, then the above inequality becomes an equality, and is called the parallelogram law: in a parallelogram, the sum of the squares of the lengths of the two diagonals is equal to the sum of the squares of the lengths of the four sides. y m x

z

Remark 1.1.10 If (X, d) is uniquely geodesic, then for given x, y there is at most one m that satisfies the inequality of Definition 1.1.8, namely the midpoint c(1/2) of the unique geodesic c joining x and y, see the following Lemma. Furthermore, the inequality of Definition 1.1.8 has the geometric interpretation of pinching of triangles relative to Euclidean “reference” triangles, see [BH99, Figure II.1.1, Proposition II.1.7(1),(2) and Exercise II.1.9(1)(a,c)]. Lemma 1.1.11 A non-positively curved geodesic space is uniquely geodesic. The midpoint of the unique geodesic connecting x and y is the unique point m satisfying the inequality of Definition 1.1.8. Proof Let (X, d) be a non-positively curved geodesic space, let x, y ∈ X and let c be a geodesic connecting x and y. Let m = c(1/2) be the midpoint. Then d(x, m) = (1/2)d(x, y) = d(y, m). If m  is any point satisfying the inequality, we can take z = m and see (1/2)d(x, y)2  2d(m , m)2 + (1/2)d(x, y)2 , thus m  = m. If c  is another geodesic joining x and y we see that c(1/2) = c (1/2). Further bisections produce a dense set of t ∈ [0, 1] such that c(t) = c (t), hence  c = c . Definition 1.1.12

Let (X, d) be a uniquely geodesic space.

(1) A subset M ⊂ X is called convex if given x, y ∈ M the (image of the) unique geodesic joining x and y lies in M. (2) The radius of a convex set M is r(M) := inf{diam(x, M)| x ∈ M }. (3) A barycenter of a convex set M is a point x ∈ M such that diam(x, M) = r(M). Lemma 1.1.13 Let (X, d) be a non-positively curved metric space. Each closed ball B(z, r) = {y ∈ X | d(x, y)  r } is convex. Proof Let x, y ∈ B(z, r). Let c be the unique geodesic joining x and y and let m = c(1/2) be its midpoint. Definition 1.1.8 implies d(m, z)  r, hence

6

Affine Root Systems and Abstract Buildings

m ∈ B(z, r). The two halves of c are the geodesics joining x with m and m with y. Proceeding inductively we obtain a dense set of t ∈ [0, 1] such that  c(t) ∈ B(z, r). Definition 1.1.14 Let (X, d) be a non-positively curved metric space and M ⊂ X a non-empty bounded subset. The closed convex hull of M is the intersection of all closed bounded convex subsets of X containing M. We note that Lemma 1.1.13 implies the existence of a closed bounded convex subset of X containing M, therefore the closed convex hull of M exists and is by construction a closed, bounded, convex set containing M. Theorem 1.1.15 space.

Let (X, d) be a complete non-positively curved geodesic

(1) A non-empty bounded closed convex subset M ⊂ X has a unique barycenter. It is invariant under all isometries of X that map M to M. (2) If M ⊂ X is a non-empty bounded subset, the stabilizer of M in the group of isometries of X has a fixed point in X. Proof The second point follows by applying the first point to the closed convex hull of the bounded subset M in the sense of Definition 1.1.14. In the first point, the fact that the unique barycenter is invariant under all isometries preserving M is obvious, since the notion of barycenter is defined in terms of the metric. It remains to prove the existence and uniqueness of a barycenter. Assume now that M is bounded, closed, and convex. To prove existence of a barycenter, let f (x) = diam(x, M) for x ∈ M and let r = inf{ f (x) | x ∈ M }. Let  be a positive real number and let x, y ∈ M be such that f (x) < r +  and f (y) < r + . We claim that x and y are close to each other. More precisely, we claim d(x, y)2 < 16r. To see this, let m be the midpoint of the geodesic joining x and y. Since M is convex, m ∈ M. Thus f (m)  r. Therefore there exists z ∈ M such that d(m, z) > r − . Applying the non-positive curvature property to x, y, m, z and using that both d(x, z) and d(y, z) are less than r + , whereas d(m, z) > r − , we obtain the inequality d(x, y)2 < 16r, proving the claim. Now let {xi } be a sequence in M such that f (xi ) = r + i , where {i } is a decreasing sequence of positive real numbers that converges to 0. For i < j we have d(xi , x j )2 < 16ri . The sequence {xi } is thus Cauchy and converges to a point c ∈ X by completeness of X. Since M is closed, c ∈ M. By construction f (c) = r, proving existence of c. The uniqueness of c follows at once from the claim: if c1 and c2 are two

1.1 Metric Spaces

7

barycenters, then f (c1 ) = r = f (c2 ) implies d(c1, c2 ) < 16r for all  > 0, thus  c1 = c2 . Remark 1.1.16 The second point of Theorem 1.1.15 is called the Bruhat– Tits Fixed Point Lemma. The proof presented here is due to Serre. The original statement due to Bruhat–Tits is actually more general, as it does not assume that the metric space is geodesic. This will not be relevant to us, since the Bruhat– Tits building is a geodesic space. For the proof of the more general statement we refer the reader to [BT72, Lemma 3.2.3], whose proof is self-contained as long as one takes [BT72, Lemma 3.2.1] as a definition. Lemma 1.1.17 For a set A of isometries of a non-empty metric space (X, d) the following are equivalent. (1) For every x ∈ X the set {g · x |g ∈ A} is bounded. (2) There exists x ∈ X for which the set {g · x | g ∈ A} is bounded. Proof Let x ∈ X such that the set {g · x |g ∈ A} is of bounded diameter. For any y ∈ X, we have d(gy, y)  d(gy, gx) + d(gx, x) + d(x, y) = 2d(x, y) + d(gx, x) and hence d(gy, y) is also bounded.



Definition 1.1.18 A set A of isometries of (X, d) that satisfies the equivalent conditions of Lemma 1.1.17 is said to have bounded action on X. Corollary 1.1.19 Let (X, d) be a non-empty complete non-positively curved metric space. A group A of isometries of (X, d) that has bounded action fixes a point of X. Proof

Any orbit of A in X is non-empty and bounded.



Proposition 1.1.20 Let (X, d) be a complete non-positively curved geodesic space. Let Y ⊂ X be a closed convex subset. (1) Given x ∈ X there exists among all y ∈ Y a unique one whose distance to x is minimal. (2) The function π : X → Y , defined so that π(x) ∈ Y is the unique point of Y closest to X, is continuous. (3) The function π : X → Y of (2) is equivariant with respect to any isometry of X that preserves Y . Proof (1) The proof is very similar to that of Theorem 1.1.15(1) so we only give a sketch. Let r = inf{d(x, y)| y ∈ Y }. Existence and uniqueness are reduced to the claim that for any  > 0 and y1, y2 ∈ Y such that d(x, y1 ) < r +  and

8

Affine Root Systems and Abstract Buildings

d(x, y2 ) < r +  we have d(y1, y2 )2 < 16r. This in turn is proved by taking m to be the midpoint of the geodesic joining y1, y2 , observing m ∈ Y by convexity of Y , concluding d(x, m)  r, hence d(x, m) > r −, and applying the non-positive curvature property to y1, y2, m, x. (2) Let (xn ) be a sequence of points of X that converges to x ∈ X. We need to show that the sequence (π(xn )) converges to π(x). We have d(x, π(x))  d(x, π(xn ))  d(x, xn ) + d(xn, π(xn ))  d(x, xn ) + d(xn, π(x)), the first inequality by definition of π(x), the second by the triangle inequality, and the third by definition of π(xn ). The limit for n → ∞ of the right-most term equals d(x, π(x)). (3) Let f : X → X be an isometry that preserves Y . Then d( f (x), π( f (x)))  d( f (x), f (π(x))) = d(x, π(x)), for all x ∈ X. Applying the same argument to f −1 gives the opposite inequality. We conclude that d( f (x), π( f (x))) = d( f (x), f (π(x))). The uniqueness of π( f (x)) implies π( f (x)) = f (π(x)).



Proposition 1.1.20 can be strengthened when X has more structure, see Remark 4.2.19.

1.2 Affine Spaces Let W be a vector space over a field k and let V ⊂ W be a subspace. The quotient W/V consists of the orbits in W for the action of V by translation. In some sense all orbits look the same, with the following exception. The orbit through 0, i.e. the subspace V itself, is special, because it has a distinguished element, namely 0. No other orbit has a special point. In this section we will recall the concept of an affine space, which is a formalization of this basic example. This will be important in the development of Bruhat–Tits theory, because apartments in Bruhat–Tits buildings are affine spaces. The simplest example of this is discussed towards the end of §3.1. Let V be a vector space over a field k. Definition 1.2.1 An affine space over V is a non-empty set A equipped with a simply transitive action of the additive group of V. We declare dim(A) := dim(V). More generally, an affine space over k is a pair (V, A) consistsing of a k-vector space V and an affine space A over V.

1.2 Affine Spaces

9

In particular, every v ∈ V gives a map Tv : A → A, usually called translation by v, and one often writes Tv (x) = x+v. For any x, y ∈ A one writes y−x ∈ V for the unique v ∈ V such that y = x+v. Then one has the rule (z−y)+(y−x) = z−x for x, y, z ∈ A. Example 1.2.2 The vector space V is tautologically an affine space over V. More generally, if V is a subspace in a vector space W, then every fiber of the projection map W → W/V is an affine space over V. We will see in Proposition 1.2.10 that every affine space over V arises in this way in a canonical manner. For any x ∈ A the map ix : V → A, v → x + v is a bijection that translates the action of V on A to the action of V on itself by translation. Thus, one may intuitively think of A as being V, but “after one has forgotten where the origin is.” The inverse ix−1 : A → V is given by ix−1 (y) = y − x. Definition 1.2.3 Let A be an affine space over V. An affine subspace B ⊂ A is a non-empty subset of A having the property that W = {y − x | x, y ∈ B} is a vector subspace of V. We may call W the derivative of B, and write f = ∇F. Clearly B is an affine space over W. Under the bijection ia the set of affine subspaces of A is identified with the set of subsets of V of the form v + W for an element v ∈ V and a vector subspace W ⊂ V. Definition 1.2.4 Let A and A be affine spaces over the vector spaces V and V  respectively. A map F : A → A is called affine if there exists a linear map f : V → V  such that F(x + v) = F(x) + f (v) for all x ∈ A and v ∈ V. We call f the derivative of F. Note that f is uniquely determined by F, namely via f (v) = F(x + v) − F(x) for some fixed x. If G : A → A is another affine map, with derivative g, then G ◦ F : A → A is also affine and its derivative is g ◦ f . If F1, F2 : A → A have the same derivative, then the vector v = F2 (x) − F1 (x) ∈ V  is independent of x ∈ A and thus F2 (x) = F1 (x) + v for all x ∈ A. We may write v = F2 − F1 . Example 1.2.5 (1) The constant map F : A → A given by F(x) = x  for a   fixed x ∈ A is affine. Its derivative of F is the zero linear map. (2) The translation Tv : A → A for v ∈ V is affine. Its derivative is the identity map on V. If we fix origins x ∈ A and x  ∈ A, then F → ix−1 ◦ F ◦ ix identifies the set of affine maps A → A with the set of maps V → V  of the form v → f (v) + v , where f : V → V  is linear and v  ∈ V . Remark 1.2.6 One checks immediately that an affine map is an isomorphism

10

Affine Root Systems and Abstract Buildings

if and only if it is bijective, or equivalently that its derivative is an isomorphism of vector spaces. It is clear that if A and A are affine spaces over the same vector space V, then there exists an affine isomorphism A → A. The set of affine isomorphisms whose derivative is the identity is again an affine space over V. If A = A, then that affine space has a distinguished point, namely id A, and is hence naturally identified with V. Given an affine space A over V let Aff(A) denote the set of affine isomorphisms A → A. Composition turns Aff(A) into a group. Taking derivative produces a group homomorphism ∇ : Aff(A) → Aut(V). This homomorphism is surjective, for given f ∈ Aut(V) we can choose x ∈ A and then ix ◦ f ◦ ix−1 ∈ Aff(A) has derivative f . We thus obtain the exact sequence 0 → V → Aff(A) → Aut(V) → 1 where the inclusion V → Aff(A) maps v to the translation Tv . This exact sequence splits non-canonically; a choice of x ∈ A gives the splitting f → ix ◦ f ◦ ix−1 . Definition 1.2.7 Let Γ be a group. An affine action of Γ on A is a group homomorphism τ : Γ → Aff(A). Clearly an affine action τ of Γ on A leads to a linear action ∇ ◦ τ of Γ on V. Assume now that the action τ is faithful, so we can identify Γ with its image under τ, a subgroup of Aff(A). There are two extreme cases that are useful to keep in mind. If Γ is finite and k is infinite, then V ∩ Γ = {0} and ∇ restricts to an isomorphism Γ → ∇Γ. If Γ contains V, then we obtain the exact sequence 0 → V → Γ → ∇Γ → 1 which is again split, a splitting being given by a choice of a ∈ A as above. Definition 1.2.8 An affine functional on A is an affine map ψ : A → k. We will write ψ := ∇ψ ∈ V ∗ and Hψ = Aψ=0 = {x ∈ A | ψ(x) = 0}. If ψ is not constant, the subset Hψ is a hyperplane in A. We have Hψ = Hη if and only if η = rψ with r ∈ k × . If ψ is an affine functional, then so is −ψ. It has the properties ∇(−ψ) = −∇ψ and H−ψ = Hψ , and these properties characterize −ψ uniquely. The set A∗ of all affine functionals A → k is a vector space over k with respect to pointwise addition and scalar multiplication. The map ∇ : A∗ → V ∗ is linear and surjective and its kernel is the subspace of constant affine linear

1.2 Affine Spaces

11

functionals, which is naturally identified with k. In this way we obtain the exact sequence of k-vector spaces 0 → k → A∗ → V ∗ → 0.

(1.2.1)

The map ψ → Hψ is a bijection between the set of lines in A∗ , without the constant line, and the set of affine hyperplanes in A. Example 1.2.9 In the special case A = V, an affine functional has the form ψ(v) = λ(v) + c, where λ ∈ V ∗ and c ∈ k. Thus A∗ = V ∗ ⊕ k, that is, the above exact sequence is canonically split. For general A, the above exact sequence is non-canonically split; choosing x ∈ A and using the identification ix : V → A induces a splitting. If x ∈ A is replaced by x + v for v ∈ V, then the splitting changes by the automorphism of V ∗ ⊕ k sending λ + c to λ − λ(v) + c. We will now show that any affine space over a vector space V arises naturally as an orbit of the action of V on a larger vector space W. For this, consider the linear dual space A∗∗ of A∗ . The exact sequence (1.2.1) dualizes to 0 → V → A∗∗ → k → 0.

(1.2.2)

In concrete terms, the embedding V → A∗∗ is given by the pairing ψ, v =  v for ψ ∈ A∗ and v ∈ V. The map A∗∗ → k in this sequence is the ψ, linear functional on A∗∗ corresponding to the element of A∗ that is the affine functional on A with constant value 1. Let us write 1 A for it. We have the natural embedding A → A∗∗ via the natural pairing ψ, a = ψ(a). This embedding identifies A with the fiber over 1 ∈ k of the linear functional 1 A : A∗∗ → k. Proposition 1.2.10 Consider the category Aff whose objects are all affine spaces over k and whose morphisms are affine transformations between affine spaces. Consider the category fVect whose objects are pairs (W, λ) consisting of a k-vector space W and a non-zero linear functional 0  λ ∈ W ∗ , and whose morphisms are linear maps of vector spaces that respect the given functionals. Then the functors F : Aff → fVect,

(V, A) → (A∗∗, 1 A), f → f ∗∗

and G : fVect → Aff,

(W, λ) → (λ−1 (0), λ−1 (1)), F → F |λ−1 (1)

are mutually inverse equivalences of categories. Proof We need to exhibit natural transformations η : idAff → G ◦ F and  : F ◦ G → idfVect . Given an affine space (V, A) set W = A∗∗ and λ = 1 A. The discussion before the statement of this proposition provides isomorphisms V → λ−1 (0) and A → λ−1 (1). These isomorphisms comprise η(V , A) . Conversely,

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given a vector space W and a non-zero linear functional λ, set V = λ−1 (0) and A = λ−1 (1). Restriction from W to A provides a linear map W ∗ → A∗ . We will prove that it is an isomorphism. Admitting this, we obtain dually an isomorphism A∗∗ → W which tautologically identifies 1 A with λ, and hence provides the desired (W ,λ) . To prove that W ∗ → A∗ is an isomorphism we note that the kernel of any element of W ∗ is a hyperplane in W containing zero and hence cannot contain the affine hyperplane λ−1 (1). Therefore W ∗ → A∗ is injective. When dim(W) < ∞ this is enough. In general let μ ∈ A∗ . Choose an element μ1 ∈ V ∗ that extends μ ∈ V ∗ . The restriction μ1 | A is an affine functional with the same derivative as μ. Therefore μ − μ1 | A is a constant n ∈ k, and we  see μ = (μ1 + nλ)| A. Definition 1.2.11 For x ∈ A define A∗x = {ψ ∈ A∗ | ψ(x) = 0}. It is clear that restricting ∇ to A∗x provides an isomorphism ∇ : A∗x → V ∗ . Definition 1.2.12 Let A be an affine space over V and let W ⊂ V be a subspace. The quotient A/W is the set of orbits in A for the action of W. It is clear that A/W is an affine space over V/W. Pulling back affine functionals under the quotient map A → A/W gives an injection (A/W)∗ → A∗ which identifies (A/W)∗ with the subspace {ψ ∈ A∗ | ψ ∈ W ⊥ }, where W ⊥ ⊂ V ∗ is the annihilator of W. Definition 1.2.13 For a non-empty subset Ω ⊂ A let Ω ⊂ A be the smallest affine space containing Ω, and let A∗Ω be the subspace of A∗ consisting of those affine functionals that vanish identically on Ω. Note that when Ω = {x} we obtain A∗Ω = A∗x . Fact 1.2.14 Let Ω ⊂ A be non-empty. (1) An affine functional vanishes on Ω if and only if it vanishes on Ω . (2) If W ⊂ V is the derivative space of Ω , then the map ∇ : A∗Ω → W ⊥ is an isomorphism. (3) The bijection Hψ ↔ k × · ψ restricts to a bijection between the set of affine hyperplanes in A containing Ω and the set of lines in A∗Ω , hence the set of lines in W ⊥ . Let A and A be affine spaces over V and V  and let f : A → A be an affine map. Its dual f ∗ : (A)∗ → A∗ is defined by f ∗ (ψ ) = ψ ◦ f . Then f ∗ is a linear map. 1.2.15 Given affine spaces A1, A2 over vector spaces V1,V2 we can form the affine space A = A1 × A2 over the vector space V = V1 × V2 . The projection

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13

A → Ai dualizes to an injection A∗i → A∗ that identifies A∗i with those elements of A∗ whose derivative lies in Vi∗ ⊂ V ∗ . The sum of these two injections fits into the exact sequence 0 → k → A∗1 ⊕ A∗2 → A∗ → 0,

(1.2.3)

n where k is embedded anti-diagonally into A∗1 ⊕ A∗2 . More generally, if { Ai }i=1 is a finite collection of affine spaces over vector spaces Vi and we consider the   affine space A = Ai over the vector space i Vi , we have the surjective map  ∗  ∗ Ai consisting of those Ai → A∗ whose kernel is the hyperplane in k n ⊂ tuples whose coordinates sum to 0.

1.2.16 Let A be an affine space over the vector space V and assume given a direct product decomposition V = V1 × V2 . This decomposition induces a direct product decomposition A = A1 × A2 as follows. Let A1 = A/V2 and A2 = A/V1 . Then Ai is an affine space over Vi . The product of the two projections A → A1 × A2 is equivariant for the translation action of V = V1 ×V2 and injective, hence bijective, and therefore an isomorphism of affine spaces over V. Assume from now on that k = R. Definition 1.2.17 (1) For an affine functional ψ we will write Aψ>0 = {x ∈ A | ψ(x) > 0}. Analogously we define Aψ0 , Aψ 0 (respectively ψ(x) < 0) for all x ∈ C. We denote by Ψ(C)+ and Ψ(C)− the set of positive and negative affine roots, respectively. The subsets Ψ(C)+ ∩ Ψnd and Ψ(C)− ∩ Ψnd will be denoted by Ψ(C)nd,+ and Ψ(C)nd,− respectively. Since no Hψ meets C, every ψ ∈ Ψ is either positive or negative, and −ψ is negative if and only if ψ is positive. Thus we have the disjoint union Ψ = Ψ(C)+ ∪ Ψ(C)− . Definition 1.3.19 Let C be a chamber. The set Ψ(C)0 consisting of those indivisible ψ ∈ Ψ(C)+ for which Hψ is a wall of C is called a basis of Ψ, and its elements are called simple affine roots. Note that C is uniquely determined by Ψ(C)0 , namely as the intersection of the half-spaces Aψ>0 for ψ ∈ Ψ(C)0 . Proposition 1.3.20 Let C ⊂ A be a chamber and let Ψ(C)0 ⊂ Ψ be the corresponding set of simple affine roots. Let S ⊂ W(Ψ) be the set of reflections along the elements of Ψ(C)0 . Then (W(Ψ), S) is a Coxeter system. Proof

This is [Bou02, Chapter V, §3, no.2, Theorem 1(i)].



Proposition 1.3.21 Every affine root system is in a natural way the direct sum of irreducible affine root systems. Proof Let W = W(Ψ). Choose a chamber C and let S ⊂ W be the set of reflection along the associated simple affine roots. Then (W, S) is a Coxeter system by Proposition 1.3.20. Write S = S1 ∪ · · · ∪ Sn such that the Si are pairwise orthogonal as in Definition 1.3.13 and n is maximal with this property. Let Wi ⊂ W be the subgroup generated by Si , let Vi⊥ be the subspace of V ∗ fixed by ∇Wi , and let Vi be the annihilator of Vi⊥ in V. According to [Bou02, Chapter V, §3, no. 7, Proposition 5] and the discussion preceding it, the subgroups Wi commute with each other, W = W1 ×· · ·×Wn , the set of subgroups {W1, . . . , Wn } is independent of the choice of chamber C, and V = V1 ⊕ · · · ⊕ Vn . Each Vi is stable under ∇W. As discussed in 1.2.16, this leads to the decomposition A = A1 × · · · × An , where Ai = A/Vi and Vi = V1 ⊕ · · · ⊕ Vi−1 ⊕ {0} ⊕ Vi+1 ⊕ · · · ⊕ Vn . Let Ψi ⊂ Ψ consist of those ψ ∈ Ψ such that ψ ∈ Vi∗ . Then Ψi ⊂ A∗i (cf. 1.2.15) and according to [Bou02, Chapter VI, §1, no. 2, Proposition 5] ∇Ψi is an irreducible root system and ∇Ψ = ∇Ψ1 ⊕ · · · ⊕ ∇Ψn . In particular, Ψ = Ψ1 ∪ · · · ∪ Ψn . Proposition 1.3.12 applied to both Ψ and Ψi implies that Ψi is an affine root system in A∗i , and Lemma 1.3.14 shows that it is irreducible. Its Weyl group is

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23

Wi by construction, and the isomorphism A → A1 × · · · × An identifies Ψ with  the direct sum Ψ1 ⊕ · · · ⊕ Ψn . Proposition 1.3.22 (1) (2) (3) (4)

Let C be a chamber and Δ = Ψ(C)0 .

The group W(Ψ) is generated by S = {rψ |ψ ∈ Δ}. W(Ψ) acts simply transitively on the set of bases of Ψ. Ψnd = W(Ψ) · Δ. Every ψ ∈ Ψ(C)+ is a non-negative integral linear combination of elements of Δ.

Assume that Ψ is irreducible. (5) Δ is a basis of A∗ . In particular the combination in (4) is unique. (6) The vertices {x0, . . . , x } of C are in bijection with Δ specified by ψi (x j ) = 0 if i  j and ψi (xi ) > 0. Proof (1) is [Bou02, Chapter V, §3, no. 1, Lemma 2]. (2) follows from Lemma 1.3.17 and the bijection between bases and chambers. (3) Let ψ ∈ Ψnd . Choose a chamber C that has Hψ as a wall. Choose w ∈ W(Ψ) such that wC = C by (2). Then Hwψ is a wall of C, so wψ ∈ Δ. Now assume that Ψ is irreducible. (5), (6) C is a simplex in A as discussed in Remark 1.3.16, therefore Δ is a basis of A∗ . Moreover, a vertex of C is the intersection of all walls of C except the one opposite to the vertex. (4) While this point is stated without assuming that Ψ is irreducible, it reduces to this case by Proposition 1.3.21. Let L denote the Z-lattice in A∗ spanned by  ψ ∨ )ψ and AR 3 implies that Δ. For any ψ ∈ Δ and η ∈ L we have rψ (η) = η − η( this lies in L. Now (1) implies that L is stable under W(Ψ) and (3) implies that L coincides with the lattice spanned by Ψ. This shows that every ψ ∈ Ψ(C)+ is an integral linear combination of elements of Δ. To show that it is non-negative, we evaluate ψ at each vertex of C and apply (6).  The possible non-uniqueness in Proposition 1.3.22(4) is explained as follows. Recall from Remark 1.3.16 that when Ψ = Ψ1 ⊕ Ψ2 and chamber C decomposes as C1 × C2 .    Ci . Then Ψ(C)0 = Ψi (Ci )0 . Lemma 1.3.23 Let Ψ = Ψi and C = Then ψ ∈Ψ(C)0 nψ ψ = 0 if and only if there exist c1, . . . , cn ∈ R such that ci = 0. ψ ∈Ψi (Ci )0 nψ ψ = ci and Proof

The “if” statement is clear. For the converse, taking the derivative of

24 Affine Root Systems and Abstract Buildings  ∗ ∗ Vi we see 0 n ψ = 0 and using the direct sum decomposition V = ψ ∈Ψ(C) ψ   ψ ∈Ψi (Ci )0 nψ ψ = 0, whence the claim. Remark 1.3.24 Consider two irreducible affine root systems Ψ1, Ψ2 with bases Δ1, Δ2 . Then the constant functional 1 in A∗i has the unique expressions a ψ = 1. Therefore, 1 ∈ (A1 × A2 )∗ has the two distinct expressions ψi ∈Δi ψi i ψi ∈Δi aψi ψi for i = 1, 2 in terms of the basis Δ = Δ1 ∪ Δ2 . In particular, we see that Ψ(C)0 is a basis of the vector space A∗ when Ψ is irreducible, but when Ψ is not irreducible then Ψ(C)0 is only generating, but not linearly independent. Next is the affine analog of [Bou02, Chapter VI, §1, no. 6, Corollary 1]. Proposition 1.3.25 Let C ⊂ A a chamber and α ∈ Ψ(C)0 . The affine reflection rα permutes the elements of Ψ(C)+ which are not proportional to α. Proof We apply Proposition 1.3.21 and see that we can replace Ψ by the irreducible factor that contains α, because all other irreducible factors are fixed by rα . Therefore we may assume that Ψ is irreducible.  α ∨ )α. Now since by Proposition For any β ∈ Ψ(C)0 , β  α, rα (β) = β − β( + 1.3.22(5), every root in Ψ(C) (respectively, Ψ(C)− = −Ψ(C)+ ) is a unique non-negative (respectively, non-positive) integral linear combination of roots  in Ψ(C)0 , the proposition is obvious. Proposition 1.3.26 Let C ⊂ A be a chamber and let Ψ(C)0 ⊂ Ψ be the corresponding set of simple affine roots. Let S ⊂ W(Ψ) be the set of reflections along the elements of Ψ(C)0 . Denote the length function of (W(Ψ), S) by . Then (1) Let w ∈ W(Ψ) and let s ∈ S be the reflection along α ∈ Ψ(C)0 . Then (sw) > (w) is equivalent to w −1 α ∈ Ψ(C)+ . (2) (w) = #(wΨ(C)nd,+ ∩ Ψ(C)nd,− ) for all w ∈ W(Ψ). (3) If w = s1 · · · sq is a reduced expression, with si the reflection along αi ∈ Φ(C)0 , then the set of roots of Ψ(C)nd,+ that are mapped onto negative roots by w is {sq · · · si+1 (αi ) | i = 1, . . . , q}. Proof (1) According to [Bou02, Chapter V, §3, no.2, Theorem 1(ii)] the condition (sw) > (w) is equivalent to the condition that the chambers C and wC are on the same side of the vanishing hyperplane Hα of α. This condition is in turn equivalent to the condition that α|wC > 0. Thus (sw) > (w) if and only if w −1 α ∈ Ψ(C)+ . (2) We induct on (w). When (w) = 1 then w is the reflection along some α ∈ Ψ(C)0 and the claim follows form Proposition 1.3.25. For (w) > 1 write

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25

w = sw  with (w ) = (w) − 1 and s ∈ S the reflection along some α ∈ Ψ(C)0 . Proposition 1.3.25 implies 



# wΨ(C)nd,+ ∩ Ψ(C)nd,− = # w  Ψ(C)nd,+ ∩ sΨ(C)nd,−

 = # w  Ψ(C)nd,+ ∩ ({α} ∪ (Ψ(C)nd,− − {−α}))

 = # w  Ψ(C)nd,+ ∩ Ψ(C)nd,− + 1 where the final equality comes from (1). (3) is an immediate consequence of (1) and (2).



The following is the main construction of reduced affine root systems. Construction 1.3.27 Let Φ be a (possibly non-reduced) finite root system in V ∗ . Take A = V and define ΨΦ = {a + n | a ∈ Φ, n ∈ Ia }, where Ia = Z if a is non-divisible, and Ia = 2Z + 1 if a is divisible. Then Ψ is an affine root system according to Proposition 1.3.12, reduced by construction. It is clear that ∇ΨΦ = Φ. In particular, ∇Ψ need not be reduced even if Ψ is reduced. In other words, the inclusion (∇Ψ)nd ⊂ ∇(Ψnd ) above can be proper. Example 1.3.28 We now give an example of a non-reduced affine root system. Let V = Rn with the standard scalar product. Let {e1, . . . , en } be the standard basis. Identifying V ∗ with V via this scalar product we consider the non-reduced root system BCn . It is the set of elements Φ = {±ei | i = 1, . . . , n} ∪ {±2ei | i = 1, . . . , n} ∪ {±ei ± e j | 1  i  j  n}. Let Ψ = {a + k | a ∈ Φ, k ∈ Z}. Then Ψ is an affine root system by Proposition 1.3.12 and is obviously non-reduced. The following rather innocuous construction will turn out to be quite useful when discussing isomorphisms. Construction 1.3.29 Let Ψ ⊂ A∗ be an affine root system and let s ∈ R× . Then sΨ ⊂ A∗ defined by sΨ = {sψ | ψ ∈ Ψ} and ∇(sψ)∨ = s−1 ψ ∨ , is also an affine root system. We say that sΨ is obtained from Ψ by rescaling. Note that Hψ = Hsψ and rψ = rsψ . Thus Ψ and sΨ share the same hyperplane arrangement and the same affine and extended affine Weyl groups. Construction 1.3.30 Let −, − be a scalar product on V that is invariant under W(∇Ψ). For example, we can take the canonical one [Bou02, Chapter VI, §1, no. 1, Proposition 3]. Identifying V with V ∗ via this scalar product, we  η .  It is degenerate. A obtain a symmetric bilinear form on A∗ by ψ, η = ψ,

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vector ψ ∈ A∗ is isotropic if and only if it is a constant functional. In particular, each ψ ∈ Ψ is anisotropic, so we can define ψ ∨ = 2ψ/ ψ, ψ . Then ∇(ψ ∨ ) is equal to the element ψ ∨ of Axiom AR 2. Note that Hψ∨ = Hψ . The reflection rψ | A is the orthogonal reflection along the hyperplane Hψ . Proposition 1.3.31 If Ψ is an affine root system, then so is Ψ∨ = {ψ ∨ | ψ ∈ Ψ}, called the dual affine root system. We have rψ∨ = rψ , hence W(Ψ) = W(Ψ∨ ), and also ∇(Ψ∨ ) = (∇Ψ)∨ . Proof

Immediate from Proposition 1.3.12.



Remark 1.3.32 We alert the reader that (ΨΦ )∨ and ΨΦ∨ are distinct affine root systems when Φ is reduced and not simply laced. More precisely, assuming that Φ is reduced and irreducible, we have (ΨΦ∨ )∨ = {a + n | a ∈ Φ, n ∈ Ia }, where Ia = Z when a is long, and Ia =  −1 Z when a is short, where  is the integer ratio of the squares of the two different root lengths in Φ. The definition of Ψ∨ involves a scalar product because it is not clear how to interpret A∗∗ as the dual of an affine space in a natural way. This is different from the case of finite root systems. One can canonify Ψ∨ by using the canonical scalar product of [Bou02, Chapter VI, §1, no. 1, Proposition 3]. If one uses an arbitrary scalar product, then Ψ∨ is independent of that choice up to rescaling (Construction 1.3.29) when Ψ is irreducible. When Ψ is reducible, the dual of each irreducible component would be well-defined up to rescaling. We will see below that the isomorphism class of Ψ∨ is independent of the choice of scalar product. Remark 1.3.33 Consider an affine root system Ψ and its dual Ψ∨ . The set of vanishing hyperplanes for Ψ∨ is the same as that for Ψ. In particular, a chamber C for Ψ is also a chamber for Ψ∨ . If Ψ is reduced and Ψ(C)0 is the corresponding basis, then Ψ∨ (C)0 = {ψ ∨ | ψ ∈ Ψ(C)0 }. Proposition 1.3.3 gave one way of obtaining a finite root system from an affine root system, namely by taking the derivative. The following proposition gives a different way, by looking at a neighborhood of a point in A. Notation 1.3.34 If Ω and Ω  are two subsets of a topological space and Ω is contained in the closure of Ω , we will write Ω ≺ Ω . Proposition 1.3.35

Let x ∈ A. We set Ψx = {ψ ∈ Ψ | ψ(x) = 0}.

(1) Ψx is a finite root system in the subspace of A∗x that it generates.

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27

(2) The map ∇ restricted to Ψx is injective. (3) Ψx depends only on the facet F containing x and may thus be denoted ΨF . (4) The subset {ψ ∨ | ψ ∈ Ψx } ⊂ Ψ∨ is identified with the root system dual to Ψx . (5) Let W(Ψ)x be the stabilizer of x in W(Ψ). The action of W(Ψ)x on Ψx identifies W(Ψ)x with W(Ψx ). (6) If C is a chamber whose closure contains F, then Ψ(C)0 ∩ ΨF is a basis for ΨF . (7) The set of chambers C whose closure contains F is in bijection with the set of Weyl chambers in ΨF , the bijection being given by C → ΨF (C)+ = ΨF ∩ Ψ(C)+ . (8) More generally, the set of facets whose closure contains F is in bijection with the set of parabolic subsets of ΨF , the bijection being given by F  → ΨF (F )+ = {ψ ∈ ΨF |ψ(F )  0}. If F1 ≺ F2 then ΨF (F2)+ ⊂ ΨF (F1). It is obvious that if Ψ is reduced then so is Ψx . Proof (2) If ψ, η ∈ Ψ have equal derivative, then there exists a c ∈ R such that η = ψ + c. Hence, unless c = 0, that is η = ψ, both of these affine roots cannot vanish at x. This shows that ∇ restricted to Ψx is injective. As Φ = ∇Ψ is finite by Proposition 1.3.3, we conclude that Ψx is also finite. (1) The axioms in the definition of finite root systems for Ψx follow from the axioms in Definition 1.3.1 for Ψ and the finiteness established in (2). (3) If ψ ∈ Ψ vanishes on x, it vanishes on F. Thus Ψx only depends on F and is contained in A∗F . (4) is immediate. (5) follows from Lemma 1.3.17(5). (6) We will use [Bou02, Chapter VI, §1, no. 7, Corollary 3]. Let Ψx (C)± = Ψx ∩Ψ(C)± and Ψx (C)0 = Ψx ∩Ψ(C)0 . All elements of Ψx (C)0 are indivisible by construction. By Proposition 1.3.22(4) every element of Ψx (C)+ can be written as a non-negative integral linear combination nψ ψ with ψ ∈ Ψ(C)0 . Each such ψ evaluates non-negatively at x, while the linear combination vanishes at x. This shows nψ = 0 if ψ(x)  0. To see that Ψx (C)0 is linearly independent we use Lemma 1.3.23. Writing C = C1 × · · · × Cn and x = (x1, . . . , xn ), a linear relation among Ψx (C) would imply an affine relation among Ψi,xi (Ci ), but since all members of the latter vanish at xi , that relation is in fact linear, and by Proposition 1.3.22(5) it must be trivial. (7) Any other basis of ΨF is obtained by applying an element of W(ΨF ) = W(Ψ)F to Ψx (C)0 . Applying this element to C produces another chamber containing F in its closure. Conversely, any other chamber containing F in its

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closure is obtained by applying an element w ∈ W(Ψ) to C. Then F, w −1 F are both contained in the closure of C, so Lemma 1.3.17 implies w ∈ W(Ψ)F . (8) Let now F  be a facet whose closure contains F. It is immediate that ΨF (F )+ is a parabolic subset. It is also immediate that if F  ≺ F  then ΨF (F )+ ⊂ ΨF (F )+ . Conversely let P ⊂ ΨF be a parabolic subset. According to [Bou02, Chapter VI, §1, no. 7, Proposition 20] and the previous point, there exists a chamber C and a subset S ⊂ ΨF (C)0 such that P is the union of ΨF (C)+ and the subset of ΨF consisting of linear combinations of elements of S with non-positive integer coefficients. Then Hψ ∩ C F  := ψ ∈S

is a facet contained in the closure of C. We have ΨF (F )+ = ΨF (C)+ ∪ (ΨF (C)− ∩ ΨF (F )+ ). An element of ΨF (C)− ∩ ΨF (F )+ is a non-negative linear combination of elements of ΨF (C)0 that vanishes on F , thus a non-negative linear combination  of elements of S. We conclude P = ΨF (F )+ . The subset Φx := ∇Ψx of Φ = ∇Ψ is a subsystem and the isomorphism ∇ : A∗x → V ∗ identifies Ψx with Φx . The relation ∇(ψ ∨ ) = (∇ψ)∨ implies that {a∨ ∈ Φ∨ | a ∈ Φx } is the dual of Φx . Corollary 1.3.36

Let x ∈ A. The subset Φx := Φnd ∩ (Φx ∪ 12 Φx )

is a reduced root system in V ∗ with the same Weyl group as Ψx , where (−)nd denotes the set of non-divisible roots. Proof This is immediate from the fact that Ψx is a root system.



Remark 1.3.37 When Φ is reduced then Φx = Φx . In general neither of Φx and Φx is contained in the other. Both Φx and Φx will be relevant for Bruhat–Tits theory. We will see (cf. Theorem 8.4.10) that Ψx  Φx is the root system of the maximal reductive quotient of the special fiber of the parahoric group scheme associated to x, while the root system Φx is the set of a ∈ Φnd for which the filtration of the root subgroup Ua (k) associated to the point x has a break at 0. Example 1.3.38 Let Φ = {−2a, −a, a, 2a} be the root system of type BC1 and let Ψ = {−a, a} × Z ∪ {−2a, 2a} × (2Z + 1)

1.3 Affine Root Systems

29

as in Construction 1.3.27. Let x, y ∈ A = V = R be determined by a(x) = 0 and a(y) = 1/2. Then Φx = Φx = Φy = {−a, a}, while Φy = {−2a, 2a}. Let Φ be the root system of type BC2 and let Ψ = {±a, ±b, ±(a + b), ±(a + 2b)} × Z ∪ {±2b, ±(2a + 2b)} × (2Z + 1) be as in Construction 1.3.27. Let {a, b} be a set of simple roots so that the corresponding positive roots are {a, b, a + b, 2b, a + 2b, 2a + 2b}. Let x, y ∈ A = V = R2 be the elements specified by a(x) = b(x) = 0, a(y) = 1/2, b(y) = 0. + + Then Φ+x = Φ+ x = {a, b, a + b, a + 2b}, Φy = {b, 2a + 2b}, Φy = {b, a + b}. These examples show that Φx need not be a closed subsystem of Φ. For further examples we point the reader to Figure 1.3.1, from which one sees visually that, if x is a vertex, the following cases occur: when Ψ is of type A2 , then Ψx is always of type A2 ; when Ψ is of type C2 , then Ψx is either of type C2 or A1 × A1 ; when Ψ is of type G2 , then Ψx is of type G2 or A2 or A1 × A1 ; if Ψ is of type BC2 , then Ψx is of type C2 ⊂ BC2 , or B2 ⊂ BC2 , or A1 × A1 . We now recall the concept of special points. There are in fact two different definitions of this notion. Definition 1.3.39

A point x ∈ A is called

(1) special, if for each ψ ∈ Ψ there exists ψ  ∈ Ψx such that Hψ and Hψ are parallel, and (2) extra special, if there exist ψ1, . . . , ψ ∈ Ψx such that {ψ 1, . . . , ψ  } is a basis of Φ = ∇Ψ. Remark 1.3.40 There is some discrepancy in the literature regarding the concept of a “special” point. We have decided to follow the convention used in [Bou02], [BT72], and [Tit74]. On the other hand, [Mac72] calls “special” what we have called here “extra special.” Remark 1.3.41 The notions of “special” and “extra special” do not change if we replace Ψ by Ψnd , cf. Remark 1.3.5. Therefore we can always reduce considerations to the case that Ψ is reduced. If Ψ = Ψ1 ⊕ Ψ2 , then a point x = (x1, x2 ) is (extra) special if and only if xi is such for the affine root system Ψi . This allows one to reduce considerations to the case that Ψ is irreducible. Lemma 1.3.42 Let x ∈ A. The map ∇ : W(Ψ) → W(Φ) restricts to an injection W(Ψ)x → W(Φ), which is surjective if and only if x is special, in which case it realizes W(Ψ) as the semi-direct product of W(Φ) with the subgroup of W(Ψ) consisting of translations.

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Proof This is [Bou02, Chapter V, §3, no. 10, Proposition 9].



Proposition 1.3.43 (1) Extra special points exist. (2) An extra special point is special. (3) A (extra) special point is a vertex. (4) If Φ(= ∇Ψ) is reduced, then a point is special if and only if it is extra special. (5) If C is a chamber, then there is a vertex of C which is extra special. (6) If x is extra special, then ∇ identifies (Ψx )nd with Φnd . (7) If Ψ is irreducible, a given basis can be enumerated {ψ0, . . . , ψ } so that {ψ 1, . . . , ψ  } is a basis of Φ. Proof (1) Let a1, . . . , a be a basis of Φ. Choose ψi ∈ Ψ with ψ i = ai . The linear independence of ψ i s implies that the hyperplanes Hψi intersect in a single point of A. This point is extra special. (2) Let ψ1, . . . , ψ ∈ Ψx be such that {ψ 1, . . . , ψ  } is a basis of Φ. The map ∇ : W(Ψ) → W(Φ), which is surjective by Proposition 1.3.3, must then remain surjective when restricted to W(Ψx ) ⊂ W(Ψ). From Proposition 1.3.35(5) we have W(Ψx ) = W(Ψ)x . The claim follows from Lemma 1.3.42. (3) If the point x ∈ A is special or extra special there exist ψ1, . . . , ψ ∈ Ψx whose derivatives are linearly independent, so x is a vertex. (4) Assume that Φ is reduced and that x is special. Let Δ ⊂ Φ be a basis. For each a ∈ Δ there exists ψ ∈ Ψ with ψ = a. Let ψ  ∈ Ψx be such that Hψ and  Since −ψ  ∈ Ψx we are done. Hψ are parallel. Since Φ is reduced, ψ  = ±ψ. (5) From (1) and (3) we know that there exists a chamber C that has an extra special vertex. Since the property of being an extra special vertex is preserved under the action of W(Ψ), the claim follows from Lemma 1.3.17. (6) We have already noted that ∇ is an isomorphism Ψx → Φx . By assumption Φx contains a basis of Φ. Since a basis consists of indivisible roots, it lies nd in Φnd x and hence comes from (Ψx ) . (7) A basis corresponds to a chamber, and this chamber has an extra special vertex x according to (5). Enumerate the basis as {ψ0, . . . , ψ }, with ψ0 corresponding to x. By Proposition 1.3.35(6) {ψ1, . . . , ψ } is a basis of Ψx and the claim follows from (6).  The following example shows the existence of a point that is special but not extra special. Example 1.3.44 Let A = V = R. Let a ∈ A∗ be the identity function and Φ = {−2a, −a, a, 2a} be the root system of type BC1 . Let Ψ = ΨΦ . Thus Ψ = {−2a + 2Z + 1, −a + Z, a + Z, 2a + 2Z + 1}. The point 0 ∈ A is extra special. Since Ψ1/2 = {−2a, 2a}, the point 1/2 ∈ A is special, but not extra special.

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In type BC2 , the difference between special and extra special vertices can be seen visually in Figure 1.3.1: the extra special vertices are those where 4 solid lines meet, while the special but not extra special vertices are those where 2 solid and 2 dotted lines meet. Lemma 1.3.45 Let Ψ be an affine root system and let C be a chamber. For all  η∨ )  0. ψ, η ∈ Ψ(C)0 , ψ  η, we have ψ( Proof

This is [Bou02, Chapter V, §3, no 4, Proposition 3].



Lemma 1.3.46 Let Φ be a finite root system. If a ∈ Φ is divisible in Z[Φ], it is divisible in Φ. Proof Assume that a is indivisible in Φ. Then there exists a basis Δ of Φ containing a. Since Z[Φ] is freely generated by Δ, a cannot be divisible in Z[Φ].  Proposition 1.3.47 Assume that Ψ is irreducible and let Δ be a basis corresponding to a chamber C. (1) There exists a collection of non-negative integers (nψ )ψ ∈Δ without common denominator such that  nψ ψ (1.3.1) c(Ψ) := ψ ∈Δ

is a constant functional. (2) This collection is uniquely determined, all integers are positive, and the constant c(Ψ) is positive. (3) A vertex x of C is extra special if and only if it is special and the integer nψ , with ψ ∈ Δ corresponding to the vertex x (cf. Proposition 1.3.22(6)), is equal to 1. (4) If a finite integral linear combination of elements of Ψ is a constant, then this constant is an integral multiple of c(Ψ). (5) c(Ψ) is independent of C. Proof (1) Let x be an extra special vertex of C. Enumerate the simple affine roots Ψ(C)0 = {ψ0, . . . , ψ } so that ψ0 (x) > 0, cf. Proposition 1.3.22(6). Then, according to Lemma 1.3.45, −ψ 0 is a positive root in Φ. So there exists a collection of non-negative integers n0, . . . , n without a common denominator such that −ψ 0 = i=1 ni ψ i . Thus c(Ψ) = ψ0 + i=1 ni ψi has zero derivative, and is therefore constant. (2) The value of the linear functional c(Ψ) at x equals ψ0 (x) > 0, which shows that c(Ψ) is positive. Evaluating c(Ψ) at every other vertex of C shows that each ni is positive. Since Ψ(C)0 is a basis of A∗ by Proposition 1.3.22(5)

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and since the constant linear functionals form a 1-dimensional subspace of A∗ we see that the collection (ni ) is uniquely determined by the properties that it consist of positive integers without common denominator. (3) The construction in (1) had the byproduct that nψ0 = 1 for ψ0 ∈ Δ corresponding to the extra special vertex x that was used to produce the collection (nψ ). The uniqueness statement of (2) shows that the collection (nψ ) is independent of the choice of x and we conclude that nψ = 1 for ψ corresponding to any extra special vertex of C. Conversely, let x be a special vertex of C with nψ = 1, where ψ ∈ Ψ(C)0 is the affine root associated to x. We numerate Ψ(C)0 = {ψ0, . . . , ψ } so that ψ0 = ψ. Then {ψ1, . . . , ψ } is a basis of Ψx by Proposition 1.3.35(6). We will show that {ψ 1, . . . , ψ  } is a basis of Φ. According to [Bou02, Chapter VI, §1, no. 7, Corollary 3], it is enough to show that every element of a ∈ Φ can be expressed as an integral linear combination of {ψ 1, . . . , ψ  } with the same sign. Since x is special, there exists b ∈ Φx that is proportional to a, i.e. a = r b with r ∈ {±1, ±2, ±1/2}. If b is divisible in Φx , after replacing it with b/2, we assume that b is not divisible in Φx . It is enough to show that r  ±1/2, i.e. that b is indivisible in Φ. For this, the fact that a is an integral linear combination of {ψ 0, . . . , ψ  } and that nψ0 = 1 shows that a is an integral linear combination of {ψ 1, . . . , ψ  }. If r = ±1/2, then b would be an integral linear combination of {2ψ 1, . . . , 2ψ  } and would therefore be a divisible element of the root lattice of Ψx . This is a contradiction to our choice by Lemma 1.3.46. (4) Assume that c  is a constant that is a finite integral linear combination of elements of Ψ. Using Proposition 1.3.22(5) we see that c  is a unique integral linear combination of elements of Ψ(C)0 . Thus c  = ni ψi with ni ∈ Z, and we find that ni/ni = c /c(Ψ) for all i. Taking i so that the vertex xi is extra special, we see from (3) that c /c(Ψ) = ni ∈ Z. (5) By (4), c(Ψ) is the smallest positive constant that is an integral linear combination of elements of Ψ, and hence it does not depend on C.  Remark 1.3.48 Let Φ be an irreducible reduced finite root system and Ψ = ΨΦ . If a1, . . . , a is a basis for Φ, a0 is the highest root, and we set ψ1 = a1, . . . , ψ = a , and ψ0 = 1 − a0 , then ψ0, . . . , ψ is a basis for Ψ. The integers n0, . . . , n of (1.3.1) are specified by n0 = 1 and a0 = i=1 ni ai . Consider now the dual affine root system Ψ∨ . By Remark 1.3.33 we know that ψ0∨, . . . , ψ∨ is a basis for Ψ∨ . Thus i=0 21 ni ψi , ψi ψi∨ is constant. So the integers (1.3.1) for Ψ∨ are obtained by taking the sequence n0 ψ0, ψ0 , . . . , n ψ , ψ and dividing each term in it by the greatest common divisor of the sequence.

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33

This procedure produces the integers (1.3.1) for all irreducible reduced affine root systems except BCn , in which case one can compute them by hand. The results are compiled in Table 1.3.5. The following proposition will be used in the study of isomorphisms (cf. Definition 1.3.50) as well as the classification of affine root systems. Proposition 1.3.49 Let Ψ be an affine root system. For each ψ ∈ Ψ let uψ ∈ R be the smallest positive number such that ψ  = ψ + uψ ∈ Ψ. Let tψ = rψ rψ . Then  ψ ∨ ). (1) For x ∈ A and η ∈ A∗ we have tψ (x) = x −uψ ψ ∨ and tψ (η) = η+uψ η( (2) For any ψ ∈ Ψ and r ∈ R, ψ +r ∈ Ψ if and only if r is an integral multiple of uψ . (3) uψ depends only on the Weyl orbit of the derivative of ψ; write ua with  a = ψ. (4) uψ is a positive integral multiple of c(Ψ). Proof (1) is an immediate computation. (2) Let m ∈ Z and r ∈ R. Then (1) implies tψm (ψ + r) = ψ + r + 2uψ m. Taking r = 0 and r = uψ we see ψ + muψ ∈ Ψ for all m ∈ Z. Conversely, if ψ + r ∈ Ψ, then ψ + r + 2uψ m = tψm (ψ + r) ∈ Ψ for all m ∈ Z. Choosing m appropriately we obtain an affine root ψ + r  with −uψ < r   uψ . It is enough to show that r  = 0 or r  = uψ . If that were not the case, then either r  ∈ (−uψ , 0) or (0, uψ ). In the first case we apply the translation rψ rψ+r  to ψ + r  to obtain the affine root ψ − r , which reduces to the second case, namely r  ∈ (0, uψ ). That is however a contradiction to the minimality of uψ . (3) Since the difference of any two affine roots with equal derivative is a constant, it follows from (2) that uψ depends only on the derivative of ψ. The surjectivity of W(Ψ) → W(∇Ψ) due to Proposition 1.3.3 reduces to showing uwψ = uψ with w ∈ W(Ψ). However w(ψ + ua ) = wψ + ua and (2) implies uwψ |uψ . Replacing ψ by wψ and w by w −1 we see the opposite divisibility relation, hence uwψ = uψ . (4) This follows from Proposition 1.3.47(4).  We will now introduce and study the concept of isomorphisms of affine root systems, as a preparation for the classification of affine root systems. Definition 1.3.50 Let Ai be an affine space under the R-vector space Vi , Ψi ⊂ A∗i an affine root system, for i = 1, 2. (1) A strong isomorphism Ψ1 → Ψ2 is an isomorphism f : A2 → A1 of

34

Affine Root Systems and Abstract Buildings affine spaces such that f ∗ : A∗1 → A∗2 induces a bijection Ψ1 → Ψ2 such that for η := ( f ∗ )−1 (ψ), (∇ f )(ψ ∨ ) = η∨ for all ψ ∈ Ψ2 .

(2) An isomorphism Ψ1 → Ψ2 is a set theoretic bijective map ϕ : Ψ1 → Ψ2 with the following property. If ψ, η ∈ Ψ1 and r, s are integers such that r ψ + sη ∈ Ψ1 , then ϕ(r ψ + sη) = r ϕ(ψ) + s ϕ(η). The analogous property is required of ϕ−1 . Remark 1.3.51 A strong isomorphism Ψ1 → Ψ2 is uniquely determined by the bijective map Ψ1 → Ψ2 that it induces, because Ψ1 generates A∗1 and an affine map A2 → A1 is uniquely determined by its dual A∗1 → A∗2 . Therefore, being a strong isomorphism is a property of a bijection Ψ1 → Ψ2 . It is clear that this property is stronger than the property of being an isomorphism as in (2) of the above definition. A strong isomorphism is thus an example of an isomorphism. Another example of an isomorphism is the natural bijection Ψ → sΨ for any s ∈ R× , cf. Example 1.3.55 below. We will show in Proposition 1.3.54 below that a general isomorphism Ψ1 → Ψ2 arises as the composition of those two examples applied to each individual irreducible factor of Ψ1 , equivalently Ψ2 . Proposition 1.3.52 Assume that Ψ1, Ψ2 are irreducible affine root systems. Any isomorphism ϕ : Ψ1 → Ψ2 extends uniquely to a vector space isomorphism f ∗ : A∗1 → A∗2 that sends the line of constants R ⊂ A∗1 to the line of constants R ⊂ A∗2 and hence descends to vector space isomorphism V1∗ → V2∗ . Proof Since Ψ1 generates A∗1 , a linear extension A∗1 → A∗2 of ϕ is necessarily unique. To show that it exists, fix a basis Δ of Ψ1 . According to Proposition 1.3.22(5) Δ is also a basis of the vector space A∗1 . Let f ∗ : A∗1 → A∗2 be the linear map extending ϕ|Δ . We will now show that f ∗ extends ϕ. For this purpose, let Θ be the subset of Ψ1 consisting of ψ ∈ Ψ1 such that f ∗ (ψ) = ϕ(ψ). Obviously, Δ ⊂ Θ. We claim that for all ψ, η ∈ Θ, rψ (η) ∈ Θ. We have the following:  ψ ∨ )ψ) = f ∗ (η) − η(  ψ ∨ ) f ∗ (ψ) = ϕ(η) − η(  ψ ∨ ) ϕ(ψ) f ∗ (rψ (η)) = f ∗ (η − η( = ϕ(rψ (η)). This proves that rψ (η) ∈ Θ. Now since Δ ⊂ Θ, and the rα , for α ∈ Δ, generate the affine Weyl group W(Ψ1 ), we see that Θ is stable under the action of W(Ψ1 ). As W(Ψ1 ) · Δ = Ψ1nd (Proposition 1.3.22(3)), and for a multipliable root ψ ∈ Ψ1 , ϕ(2ψ) = 2ϕ(ψ), we conclude that Θ = Ψ1 , that is, f ∗ indeed extends ϕ. To see that f ∗ is an isomorphism, we apply the same argument to ϕ−1 and obtain a linear extension g ∗ : A∗2 → A∗1 . The compositions f ∗ ◦ g ∗ and g ∗ ◦ f ∗

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35

are equal to the identity when restricted to a basis of Ψ2 resp. Ψ1 , and hence are equal to the identity on A∗2 resp. A∗1 . It remains to prove that f ∗ (1 A1 ) ∈ R · 1 A2 . Choose any affine root ψ1 ∈ Ψ1 and let ψ2 = f ∗ (ψ1 ). Consider the sequence ψ1 + nuψ1 ∈ Ψ1 for n = 1, 2, 3, . . ., cf. Proposition 1.3.49. The image sequence f ∗ (ψ1 + nuψ1 1 A1 ) consists of elements of Ψ2 . Since according to Proposition 1.3.3, the set of derivatives of elements of Ψ2 is finite, there exist n1 < n2 such that f ∗ (ψ1 + n1 uψ1 1 A1 ) and f ∗ (ψ1 + n2 uψ1 1 A1 ) have equal derivative. Hence, f ∗ (ψ1 + n2 uψ1 1 A1 ) − f ∗ (ψ1 + n1 uψ1 1 A1 ) = c1 A2 , where c ∈ R. On the other hand, c1 A2 = f ∗ ((ψ1 + n2 uψ1 1 A1 ) − (ψ1 + n1 uψ1 1 A1 )) = (n2 − n1 )uψ1 f ∗ (1 A1 ), so f ∗ (1 A1 ) = (c/((n2 − n1 )uψ1 ))1 A2 .



Lemma 1.3.53 Let ψ, η ∈ Ψ. Assume that their derivatives ψ and η are linearly independent. Let r and s be the largest non-negative integers such that  η∨ ) = r − s. ψ − rη, ψ + sη ∈ Ψ. Then ψ( Proof Since ψ and η have been assumed to be linearly independent, there is a x ∈ A where both ψ and η vanish. So ψ − rη, ψ + sη ∈ Ψx , which is a finite root system according to the Proposition 1.3.35(1). The claim now follows from [Bou02, Chapter VI, §1, no. 3, Proposition 9].  Proposition 1.3.54 Let Ψ1 and Ψ2 be irreducible affine root systems and ϕ : Ψ1 → Ψ2 be an isomorphism . Let f ∗ : A∗1 → A∗2 be the extension of ϕ as in Proposition 1.3.52. (1) Let α1 ∈ A∗1 and η1 ∈ Ψ1 . We denote f ∗ (α1 ) by α2 and ϕ(η1 ) by η2 . Then α 1 (η1∨ ) = α 2 (η2∨ ). (2) There exists an ε ∈ {±1} such that f ∗ (c(Ψ1 )1 A1 ) = εc(Ψ2 )1 A2 . (3) If c(Ψ1 ) = c(Ψ2 ), and ε = +1, then ϕ is a strong isomorphism. Proof (1) Since Ψ1 spans A∗1 it is enough to assume α1 ∈ Ψ1 . By Proposition 1.3.52, f ∗ descends to an isomorphism V1∗ → V2∗ . Therefore, if α 1 and η1 are linearly independent, then so are α 2 and η2 , and the claim then follows from Lemma 1.3.53. If α 1 and η1 are linearly dependent, then α 1 = c η1 for some c ∈ R (in fact, c ∈ {±1, ±2, ± 12 }), and then α 2 = c η2 , implying α 1 (η1∨ ) = 2c = α 2 (η2∨ ). (2) To see that f ∗ (c(Ψ1 )1 A1 ) = ε c(Ψ2 )1 A2 , let (nψ )ψ ∈Δ be positive inte gers such that ψ ∈Δ nψ ψ = c(Ψ1 )1 A1 . Then according to Proposition 1.3.52,

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f ∗ (c(Ψ1 )1 A1 ) is a constant functional on A2 . But  nψ ϕ(ψ), f ∗ (c(Ψ1 )1 A1 ) = ψ ∈Δ

hence according to Proposition 1.3.47(4), there exists an integer n2 such that f ∗ (c(Ψ1 )1 A1 ) = n2 c(Ψ2 )1 A2 . Applying the same reasoning to ϕ−1 , we conclude that there exits an integer n1 such that ( f ∗ )−1 (c(Ψ2 )1 A2 ) = n1 c(Ψ1 )1 A1 . Hence, c(Ψ1 )1 A1 = ( f ∗ )−1 f ∗ (c(Ψ1 )1 A1 ) = ( f ∗ )−1 (n2 c(Ψ2 )1 A2 ) = n1 n2 c(Ψ1 )1 A1 . So, n1 n2 = 1, implying that n1 = n2 = ±1. (3) If c(Ψ1 ) = c(Ψ2 ), and ε = +1, then (2) implies f ∗ (1 A1 ) = 1 A2 . Now according to Proposition 1.2.10, f ∗ is the dual of an affine isomorphism  f : A2 → A1 . Example 1.3.55 Let Ψ ⊂ A∗ be an irreducible affine root system. Let s ∈ R× , sΨ be the affine root system as in 1.3.29, and Δ be a basis of Ψ. Then |s|Δ := {|s| ψ | ψ ∈ Δ} is a basis of sΨ. So c(sΨ) = |s| c(Ψ). Now let ϕ : Ψ → sΨ be the natural isomorphism ψ → sψ, for ψ ∈ Ψ. Then the extended isomorphism f ∗ (as in Proposition 1.3.52) is clearly the automorphism of A∗ defined by x → s x for x ∈ A∗ . Hence, f ∗ (c(Ψ)1 A∗ ) = sc(Ψ)1 A∗ = εc(sΨ)1 A∗ , where, ε = s/|s|. Thus ε = +1 if and only if s is positive. Proposition 1.3.56 Let ϕ : Ψ → Ψ  be an isomorphism of affine root systems. Then there exists an affine root system Ψ  ⊂ A∗ such that (1) If Ψ = Ψ1 ⊕ · · · ⊕ Ψn is the decomposition of Ψ into irreducible pieces and A = A1 × · · · × An is the corresponding decomposition of the affine space A, then Ψ  = r1 Ψ1 ⊕ · · · ⊕ rn Ψn for some r1, . . . , rn ∈ R× . (2) The composition ϕ  : Ψ  → Ψ  of ϕ with the obvious bijection Ψ  → Ψ is a strong isomorphism.   Proof Let Ψi = ϕ(Ψi ) ⊂ Ψ . As Ψ = i Ψi , we see that Ψ  = i Ψi. Hence, Ψ  = Ψ1 ⊕ · · · ⊕ Ψn . For each i, ϕ | Ψi : Ψi → Ψi is an isomorphism. Let fi∗ : A∗i → Ai ∗ be its extension and let ri ∈ R such that fi∗ (c(Ψi )1 Ai ) = ri c(Ψi)1 A , and let Ψi = i ri Ψi . Then Proposition 1.3.52 shows that the composition of ϕ|Ψi with the  natural bijection Ψi → Ψi is a strong isomorphism.

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Corollary 1.3.57 Let ϕ : Ψ → Ψ  be an isomorphism. There exists an isomorphism f : A → A that identifies the hyperplane arrangement of Ψ  with that of Ψ, and in particular is equivariant for the action of the extended affine Weyl groups. Proof Using Proposition 1.3.21 decompose Ψ = Ψ1 ⊕ · · · ⊕ Ψn . Let Ψ  = r1 Ψ1 ⊕ · · · ⊕ rn Ψn as in Proposition 1.3.56(1). The hyperplane arrangement of Ψ is the same as that of Ψ , and that in turn is the same as that of Ψ  due to Proposition 1.3.56(2).  Remark 1.3.58 Our notion of a strict isomorphism is the analog of the notion of isomorphism introduced in [Mac72, §2]. On the other hand, in [Mac72, §3] the notion of “similarity” is introduced, according to which Ψ, Ψ  are called similar, if there exist irreducible affine root systems Ψ1, . . . , Ψn and non-zero real numbers r1, . . . , rn such that Ψ  Ψ1 ⊕ · · · ⊕ Ψn and Ψ   r1 Ψ1 ⊕ · · · ⊕ rn Ψn . Corollary 1.3.57 shows that our notion of isomorphism of Definition 1.3.50(2) recovers Macdonald’s notion of similarity. We now turn to the problem of classifying affine root systems. We will see in Proposition 1.3.67 below that every irreducible reduced affine root system is isomorphic to either ΨΦ or ΨΦ∨ of Construction 1.3.27 for some irreducible (possibly non-reduced) finite root system Φ. The non-reduced affine root systems can be easily enumerated separately. As before, we denote the finite root system ∇Ψ by Φ in what follows. Lemma 1.3.59 Let Ψ be a reduced affine root system. If ψ ∈ Ψ and r ∈ R are such that 2ψ + r ∈ Ψ, then r = muψ for an odd integer m. In particular,  u2a = 2ua for a = ψ. Proof Let μ ∈ R be the smallest non-negative number such that 2ψ + μ ∈ Ψ. We claim that μ = ua . Since Ψ is reduced we know μ > 0. We have r2ψ+μ (ψ) = −(ψ + μ), thus ψ + μ ∈ Ψ, so Proposition 1.3.49(2) implies μ = r ua with some positive integer r. If r = 2 then both ψ + ua and 2ψ + 2ua are affine roots, which contradicts the assumption that Ψ is reduced. If r  3 there exists an integer r/4  r  < r/2. Then −rψ+r  ua (2ψ + r ua ) = 2ψ + (4r  − r)ua ∈ Ψ. But then 0 < (4r  − r)ua < r ua = μ contradicts the minimality of μ. Therefore r = 1 is the only possibility, confirming μ = ua . From the definition of μ it follows that there is no affine root between 2ψ

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and 2ψ + μ. But −rψ (2ψ + μ) = 2ψ − μ ∈ Ψ and we see that there is no affine root between 2ψ − μ and 2ψ. Thus there is no affine root between 2ψ − μ and  2ψ + μ and Proposition 1.3.49(2) shows that u2a = 2μ = 2ua . ∗ Proposition 1.3.60 Let Ψ = {u−1 a a | a ∈ Φ}. Then Ψ is a root system in V ∨ with the same Weyl group as Φ and its dual is {ua a | a ∈ Φ} ⊂ V. If Ψ is nd reduced, then Ψ = {u−1 a a | a ∈ Φ }, so Ψ is also reduced.

Proof We verify [Bou02, Chapter VI, §1, no. 1, Definition 1]. By construction Ψ is finite, does not contain 0, and spans V ∗ , hence (RSI ). It is stable under W(Φ) according to Proposition 1.3.49(3). The reflection associated to the vector ∨ ∨ u−1 a a and the covector u a a is the same as the reflection associated to a and a , −1 ∨ −1 ∨ which is ra , hence (RSII ). Finally ua a(ub b ) = ua ub a(b ). Choose α, β ∈ Ψ with gradients a, b respectively. By Proposition 1.3.49(1) we have  β∨ ) = α + ub a(b∨ ) Ψ  tβ (α) = α + ub α( and Proposition 1.3.49(2) implies ub a(b∨ ) ∈ ua Z, hence (RSIII ). Assume now that Ψ is reduced. If Φ is also reduced the statement is immedi−1 a = u−1 a.  ate. Otherwise Lemma 1.3.59 shows that u2a = 2ua , hence 2u2a a Proposition 1.3.61 Assume that Φ is reduced. Then the translation subgroup ∨ in W(Ψ) is given by the lattice in V spanned by Ψ . Proof Let T ⊂ V be the translation subgroup of W(Ψ) and let T  be the lattice ∨ spanned by Ψ .  Then tψ = −ua a∨ by Proposition 1.3.49(1), hence Let ψ ∈ Ψ and let a = ψ.  T ⊂ T. Conversely, fix a special vertex x and apply Lemma 1.3.42 to write W(Ψ) = T  W(Φ). Since T  is stable under W(Φ) we can form the subgroup W(Ψ) := T   W(Φ) of W(Ψ). It is enough to show W(Ψ) = W(Ψ). In turn, it is enough to show rψ ∈ W(Ψ) for all ψ ∈ Ψ. By Proposition 1.3.43 we have the isomorphism ∇ : Ψx → Φ, so there exists ψ  ∈ Ψx with ψ  = a. By Proposition 1.3.49 we have ψ = ψ  + nu a with n ∈ Z. Then rψ = t−nua a∨ rψ , where t−nua a∨ is the translation by the vector −nua a∨ .  Therefore rψ ∈ W(Ψ) as claimed. Lemma 1.3.62 Proof

∨

Ψ = Ψ.

 Then Let ψ ∈ Ψ and n ∈ Z. Write a = ψ. (ψ + nua )∨ = ψ ∨ + 2nua a, a −1 ; hence, ua∨ = 2ua a, a −1 .

Therefore, ∨

∨ ∨ ∨ −1 Ψ = {u−1 a∨ a | a ∈ Φ } = {u a a | a ∈ Φ} = Ψ.

1.3 Affine Root Systems

39 

Theorem 1.3.63 Let Ψ be an irreducible reduced affine root system. Then it is isomorphic to a system obtained from Construction 1.3.27, or to its dual. Proof We distinguish the following cases. (1) The function a → ua is constant on Φnd : Up to rescaling Ψ we can assume ua = 1 for all a ∈ Φnd . Let x ∈ A be an extra special vertex. Then Proposition 1.3.49 and Lemma 1.3.59 show that every ψ ∈ Ψ is of the form ψx + n for some ψx ∈ Ψx = Φnd and n ∈ Z, or to 2ψx + 2n + 1. Thus Ψ is obtained by applying Construction 1.3.27 to the finite root system Φ. (2) Φ is reduced and there are at least two different values of ua for a ∈ Φ: By Proposition 1.3.49(3) there must exist exactly two root lengths in Φ. Let us again rescale Ψ to acheive ua = 1 when a ∈ Φ is short. When a ∈ Φ is long, then −1 ua  1, so u−1 a a has a different length from a. In order for Ψ = {u a a | a ∈ Φ} to be a root system, a short a ∈ Φ must become a long a ∈ Ψ and a long a ∈ Φ must become a short u−1 a a ∈ Ψ. This implies u a = a, a when a ∈ Φ is long, where −, − is rescaled so that a, a = 1 when a ∈ Φ is short. A simple computation shows that ua∨ = 2ua / a, a for all a ∈ Φ. Therefore the dual system Ψ∨ falls under case (1) and we see that it is obtained by applying Construction 1.3.27 to the finite root system Φ∨ . (3) Φ is non-reduced and there are at least two different values of ua for a ∈ Φnd : In this case, Φ must be of type BCn with n  1. As in (2) we rescale Ψ so that ua = 1 when a ∈ Φnd is short and conclude that ua = 2 when a ∈ Φnd is long. Choose an extra special vertex x ∈ A. Then ∇ : Ψxnd → Φnd is an isomorphism by Proposition 1.3.43(6). Let a, b ∈ Φnd be orthogonal short roots so that a ± b, 2a, 2b ∈ Φ. Let α, β ∈ Ψxnd have gradients a, b respectively. Then α ± β ∈ Ψx . By Lemma 1.3.59 also 2β + 1 ∈ Ψ. Then r2β+1 (α + β) = α − β + 1, showing ua−b  1. But ua−b = 2 as a − b ∈ Φnd is long. Thus case (3) cannot exist.  Construction 1.3.64 Let Ψ be an irreducible affine root system. One can  as follows. Let C be a chamber associate to it an affine Dynkin diagram D 0 and let Ψ(C) be the corresponding basis as in Definition 1.3.19. The nodes of the affine Dynkin diagram are the elements of Ψ(C)0 , and the bonds and arrows are inserted according to the same rules as for finite root systems: two ψ, η ∈ Ψ(C)0 , such that ψ is not divisible in Φ, are joined by a bond with  η∨ )· η(  ψ ∨ ). Hence the only possible values for f (ψ, η) multiplicity f (ψ, η) := ψ( are 0, 1, 2, 3, 4 according to [Bou02, Chapter VI,§1, no. 3]. This is just like for finite root systems, except for the possibility of the value 4, which according  Both of these possibilities do to loc. cit. occurs if either ψ = ± η or ψ = ±2, η.

40

Affine Root Systems and Abstract Buildings

occur, the first for ΨΦ with Φ of type A1 , and the second with Φ of type BC1 . The value 0 indicates the absence of a bond. An arrow is placed on the bond if  η∨ ). In that case one of these two numbers has absolute  ψ ∨ )  ψ( and only if η(  ψ ∨ ), and the arrow points towards η. value 1, say without loss of generality η( One can interpret most of this recipe also in terms of a Weyl group invariant scalar product (−, −) on V ∗ . If the bond between η and ψ has an arrow on it, then the arrow points towards the shorter root (i.e. the root whose norm in terms of the scalar product is smaller), and the multiplicity of the bond equals (ψ, ψ)/(η, η), assuming η is the shorter root. This however does not apply to bonds of multiplicity 4 without an arrow, because then (ψ, ψ)/(η, η) is equal to 1, rather than 4. Note that if Ψ is not reduced, the affine Dynkin diagram is the same as for the reduced subsystem Ψnd = {ψ ∈ Ψ | ψ/2  Ψ}, because Ψ(C)0 ⊂ Ψnd . Fact 1.3.65 The affine Dynkin diagram of the dual system Ψ∨ is obtained from the affine Dynkin diagram of Ψ by inverting all arrows. Fact 1.3.66 Let Ψ be an irreducible reduced affine root system, C a chamber, and F a facet contained in its closure. The Dynkin diagram of ΨF is obtained from the Dynkin diagram of Ψ by removing all vertices of the facet F and all edges emanating from them. Proposition 1.3.67

Let Ψ be an irreducible reduced affine root system.

(1) The Dynkin diagram of Ψ is among those given in Table 1.3.4, and each diagram in this table is the diagram of some Ψ. (2) The isomorphism class of Ψ is determined by its affine Dynkin diagram. The label of the diagram is called the type of Ψ. Proof (1) Theorem 1.3.63 shows that either Ψ or Ψ∨ is produced by Construction 1.3.27. Computing the resulting Dynkin diagram is a simple exercise left to the reader. (2) Removing a node from the Dynkin diagram of Ψ produces the Dynkin diagram of the finite root system Ψx , where x is the vertex of the chamber C corresponding to the removed node, cf. Fact 1.3.66. Since Ψx is determined by its Dynkin diagram up to isomorphism, the same is true for Ψ.  Remark 1.3.68 Another way to classify the possible affine Dynkin diagrams is as follows. Let S be the set of simple reflections corresponding to the basis Ψ(C)0 of Ψ and let W = W(Ψ). Since (W, S) is a Coxeter system by Proposition 1.3.20, one can appeal to the classification of Coxeter graphs in [Bou02, Chapter VI, §4, no. 3, Theorem 4]. One has to only replace a bond with label 4 with a

1.3 Affine Root Systems

41

double edge with orientation, and a bond with label 6 with a triple edge with orientation. Theorem 1.3.69 Let Ψ be an irreducible non-reduced affine root system. Consider the reduced subsystems Ψnd and Ψnm . The pair consisting of their types determines the isomorphism class of Ψ. The possibilities are given in Table 1.3.3. Proof The finite root system Φ = ∇Ψ is irreducible and non-reduced, hence of type BCn . Thus Φnd and Φnd are the subsystems of type Bn and Cn , respectively. Moreover, we have Φnd ⊂ ∇(Ψnd ) ⊂ Φ and Φnm ⊂ ∇(Ψnm ) ⊂ Φ. Therefore ∇(Ψnd ) is of type Bn or BCn , showing that Ψnd is of type Bn , Cn∨ , or BCn . In the same way, ∇(Ψnm ) is of type Cn or BCn , showing that Ψnm is of type Bn∨ , Cn , or BCn . But since the Weyl groups of Ψnd and Ψnm agree, the only possible options are those listed in Table 1.3.3. 

Table 1.3.3 The affine Dynkin diagrams of the non-reduced irreducible affine root systems Label

Diagrams

(BCn, Cn )

(n  1)

(Cn∨, BCn )

(n  1)

(Bn, Bn∨ )

(n  2)

(Cn∨, Cn )

(n  1)

,

,

,

, ,

,

,..., ,

,

,...,

,..., ,...,

Remark 1.3.70 In Table 1.3.3 we have listed the types of the non-reduced irreducible affine root systems as discussed in Theorem 1.3.69. Thus the type is a pair (X,Y ), with X the Dynkin type of Ψnd and Y the Dynkin type of Ψnm . Since the affine Dynkin diagram is the same as that for Ψnd , and the special and extra special vertices are also the same, instead of recording those, we have recorded in Table 1.3.3 the information about which simple root is multipliable: the non-multipliable simple roots are labeled by a solid node ( ), while the multipliable simple roots are labeled by a solid node with a circle around it. In Table 1.3.4 we have recorded the special and extra special vertices as follows. According to Proposition 1.3.22(6) the vertices of a chamber are in

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Affine Root Systems and Abstract Buildings

bijection with the corresponding set of simple roots. We have labeled by an empty node ( ) those simple roots that correspond to extra special vertices. We have labeled by a crossed node ( ) those simple roots that correspond to vertices that are special, but not extra special. By Proposition 1.3.43 such vertices exist only if the derivative root system Φ is non-reduced. The simple roots that correspond to non-special vertices are labeled by a solid node ( ).

Table 1.3.4 The affine Dynkin diagrams of the reduced irreducible affine root systems Label

Diagrams

An

(n  1)

Bn

(n  3)

,

,...,

Bn∨

(n  3)

,

,...,

Cn

(n  2)

,

,...,

Cn∨

(n  2)

,

,...,

BCn

(n  1)

Dn

(n  4)

E6 E7 E8 F4 F4∨ G2 G∨ 2

,

,...,

,

, ,

,..., ,...,

1.3 Affine Root Systems

43

Table 1.3.5 The integers (1.3.1) for the reduced irreducible affine root systems Label

Diagram 1

An

(n  1)

Bn

(n  3)

1

1

1

1

1 2

2

2

2

2

2

2

2

2

2

2

1

1 1

Bn∨

(n  3) 1

Cn

(n  2)

Cn∨

(n  2)

BCn

(n  1)

Dn

(n  4)

1

2

2

2

2

1

1

1

1

1

1

1

1

2

2

2

2

2

2

1

1 2

2

2

2

2 1

1 1

2

E6

1

2

3

2

1

3

4

3

2

1

3

2

2

E7

1

2 3

E8 F4 F4∨ G2 G∨ 2

2

4

6

5

4

1

2

3

4

2

1

2

3

2

1

1

2

3

1

2

1

1

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Affine Root Systems and Abstract Buildings

Definition 1.3.71 The extended affine Weyl group of Ψ is the group W(Ψ)ext consisting of those automorphisms of the affine space A that preserve Ψ and whose derivative is an element of W(Φ). 1.3.72 If C ⊂ A is any chamber and ΞC is the stabilizer of C in W(Ψ)ext , then Lemma 1.3.17 implies W(Ψ)ext = W(Ψ)  ΞC . Remark 1.3.73 Consider an irreducible affine root system Ψ. It is clear from Table 1.3.5 that the set of vertices, the information whether or not two vertices are linked, and the sequence of integers (1.3.1), completely determines the affine Dynkin diagram of the reduced subsystem Ψnd . Table 1.3.3 shows that, if one adds to this the information about which affine simple root is multipliable, then this completely determines both Ψnd and Ψnm , hence also Ψ. Lemma 1.3.74

Assume that Ψ is reduced.

(1) The translation subgroup W(Ψ)ext ∩ V is the lattice {v ∈ V | v, a ∈ ua Z for all a ∈ Φnd }. (2) The translation subgroup W(Ψ)aff ∩ V is the lattice generated by the (ua /δa )a∨ for a ∈ Φnd , where δa = 1 if a is non-multipliable, and δa = 2 if a is multipliable. Proof (1) follows from Proposition 1.3.49 and Lemma 1.3.59. (2) follows from Proposition 1.3.61 when Φ is reduced. Since the statement respects direct sums, the only remaining case to check is that of type BCn , which can be checked by hand.  Lemma 1.3.75 Let Ψ be an irreducible affine root system. Assume that it is either reduced, or as in Example 1.3.28. Let C be a chamber. Then Aut(C) ⊂ W(Ψ)ext acts simply transitively on the set of extra special vertices of C. Proof When Φ is reduced this is [Bou02, Chapter VI, §2, no. 3, Proposition 6], where 0 in loc. cit. is by construction a special, hence also extra special, vertex. When Φ is not reduced one checks directly that there is a unique extra  special vertex. At the same time W(Ψ)aff = W(Ψ)ext , so Aut(C) = {1}. Remark 1.3.76 Consider an irreducible affine root system Ψ ⊂ A∗ and a chamber C ⊂ A. Recall the stabilizer Ξ = ΞC of C in W(Ψ)ext . The action of Ξ on C is faithful, because the action of W(Ψ)ext on A is faithful by definition of W(Ψ)ext as a subgroup of affine transformations of A. Since there is a bijection between the set of vertices of C and the set of simple affine roots (cf. Proposition 1.3.22(6)), hence the set of vertices of the affine Dynkin diagram, one obtains

1.3 Affine Root Systems

45

a faithful action of Ξ on the affine Dynkin diagram. With respect to this action, Ξ is realized as a normal subgroup of the full symmetry group of the affine  Let us describe the two groups Dynkin diagram, which we shall call Aut(D).  It is enough to assume that Ψ is reduced, since in the non-reduced Ξ ⊂ Aut(D).  More generally, if the irreducible root system Φ cases we have Ξ = Aut(D). contains roots of two different lengths or if it is of type A1 , E7 or E8 , then it does not admit a non-trivial automorphism that preserves a basis, and in these  (see (1.3.2)). cases we have Ξ = Aut(D)  = Z/2Z. (1) If Ψ is of type A1 , then Ξ = Aut(D)  = Z/(n+1)ZZ/2Z is the dihedral (2) If Ψ is of type An , n > 1, then Aut(D) group of order 2(n + 1), and Ξ is the subgroup of index 2 consisting of all rotations.  = Z/2Z. (3) If Ψ is of type Bn , Bn∨ , Cn , Cn∨ , then Ξ = Aut(D)  (4) If Ψ is of type BCn then Ξ = Aut(D) = {1}.  = S4 , and Ξ is the unique Sylow-2 (5) If Ψ is of type D4 , then Aut(D) subgroup of A4 , hence isomorphic to (Z/2Z)2 . One can represent two of its generators as follows:

 = (Z/2Z)2  Z/2Z is (6) If Ψ is of type D2n with n > 2, then Aut(D) generated by the following three automorphisms of order 2: switch the two left nodes and fix all others, switch the two right nodes and fix all others, switch the left and right branches. In fact, there are two distinct automorphisms of order 2 that switch the left and right branches; they commute and their product is the unique central element in the symmetry group of the affine Dynkin diagram. These two automorphisms generate the subgroup Ξ, which is hence isomorphic to (Z/2Z)2 . One can represent two of its generators as follows:

 = (Z/2Z)2  Z/2Z with (7) If Ψ is of type D2n−1 , n > 2, then again Aut(D) the same description as for D2n . The subgroup Ξ is the unique subgroup isomorphic to Z/4Z. One can represent a generator of it as follows:

46

Affine Root Systems and Abstract Buildings

 = S3 = (Z/3Z)  Z/2Z is the dihedral (8) If Ψ is of type E6 , then Aut(D) group of order 6 and Ξ is the unique subgroup Z/3Z. One can represent a generator of it as follows:

 = Z/2Z. (9) If Ψ is of type E7 , then Ξ = Aut(D) ∨ ∨  = {1}. (10) If Ψ is of type E8 , F4 , F4 , G2 , G2 , then Ξ = Aut(D) More information about Ξ can be found in [Ree10, §3.6] and [Bou02, Chapter VI, §2, no. 3]. Note in particular that, when Ψ = ΨΦ for an irreducible reduced finite root system Φ, and x is a special node in the affine Dynkin diagram, then the map sending ω ∈ Ξ to ωx is a bijection between the group Ξ and the set  the node x and all edges emanating from of special nodes. Removing from D  x is the stabilizer of x in x we obtain the Dynkin diagram D of Φ. If Aut(D)  = Ξ  Aut(D)  x . The extension  then Aut(D) = Aut(D)  x and Aut(D) Aut(D),  → Aut(D) → 1 1 → Ξ → Aut(D)

(1.3.2)

obtained in this way does not depend on the choice of x, and the choice of x gives  → Aut(D), a splitting of this extension. Another description of the map Aut(D) which does not involve the choice of x, can be given as follows. The group  is the quotient of the automorphism group of Ψ by the affine Weyl Aut(D) group W(Ψ); the group Aut(D) is the quotient of the automorphism group of Φ = ∇Ψ by the finite Weyl group W(Φ); the derivative map Aut(Ψ) → Aut(Φ)  → Aut(D). induces the homomorphism Aut(D) Recall from [Bou02, Chapter VI, §1, no. 7, Definition 4] the notions of a closed, parabolic, and symmetric subset X of a root system Φ: it is symmetric if −X = X; closed if a, b ∈ X and a + b ∈ Φ implies a + b ∈ X; and parabolic if it is closed and X ∪ −X = Φ. A closed symmetric subset is the same as a closed subroot system; cf. [Bou02, Chapter VI, §1, no. 7, Proposition 23]. Lemma 1.3.77 Let A be an affine space over V, Ψ ⊂ A∗ an affine root system, and Φ ⊂ Φ a closed symmetric subset. Let W ⊂ V be the subspace annihilated by Φ. Then Ψ  = {ψ ∈ Ψ | ψ ∈ Φ } is an affine root system in (A/W)∗ . It is reduced if Ψ is. Proof Since Ψ  ⊂ Ψ we know that Ψ  does not contain 0. The subspace of V ∗ generated by Φ is W ⊥ = (V/W)∗ . For each a ∈ Φ the set {ψ ∈ Ψ | ψ = a} is a free abelian group of rank 1 (cf. [Mac72, Proposition 6.9]), therefore Ψ  generates (A/W)∗ , hence satisfies Axiom AR 1. For ψ ∈ Ψ  the reflection rψ

1.4 Tits Systems

47

preserves Φ, therefore Axiom AR 2 for Ψ  follows from Axiom AR 2 for Ψ.  The remaining axioms for Ψ  follow immediately from those for Ψ.

1.4 Tits Systems In this section we review the notion of a Tits systems and some of its properties. Tits systems are very closely related to Tits buildings, a notion reviewed in the next section. Definition 1.4.1 A Tits system is a tuple (G, B, N, S) consisting of a group G, two subgroups B and N of G, and a subset S of N/(B ∩ N), subject to the following axioms. TS 1 The set B ∪ N generates G and B ∩ N is a normal subgroup of N. TS 2 The set S generates the group W = N/(B ∩ N) and consists of elements of order 2. For w ∈ W and n any lift of w in N, we define wB := nB and Bw := Bn. These are well-defined cosets of B in G. TS 3 Given s ∈ S and w ∈ W one has sBw ⊂ BwB ∪ BswB. TS 4 Given s ∈ S one has sBs  B. The system is called saturated, if in addition the following axiom holds.

TS 5 n∈N nBn−1 = B ∩ N. Remark 1.4.2 Set T = B ∩ N. The group W = N/T is called the Weyl group of the Tits system. According to [Bou02, Chapter IV,§2,no. 5, Corollary] the set S consists precisely of those non-trivial elements of W for which the set B ∪ BwB is a subgroup of G. Since G will usually be fixed, we may also refer to the pair (B, N) as a Tits system. Sometimes this pair is called a BN-pair, but this can cause confusion when its members are not called B and N. Example 1.4.3 If G is a connected reductive group over a field k, P is a minimal parabolic k-subgroup, and N the normalizer of a maximal k-split torus S contained in P, then (P(k), N(k)) is a Tits system in G(k) with finite Weyl group; see [Bor91, Theorem 21.15]. The role of T is then played by M(k), where M is the centralizer of S in P, equivalently in G; it is a Levi k-subgroup of P. This is usually called the standard Tits system of G(k). We may call it the spherical Tits system; see Example 1.5.11. Remarkably, when k is infinite, any Tits system in G(k) with finite Weyl group that satisfies a mild natural condition is the spherical Tits system for some choice of P and N; see [Pra14, Theorem B].

48

Affine Root Systems and Abstract Buildings

In this book we will be primarily concerned with another fundamental example of a Tits system. For this the abstract group G will be a certain subgroup G(k)0 of the group G(k) of k-points of a connected reductive group G over a discretely valued Henselian field k, and the role of B will be played by a certain bounded subgroup of G(k)0 , called an Iwahori subgroup. The Weyl group of this Tits system will be infinite. Definition 1.4.4

Let (G, B, N, S) be a Tits system.

(1) For any subset X ⊂ S let WX ⊂ W be the subgroup generated by X and let G X = BWX B. The group G X is called a standard parabolic subgroup. (2) Any subgroup of G containing a conjugate of B is called a parabolic subgroup. We have the following properties of Tits systems; see [Bou02, Chapter IV, §2] and the summary in [Tit74, §3.2]. Proposition 1.4.5 (1) (Bruhat Decomposition) The map w → BwB is a bijection from the Weyl group W to the set of B-double cosets in G. In particular, G = BW B. (2) Any parabolic subgroup is conjugate to a unique standard parabolic subgroup. (3) Each parabolic subgroup is equal to its normalizer. (4) Let Q be a subgroup of G that contains two parabolic subgroups Q1 and Q2 of G. Then any g ∈ G such that gQ1 g −1 = Q2 belongs to Q. Definition 1.4.6 Let P ⊂ G be a parabolic subgroup. The subset X ⊂ S such that P is conjugate to G X = BWX B is called the type of P. According to [Bou02, Chapter IV, §2, no. 4, Theorem 2], the tuple (W, S) is a Coxeter system in the sense of [Bou02, Chapter IV, §1, no. 3, Definition 3]. Recall from the end of [Bou02, Chapter IV, §1] that a Coxeter system (W, S) is called irreducible if one cannot write S as a disjoint union S = S1 ∪ S2 of two non-empty subsets such that each element of S1 commutes with each element of S2 ; equivalently the Coxeter graph of (W, S) is connected and non-empty. We call (G, B, N, S) irreducible if its Coxeter system is irreducible. More generally we will be interested in Tits systems for which (W, S) may not be irreducible, but S is finite. There exists a unique smallest disjoint union decomposition S = S1 ∪ · · · ∪ Sn such that for i  j each element of Si commutes with each  element of S j . Then W = i Wi , where Wi is the subgroup of W generated by Si ; see [Bou02, Chapter IV, §1, no. 9, Proposition 8]. Each (Wi , Si ) is an irreducible Coxeter system, and the graphs of (Wi , Si ) are the irreducible components of the graph of (W, S). The groups G, B, and N, need not have an analogous direct

1.4 Tits Systems

49

product decomposition. However, the set of parabolic subgroups of (G, B, N, S) does have such a decomposition, as we will now discuss. Let (G, B, N, S) be a Tits system and let S = S1 ∪ S2 be a disjoint union such that each element of S1 commutes with each element of S2 . Write Ni for the preimage of Wi in N. Then N1 and N2 normalize each other, their intersection is T, and their product is N. Lemma 1.4.7 G Si ∩ N = Ni . Proof By construction G Si = BNi B, so Ni ⊂ G Si ∩ N. At the same time, G S1 ∩ N = G S1 ∩ (N1 · N2 ) = N1 · (G S1 ∩ N2 ) ⊂ N1 · (G S1 ∩ G S2 ∩ N) =  N1 · (B ∩ N) = N1 and analogously G S2 ∩ N ⊂ N2 . Lemma 1.4.8

(G, G S1 , N, S2 ) is a Tits system with Weyl group W2 .

Proof Since B ∪ N generates G, so does G S1 ∪ N. The group G S1 ∩ N equals N1 by Lemma 1.4.7 and is thus normal in N. The quotient N/N1 is isomorphic to N2 /(N1 ∩ N2 ) = N2 /T = W2 . To verify Axiom TS 3 we first claim that for any w ∈ W we have W1 Bw ⊂ BW1 wB. Since S1 generates W1 this is equivalent to s1(1) · · · s1(n) Bw ⊂ W1 wB for s1(1), . . . , s1(n) ∈ S1 . We work by induction on n starting with the trivial case n = 0. For the induction step we compute using Axiom TS 3 for the system (G, B, N, S) and the induction hypothesis that s1(1) · · · s1(n) Bw ⊂ (s1(1) · · · s1(n−1) BwB) ∪ (s1(1) · · · s1(n−1) Bs1(n) wB) ⊂ BW1 wB ∪ BW1 s1(n) wB = BW1 wB. The claim is proved. We now check Axiom TS 3 for (G, G S1 , N, S2 ) by taking s2 ∈ S2 , w2 ∈ W2 , and computing s2 G S1 w2 = s2 BW1 Bw2 ⊂ s2 BW1 w2 B = s2 Bw2W1 B ⊂ Bw2W1 B ∪ Bs2 w2W1 B ⊂ G S1 w2 G S1 ∪ G S1 s2 w2 G S1 . Finally let s ∈ S and let n ∈ N be a lift. If nG S1 n−1 = G S1 then Proposition 1.4.5 and Lemma 1.4.7 imply n ∈ N1 . Thus if s ∈ S2 then sG S1 s  G S1 , hence  Axiom TS 4 holds for (G, G S1 , N, S2 ). Let Sic = S − Si . Of course we have S1c = S2 and S2c = S1 . Let P be the set of parabolic subgroups of the Tits system (G, B, N, S) and let Pi be the set of parabolic subgroups of the Tits system (G, G Sic , N, Si ). There is a tautological order-preserving G-equivariant inclusion ιi : Pi → P, defined by ιi (Pi ) = Pi . We have typeP (ιi (Pi )) = typePi (Pi ) ∪ Sic ,

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Lemma 1.4.9 Let P ∈ P. Then P ⊂Pi ∈Pi Pi is an element of Pi , which we shall call πi (P). If P = G X with X ⊂ S then πi (P) = G X∪Sic . In particular, typePi (πi (P)) = typeP (P) ∩ Si . The map πi is an order-preserving G-equivariant map P → Pi that is a section of the inclusion ιi : Pi → P. Proof To each P ∈ P we can assign the set of all elements of Pi that contain P. The assignment of this set to P is equivariant for the action of G by conjugation on P and Pi , respectively. Therefore we may assume that P is standard, say P = G X for some X ⊂ S. We claim that any Pi ∈ Pi containing G X must also contain G X∪Sic . Indeed, Pi ∈ Pi is equivalent to the existence of g ∈ G such that gG Sic g −1 ⊂ Pi . Therefore gBg −1 ⊂ Pi . At the same time B ⊂ G X ⊂ Pi . Thus B is contained in both Pi and g −1 Pi g. Proposition 1.4.5 implies g ∈ Pi . Therefore PSic ⊂ Pi and G X ⊂ Pi , from which the claim follows. But the claim immediately implies that G X∪Sic is the intersection of all elements of Pi containing G X .  Proposition 1.4.10 The map π : P → P1 × P2 sending P to (π1 (P), π2 (P)) is an order-preserving G-equivariant bijection. It satisfies typeP1 (π1 (P)) ∪ typeP2 (π2 (P)) = typeP (P). Its inverse is given by (P1, P2 ) → P1 ∩ P2 . Proof The G-equivariance is clear. Using it, injectivity is reduced to the claim G X = G X∪S1c ∩ G X∪S2c . But for any two subsets X1, X2 ⊂ X we have G X1 ∩ G X2 = G X1 ∩X2 , and the claim is immediate. To prove surjectivity and the claim about the inverse, consider a pair (P1, P2 ). We claim that P1 ∩ P2 ∈ P. Again by G-equivariance we are free to conjugate both P1 and P2 by the same element of G. Since both P1, P2 are parabolic subgroups of (G, B, N, S) we may assume, after conjugating both by an element of G, that at least P1 contains B, therefore P1 = G X1 for some S1c ⊂ X1 ⊂ S. Let g ∈ G be such that gP2 g −1 contains B and thus equals G X2 for some S2c ⊂ X2 ⊂ S. Using Proposition 1.4.5 write g = b1 nb2 with b1, b2 ∈ B and n ∈ N. −1 Write n = n1 n2 with ni ∈ Ni . Then P2 = g −1 G X2 g = b−1 2 n2 G X2 n2 b2 contains −1 −1 −1 b2 n2 Bn2 b2 . Since n2, b2 ∈ G X1 , the latter also contains b−1 2 n2 Bn2 b2 . We conclude that P1 ∩ P2 contains a conjugate of B and therefore lies in P, and the claim is proved. Let now P = P1 ∩ P2 . We want to show Pi = πi (P). Again by G-conjugation we may assume that P, hence also P1 and P2 , contain B. Thus Pi = G Xi with Sic ⊂ Xi ⊂ S and then P = G X1 ∩X2 , while πi (P) = G(X1 ∩X2 )∪Sic . We want to show (X1 ∩ X2 ) ∪ Sic = Xi . The case i = 2 is shown by (X1 ∩ X2 ) ∪ S1 ⊂ X2 =  S1 ∪ (S2 ∩ X2 ) ⊂ S1 ∪ (X1 ∩ X2 ) and the case i = 1 is entirely analogous. Example 1.4.11 Let (Gi , Bi , Ni , Si ) for i = 1, 2 be two Tits systems. Let Pi be the corresponding sets of parabolic subgroups. Set G = G1 × G2 , B = B1 × B2 ,

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N = N1 × N2 , S = S1 ∪ S2 . Then (G, B, N, S) is also a Tits system. The set P of parabolic subgroups of that Tits system is {P1 × P2 | Pi ∈ Pi }  P1 × P2 . The Tits system (G, G S1 , N, S2 ) of Lemma 1.4.8 is given by (G1 × G2, G1 × B2, N1 × N2, S2 ). As above let P2 be the set of parabolic subgroups of that Tits system. The map P2 → G1 × P2 is an order-preserving bijection ψ2 : P2 → P2 . The map π2 : P → P2 of Lemma 1.4.9 sends P1 × P2 to G1 × P2 . Thus ψ2 ◦ π2 : P = P1 × P2 → P2 is the natural projection. The bijection π : P → P1 × P2 of Proposition 1.4.10 sends P = P1 × P2 to (P1 × G2, G1 × P2 ). Therefore the composition (ψ1, ψ2 ) ◦ π : P → P1 × P2 is the identity map. Next we will discuss ways to modify a Tits system while preserving the set of parabolic subgroups. Our guiding example is that of the spherical Tits system for a connected reductive group G and the maps G → Gad and Gsc → G, where Gad is the adjoint group of G and Gsc is the simply connected cover of the derived subgroup of G. Lemma 1.4.12

Let (G, B, N, S) be a Tits system.

(1) Let Z ⊂ T be a subgroup that is normal in G. Set G  = G/Z, B  = B/Z, N  = N/Z. Then (G , B , N , S) is a Tits system with the same Weyl group as (G, B, N, S). It is saturated if (G, B, N, S) is. (2) Let G → G  be an inclusion with normal image, T  ⊂ G  a subgroup normalizing B and N and normalized by N such that G  = GT  and T  ∩ G = T. Set B  = BT  and N  = NT . Then (G , B , N , S) is a Tits system with the same Weyl group as (G, B, N, S). It is saturated if (G, B, N, S) is. Proof Consider (1). It is immediate that B/Z and N/Z generate G/Z and that B/Z ∩ N/Z = T/Z is normal in N/Z. We have (N/Z)/(B/Z ∩ N/Z) = (N/Z)/(T/Z) = N/T. The inclusion s(B/Z)w ⊂ (B/Z)w(B/Z) ∪ (B/Z)sw(B/Z) is also immediate. If we assume s(B/Z)s = B/Z, then taking preimage in G we obtain sBs = B, a contradiction. Assume (G, B, N, S) is saturated. Since

−1 is the image of nBn−1 under G → G/Z, it equals (B ∩ n n(B/Z)n n N)/Z = T/Z = B/Z ∩ N/Z. Consider (2). It is immediate that B  and N  generate G . Using G ∩ T  = T we see that for any collection (Ai )i of subgroups T ⊂ Ai ⊂ G we have

  i (Ai T ) = ( i Ai )T . Indeed, an element of the left-hand side is given by a collection ai ∈ Ai and ti ∈ T  such that ai ti = a j t j for all i, j. Fix one index i.

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 −1 ∈ G ∩ T  = T, so we For each index j we have ai ti = a j t j , thus a−1 j ai = t j ti    can write t j = t j ti with t j ∈ T, and hence a j t j = (a j t j )ti. Replacing each a j by a j t j we obtain ai = a j for all j, thus an element of the right-hand side. In particular B  ∩ N  = (B ∩ N)T  = TT  = T , normal in N . We have N /(B  ∩ N ) = NT /T  = N/(N ∩ T ) = N/T = W. We have sB  w = sBT  w = sBwT  ⊂ BwBT  ∪ BswBT  ⊂ B  wB  ∪ B  swB . If sB  s = B , then sBsT  = BT . Intersecting with G and using G ∩ T  = G we obtain

sBs = B, a contradiction. Assume (G, B, N, S) is saturated. We have nB  n−1 =

 ( n nBn)T  = TT  = T .

Lemma 1.4.13 Let (G, B, N, S) and (G , B , N , S ) be Tits systems. Assume that (G, B, N, S) is saturated. Let f : G → G  be a group homomorphism mapping B to B  and N to N . Assume that (1) (2) (3) (4)

ker( f ) is contained in T, f (G) is normal in G  and G / f (G) is abelian, T  normalizes f (B), B  = f (B) · T , N  = f (N) · T .

Then (1) (2) (3) (4)

G  = T  f (G) = f (G)T , f −1 (T ) = T, T  normalizes f (BwB) for any w ∈ W, the map W → W  induced by f is an isomorphism that carries S bijectively onto S .

Proof Since B  = f (B)T  and N  = f (N)T  generate G , so do T  and f (G). Moreover, since f (G) is normal in G , this reduces to G  = T  f (G) = f (G)T . We claim that f −1 (T ) ⊂ B. Let g ∈ G be such that f (g) ∈ T . Since T  normalizes f (B) we have f (B) = f (gBg −1 ). Since the kernel of f is contained in B and is normal in G, it is contained in both B and gBg −1 , hence we have B = gBg −1 , which implies g ∈ B and the claim is proved. Together with B  = f (B)T  this implies f −1 (B ) = B. We can apply this argument to the Tits system

(G, nBn−1, N, nSn−1 ) for any n ∈ N and conclude f −1 (T ) ⊂ n∈N nBn−1 . In particular, if (G, B, N, S) is saturated, then f −1 (T ) = T. The surjectivity of the map W → W  induced by f is immediate from the assumption N  = f (N)·T , while its injectivity is immediate from f −1 (T ) = T. Next we claim that for any w ∈ W the subset f (BwB) of G  is normalized by T . Indeed, letting n  ∈ N  represent f (w) and taking t  ∈ T  we see t  f (B)n  f (B)t −1 = f (B)(t  n  t −1 n −1 )n  f (B), using that t  normalizes f (B). The commutator t  n  t −1 n −1 vanishes in G / f (G) since that quotient is assumed abelian, and hence lies in T  ∩ f (G) = f (T), and the claim is proved.

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Next we show that the isomorphism W → W  maps S bijectively to S . Given s ∈ S we know that B ∪ BsB is a subgroup of G. Let s  = f (s). Then f (B ∪ BsB) = f (B) ∪ f (BsB) is a subgroup of G . Since T  normalizes both f (B) and f (BsB), we have that T ( f (B) ∪ f (BsB)) = T  f (B) ∪ T  f (B)s  f (B) = B  ∪ B  s  B  is a subgroup of G , hence s  ∈ S . Conversely, let w ∈ W be such that its image in W  lies in S . Thus B  ∪B  w  B  is a subgroup of G  and by the above equation equals T ( f (B) ∪ f (BwB)). The following two elementary facts, valid for any homomorphism f : G → G  of groups, imply that B ∪ BwB is a subgroup of G, hence w ∈ S. Fact 1: If a subset X ⊂ G is stable under left multiplication by ker( f ) and f (X) is a subgroup of G , then X is a subgroup of G. Fact 2: If a subset X  ⊂ f (G) is normalized by a subgroup T  ⊂ G  and stable under left multiplication by T  ∩ f (G), and if T  · X  is a subgroup of G , then X  is a subgroup of G . The proofs of these facts are immediate and left to the reader.  Lemma 1.4.14 Let (G, B, N, S) and (G , B , N , S ) be Tits systems. Let f : G → G  be a group homomorphism satisfying the assumptions of Lemma 1.4.13. If P ⊂ G is a parabolic subgroup, then so is P  = NG  ( f (P)) = f (P) · T . The maps P → f (P) · T ,

P  → f −1 (P )

are type-preserving, order-preserving, f -equivariant, mutually inverse bijections between the sets of parabolic subgroups. Proof First we prove that the two maps P → T  f (P) and P  → f −1 (P ) induce mutually inverse type-preserving bijections between the sets of standard parabolic subgroups. In fact, the bijection S → S  induced by f as in Lemma 1.4.13 already establishes such a bijection, under which the standard parabolic subgroups BWX B of G and B WX  B  of G  correspond, when X ⊂ S and X  ⊂ S  correspond to each other. So we just need to check that the above maps recover the two directions of this bijection, which is immediate from T  f (BWX B) = T  f (B)WX  f (B) = B WX  B  and f −1 (T  f (P)) = f −1 (T )P = T P = P, the latter by Lemma 1.4.13. That these bijections are order reversing is then also clear. Let us now check that T  f (P) = NG  ( f (P)). Lemma 1.4.13 states that G  =  T f (G) and that T  normalizes f (P), which in turn implies that NG  ( f (P)) = NT  f (G) ( f (P)) = T  N f (G) ( f (P)). Now f (P) is a parabolic subgroup for the Tits system ( f (G), f (B), f (N), S), hence N f (G) ( f (P)) = f (P), and we conclude NG  ( f (P)) = T  f (P) as desired.

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Finally, since B  and N  generate G , the assumptions B  = f (B)T  and = f (N)T  imply that T  and f (G) generate G , which by the normality of f (G) in G  implies G  = f (G)T . Therefore any G -conjugate of B  is also an f (G)-conjugate of B . Since the two maps P → NG  ( f (P)) and P  → f −1 (P ) are equivariant under conjugation by G and f (G), respectively, the proof is complete.  N

1.5 Abstract Buildings In this section we review the notion of a Tits building, which will be essential for our construction of the Bruhat–Tits building of a reductive group defined over a discretely valued Henselian field. Tits buildings are very closely related to Tits systems. This relationship is explored in [Tit74, §3.2], [BT72, §2], and the exercises to Chapter IV in [Bou02]. We give here just a brief summary. We alert the reader that for a given Tits system one can define two buildings. They are closely related, but often distinct. The usual building, introduced by Tits, is reviewed here in Proposition 1.5.6. It is always a simplicial complex, even when the Tits system is not irreducible. A slight variant of it, which we call the “restricted building,” is given in Proposition 1.5.18. The two buildings coincide when the Tits system is irreducible, but not otherwise. The restricted building of a Tits system that is not irreducible is a polysimplicial complex. The Bruhat–Tits building of a reductive group will be the restricted building associated to a particular Tits system. Definition 1.5.1 (1) A simplicial complex is a pair (V, B) consisting of a non-empty set V and a non-empty set B of non-empty finite subsets of V. We call V the set of vertices. We require {x} ∈ B for all x ∈ V, and further that ∅  A ⊂ B ∈ B implies A ∈ B. Abusing notation, we will refer to B as the simplicial complex, and to V as the underlying set of vertices; see Remark 1.5.4. (2) A polysimplicial complex B is a tuple (B1, . . . , Bn ) of simplicial complexes. We set V = V1 × · · · ×Vn and B = B1 × · · · × Bn . Abusing notation, we refer to B as the polysimplicial complex. Let B be a simplicial or polysimplical complex. (3) An element of B is called a facet. If B is a simplicial complex, it is also called a simplex. (4) If A, B ∈ B and A ⊂ B, then A is called a face of B. (5) If A is a face of B, we define codim(A, B) to be the largest n for which there exists a chain A = A0  A1 ⊂ · · ·  An = B and Ai ∈ B. (6) For B ∈ B we define dim(B) to be sup{codim(A, B)| A ∈ B, A ⊂ B}.

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(7) A subcomplex of B is a subset B ⊂ B such that if A ∈ B, B ⊂ A, and B ∈ B, then B ∈ B. (8) The (open) star of A ∈ B is the complex consisting of all facets of B that contain A. It need not be a subcomplex. (9) A chamber complex is a polysimplicial complex in which every element is contained in a maximal element, and given two maximal elements C, C  there exists a sequence C = C1, . . . , Cn = C  such that Ci ∩ Ci+1 ∈ B and codim(Ci ∩ Ci+1, Ci ) = codim(Ci ∩ Ci+1, Ci+1 ) = 1 for all i = 1, . . . , n − 1. (10) The maximal elements in a chamber complex are called chambers, and a sequence C = C1, . . . , Cn = C  as above is called a gallery joining C and C . (11) A chamber complex is called thick, if each facet of codimension 1 is the face of at least three chambers. It is called thin, if each facet of codimension 1 is the face of exactly two chambers. (12) An isomorphism (V1, B1 ) → (V2, B2 ) of simplicial complexes is a bijection f : V1 → V2 such that f (A1 ) ∈ B2 for all A1 ∈ B1 and f −1 (A2 ) ∈ B1 for all A2 ∈ B2 . (13) An isomorphism (B1, . . . , Bn ) → (B1 , . . . , Bn ) of polysimplicial complexes is a tuple (σ, f1, . . . , fn ), where σ is a permutation of {1, . . . , n}  is an isomorphism of simplicial complexes. and fi : Bi → Bσ(i) (14) An isomorphism of chamber complexes is an isomorphism of (poly) simplicial complexes that maps chambers to chambers. Remark 1.5.2 If B is a simplicial complex then dim(A) = #A − 1 and codim(A, B) = #B − #A. Remark 1.5.3 We have specifically required that the empty subset of V not be an element of a simplicial complex. This is not always done in the literature. This choice is more convenient for our purposes. Remark 1.5.4 Let (V, B) be a (poly)simplicial complex. Then the inclusion of subsets of V endows the set B with an order. One can recover V from the set B and that order relation: if we identify v ∈ V with the singleton set {v} then V is the subset of minimal elements in B. This gives another way to think of a (poly)simplicial complex, as an ordered set subject to certain axioms, namely those translated from the axioms imposed on the pair (V, B) above. Definition 1.5.5 A building is a chamber complex B equipped with a collection of subcomplexes, called apartments, satisfying the following axioms. BL 1 B is a thick chamber complex. BL 2 Each apartment is a thin chamber complex. BL 3 Any two chambers belong to an apartment.

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BL 4 Given two apartments A1, A2 and two facets F1, F2 ∈ A1 ∩ A2 , there exists an isomorphism A1 → A2 that leaves invariant F1 , F2 , and all of their faces. Note that in Axiom BL 4 it is not assumed that the isomorphism A1 → A2 is the restriction of an automorphism of B. This will however be the case for the buildings coming from Tits systems, cf. Proposition 1.5.13. The relationship between Tits systems and buildings is expressed in the following two propositions (1.5.6 and 1.5.28) due to Tits, see [Tit74, Theorem 3.2.6, Proposition 3.11]. Proposition 1.5.6 Let (G, B, N, S) be a Tits system with S finite. Let V be the set of all maximal proper parabolic subgroups of the Tits system. Let B the set of those finite sets {P0, . . . , Pn } of maximal proper parabolic subgroups

n Pi is itself a parabolic subgroup. Endow B with the action of such that i=0 G defined by g{P0, . . . , Pn } = {gP0 g −1, . . . , gPn g −1 }. Let C ⊂ B consist of all subsets of the set of standard maximal proper parabolic subgroups. Let A ⊂ B be the union of all N-conjugates of C. (1) The pair (V, B) is a simplicial complex.

n Pi . The map F → PF (2) Given a facet F = {P0, . . . , Pn } ∈ B, let PF = i=0 is a G-equivariant bijection from the set of facets of B to the set of proper parabolic subgroups, which translates the face relation between facets to the opposite of the inclusion relation between parabolic subgroups. Thus the maximal facets (called chambers) correspond to the minimal parabolic subgroups of G. These minimal parabolic subgroups are gBg −1 , for g ∈ G. (3) The subset C is a chamber, called the standard chamber. (4) If g ∈ G stabilizes a facet, then it fixes each of its vertices. (5) Given two faces F and F  of the standard chamber C and elements n ∈ N and g ∈ G, if gA contains F and nF  , then there is an element of G that carries A to gA and fixes every vertex of F and nF  (note that F and nF  are contained in A). (6) The complex B is a building whose set of apartments is {gA | g ∈ G}. It is called the Tits building of the Tits system. The subset A is called the standard apartment. (7) The group G acts transitively on the set of pairs consisting of an apartment and a chamber contained in it. Note that the building B depends only on the G-conjugacy class of the pair (B, N). The standard chamber C depends on B, and the standard apartment

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A depends on T = B ∩ N. It is often, but not always, true that the standard apartment consists of those proper parabolic subgroups that contain T. Proof (1) The fact that any subgroup of G that contains a parabolic subgroup is itself a parabolic subgroup implies (1). (2) Since PF is a parabolic subgroup, it contains a conjugate, say gBg −1 of B. So g −1 Pi g contains B, hence it is the standard parabolic G Xi , for a

n

Xi . Then G Xi = G X . Therefore, maximal proper subset Xi of S. Let X = i=0 PF = gG X g −1 . Conversely, given a g ∈ G, and a proper subset X of S, let Xi for 0  i  n be the proper maximal subsets of S that contain X and let Pi = gG Xi ng −1 . Then F = {P0, . . . , Pn } is the unique facet of B corresponding to the parabolic subgroup gG X g −1 . It is obvious that the maximal facets of B correspond to the minimal parabolic subgroups gBg −1 , for g ∈ G, since G∅ = B. The maximal facets are called chambers of B. As every facet of B is clearly a face of a chamber, B is a chamber complex. (3) Let P0, . . . , Pr−1 , r = #S, be the standard maximal proper parabolic

subgroups. Then r−1 i=0 Pi = B, hence C := {P0, . . . , Pr−1 } is a maximal facet. (4) Suppose F is a facet that is stable under the action of g ∈ G. Then the corresponding parabolic subgroup P is normalized by g. As the normalizer of P is itself, g lies in P, and therefore it normalizes all the subgroups of G containing P. This implies that every face of F, so in particular every vertex of F, is fixed by g. (5) We will now establish the assertion (5). Let P = G X and P  = G X  , with X, X  ⊂ S, be the standard parabolic subgroups corresponding to the facets F and F  respectively. Let Y and Y  be the subgroups of W generated by X and X  respectively. Then P = G X = BY B and P  = G X  = BY  B. Since F ⊂ gA, using Proposition 1.4.5(2) we see that P = gn0 Pn0−1 g −1 for some n0 ∈ N. We replace g with gn0 to assume that P = g Pg −1 . As P is equal to its own normalizer in G, g ∈ P. Again using Proposition 1.4.5(2), we see that the condition nF  ⊂ gA implies that nP  n−1 = gn  P  n −1 g −1 for some n  ∈ N. So n −1 g −1 n normalizes P  and hence it belongs to P . As g ∈ P, we infer that n ∈ Pn  P . Let w, w  be the images of n, n  in W. From the fact that n ∈ Pn  P , using Axiom TS 3 of Definition 1.4.1 we see that w ∈ Y w Y . Let w1 ∈ Y be such that w ∈ w1 w Y , and let n1 be a representative of w1−1 in N. Then n1 is in P and n1 n ∈ n  P . Therefore, gn1 ∈ P, gn1 nP  = gn  P  = nP  and gn1 A = gA, so the left multiplication by gn1 is the desired isomorphism. Thus we have shown that (5) holds. (6) Now we will show that B is a building by verifying the axioms listed in Definition 1.5.5. Axiom BL 4 is just (5). To prove that the apartments are thin, it is enough to show that A is thin. For

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this, it suffices to show that given a face P of C of codimension 1 (such faces are called panels), there is a unique chamber C   C in A that shares P. The faces of codimension 1 of C correspond to the parabolic subgroups G {s } for s ∈ S. Hence, chambers in A that share the panel corresponding to G {s } = B ∪ Bs B are the chambers corresponding to nBn−1 , for n ∈ N, such that nBn−1 ⊂ G {s } . But the minimal parabolic subgroups B and nBn−1 ( B) that are contained in G {s } are conjugate to each other under G {s } . Thus, if nBn−1  B, there is a b ∈ B such that nBn−1 = bs Bs−1 b−1 . Then n ∈ BsB and hence, by the Bruhat decomposition, n ∈ s(B ∩ N). This implies that the only chambers in A that share the panel corresponding to the parabolic subgroup G {s } are the chambers C (which corresponds to B) and the chamber sC (which corresponds to the parabolic subgroup s Bs−1 = s Bs). Thus A is thin. We will now show that B is thick by determining all the chambers in it that share the panel corresponding to G {s } . The set of such chambers is in natural bijective correspondence with the set of conjugates of B in G {s } = B ∪ Bs B. As we saw in the preceding paragraph, besides B itself, its other conjugates in G {s } are bs Bs−1 b−1 , with b ∈ B. Moreover, bs Bs−1 b−1 = b s Bs−1 b−1 for b, b ∈ B, if and only if b−1 b ∈ s Bs−1 = s Bs. Therefore, for any b ∈ B − s Bs, the conjugates s Bs and bs Bs−1 b−1 are different. This shows that B is thick. We will now show that given two chambers C1 and C2 , there is an apartment that contains both of them. Let gi C = C i , for gi ∈ G. By the Bruhat decomposition, g1−1 g2 = b1 nb2 , with bi ∈ B and n ∈ N. Then (C1, C2 ) = g1 (C, g1−1 g2 C) = g1 (C, b1 nb2 C) = g1 b1 (C, nC). Thus C1 and C2 are contained in the apartment g1 b1 A. We will now show that given any two chambers, there is a gallery in B joining them. In view of the result in the preceding paragraph, it is enough to show that for any n ∈ N, the chambers C and nC can be joined by a gallery. We denote by w, the image of n in W, and let w = s1 s2 · · · sn , with si ∈ S for i  n. Let w0 = 1, and for j > 0, let w j = s1 · · · s j and C j = w j C. Then C = C0, C1, . . . , Cn = w C is a gallery in A joining C to w C. Thus we have verified all the axioms in the definition of buildings for B, and the apartments gA, g ∈ G. Hence B is a building. (7) Observe that, given a pair (A, C ) consisting of an apartment A = gA and a chamber C  in it, there is an element h ∈ G such that h(A, C) = (A , C ). As g −1 C  is a chamber of A, we see that g −1 C  = nC for an n ∈ N. Hence, C  = gnC. So the given pair (A, C) is (gA, gnC) = gn(A, C). Setting h = gn,  we see that the pair (A, C ) = h(A, C) with h ∈ G. This implies (7). Remark 1.5.7

Using the bijection F → PF one can identify the simplices

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of B with the proper parabolic subgroups of G. Under this identification, the face relation becomes the opposite of the inclusion relation between proper parabolic subgroups. In fact, following Remark 1.5.4, we can also interpret B as the set of all parabolic subgroups of the Tits system together with the opposite of the inclusion order. Definition 1.5.8

A Tits system (G, B, N, S) is called

(1) spherical, if each apartment is the triangulation of a sphere; (2) affine, if each apartment is the triangulation of a Euclidean space. Remark 1.5.9 It can be shown that a Tits system is spherical if and only if its Weyl group W = N/(B ∩ N) is finite. Example 1.5.10 The simplest example of an affine building is a tree (cf. [Ser03]), provided each vertex has at least three edges emanating from it. An apartment is an infinite path through the tree, thus a simplicial subcomplex whose geometric realization is a line. The chambers are the edges of the tree. Figure 1.5.1 illustrates the case of a 3-regular tree. We will see in §3.1 that this is the (affine) Bruhat–Tits building associated to the group SL2 /Q2 . Non-regular trees also occur as (affine) Bruhat–Tits buildings of reductive groups over local fields. This is the case with the group SU3 /Q p associated to an unramified quadratic extension of Q p , cf. §3.2.

Figure 1.5.1 The 3-regular tree.

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Example 1.5.11 Consider the Tits system of Example 1.4.3. The building associated to it by Proposition 1.5.6 is called the spherical building of G. Its facets are in 1–1 correspondence with the proper parabolic k-subgroups of G; see [Tit74, Theorem 5.2]. Example 1.5.12 Let k be any field. The simplest examples of a spherical building is the set P1 (k), seen as a 0-dimensional simplicial complex. The vertices, which are also the chambers, are the elements of that set. Every pair of points is an apartment. This building arises as the spherical building of the spherical Tits system of the reductive group SL2 /k. Proposition 1.5.13 Proposition 1.5.6.

Let (G, B, N, S) be a Tits system and B the building of

(1) Given two apartments A1, A2 , both containing a facet F of B, there exists a g ∈ G that transports A1 to A2 and fixes all the vertices of F. (2) Given a facet F ∈ B, its stabilizer {g ∈ G | gF = F} is equal to the parabolic subgroup PF . (3) The fixed point set of PF in B is the set of faces of F (including F itself). (4) If F1, F2 are two facets of the same chamber and g ∈ G satisfies gF1 = F2 , then F1 = F2 and g ∈ PF1 . (5) Two facets F1, F2 ∈ A are conjugate under G if and only if they are conjugate under N. Proof (1) For i = 1, 2, let Ci be a chamber in Ai such that F is a face of Ci . We fix an apartment A that contains both C1 and C2 . Then Ci is contained in both A and Ai . Now Proposition 1.5.6 (7),(4) imply, for i = 1, 2, that there is a gi ∈ G that transports A to Ai and fixes every vertex of the chamber Ci . Then g := g2 g1−1 (∈ G) transports A1 to A2 and fixes every vertex of F. (2) Since proper parabolic subgroups of G and facets of B are the same, the second assertion is equivalent to the statement that each proper parabolic subgroup is equal to its own normalizer, which is part of Proposition 1.4.5. (3) The third assertion follows from the order-reversing bijective correspondence between the facets of B and the parabolic subgroups of G and the fact that each parabolic subgroup equals its own normalizer in G. (4) To prove the fourth assertion, we consider the parabolic subgroups P1 := PF1 and P2 := PF2 . Both are standard with respect to the chamber of which F1 and F2 are assumed to be faces. Then gP1 g −1 = P2 implies via Proposition 1.4.5 that P1 = P2 and g ∈ P1 . (5) To prove the fifth assertion, choose a chamber C in A of which F1 is a face and n ∈ N so that n−1 F2 is also a face of C. Let g ∈ G be such that gF1 = F2 . Then (4) implies that F1 = n−1 F2 and n−1 g ∈ PF1 . Thus nF1 = gF1 = F2 . 

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Proposition 1.5.14 Let F1, F2 ∈ B be two facets. (1) There exists an apartment A containing both F1 and F2 . (2) Assuming A is the standard apartment, the mixed Bruhat decomposition G = PF1 · W · PF2 holds. Proof Without loss of generality we may replace Fi with a chamber C i containing Fi in its closure and we may further assume that C1 is the standard chamber. Let g ∈ G be such that C2 = gC1 . Write g = bnb ∈ BW B according to the Bruhat decomposition, Proposition 1.4.5. Thus both C2 = bnC1 and C1 = bC1 are contained in the apartment bA. Assuming now that bA = A, we have PC1 = B and PC2 = nBn−1 and the Bruhat decomposition implies G = BN B = PC1 N PC2 n, hence G = PC1 N PC2 .  Remark 1.5.15 With B as in Proposition 1.5.13, using the bijection F → PF between facets and parabolic subgroups, we can associate via Definition 1.4.6 to each facet of B a subset of S, called its type. Thus the type of F is the subset X ⊂ S such that PF is conjugate to G X = BWX B. Remark 1.5.16 We have made a very minor change in the definition of the building of a Tits system as compared to [Tit74] and other sources, by considering only proper parabolic subgroups, that is, excluding G from consideration. This corresponds to excluding the empty subset of V from the definition of a simplicial complex. We will find it useful to make a further change. Namely, consider the basic example where G is (the set of k-points of) a connected linear algebraic group defined over an algebraically closed field k, B is a Borel subgroup, and N is the normalizer of a maximal torus. As defined so far, the building is the set of all proper parabolic subgroups. It is a simplicial complex. Consider now the situation where G = G1 × G2 for two connected algebraic groups G1 and G2 . If P1 ⊂ G1 is a proper parabolic subgroup, then P = P1 × G2 is a proper parabolic subgroup of G. For our future purposes we would like to have a variant of the building that excludes such parabolic subgroups from consideration, and instead only contains parabolic subgroups of the form P1 × P2 , where Pi ⊂ Gi is a proper parabolic subgroup. Definition 1.5.17 Let (G, B, N, S) be a Tits system with S finite and non-empty. Write S = S1 ∪ · · · ∪ Sn with Si pairwise commuting and irreducible. (1) A subset X ⊂ S is called admissible if Si ∩ X is a proper subset of Si for all i.

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(2) A facet (respectively a parabolic subgroup) is called admissible if its type is admissible. Proposition 1.5.18 Let (G, B, N, S) be a Tits system with S finite and nonempty. Let B be its Tits building of Proposition 1.5.6. (1) The subset B of B consisting of all admissible facets forms itself a building, where the apartments are given by A = B ∩ A for apartments A ⊂ B. (2) Every chamber in B is admissible and hence lies in B. (3) B is a polysimplicial complex, invariant under the action of G. (4) B = B if and only if S is irreducible. Proof Using Remark 1.5.7 we interpret B as the set of parabolic subgroups of (G, B, N, S) with the opposite inclusion order. Write S = S1 ∪ · · · ∪ Sn as a disjoint union of mutually commuting subsets, with each Si irreducible. Let  Sic = S − Si . Then Proposition 1.4.10 gives a bijection B → i Bi , where Bi is the building of the Tits system (G, PSic , N, Si ). This is a bijection of sets that preserves the order relation, hence endows the ordered set B with the structure of a polysimplicial complex. The remaining claims are immediate.  Remark 1.5.19 The vertices in B are the facets of B of type X1 ∪ · · · ∪ Xn , where Xi ⊂ Si is a maximal proper subset. On the other hand, the vertices in B are the facets of B of type X1 ∪ · · · ∪ Xn , where for some i0  n, Xi0 is a maximal proper subset of Si0 and for all i  i0 , Xi = Si . Thus, unless S is irreducible, the vertices in B do not lie in B, and the vertices of B are facets of B whose dimension is greater than 1. Definition 1.5.20 We will call the building B of Proposition 1.5.18 the restricted building. Remark 1.5.21 Except for not necessarily being a simplicial complex, all properties of B stated in Proposition 1.5.6 and Proposition 1.5.13 immediately carry over to B. Definition 1.5.22 A panel is face of codimension 1 of a chamber of B. Two chambers are said to be adjacent if they share a common panel. Given two facets F and F  in B, a gallery of length n joining them is a sequence C0, C1, . . . , Cn of chambers of B such that for i < n, C i is adjacent to Ci+1 and F is a face of C0 and F  is a face of Cn . A gallery of length n joining F and F  is said to be minimal if there is no gallery of length smaller than n joining F and F .

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Lemma 1.5.23 Let A be an apartment of B and C a chamber in A. If f is an automorphism of A that keeps every vertex of C fixed, then f is trivial. Proof Let C  be a chamber in A and let C = C0, C1, . . . , Cn = C  be a gallery in A joining C to C  such that for all i < n, C i  Ci+1 . We will prove by induction that the vertices of C i , for i  n, are fixed under f . This will prove the lemma. Let us assume that for some j < n, every vertex of C j is fixed under f and let P be the panel shared by C j and C j+1 . Then as C j and C j+1 are the only chambers in A that have P as a face, and as f fixes C j , we infer that it fixes C j+1 also. Since the vertices of P are fixed under f , the remaining vertex of C j+1 is also fixed under f . Now by induction, we see that for all i  n, every vertex of C i is fixed under f .  Proposition 1.5.24 Let A ⊂ B be an apartment and C ⊂ A a chamber. There exists a unique polysimplicial map ρ = ρA,C : B → A with the following properties. (1) ρ|A is the identity. (2) For every apartment A that contains C, the map ρ|A : A → A is a polysimplicial isomorphism. (3) For any vertex x of C, ρ−1 (x) = {x}. Proof Consider an arbitrary facet F of B. Choose an apartment A containing F and C; such an apartment exists by Definition 1.5.5. By BL 4 of Definition 1.5.5 and Lemma 1.5.23 there exists a unique isomorphism σA : A → A that fixes every vertex of C. Define ρ(F) = σA (F). We claim that ρ(F) does not depend on the choice of A. Let A be another apartment containing F and C. By BL 4 of Definition 1.5.5, there exists an isomorphism τ : A → A that fixes every vertex of F and C. It is obvious that σA : A → A is just σA ◦ τ. This implies that ρ(F) is independent of the choice of A. Consider now a vertex y of B such that x := ρ(y) is a vertex of C, and let A be an apartment containing y and C. Then since ρ|A : A → A is an isomorphism, that maps x, y to x, we conclude that y = x.  Definition 1.5.25 The polysimplicial map ρA,C : B → A constructed in the preceding proposition is called the retraction of B onto A with center C. Example 1.5.26 In Example 1.5.10, where B is a 3-regular tree, the retraction to an apartment centered at a fixed edge is depicted in Figure 1.5.2. An intuitive way to describe it may be to imagine holding the tree with your hand at the fixed edge and shaking it until all branches collapse onto a single line (the apartment onto which the building is being retracted).

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Figure 1.5.2 The retraction of a 3-regular tree.

Corollary 1.5.27 Let A be an apartment of B, C  a chamber in A and F a facet in A. Let G := C0, C1, . . . , Cn = C  be a minimal gallery joining F to C . Then all the chambers C i are contained in A. Proof Assume, if possible, that not all chambers in G are contained in A. Let j < n be the largest integer such that the chamber C j is not contained in A. Let P be the common panel of C j and C j+1 and C be the unique chamber in A different from C j+1 that has P as a face. According to the last assertion of Proposition 1.5.24, the retraction ρ := ρA,C does not map C j onto C. Hence, ρ(C j ) = C j+1 . Therefore, the gallery ρ(G ) := ρ(C0 ), ρ(C1 ), · · · , ρ(Cn ) = C  is a gallery joining F and C  contained in A and as ρ(C j ) = C j+1 = ρ(C j+1 ) since  C j+1 is contained in A, the gallery G is not minimal. A contradiction. The following proposition is a converse of Proposition 1.5.6. Proposition 1.5.28 Let G be a group that operates on B by simplicial automorphisms. We assume that this action has the following two properties: (1) Given two pairs (A, C ) and (A, C ) consisting of an apartment of B and a chamber in it, there exists an element g ∈ G that carries the first pair onto the second pair, and fixes every vertex common to C  and C . (2) If an element of G fixes a chamber, then it fixes all its vertices. We choose a pair (A, C) and let B be the subgroup consisting of all elements of G that keep C stable, and N be the subgroup consisting of all elements of G

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which keep A stable. Then (B, N) is a saturated Tits system (Definition 1.4.1) in G. The Weyl group of this Tits system is N/B ∩ N. A simplicial automorphism of B, or A, which has the property that if it stabilizes a chamber, then it fixes all its vertices, will be called a special automorphism. Proof From conditions (1) and (2) it is obvious that the subgroup N acts transitively on the set of chambers in A and it maps onto the group W of all special automorphisms of A. Using Lemma 1.5.23 we see that the kernel of the homomorphism N → W is T := B ∩ N, so T (= B ∩ N) is a normal subgroup of N and N/T  W. On the other hand, in view of (2), an element of G acts trivially on A if and only if it keeps every chamber in A stable, thus the kernel

of N → W is n∈N nBn−1 . Hence, T = n∈N nBn−1 . Now let g ∈ G and C  := gC. Let A be an apartment that contains both C and C . Then there is an element b ∈ B that carries the pair (A, C) to the pair (A, C). In particular, bgC is a chamber in A. Hence there is n ∈ N such that bgC = nC, which implies that n−1 bg ∈ B, and so g ∈ BN B. Thus we have shown that G = BN B and condition TS 1 of 1.4.1 has been verified. Given a panel of C, let C  be the other chamber in A that shares this panel. Then there is an element in N that carries C to C . Let s be its image in W. Then it is clear that s carries C  back to C; condition (2) and Lemma 1.5.23 imply that s2 = 1, that is, s is a reflection. By considering all the panels of C we obtain a set S of reflections. We will now show that S generates W. For w ∈ W, we define its length (w) to be the length of a minimal gallery joining C to wC. Let C = C0, C1, . . . , Cm = wC be a minimal gallery (so (w) = m). Let P be the panel shared by C = C0 and C1 and s be the reflection in P. Then sC1 = C, hence applying s to the gallery C1, . . . , Cm (= wC), we obtain the gallery C, . . . , sCn = swC of length m − 1 joining C to swC, therefore, (sw)  n − 1. So by induction on length, we conclude that sw lies in the group generated by S, and hence so does w. This proves that S generates W and we have verified condition TS 2 of Definition 1.4.1. We will now verify TS 3. Let w ∈ W, s ∈ S and b ∈ B. We fix n ∈ N that maps onto w. The chambers C, sC, and bsC share a panel P. Hence, nC, nsC and nbsC share the panel nP. Let C = C0, C1, . . . , Cm be a minimal gallery joining C to nP, and A be an apartment containing C and nbsC. Let b ∈ B be an element that carries the pair (A, C) to the pair (A, C). Since A contains nP, according to Corollary 1.5.27 this apartment contains C0, C1, . . . , Cm . By induction on j we see that b leaves invariant C j for all j  m. It follows that b nP = nP. Therefore , the chamber b nbsC contains the panel nP and so it is either nC or

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nsC. Hence, b nbsB = nB or nsB, which implies that nBsB ⊂ BnB ∪ BnsB. This completes the verification of TS 3. To verify TS 4, let s ∈ S and P be the panel shared by C and sC. As the building B is thick, there is a chamber C  different from C and sC, which shares the panel P. Let A be an apartment that contains C and C . Then an element of G that carries (A, C) to (A, C) belongs to B but not to sBs since it does not  fix the chamber sC (in fact, b sC = C ). Lemma 1.5.29 In the setting of Proposition 1.5.28, let A be an apartment of B and C a chamber of A, and let ρ : B → A be the retraction with center C. For each apartment A containing C there exists g ∈ G such that ρ(x) = gx for all x ∈ A. In particular, for a facet F ⊂ A and g ∈ GC one has ρ(gF) = F. Proof By Proposition 1.5.28 there exists an element g ∈ G that fixes C and maps A to A. The composition of the action of g −1 with ρ is a polysimplicial automorphism of A that fixes C pointwise. By Lemma 1.5.23, this automorphism is trivial. We conclude that ρ|A : A → A coincides with the action of  g. To prove the second statement, take A = gA.

 1.6 The Monoid R This section does not belong to the discussion of affine root systems and abstract buildings; it introduces useful notation that will be applied throughout the book. Consider a group X equipped with a descending filtration Xr indexed by real  numbers r ∈ R. It is oftentimes useful to consider the group Xr+ = s>r Xs , which is contained in Xr and could be a proper subgroup. We can think of r+ as a number that is infinitesimally larger than r; it is larger than r, but smaller

than any real number that is larger than r. Moreover, we can set X∞ = r Xr . The filtration is called separated if X∞ = {1}.  = (R × {0, 1}) ∪ {∞}. We will write This leads us to introduce the set R r in place of (r, 0) and r+ in place of (r, 1), and we think of r+ as a number infinitesimally larger than r.  is made into a totally ordered commutative monoid that contains The set R R as an ordered submonoid via the following rules. (1) (2) (3) (4)

r + (s+) = (r+) + s = (r+) + (s+) = (r + s)+. r + ∞ = (r+) + ∞ = ∞. ∞ > r+ > r for any r ∈ R. r+ > s+ if r > s.

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 by setting (r+)+ = r+ and ∞+ = ∞. We define an operation  r →  r + on R  is as the monoid of intervals of Another way to think about the monoid R R of the form [r, ∞) or (r, ∞) for −∞ < r  ∞. Then r ∈ R corresponds to [r, ∞), r+ corresponds to (r, ∞), and ∞ corresponds to the empty interval. Addition corresponds to pointwise addition of intervals. The operation  r → r+ corresponds to taking the interior. The order is the opposite of the inclusion order. A descending filtration Xr of a group X indexed by R extends, as was just  in a natural way. A filtration discussed, to a descending filtration indexed by R 0 . indexed by R0 can be extended in the same way to R

2 Algebraic Groups

2.1 Henselian Fields Definition 2.1.1 A field k equipped with a non-trivial valuation ω : k × → R is called Henselian if the valuation ω extends uniquely to any algebraic extension of k. We make the convention ω(0) = ∞. Such fields are called quasicomplete in [Ber93, Definition 2.3.1]. We list here some of their properties following the summary given in [Ber93, §§2.3–2.4]. Fact 2.1.2 Let k be a field endowed with a discrete valuation ω. We assume that k is Henselian. Let  be an algebraic extension of k equipped with the unique extension of ω. Then  is a Henselian field. Proposition 2.1.3 The following statements for a field k equipped with a valuation ω : k × → R are equivalent. (1) k is Henselian. (2) The local ring o = {x ∈ k | ω(x)  0} is Henselian in the sense that the statement of Hensel’s lemma holds. Reference.

[Ber93, Proposition 2.4.3].



Example 2.1.4 If k is complete with respect to the topology induced by the valuation, then it is Henselian. Example 2.1.5 Let k be a Henselian field under a valuation ω : k × → R. Let k  be a subfield of k such that the restriction of the valuation ω to it is non-trivial. It is easily seen that the separable closure  of k  in k is Henselian. Thus the algebraic closure  of Q in Q p is an incomplete countable Henselian field with a finite residue field. Denote by m = {x ∈ o | ω(x) > 0} the unique maximal ideal of o and by 68

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f the residue field o/m of k. Denote by  k (respectively o) the completion of k (respectivelyo) with respect to ω. Let k s be a separable closure of k. Proposition 2.1.6 If /k is a finite separable extension, then  ⊗k  k →  is an isomorphism. Upon embedding k s into ( k)s over k →  k, the correspondence  → induces an equivalence between the categories of finite separable extensions of k and of  k. In particular, there is a canonical identification k)s / k). Gal(k s /k) → Gal(( Reference.

[Ber93, Proposition 2.4.1].



Let /k be a Galois extension, finite or infinite. Let f and f be the residue fields of  and k, respectively. Any σ ∈ Gal(/k) preserves ω and hence also the local ring o and its maximal ideal, and induces an action on the residue field f , which we denote by σ | f . Let I(/k) = {σ ∈ Gal(/k)|σ | f = 1} be the inertia subgroup and W(/k) = {σ ∈ Gal(/k) | ω(x − σ(x)) > ω(x) for all x ∈  × } the wild inertia subgroup. Proposition 2.1.7 (1) I(/k) and W(/k) are normal subgroups of Gal(/k) and W(/k) is contained in I(/k). (2) The extension f /f is Galois with Galois group Gal(/k)/I(/k). (3) W(/k) is a pro-p group, where p = char(f) (trivial when p = 0). (4) There is a canonical isomorphism I(/k)/W(/k) → Hom(ω( × )/ω(k × ), f× ). In particular, I(/k)/W(/k) has pro-order prime to p when p > 0. Reference.

[Ber93, Proposition 2.4.4].



Applying this to a separable closure k s we obtain the absolute inertia and wild inertia groups I and W. Then  W is a pro-p group and there is a canonical isomorphismI/W → Hom( ω(k × )/ω(k × ), fs× ), where fs is the separable closure of f and ω(k × ) is the divisible subgroup of R generated by ω(k × ). Proposition 2.1.8 Let /k be a finite separable extension. Then /k is unramified (respectively tamely ramified) if the following conditions hold. (1) ω( × ) = ω(k × ) (respectively p  #(ω( × )/ω(k × ))). (2) [ : k] = [f : f] · #(ω( × )/ω(k × )). (3) f /f is separable. Reference. [Ber93, Proposition 2.4.7].



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In this book we will assume, with a few exceptions like §2.2, that the valuation ω is discrete and we will most often normalize it so that ω(k) = Z. Then × × I/W  Hom(Q/Z, f ) = lim μn (f ) is pro-cyclic, where the limit is taken over ←−− all natural numbers n not divisible by the characteristic of f and the transition map μm → μn is the (m/n)-power map. In particular, any finite totally tamely ramified Galois extension /k is cyclic. We will also assume that f is perfect. We will denote by K the maximal unramified extension of k contained in k s .

2.2 Bounded Subgroups of Reductive Groups In this section, k is a field equipped with a valuation ω and o is its ring of integers. We do not assume that ω is discrete. For now we also do not assume that k is Henselian, although we will do so further below. Let G be a connected reductive k-group. Definition 2.2.1 Let X be an affine k-scheme of finite type. A subset B of X(k) is said to be bounded if for every f ∈ k[X], the set {ω( f (b)) | b ∈ B} is bounded below. Remark 2.2.2 If we choose a closed embedding X → An into affine space, then B ⊂ X(k) is bounded if and only if its image in An (k) = k n is bounded. Using such an embedding, one can endow X(k) with an analytic topology inherited from k by taking the subspace topology of the analytic topology on An (k) = k n . We refer to [Con12] for details and generalization to the case of not necessarily affine varieties, as well as adèles. Fact 2.2.3 If k is locally compact and B ⊂ X(k) is closed, then B is bounded if and only if B is compact with respect to the analytic topology of X(k). Fact 2.2.4 If f : X → Y is a morphism of affine k-schemes of finite type, then f carries bounded subsets of X(k) to bounded subsets of Y (k). Lemma 2.2.5 If /k is a finite extension endowed with a valuation extending ω and X is an affine -scheme of finite type, then under the identification (R/k X)(k) = X() bounded subsets correspond to bounded subsets. Proof A closed embedding X → An realizes R/k X as a closed subscheme of R/k An and reduces the problem to X = An . In that case, one notes that the topology on  induced from the given valuation extending ω coincides with the topology on  as a finite-dimensional k-vector space.  Lemma 2.2.6 If X is an affine algebraic k-group and B, B  are two non-empty

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subsets of X(k), then BB  := {bb | b ∈ B, b ∈ B  } is bounded if and only if both B and B  are bounded. Proof If BB  is bounded, then so are the subsets bB  and Bb for fixed b ∈ B, b ∈ B , and thus B and B  are bounded. If conversely B and B  are bounded, then so is the subset B × B  of (X × X)(k), and also its image BB  under the multiplication map X × X → X.  Lemma 2.2.7 A subgroup B ⊂ GLn (k) is bounded if and only if for each 1  i, j  n the function b → ω(bi, j ) is bounded below, where bi, j is the matrix entry of b at the coordinates i, j. Proof The ring of regular functions of GLn is generated by the matrix entries g → gi, j and the function det−1 . Since B is a subgroup, ω(det−1 (b)) = ω(det(b−1 )) is bounded below if and only if ω(det(b)) is bounded below. The latter is implied by ω(bi, j ) being bounded below, since det is a polynomial in  the gi, j . Lemma 2.2.8 Let C be a k-split torus and C → GL(V) a k-rational representation of C. Let v ∈ V be a vector such that the orbit C(k) · v is bounded. Then C fixes v. Proof Let Θ be the set of non-zero weights of C on V and λ : Gm → C be a 1parameter group such that for θ ∈ Θ, θ, λ  0. Let Θ+ = {θ ∈ Θ | θ, λ > 0}, and Θ− = Θ − Θ+ . We write V = V − ⊕ V 0 ⊕ V + , where V − (respectively V + ) is the sum of the weight spaces for weights in Θ− (respectively Θ+ ), and V 0 is the subspace of V consisting of vectors fixed under the action of λ, and hence fixed under C. We use this decomposition of V to write v = v − + v 0 + v + . The set {λ(t) · v = λ(t) · v − + v 0 + λ(t) · v + | t ∈ k × } is bounded by the hypothesis. But for {t ∈ k × | ω(t) → −∞}, λ(t) · v + → ∞ and λ(t) · v − → 0, so {λ(t) · v | t ∈ k ×, ω(t) → −∞} is unbounded unless v + = 0. Similarly, considering {t ∈ k × | ω(t) → +∞}, we see that {λ(t) · v | t ∈ k ×, ω(t) → ∞} is unbounded unless v − = 0. Therefore, we conclude that as the set {λ(t)·v |t ∈ k × }  is bounded, v = v 0 , that is, C fixes v. The group of k-rational characters on G will be denoted by X∗k (G). The following theorem is due to Bruhat, Tits, and Rousseau. An elementary proof was given in [Pra82], which we recall here for the reader’s convenience. In the rest of this section we will assume that k is Henselian. For any algebraic extension  of k, the unique valuation of , extending the given valuation on k, will also be denoted by ω. Theorem 2.2.9 G(k) is bounded if and only if G is anisotropic over k.

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Thus if k is a locally compact field whose topology is not the discrete topology, then G(k) is compact if and only if G is k-anisotropic. To prove the above theorem we will use the following two lemmas. Lemma 2.2.10 If f : X → Y is a finite k-morphism between affine k-schemes of finite type and B is a bounded subset of Y (k), then the subset f −1 (B) of X(k) is bounded. Proof Since k[X] is module-finite over k[Y ], we can pick a finite set of generators of k[X] as a k[Y ]-module (so also as a k[Y ]-algebra), and each satisfies a monic polynomial over k[Y ]. Hence, this realizes X as a closed subscheme of the closed subscheme Z ⊂ Y × An defined by n monic 1-variable polynomials f1 (t1 ), . . . , fn (tn ) over k[Y ], so it remains to observe that when one has a bound on the coefficients of a monic 1-variable polynomial over k of known degree (e.g., specializing any f j at a k-point of Y ) then one gets a bound on its possible k-rational roots depending only on the given coefficient bound and the degree of the monic polynomial.  Lemma 2.2.11 Let G be an unbounded subgroup of G(k) which is dense in G in the Zariski topology. Let ϕ : G → GL(V) be a finite-dimensional k-rational representation of G with finite kernel. Then G contains an element g which has an eigenvalue ζ ∈ k (for the action on V) with ω(ζ) < 0. An interesting source of subgroups G of G(k) that are Zariski dense in G are those subgroups of G(k) that are open in the analytic topology of G(k). Proof

Let k ⊗k V =: V0 ⊃ V1 ⊃ · · · ⊃ Vs ⊃ Vs+1 = {0}

be a flag of G k -invariant subspaces such that for 0  i  s, the induced  representation ρi of G k on Wi := Vi /Vi+1 is irreducible. Let ρ = i ρi be  W . The kernel of ρ is finite since it is an the representation of G k on i i extension of the (finite) unipotent normal k-subgroup scheme of the reductive group ϕ(G)k , by the finite kernel of ϕ. Now as G is an unbounded subgroup of G(k), Lemma 2.2.10 implies that ρ(G) is an unbounded subgroup of ρ(G(k)). Hence, there is a non-negative integer a  s such that ρa (G) is unbounded. Since Wa is an irreducible G k -module, and G is dense in G in the Zariski topology, ρa (G) spans Endk (Wa ). We fix {gi } ⊂ G so that {ρa (gi )} is a basis of Endk (Wa ). Let { fi } ⊂ Endk (Wa ) be the basis which is dual to the basis {ρa (gi )} with respect to the trace-form. Then Tr( fi · ρa (g j )) = δi j , where δi j is the Kronecker delta. Now assume, if possible, that the eigenvalues of

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all the elements of G lie in the ring of integers ok of k, then for all x ∈ G, Tr(ρa (x)) is contained in ok . For g ∈ G, if ρa (g) = i ci fi , with ci ∈ k, then Tr(ρa (g · g j )) = i ci Tr( fi · ρa (g j )) = c j . As Tr(ρa (g · g j )) ∈ ok , we conclude that c j belongs to the ring of integers ok for all j (and all g ∈ G). This implies  that ρa (G) is bounded, a contradiction. Proof of Theorem 2.2.9 As Gm (k) = k × is unbounded, we see that if G is kisotropic, then G(k) is unbounded. We will now assume that G(k) is unbounded and prove the converse. It is well known that G(k) is dense in G in the Zariski topology [Bor91, 18.3], hence according to Lemma 2.2.11, there is an element g ∈ G(k) such that Adg has an eigenvalue ζ with ω(ζ)  0. Now in the case that k is of positive n characteristic p, after replacing g by g p for some integer n  0, we assume that g is semi-simple. On the other hand, in the case that k is of characteristic zero, let g = s · u = u · s be the Jordan decomposition of g with s ∈ G(k) semi-simple and u ∈ G(k) unipotent. Then the eigenvalues of g are same as those of s. So, after replacing g with s, we may (and do) again assume that g is semi-simple. There is a maximal k-torus T of G such that g ∈ T(k), which one sees by applying [Bor91, Theorem 18.2] to the neutral component of the centralizer of g, which contains g by [Bor91, Corollary 11.12] and is moreover defined over k, since g is. Since any absolutely irreducible representation of a torus is 1-dimensional, there exists a finite Galois extension  of k and a character χ of T such that χ(g) = ζ. Then    γ  ω χ (g) = mω( χ(g)) = mω(ζ)  0; γ ∈Gal(/k)

where m = [ : k]. Thus the character γ ∈Gal(/k) γ χ is non-trivial. On the other hand, this character is obviously defined over k. Hence, T admits a non-trivial character defined over k and therefore it contains a non-trivial k-split subtorus. This proves that if G(k) is unbounded, then G is isotropic over k.  Proposition 2.2.12 We assume that the derived subgroup Gder of G is kanisotropic. Then G(k) contains a unique maximal bounded subgroup G(k)b , which has the following description: G(k)b = {g ∈ G(k) | χ(g) ∈ o× for all χ ∈X∗k (G)}. Proof Let G a be the inverse image of the maximal k-anisotropic subtorus of the k-torus G/G  under the natural homomorphism G → G/Gder . Then G a is the maximal connected normal k-anisotropic subgroup of G. Let S be the maximal k-split central torus of G. Then G = S ·G a (almost direct product). Let

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C = S ∩ G a ; C is a finite central k-subgroup scheme, so G a /C is k-anisotropic, as k-rank is unaffected by central k-isogeny. Let f : G → (G/G a ) × (G/S) = S/C × G a /C be the natural surjective homomorphism with kernel isomorphic to C. The image of the induced homomorphism f ∗ : X∗k ((S/C) × (G a /C)) → X∗k (G) is of finite index. It is obvious that as (G a /C)(k) is bounded (by Theorem 2.2.9), the proposition is true for the direct product (S/C) × (G a /C). Now using Lemma 2.2.10 we conclude that the proposition holds for G.  When Gder is not anisotropic there are many maximal bounded subgroups in G(k). One of the goals of Bruhat–Tits theory is to describe these, cf. Theorem 4.2.15. Note that the subgroup of G(k) that was denoted by G(k)b in Proposition 2.2.12 can be defined without assuming that Gder is anisotropic, but it will not be bounded. The maximal bounded subgroups of GL N (k) can be described easily, as the following proposition shows. Proposition 2.2.13

Let G be an open subgroup of G(k).

(1) Any bounded subgroup of G is contained in a maximal bounded open subgroup of G. (2) Assume that k is complete and its residue field is finite (equivalently, k is locally compact) of characteristic p. Then every pro-p subgroup of G is contained in a maximal open pro-p subgroup of G. (3) If V is a finite-dimensional k-vector space, a subgroup of GL(V) is maximal bounded if and only if it is the stabilizer GL(Λ) of a lattice Λ ⊂ V. The first assertion of this proposition was proved by R.Langlands using an argument different from the one given below. Proof To prove the first two assertions, we fix a faithful k-rational representation G → GL(V) on a finite-dimensional k-vector space V and let L be a lattice in V (that is, L is a bounded open o-submodule of V). (1) Let B be a bounded subgroup of G. We first show that B is contained in a bounded open subgroup of G. The image of B in GL(V) is bounded. By Lemma 2.2.7 all matrix entries of B are bounded. Also all coordinates of L are bounded. Therefore, all coordinates of the o-submodule Λ generated by {b · L | b ∈ B} are bounded, and hence Λ itself is bounded. It is also stable under the action of B. Therefore, B is contained in the stabilizer GL(Λ) ∩ G of Λ in G, which is a bounded open subgroup of G since GL(Λ) is a bounded open subgroup of GL(V). To see the boundedness of GL(Λ), note that there exists a

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non-zero a ∈ k such that L ⊂ Λ ⊂ aL, which implies that the matrix entries of the elements of GL(Λ) are bounded below, and we can apply Lemma 2.2.7. We now assume that B is a bounded open subgroup of G. To prove the first assertion of the proposition using Zorn’s Lemma, it suffices to show that the union of any increasing family {Bi } of bounded open subgroups of G containing B is bounded. If this were not true then according to Lemma 2.2.11, the union  i Bi would contain an element g which has an eigenvalue ζ with ω(ζ) < 0. But, being bounded, no Bi can contain such an element. (2) To prove the second assertion, we first prove that any pro-p subgroup P of G is contained in an open pro-p subgroup of G. The subgroup P ∩ GL(L) is an open subgroup of P and hence it is of finite index. Let Λ be the lattice spanned by {g · L | g ∈ P}. Then P ⊂ GL(Λ). Now let m denote the maximal ideal of o. We consider the following subgroups of GL(Λ): for each integer n  0, let GL(Λ)(n) denote the subgroup consisting of g ∈ GL(Λ) such that (g − I) · Λ ⊂ m n Λ. Then, since in assertion (2) the residue field of k has been assumed to be finite, we can easily see that for all n, GL(Λ)(n) /GL(Λ)(n+1) is finite, and for n  1 it is a p-group. Hence, GL(Λ)(1) is an open normal pro-p subgroup of GL(Λ). Therefore, P · (G ∩ GL(Λ)(1) ) is an open pro-p subgroup of G that contains P. Now let P be an open pro-p subgroup of G. To establish the second assertion using Zorn’s Lemma it suffices to show that the union of any increasing family {Pi } of open pro-p subgroups of G containing P is a pro-p subgroup. If this  were false, then according to Lemma 2.2.11, the union i Pi would contain an element g which has an eigenvalue ζ with ω(ζ) < 0. But, being bounded, no Pi can contain such an element. (3) As o is a principal ideal domain, every lattice in V contains a basis of V that spans the lattice as an o-module. Hence, any two lattices Λ and Λ in V are conjugate to each other under an element of GL(V), so the subgroups GL(Λ) and GL(Λ) of GL(V) are conjugate to each other. Therefore, if GL(Λ) is a maximal bounded subgroup of GL(V), then for every lattice Λ in V, GL(Λ) is also a maximal bounded subgroup of GL(V). We know from (1) that GL(V) contains maximal bounded subgroups. So to prove the third assertion, it suffices to show that given a maximal bounded subgroup B of GL(V), there is a lattice Λ such that B = GL(Λ). Let L be a lattice in V. Then the o-submodule Λ of V generated by b · L, for b ∈ B, is a lattice. Since B stabilizes Λ by construction, we have B ⊂ GL(Λ). Now maximality of the bounded subgroup B implies that B = GL(Λ).  Proposition 2.2.14 Let G be a semi-simple k-group that is k-simple. Every unbounded open subgroup of G(k) contains the normal subgroup G(k)+

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generated by the k-rational elements of the unipotent radicals of parabolic k-subgroups of G. This proposition was stated by J.Tits without proof. The first published proof is in [Pra82]. It is a complement to Proposition 2.2.13. The two statements together say that for an open subgroup G of G(k) there is a dichotomy: either G is bounded, then is contained in a maximal bounded subgroup, or G is unbounded, in which case it is almost equal to the whole group, at least when k is a local field, in the following sense. By the positive solution of the Kneser–Tits problem given in [PR85], if k is a locally compact field whose topology is not the discrete topology and G is k-isotropic and simply connected, G(k)+ = G(k). If we drop the assumption that k is a local field, the identity G(k)+ = G(k) is known when G is quasi-split and simply connected. The proof of Proposition 2.2.14 is based on the following lemma. Lemma 2.2.15 Let G be a semi-simple k-group and let H be an open subgroup of G(k). Assume that there exist an element g ∈ G(k) that normalizes H and a k-simple normal subgroup G  of G such that Ad(g) has an eigenvalue ζ on the Lie algebra of G  with ω(ζ) < 0. Then H contains G (k)+ . Proof As in the proof of Theorem 2.2.9 in the case that k is of characteristic 0, after replacing g by its semi-simple Jordan component, and in the case n that k is of characteristic p > 0, after replacing g with g p for some integer n  0, we assume that g is semi-simple. Again after replacing g by a positive integral power, we assume that the Zariski closure of the cyclic group generated by g is a k-torus S. Let S  be the maximal k-split subtorus of S. Let X∗k (S) (respectivelyX∗k (S )) be the group of characters of S (respectively S ) defined over k and consider the homomorphism X∗k (S) → Z defined by χk → ω( χk (g)) for χk ∈ X∗k (S). Since the Q-space Q ⊗Z Homk (Gm, S)(= Q ⊗Z Homk (Gm, S )) is dual to the Q-space Q ⊗Z X∗k (S)(= Q ⊗Z X∗k (S )), we see that there exists a λ : Gm → S and a positive integer s such that χk , λ = sω( χk (g)) for all χk ∈ X∗k (S). Let  be the splitting field of S and m = [ : k]. Every character of Sk is defined over ; we will denote the group of all such characters simply by X∗ (S). There is a natural homomorphism X∗ (S) → X∗k (S) defined by χ → χk := γ ∈Gal(/k) γ χ. Since for χ ∈ X∗ (S) and γ ∈ Gal(/k), ω( χ(g)) = ω(γ( χ(g))) = ω(γ χ(g)) and γ χ, λ = χ, λ we infer that s 1 ω( χk (g)) = χk , λ = χ, λ . m m In particular, for any weight χ of S in the adjoint representation on the Lie algebra of G , χ, λ = sω( χ(g)). sω( χ(g)) =

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77

In the notation of §2.11, UG  (λ) is the unipotent radical of the parabolic ksubgroup PG  (λ) and UG  (−λ) is the unipotent radical of the opposite parabolic k-subgroup PG  (−λ). Now since for all weights χ of S , χ, λ = sω( χ(g)), we see that U := UG  (λ)(k) admits the following description: U = {x ∈ G (k) | the sequence {g i x g −i } converges to 1 as i → ∞}. We see similarly that, U− := UG  (−λ)(k) admits the following description: U− = {x ∈ G (k) | the sequence {g −i x g i } converges to 1 as i → ∞}. Now we observe that as H ∩ G (k) is an open subgroup, for any u ∈ U (respectivelyu ∈ U− ), for all large positive integers n, g n ug −n (respectivelyg −n ug n ) lies in H ∩ G (k). So H ∩ G (k) contains both U = UG  (λ)(k) and U− = UG  (−λ)(k). But according to [BT73, Proposition 6.2(v)] (see alternatively [ConII, Theorem V.4.5]), UG  (λ)(k) and UG  (−λ)(k) together generate G (k)+ .  Proof of Proposition 2.2.14 We assume that G is k-simple. Let G be an unbounded open subgroup of G(k). According to Lemma 2.2.11 applied to AdG , the subgroup G, being Zariski dense in G, contains an element g such that Adg has an eigenvalue ζ with ω(ζ) < 0. Apply Lemma 2.2.15.  Proposition 2.2.16 malizer in

Let G be a connected reductive k-group. Then the nor-

G(k)1 = {g ∈ G(k) | χ(g) ∈ o× for all χ ∈X∗k (G)} of any bounded open subgroup of G(k) is bounded. Proof We denote by N the normalizer in G(k)1 of some bounded open subgroup H ⊂ G(k). We first reduce to the case that G is semi-simple. For this, let Z be the central torus of G, and consider the isogeny G → G/Gder × G/Z. According to Lemma 2.2.10 it is enough to show that the image of N under this isogeny is bounded. The image of G(k)1 in (G/Gder )(k) is clearly contained in (G/Gder )(k)1 and hence the image is bounded. The projection map G → G/Z is smooth. Therefore, G(k) → (G/Z)(k) is open. So the image H of H in (G/Z)(k) is open; it is also bounded by Fact 2.2.4. The image of N in (G/Z)(k) is contained in the normalizer of H. Hence, it is enough to show that the latter is bounded. This reduces the problem to the semi-simple group G/Z. We now assume that G is semi-simple. By way of contradiction let us assume, if possible, that the normalizer N in G(k)1 = G(k) of some bounded open

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subgroup H of G(k) is unbounded. Being an open subgroup of G(k), N is Zariski dense in G. Hence, according to Lemma 2.2.11 it contains an element g such that in the adjoint representation it has an eigenvalue ζ with ω(ζ) < 0. Now let G  be a normal k-simple subgroup of G such that ζ is an eigenvalue of Ad(g) on the Lie algebra of G . Then G  is k-isotropic, and according to Lemma  2.2.15(1), H contains the unbounded subgroup G (k)+ , a contradiction.

2.3 Fields of Dimension  1 Let F be a field and E a Galois extension of F. For any abelian group M equipped with an action of Gal(E/F) that is continuous for the discrete topology of M (usually referred to as a Gal(E/F)-module) we have the Galois cohomology groups Hi (Gal(E/F), M), which we abbreviate by Hi (E/F, M). If M is not assumed abelian, then we only have the group H0 (E/F, M) and the set H1 (E/F, M). When E = Fs is a separable closure of F we write Hi (F, M) for Hi (E/F, M). For a smooth algebraic F-group G we will write Hi (F, G) = Hi (F, G(Fs )). We refer to [Ser97] for the general theory. Here we summarize some aspects that will be important for us. When E/F is finite and i > 0, every element of Hi (E/F, M) has finite order that divides [E : F]. Therefore, every element of Hi (F, M) has finite order. If there exists an integer n such that Hi (F, M) = 0 for every torsion Gal(Fs /F)module M and every i > n, we say that cd(Gal(Fs /F))  n. If there exists an integer n such that Hi (F, M) = 0 for every Gal(Fs /F)-module M and every i > n, we say that scd(Gal(Fs /F))  n. It is known that cd(Gal(Fs /F))  scd(Gal(Fs /F))  cd(Gal(Fs /F)) + 1. Example 2.3.1

(1) When F is a finite field, cd(Gal(Fs /F)) = 1,

scd(Gal(Fs /F)) = 2.

(2) When F is a finite extension of Q p or F p ((t)), then cd(Gal(Fs /F)) = scd(Gal(Fs /F)) = 2. (3) When F = R, then cd(Gal(Fs /F)) = scd(Gal(Fs /F)) = ∞. Recall [Ser97, Chapter II, §3.1] that dim(F)  1 means that the following equivalent conditions hold. (1) cd(Gal(Fs /F))  1. In addition, if char(F) = p  0, then Br(E)(p) = 0 for every algebraic extension E/F. (2) Br(E) = 0 for every algebraic extension E/F.

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(3) For every algebraic extension E/F and every finite Galois extension L/E, the norm map L × → E × is surjective. Moreover, it is enough to check these conditions for all those E/F that are finite and separable. It is clear that if dim(F)  1, then dim(E)  1 for every algebraic extension E/F. When F is assumed perfect, then dim(F)  1 is equivalent to cd(Gal(Fs /F))  1. Recall further [Ser97, Chapter II, §3.2] that dim(F)  1 is implied by property (C1 ), which states that if f (X1, . . . , Xn ) is a homogeneous polynomial of degree d with coefficients in F, and n > d, then there is a non-zero (x1, . . . , xn ) ∈ F n such that f (x1, . . . , xn ) = 0. Example 2.3.2 The following fields F satisfy property (C1 ), and therefore also dim(F)  1, see [Ser97, Chapter II, §3.3]: (1) a finite field; (2) a discretely valued Henselian field K with algebraically closed residue  is separable over K. field such that the completion K This second example is a theorem proved by Lang in his thesis, and applies in particular to the maximal unramified extension K of a local field k, or the completion of the maximal unramified extension of any discretely valued Henselian field with a perfect residue field. We will see in this section that if K is a discretely valued Henselian field with  algebraically closed residue field, then dim(K)  1, without assuming that K is separable over K. The main reason for our interest in dim(F)  1 is the following theorem due to Steinberg. Theorem 2.3.3 Let F be a field with dim(F)  1. (1) The Galois cohomology H1 (F, G) is trivial for any connected reductive F-group G. (2) Any connected reductive F-group is quasi-split. If in addition F is assumed perfect, then the above statements hold for any connected affine algebraic F-group, without assuming that it is reductive. Reference For (1) see [Ser97, Chapter III, §2.3, Theorem 1 and Remark 1]. For (2) see [Ser97, Chapter III, §2.2, Theorem 1]. We note that the proof of the fact that assertion (i) of that theorem implies that the semi-simple group L in that proof contains a Borel subgroup defined over the base field does not require the base field to be perfect. 

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The proof of Theorem 2.3.3(1) is complicated and relies, among other things, on the existence of Steinberg sections, which is the main result of [Ste65]. When G = T is a torus the vanishing of H1 (F,T) is much more elementary, and will be used often in this book, so we give a self-contained proof below, see Lemma 2.5.4. Let now K be a discretely valued Henselian field with algebraically closed residue field. We will show that dim(K)  1. This requires some preparation. Proposition 2.3.4 Let k be a discretely valued Henselian field, o its ring of integers and  k (respectively o) be the completion of k (respectivelyo). (1) Let X be a smooth scheme over k. The image of X(k) in X( k) is dense. (2) Let X be a smooth o-scheme. Then the image of X (o) in X ( o ) is dense. Proof (1) Assertion (1) is Zariski local on X. By the Zariski local structure for smooth morphisms [EGAIV4 , 17.11.4, 18.4.6(ii)] applied to X → Spec(k), we can thereby reduce to the case that X is affine and “standard-étale” over an affine space Akn = Spec(R) for R = k[t1, . . . , tn ]. In other words, X = Spec(R[Y ]/ f )h f  for a monic f (t,Y ) ∈ R[Y ] and some h(t,Y ) ∈ R[Y ]. A point in X( k) thereby n  k corresponds to an ordered n-tuple c = (c1, . . . , cn ) ∈ k and a simple root y ∈  k n sufficiently near c, we claim that of f (c,Y ) with h(c, y)  0. For any c  ∈  k of for each simple root of f (c,Y ) in  k (e.g., y) there is a unique root y  ∈   f (c ,Y ) near y (made as close as we wish by taking c  near enough to c). Grant that claim. By taking c  near enough to c we can ensure (h f )(c , y )  0 since (h f )(c, y)  0. In particular, taking such c  in k n , we get the monic k as f (c ,Y ) in k[Y ] for which the claim provides a unique simple root y  in  close as we want to y. This root viewed in k s must be in k due to Proposition 2.1.6. The point (c , y ) in X(k) is as close as we want to the initial point (c, y) ∈ X( k), so we would be done. Now we are left with proving a result about roots over  k. More specifically, kn writing f (t,Y ) = Y d + ad−1 (t)Y d−1 + · · · + a0 (t) for a j (t) ∈ R, by taking c  ∈   near enough to c we can make each a j (c ) as close as we want to a j (c). In this way, the role of the affine space disappears and we just have to prove “continuity of separable roots” over  k for monic polynomials of a fixed degree over  k. This is a well-known fact; see for example [BGR84, p.146]. Note that, while the reference only guarantees the existence of a root in  k s , the uniqueness of that root and Galois theory implies that it must lie in  k. (2) By (1), the image of X (k) in X ( k) is dense. Now since X ( o ) is an open subset of X ( k), we see that the image of X (k) ∩ X ( o ) = X (o) is dense in X ( o ). 

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Proposition 2.3.5 For any smooth algebraic K-group G the natural map  G) is injective. In particular, for a connected reductive H1 (K, G) → H1 (K, K-group G one has H1 (K, G) = {1}. Proof The second statement follows from the first and Theorem 2.3.3. For the first statement we note that, since this is a map of sets, we must check that each fiber has at most one element. This can be reduced to checking the fiber through the trivial element using Serre twisting explained in [Ser97, §5.4]: given any element x ∈ H1 (K, G), choose a cocycle representing it and twist G  G) through by it to obtain a twisted form G x . The fiber of H1 (K, G) → H1 (K,  G x ) through the trivial x is identified with the fiber of H1 (K, G x ) → H1 (K, element. We now consider the fiber through the trivial element. Let X be a G-torsor over K representing an element of H1 (K, G) that maps to the trivial element  G). Thus X has a K-point.  Since G is smooth, so is X. Proposition of H1 (K, 2.3.4(1) implies that X has a K-point, hence it represents the trivial element of  H1 (K, G).  G) is actually bijective, see Remark 2.3.6 The map H1 (K, G) → H1 (K, [GGMB14, Proposition 3.5.3(2)]. Corollary 2.3.7

dim(K)  1.

Proof Consider a finite separable extension E/K and a finite Galois extension L/E. We need to show that the norm map L × → E × is surjective. Since E is itself Henselian and discretely valued, cf. Fact 2.1.2, Proposition 2.3.5 implies that H1 (E,T) vanishes for any E-torus T. We consider the norm map R L/E Gm → Gm and let T be its kernel. The vanishing of H1 (E,T) implies the  surjectivity of L × → E × . Corollary 2.3.8

Any connected reductive K-group G is quasi-split.

2.4 Affine Group Schemes over Perfect Fields Let F be a perfect field. We recall some basic material on affine algebraic F-group schemes. To simplify notation, we drop the word “scheme”.

(a) Homomorphisms and Kernels To an affine algebraic F-group A we can associate the closed subgroup Ared , which is a reduced, and hence smooth, closed subgroup of A, [DG70, II.5.2.3].

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In particular, A is smooth if and only if it is reduced. When char(F) = 0 then A is automatically reduced ([DG70, II.6.1.1]) but when char(F) = p > 0 there do exist non-reduced affine algebraic F-groups, such as the connected multiplicative group μ p and the connected unipotent group αp . A smooth closed subgroup B of the affine algebraic group A is uniquely determined by the subgroup B(F) of the group A(F), [DG70, II.5.4.3]. Consider a homomorphism A → B of smooth affine algebraic F-groups and let K be its kernel, that is, the scheme-theoretic fiber product of the diagram A → B ← {1}. Then K is a closed, hence affine, algebraic subgroup of A. It need not be smooth. The image of the topological space A (the underlying space of the scheme A equipped with the Zariski topology) in the topological space B is closed and we can equip it with the reduced scheme structure. It is then a smooth closed, hence affine, subgroup of B, which we call im(A → B). The homomorphism A → B factors as the composition of a faithfully flat homomorphism A → im(A → B) and a closed immersion im(A → B) → B. We have dim(A) − dim(K) = dim(im(A → B)). For all this, see [DG70, II.5.5.1]. Applying the Lie algebra functor to these homomorphisms we obtain a homomorphism Lie(A) → Lie(B) that factors as the composition of a homomorphism Lie(A) → Lie(im(A → B)) and Lie(im(A → B)) → Lie(B). The homomorphism Lie(A) → Lie(im(A → B)) need not be surjective; it is surjective if and only if K is smooth, see [DG70, II.5.5.3]. In the case when the smooth closed subgroup im(A → B) of B is normal we can form the quotient of B by that subgroup, cf. [DG70, III.3.5.6], which we will denote by cok(A → B). The homomorphism B → cok(A → B) is faithfully flat and its kernel is im(A → B). The group cok(A → B) is smooth. The following statements are equivalent. (1) K is smooth. (2) The inclusion im(Lie(A) → Lie(B)) ⊂ Lie(im(A → B)) is an equality. (3) The surjection cok(Lie(A) → Lie(B)) → Lie(cok(A → B)) is an isomorphism.

(b) Unipotent Groups Remark 2.4.1 In this remark, F is a general field. Consider a unipotent Fgroup U. When char(F) = 0, U is automatically smooth and connected, as well as split. The latter means that U has a composition series whose successive subquotients are isomorphic to Ga , cf. [Bor91, Definition 15.1]. Indeed, smoothness was already remarked in §(a) above. To see connectedness, replace

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U by π0 (U) = U/U 0 , so that now U is a smooth unipotent finite group. Apply [Bor91, 4.8 Corollary] to realize U as a closed subgroup of the group Un of unipotent upper triangular matrices in GLn for some suitable n, and note that ¯ ¯ = {1} which, since U is Un does not have F-points of finite order. Thus U(F) smooth, implies U = {1}. When char(F) = p > 0, U need not be smooth or connected. The étale group Z/pZ is a smooth disconnected unipotent group, while the group αp whose coordinate ring is F[X]/(X p ) is connected unipotent, but not smooth. When F is perfect, a smooth connected unipotent group is always split, cf. [Bor91, Corollary 15.5(ii)]. This need not be the case when F is not perfect. Examples of non-split smooth connected unipotent groups are provided by the F-wound unipotent groups, see [CGP15, Appendix B.2]. Lemma 2.4.2 Let F be a perfect field, E/F a Galois extension with Galois group Θ, and U a smooth connected unipotent F-group. Then H1 (Θ, U(E)) = {0}. If U is commutative, then Hn (Θ, U(E)) = {0} for every n  1. Proof Since F is perfect, U is split, that is, it has a composition series whose subquotients are isomorphic to the additive group Ga , cf. [Bor91, Corollary 15.5]. This reduces the proof to the case U = Ga . In that case Hn (Θ, U(E)) = lim Hn (Θ, E ), where the limit is taken over the system of −−→ finite Galois extensions E  of F inside E, and Θ = Gal(E /F). The normal basis theorem implies that the additive group E  is an induced F[Θ]-module,  hence Hn (Θ, E ) vanishes.

(c) The Reductive Quotient and Conjugacy Results Let G be a smooth connected affine algebraic F-group. Let Ru (G F ) be the unipotent radical of the base change G F = G ×F F, where F is an algebraic closure of F. Since we are assuming that F is perfect, the extension F/F is Galois, and Galois descent implies that Ru (G F ) descends to F, and hence equals Ru (G)F , cf. [Bor91, Theorem 14.4] or [CGP15, Proposition 1.1.9]. Therefore the quotient G = G/Ru (G) is a connected reductive F-group. If F is not perfect, then the base change to F of the unipotent radical Ru (G) of G can be strictly smaller than the unipotent radical Ru (G F ) of G F , so G/Ru (G) can fail to be reductive. It is pseudo-reductive, and the rich and beautiful structure of such groups is the topic of [CGP15]. In the current book we will often be working with groups that are either connected and reductive, defined over a base field that need not be perfect, or more generally smooth connected affine algebraic but defined over a perfect

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base field. Here we will collect some basic results about the structure of those latter groups. Let G be a smooth connected affine F-group. Proposition 2.4.3 The natural projection map G(F) → G(F) is surjective. Proof This follows at once from Lemma 2.4.2.



Theorem 2.4.4 ([Bor91, Theorem 15.14]) Any two maximal F-split tori of G are conjugate under G(F). Proposition 2.4.5 Consider the natural quotient map f : G → G. The maps P → f (P),

P → f −1 (P)

are mutually inverse Gal(F/F)-equivariant, G(F)-equivariant, inclusion-preserving bijections between the sets of parabolic subgroups of G and G, respectively. Proof It is clear that the maps are Gal(F/F)-equivariant. For the rest of the statement, we can extend scalars to F and hence assume F = F. Any parabolic subgroup of G contains a Borel subgroup, hence also the unipotent radical Ru (G). The claim follows from [Bor91, Corollary 11.2, Proposition 11.14].  Theorem 2.4.6

Let P1, P2 be minimal parabolic F-subgroups of G.

(1) There exists g ∈ G(F) such that gP1 g −1 = P2 . (2) The intersection P1 ∩ P2 contains the centralizer of a maximal F-split torus. (3) If S is a maximal F-split torus contained in P1 ∩ P2 , then g can be chosen to normalize S. Proof (1) follows from [Bor91, Theorem 20.9(1)] applied to G and Proposition 2.4.5. (2) is [Bor91, Theorem 20.7(1)]. (3) follows by applying Theorem 2.4.4 to the group P2 and the maximal  F-split tori S and gSg −1 . Proposition 2.4.7 There exists a proper parabolic F-subgroup of G if and only if G has a non-central F-split torus. Proof This follows from [Bor91, Theorem 20.6(ii)] applied to G and Proposition 2.4.5.  Lemma 2.4.8

Let S ⊂ G be a maximal F-split torus.

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(1) The affine F-group NG (S)/ZG (S) is finite étale. (2) The image S ⊂ G of S under the quotient map f : G → G is a maximal F-split torus and every maximal F-split torus of G arises this way. (3) The restriction of the quotient map G → G to S induces isomorphisms S → S and NG (S)/ZG (S) → NG (S)/ZG (S), and surjective maps NG (S) → NG (S) and ZG (S) → ZG (S). Proof (1) The automorphism group scheme Aut(S) is étale. The conjugation action of NG (S) on S induces a homomorphism of F-groups NG (S)/ZG (S) → Aut(S) with trivial kernel, and hence realizes NG (S)/ZG (S) as a closed subgroup of Aut(S). Therefore, NG (S)/ZG (S) is étale. Being of finite type, it is finite étale. (2) The kernel of the restriction of f to S is S ∩ Ru (G), which is trivial. Thus f restricts to an isomorphism S → S. In particular, the dimension of S is not larger than the dimension of a maximal F-split torus of G. Conversely, consider a maximal F-split torus S ⊂ G and let R be its preimage under f (as a topological subspace of the underlying topological space of G, which, being closed, we endow with the reduced scheme structure). Then R is a smooth closed subgroup of G and is an extension of S by Ru (G), hence connected and solvable. Let S be a maximal F-split torus of R. Then f restricts to an isomorphism S → S. We conclude that the dimension of a maximal F-split torus of G is not smaller than that of S. (3) In the course of proving (2) we already established that f induces an isomorphism S → S. It is clear that f (NG (S)) ⊂ NG (S) and f (ZG (S)) ⊂ ZG (S), so f does indeed induce a map NG (S)/ZG (S) → NG (S)/ZG (S). Since both source and target embed into Aut(S) = Aut(S), the map is injective. To show surjectivity, let n ∈ NG (S)(F) and choose g ∈ G(F) mapping to n. Then g normalizes the preimage R ⊂ G of S. Then both S and gSg −1 are maximal F-tori of R. According to [Bor91, Theorem 19.2] there exists an element h ∈ Ru (R) = Ru (G) such that hg ∈ NG (S). Note that f (hg) = n. We have thus shown that the map NG (S) → NG (S) is surjective. This implies that the map NG (S)/ZG (S) → NG (S)/ZG (S) is bijective on F-points. Since source and target are étale by (1), we conclude that the map is an isomorphism. The  surjectivity of ZG (S) → ZG (S) now follows. Proposition 2.4.9 Let P ⊂ G be a parabolic F-subgroup and let S ⊂ P be a maximal F-split torus in P. Then ZG (S) is contained in P. Proof Proposition 2.4.5 and Lemma 2.4.8(3) reduce to the case that G is reductive. Let U be the unipotent radical of P and let S be the image of S under P → P/U. The restriction of this quotient map to S induces an isomorphism

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S → S. Let M be a Levi factor of P defined over F, cf. [Bor91, Proposition 20.5]. The composition M → P → P/U is an isomorphism. Let S1 ⊂ M be the preimage of S. Then S1 is an F-split torus of P of the same dimension as S, hence also of the same dimension as S. Thus S1 is a maximal F-split torus of P. Therefore there exists an element of P(F) conjugating S1 to S. Conjugating M by this element we may assume that S1 = S. Then S contains the maximal F-split torus S2 of the center of M and therefore ZG (S) ⊂ ZG (S2 ). The latter equals M by [Bor91, Proposition 20.6] and is thus contained in P. 

2.5 Tori Let F be a field, Fs /F a separable closure, and Θ = Gal(Fs /F). Let T be an F-torus. We have the finitely generated free Z-modules X∗ (T) = HomFs (T, Gm ) and X∗ (T) = HomFs (Gm,T), in natural duality. We have the natural Θ-equivariant isomorphisms X∗ (T) ⊗Z (Fs )× → T(Fs ),

λ ⊗ x → λ(x)

and T(k s ) → Hom(X∗ (T), (Fs )× ),

t → ( χ → χ(t)).

(a) Induced Tori Definition 2.5.1 An F-torus T is called induced if the lattice X∗ (T), equivalently X∗ (T), has a Z-basis that is invariant under the Galois group of the splitting extension of T. If E/F is a finite separable extension then the torus T = RE/F Gm is induced. ΘF ΘF ∗ F We have X∗ (T) = IndΘ Θ E X (Gm ) = IndΘ E Z and X∗ (T) = IndΘ E X∗ (Gm ) =

F IndΘ Θ E Z. A general induced torus is a product of tori of this form, for possibly different extensions E/F. Sometimes such tori are also called quasi-split in the literature, although this does cause some confusion – any torus whatsoever is a quasi-split reductive group. If G is a quasi-split connected reductive Fgroup and T is a maximal F-torus contained in a Borel F-subgroup, and if in addition G is either simply connected or adjoint, then T is induced, since the fundamental weights or the simple roots, respectively, form a Θ-invariant basis of X∗ (T). F 2.5.2 The map IndΘ Θ E Z → Z given by sending f : ΘE \ ΘF → Z to the sum of its values is ΘF -invariant and its kernel is a primitive sublattice. Therefore,

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this map induces a smooth surjective morphism of tori RE/F Gm → Gm whose kernel is again a torus. Lemma 2.5.3

Let T be an F-torus. There exists an exact sequence R1 → R0 → T → 1

with R1 and R0 induced. In particular, ker(R0 → T) is an F-torus. Proof Let E/F be the splitting extension of T. Define R0 = RE/F (T ×F E). Then R0 is induced and the norm map R0 → T has connected kernel. This kernel being a torus, the same procedure can be applied.  Lemma 2.5.4 Let F be a field with dim(F)  1. For any F-torus T and any i > 0, Hi (F,T) is trivial. Proof

By Lemma 2.5.3, there is a short exact sequence of F-tori 1 → T → R → T → 1

with R induced. Shapiro’s lemma expresses Hi (F, R) as a product of groups of the form Hi (E, Gm ) for some finite separable extensions E/F. Hilbert’s theorem 90 implies H1 (E, Gm ) = 1, while the assumption dim(F)  1 implies dim(E)  1, hence H2 (E, Gm ) = Br(E) = 1. Finally, H3 (E, Gm ) = 1 since the strict cohomological dimension of Gal(Es /E) is at most 1 more than the cohomological dimension, and hence at most 2. Thus the vanishing of Hi (F,T) is equivalent to the vanishing of Hi+1 (F,T ). Iterating this procedure we reduce to the case i  3, which again follows from the fact that the strict cohomological dimension of Gal(Fs /F) is at most 2. 

(b) The Valuation Homomorphism We new specialize the base field to be a discretely valued Henselian field k with perfect residue field. Thus Θ = Gal(k s /k) for a fixed separable closure k s of k. Let T be a k-torus. We have the functorial homomorphism ωT : T(k) → HomZ (X∗ (T)Θ, Z),

t → ( χ → −ω( χ(t))).

Lemma 2.5.5 The restriction map along the inclusion X∗ (T)Θ → X∗ (T) induces an isomorphism HomZ (X∗ (T), Q)Θ → HomZ (X∗ (T)Θ, Q). Proof

The exact sequence 0 → X∗ (T)Θ → X∗ (T) →

X∗ (T) →0 X∗ (T)Θ

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remains exact after applying HomZ (−, Q)Θ . But Hom(X∗ (T)/X∗ (T)Θ, Q)Θ = Hom((X∗ (T)/X∗ (T)Θ )Θ, Q) and the abelian group (X∗ (T)/X∗ (T)Θ )Θ is finite,  since (X∗ (T)/X∗ (T)Θ )Θ = {0}. Composing ωT with the inclusion HomZ (X∗ (T)Θ, Z) → HomZ (X∗ (T)Θ, Q) and the inverse of the isomorphism of Lemma 2.5.5 we obtain the functorial homomorphism ωT : T(k) → (X∗ (T) ⊗Z Q)Θ = X∗ (T)Θ ⊗Z Q.

(2.5.1)

Lemma 2.5.6 Let /k be a Galois extension of finite ramification degree, equipped with the unique valuation that extends ω. Then ωT is the restriction to T(k) of ωT : T() → X∗ (T)Θ ⊗Z Q. Proof This follows at once from the fact that the formula ωT (t), χ = ω( χ(t)) for t ∈ T(k) and χ ∈ X∗ (T)Θ uniquely determines ωT (t) ∈ X∗ (T)Θ ⊗ Q and is independent of k.  Lemma 2.5.7 The image of ωT contains X∗ (T)Θ ⊂ X∗ (T)Θ ⊗Z Q as a subgroup of finite index. This index is equal to 1 if T is split. Proof If T is split, then ωT : T(k) → HomZ (X∗ (T), Z) = X∗ (T) is surjective, as is seen by reducing to the case T = Gm . For a general torus T, let S ⊂ T be the maximal split torus. Then X∗ (S) = X∗ (T)Θ and the functoriality of ωT implies that the image of ωT contains the image of ωS , which is X∗ (T)Θ . If /k is a finite Galois extension splitting T and e is its ramification degree, then the  image of ωT equals e−1 X∗ (T). The claim follows from Lemma 2.5.6. For an exact computation of the image of ωT see Corollary 11.7.3.

(c) The Maximal Bounded Subgroup We continue with a discretely valued Henselian field k with perfect residue field, and let K be the maximal unramified extension in a fixed separable closure k s . Proposition 2.5.8 Let T be a k-torus. The group T(k) has a unique maximal bounded subgroup, denoted by T(k)b or T(k)1 . It is given by T(k)b = ker(ωT ) = {x ∈ T(k) | for all χ ∈ X∗ (T)Θ, ω( χ(x)) = 0} = {x ∈ T(k) | for all χ ∈ X∗ (T), ω( χ(x)) = 0}. In particular, T(k) is bounded if and only if T is anisotropic, i.e., X∗ (T)Θ = {0}.

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Proof The second and third equalities follow from Equation (2.5.1) and Lemma 2.5.5. The rest of the statement is a special case of Proposition 2.2.12.  Proposition 2.5.9 Let f : T → T  be a morphism of k-tori. Then the map T(k) → T (k) maps T(k)b into T (k)b . If the kernel of this morphism is anisotropic (in particular, if it is finite), then T(k)b = f −1 (T (k)b ). Proof The first claim follows from Fact 2.2.4. For the second, the assumption that X∗ (ker( f ))Θ is finite implies that the image of (X∗ (T )/X∗ (cok( f )))Θ → X∗ (T)Θ is of finite index. We may replace Θ = Gal(k s /k) by any finite quotient through which the action on X ∗ (T) and X ∗ (T ) factors. Then the group H1 (Θ, X∗ (cok( f ))) is finite, and this implies that the image of X∗ (T )Θ → X∗ (T)Θ is of finite index, hence Q ⊗Z X∗ (T )Θ → Q ⊗Z X∗ (T)Θ is surjective, hence Q ⊗Z X∗ (T)Θ → Q ⊗Z X∗ (T )Θ is injective. Therefore ωT (x) = 0 is  equivalent to ωT  ( f (x)) = 0. Lemma 2.5.10 Let 1 → T1 → T2 → T3 → 1 be an exact sequence of tori. If T1 is anisotropic over K then T2 (K)b → T3 (K)b is surjective. In general, the image of T2 (K)b is of finite index in T3 (K)b . Proof Proposition 2.3.5 implies that T2 (K) → T3 (K) is surjective. The first claim follows from Proposition 2.5.9. According to Lemma 2.5.7 there exists a natural number e such that the image of ωTi ×k K lies in e−1 X∗ (Ti )Gal(ks /K) ⊂ Q ⊗Z X∗ (Ti )Gal(ks /K) for i = 1, 2, 3. We apply the snake lemma to ωTi : Ti (K) → X∗ (Ti )Gal(ks /K) to see that the index of the image of T2 (K)b in T3 (K)b equals the index of the image of T1 (K) in e−1 X∗ (T1 )Gal(ks /K) . By Lemma 2.5.7 this image contains  X∗ (T1 )Gal(ks /K) , so the index is finite.

(d) The Iwahori Subgroup Remark 2.5.11 We recall from [Ser79, Chapter V, §7, Corollary 4] that for a finite separable extension L/K the norm map O×L → O× is surjective, where O and O L denote the rings of integers of K and L, respectively. Lemma 2.5.12 Let T be a k-torus and L/K a finite Galois extension splitting T. The image of T(L)b under the norm map T(L) → T(K) is an open subgroup of T(K) that is independent of L. Proof That the image is open follows from the fact that the norm map T(L) → T(K) is the map on K-points of the algebraic norm map R L/K (T ×K L) → T,

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which in turn is smooth and surjective. For independence of L it is enough to show that if L ⊂ L  then the norm map T(L ) → T(L) maps T(L )b surjectively onto T(L)b . Since T splits over L one reduces immediately to T = Gm , which follows from Remark 2.5.11.  Definition 2.5.13 Let T(K)0 ⊂ T(K) be the image of T(L)b under the norm map T(L) → T(K) for any finite Galois extension L/K splitting T and let T(k)0 = T(K)0 ∩ T(k). These are called the Iwahori subgroup of T(K) respectively T(k). Recall from 2.5.8 that both T(k)b and T(k)1 denote the maximal bounded subgroup of T(k). Lemma 2.5.14 finite index.

The Iwahori subgroup T(k)0 lies in T(k)1 as a subgroup of

Proof Using T(k)0 = T(K)0 ∩ T(k) and T(k)1 = T(K)1 ∩ T(k) one reduces immediately to the case k = K. Lemma 2.5.10 applied to the norm map  R L/K (TL ) → T implies the claim. Remark 2.5.15 The inclusion T(k)0 ⊂ T(k)1 may be strict. For example, let k = Q p and let /k be a quadratic extension. Let T be the kernel of the norm map R/k Gm → Gm . Then T(k)1 = T(k) =  1 is the subgroup of  × consisting of elements whose norm in k equals 1. Let L be the maximal unramified extension of  in k s . The torus T splits over , hence also over L. Let σ be the non-trivial element of Gal(/k). If /k is unramified, then L = K, so T(K)0 = T(K)1 . On the other hand, if /k is ramified, then L/K × is quadratic and T(K) = L 1 = μ2 (f) × (1 + M L )1 , where μ2 (f) = {±1} ⊂ f , M L is the maximal ideal of O L and (1 + M L )1 = L 1 ∩ (1 + M L ). The group × T(L)b = O×L = f × (1 + M L ) maps to T(K) = L 1 by the map x → x/σ(x). ×

This map is trivial on f and its image lies in (1 + M L )1  T(K)1 , which shows that T(K)0 is a proper subgroup of T(K)1 . We can compute this subgroup more precisely. If 1 + z ∈ 1 + M L then     z − σ(z) 1+z −1 =ω ω = ω(z − σ(z)). σ(1 + z) σ(1 + z) If /k is tamely ramified (i.e. p  2) then there exists z for which this valuation equals 1, and one concludes that T(K)0 = (1 + M L )1 . If /k is wildly ramified (i.e. p = 2) the smallest such valuation equals s + 1, where s is the break √ of the lower ramification filtration of Gal(/k). For example, when  = Q2 ( 2), then s = 2, and T(K)0 = (1 + M2L )1 .

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Definition 2.5.16 We say that T has induced ramification if TK is an induced torus in the sense of Definition 2.5.1. Remark 2.5.17 This definition is equivalent to the requirement that X∗ (T), equivalently X∗ (T), has a basis permuted by the inertia subgroup of the Galois group. Lemma 2.5.18 If T has induced ramification, then T(k)0 = T(k)1 . Proof One reduces immediately to k = K, and then further to the case that T = R L/K Gm . Let M/K be a Galois extension containing L and thus splitting T. To compute the norm map T(M) → T(K) we note that R M/K (TM ) = R M/K (R(L ⊗ K M)/M Gm, L ⊗ K M ) and the group of K points of this torus equals (L ⊗K M)× . The norm map  (L ⊗K M)× = T(M) → T(K) = L × sends l ⊗ m to σ ∈Gal(M/K) lσ(m). We have the algebra isomorphism  M, l ⊗ m → (mτ(l))τ , L ⊗K M → τ

where τ runs over the set of embeddings of K-algebras L → M. That set is the coset space Gal(M/K)/Gal(M/L) and we see that the norm map becomes    M × → L ×, (mτ ) → γ τ −1 (mτ ). τ



γ ∈Gal(M/K) τ

We have R M/K (TM )(K)1 = τ O×M and the above map sends this surjectively  onto O×L = T(K)1 , according to Remark 2.5.11. Lemma 2.5.19

A morphism T → T  of k-tori maps T(k)0 to T (k)0 .

Proof We can choose L splitting both T and T  and obtain the commutative diagram / R L/K (T  L ) R L/K (TL )  T

 / T

from which the statement follows.



Lemma 2.5.20 Let 1 → T1 → T2 → T3 → 1 be an exact sequence of K-tori. (1) The map T2 (K)0 → T3 (K)0 is surjective. (2) If T1 is induced, then 1 → T1 (K)0 → T2 (K)0 → T3 (K)0 → 1 is exact. (3) If both T1 and T3 are induced, then T2 (K)0 = T2 (K)1 .

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Proof Let E/K be a finite Galois extension splitting the three tori. Tensoring the exact sequence 0 → X∗ (T1 ) → X∗ (T2 ) → X∗ (T3 ) → 0 with OE we obtain the exact sequence 1 → T1 (E)0 → T2 (E)0 → T3 (E)0 → 1, and hence the first point. The second point follows from Lemma 2.5.19 and T2 (K)0 ∩ T1 (K) ⊂ T2 (K)1 ∩T1 (K) = T1 (K)1 = T1 (K)0 , where the last equality is by Lemma 2.5.18. For the third point, consider t ∈ T2 (K)1 . Its image in T3 (K) lies in T3 (K)1 by Proposition 2.5.9, which equals T3 (K)0 by Lemma 2.5.18. Therefore we may modify t by an element of T2 (K)0 so that it lies in T1 (K). But then it lies in  T1 (K) ∩ T2 (K)1 = T1 (K)1 = T1 (K)0 , again by Lemma 2.5.18. Example 2.5.21 In the situation of Lemma 2.5.20, it is not always true that T2 (K)b → T3 (K)b is surjective. As an example we can take T2 = R/k Gm for a ramified separable quadratic extension /k, 1 Gm = ker(N : R/k Gm → Gm ), T3 = R/k

 and the map T2 → T3 sending x to x/x. Only when T1 is anisotropic over K one has the desired surjectivity, by Lemma 2.5.10.

(e) The Lie Algebra Construction 2.5.22 Let T be a k-torus. There is a functorial Galois-equivariant identification of the Lie algebra of Tks with the k s -vector space X∗ (T) ⊗Z k s equipped with diagonal Galois action, and hence of the Lie algebra of T with the k-vector space (X∗ (T) ⊗Z k s )Θ . Taking the derivative of an element of X∗ (T) = Homks (Gm,T) gives a Lie algebra homomorphism Lie(Gm, ks ) → Lie(Tks ). The image of the canonical element ∂tt ∈ Lie(Gm, ks ) gives an element of Lie(Tks ).

2.6 Reductive Groups In this section we will review some basic notation and results about reductive groups that will be used throughout the book. This will by no means be a complete survey. We refer the reader to [Bor91] for the basic theory, and to [CGP15] for comprehensive treatment of the more general case of pseudoreductive groups. In §§(a)–(c) we will work over an arbitrary field F, that in the body of the book will be specialized to either a discretely valued Henselian field k, or its (mostly assumed perfect) residue field f.

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(a) Basic Notation In this subsection F is any field and G is a connected reductive F-group. We denote by Gder its derived subgroup, by Gad the adjoint quotient G/ZG = Gder /ZGder , and by Gsc the simply connected cover of Gder . If T ⊂ G is a maximal torus, its preimage Tsc under the map Gsc is a maximal torus of Gsc . The intersection of T with Gder is a maximal torus Tder of Gder . The image Tad of T under the map G → Gad is a maximal torus of Gad . The natural maps between (absolute) root systems Φ(Tsc, Gsc ) → Φ(Tder, Gder ) → Φ(T, G) → Φ(Tad, Gad ) are bijective and we identify all of these root systems with Φ(T, G). For  a ∈ Φ(T, G), the natural maps Gsc → Gder → G → Gad induce isomorphisms on the corresponding (absolute) root groups, and we will denote all of them by Ua. If S ⊂ G is a maximal split torus, let Sder be the maximal split torus of Gder contained in S, Ssc the maximal split torus of Gsc whose image in Gder lies in Sder , and Sad the image of S in Gad , a maximal split torus in Gad . The natural maps between relative root systems Φ(Ssc, Gsc ) → Φ(Sder, Gder ) → Φ(S, G) → Φ(Sad, Gad ) are bijective and we identify all of these root systems with Φ(S, G). For a ∈ Φ(S, G) the natural maps Gsc → Gder → G → Gad induce isomorphisms on the corresponding relative root groups, and we will denote all of them by Ua . The following fact from Galois cohomology is often useful. Fact 2.6.1 Let F be any field. For any connected reductive F-group G and a parabolic F-subgroup P ⊂ G with Levi factor L, the natural map H 1 (F, L) → H 1 (F, P) is bijective and the natural map H 1 (F, P) → H 1 (F, G) is injective. Definition 2.6.2 A maximal F-torus T ⊂ G is called maximally split, if its maximal F-split subtorus S is a maximal F-split torus in G.

(b) The Absolute and Relative Root Datum We continue with an arbitrary field F. Let G be a connected reductive F-group and let S ⊂ G be a maximal split torus. The subset Φ(S) ⊂ X∗ (S) of weights for the action of S on the Lie-algebra of G forms a root system, and the action of the finite group W = NG (S)(F)/ZG (S)(F) on X∗ (S) identifies W with the Weyl group of that root system [Bor91, Theorem 21.6]. We will refer to Φ(S) as the relative root system of G.

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2.6.3 The kernel of a root Let a ∈ Φ(S) ⊂ X∗ (S) and let Ca = ker(a) ⊂ S. Then Ca is a closed subgroup of S, evidently of multiplicative type. Note however that Ca may not be smooth. For example, one of the roots of the split reductive group Sp4 is the homomorphism Gm × Gm → Gm which is trivial on the first component and the squaring map on the second component. The group Ca is then Gm × μ2 , and μ2 is not a smooth F-group when the characteristic of F is 2. Returning to the general case, [SGA3, Exp. VIII, Theorem 3.1] shows that a induces an isomorphism S/Ca → Gm . 2.6.4 The 1-parameter subgroup associated to a root Let a ∈ Φ(S) ⊂ X∗ (S) and let ra ∈ W be the associated reflection. The homomorphism S → S mapping s to s · ra (s)−1 factors through S/Ca . Composing this homomorphism with the inverse of the isomorphism S/Ca → Gm induced by a as explained in 2.6.3, we obtain a homomorphism Gm → S. We will show in Lemma 2.6.5 below that this homomorphism is naturally identified with the coroot associated to the root a, and hence we will denote this homomorphism by a∨ . Let V be the Q-vector space spanned by Φ(S) and let V ∗ be its dual space. There is a natural root system Φ(S)∨ ⊂ V ∗ associated to Φ(S), cf. [Bou02, Chapter VI, §1, no. 1, Proposition 2], called the dual root system. We will now show that the above construction embeds Φ(S)∨ into the cocharacter module X∗ (S), which is the integral dual of X∗ (S). Lemma 2.6.5 The integral duality between X∗ (S) and X∗ (S) identifies the set of 1-parameter subgroups constructed in 2.6.4 for all a ∈ Φ(S) with the dual root system Φ(S)∨ of Φ(S). Proof Let S  be the maximal torus contained in S ∩ Gder . The restriction map X∗ (S) → X∗ (S ) maps Φ(S) bijectively onto the relative root system of Gder with respect to S , and induces an isomorphism V → X∗ (S )Q . The integral duality between X∗ (S ) and X∗ (S ) extends to an isomorphism V ∗ = X∗ (S )Q . Let a ∈ Φ(S). We will write temporarily a† : Gm → S for the 1-parameter subgroup constructed in 2.6.4. By construction, the image of a† consists of commutators, hence takes values in Gder ∩S. Since its source is Gm , we conclude that a† factors through the closed immersion S  → S. Thus a† ∈ X∗ (S ) ⊂ V ∗ . To prove the claim it remains to show that a† satisfies the axiomatic property characterizing the coroot a∨ ∈ V ∗ , namely x − a†, x a = ra (x) for all x ∈ V. Since both sides are linear, it is enough to consider x ∈ X∗ (S). Then, for all s ∈ S, ra (x)(s) = x(ra (s)) = x(s) · (x(s/ra (s)))−1 = x(s) · x(a† (a(s)))−1 = (x − a†, x a)(s).

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Definition 2.6.6 The tuple (X∗ (S), Φ(S), X∗ (S), Φ(S)∨ ) is called the relative root datum of G with respect to S. The conjugacy of maximal split tori of G under G(F) implies that the relative root datum is unique up to isomorphism. Consider now a maximal F-torus T ⊂ G. Applying the preceding discussion to T ×F Fs and G ×F Fs , where Fs is a separable closure of F, we obtain a root system Φ(T) ⊂ X∗ (T) and its dual Φ(T)∨ ⊂ X∗ (T). The Galois group of Fs /F acts compatibly on X∗ (T) and X∗ (T) and preserves Φ(T) and Φ(T)∨ . Definition 2.6.7 The tuple (X∗ (T), Φ(T), X∗ (T), Φ(T)∨ ) is called the absolute root datum of G with respect to T. The conjugacy of maximal tori of G under G(Fs ) implies that the absolute root datum is unique up to isomorphism. However, the action of the Galois group on it depends on the choice of T. The restriction map X∗ (T) → X∗ (S) induces a map Φ(T) → Φ(S) ∪ {0} and the elements of Φ(S) are the non-zero restrictions of the elements of Φ(T), cf. [Bor91, 21.8]. The root system Φ(T) is always reduced (if r ∈ R and a, r a ∈ Φ(T), then r ∈ {±1}), but the root system Φ(S) need not be reduced (it is possible that r ∈ {±1, ±2}), cf.[Bor91, 21.7]. If a, 2a ∈ Φ(S), then a is called multipliable (or non-divisible) and 2a is called divisible (or non-multipliable). 2.6.8 Structure of root groups Given a ∈ Φ(S) there exists a unique smooth closed subgroup Ua that is normalized by S and whose Lie algebra is the direct sum of the weight spaces for the action of S on Lie(G) for weights that are positive integer multiples of a, cf. [Bor91, Proposition 21.9]. This group is split unipotent. When 2a  Φ, Ua is in fact a vector group (cf. 2.10.20), in particular abelian. When 2a ∈ Φ, then U2a ⊂ Z(Ua ) and Ua /U2a is a vector group. In particular Ua is an extension of one vector group by another. See [Bor91, Remark 21.10] or Proposition 2.11.14. We will investigate the structure of Ua more precisely when G is quasi-split. Notation 2.6.9 Ua (F)∗ = Ua (F) − {1}. Assumption 2.6.10 From now on and until the end of this subsection, we assume that G is quasi-split. Let T be a maximally split F-torus of G. Since G is assumed quasi-split, T is a minimal Levi subgroup of G. Each of S and T can be recovered from the

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Algebraic Groups

other, S being the maximal split torus in T, and T being the centralizer of S in G. In particular, all maximally split tori are conjugate under Gsc (F). Because of that, if we consider the absolute root datum (X∗ (T), Φ(T), X∗ (T), Φ(T)∨ ) for a maximally split F-torus T, it is independent of T up to a unique isomorphism, including its Galois action. This action factors through the Galois group of the minimal splitting extension of G. A Borel subgroup B containing T and defined over F corresponds to a system of simple roots Δ(T) ⊂ Φ(T) invariant under Θ. We refer to the tuple (X∗ (T), Δ(T), X∗ (T), Δ∨ (T)) as a based root datum. From S we obtain the relative root datum (X∗ (S), Φ(S), X∗ (S), Φ(S)∨ ). Note ∗ X (S) = X∗ (T)Θ,free and X∗ (S) = X∗ (T)Θ . The subset Φ(S) ⊂ X∗ (S) is the image of Φ(T) under the natural projection X∗ (T) → X∗ (S), that is, the map induced by restriction along the inclusion S → T. The kernel of X∗ (T) → X∗ (S) does not contain any element of Φ(T), since G is quasi-split. The preimage of any element of Φ(S) is a single Θ-orbit in Φ(T). We will now recall the structure of the relative root groups. Since G is assumed quasi-split, this structure takes a particularly simple form.  = Φ(T) and Φ = Φ(S). For every χ ∈ X∗ (T) Notation 2.6.11 We abbreviate Φ we write Θ χ for its stabilizer and Fχ for the fixed field of Θ χ in Fs .  and the base a ∈Φ We will often use this notation in the form k a, where  field is k. It is useful to classify the elements of Φ(S) into three types, by saying that a ∈ Φ(S) is of type (1) R1 if a is neither divisible nor multipliable, (2) R2 if a is multipliable, (3) R3 if a is divisible. These types are borrowed from the summary [KS99, §1] of Steinberg’s work [Ste68]. Even though the latter applies to algebraic automorphisms of algebraic groups, the non-algebraic automorphisms by which the Galois group acts behave in a somewhat analogous manner and we can use the same type distinction.  consists Most roots a ∈ Φ are of type R1. The preimage of such a in Φ  of either a single root or a set of pairwise orthogonal roots. For each  a ∈ Φ

belonging to that preimage we have the injection Ua → Ua , defined over Fa, a from the 1-dimensional root subgroup Ua corresponding to the absolute root  to the root subgroup Ua corresponding to the relative root a. This leads to an

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isomorphism of algebraic F-groups RFa /F Ua → Ua,

(2.6.1)

hence an isomorphism of Θ-modules IndΘ (Fs ) → Ua (Fs ), and hence the Θ a Ua isomorphism of abstract groups Ua(Fa) → Ua (F). In particular, the algebraic group Ua is commutative, and so is the abstract group Ua (F). Explicitly, this  isomorphism sends an element u ∈ Ua(Fa) ⊂ G(Fa) to σ ∈Θ/Θ a σ(u) ∈ Ua (F) ⊂ G(F). The order of the product does not matter because of the commutativity of Ua . Roots of type R2 and R3 occur only when G contains a normal subgroup whose derived subgroup has an odd special unitary group as a simply connected  has an irreducible factor cover, equivalently when the absolute root system Φ of type A2n and some element of Θ preserves and acts non-trivially on this factor. The reason for this special behavior is that there exist Galois orbits of roots whose members are not pairwise orthogonal. The prototypical example is that of the group SU3 that will be discussed in detail in §2.7. In the general case, consider a pair a, 2a ∈ Φ of relative roots, so that a is of type R2 and  has even order and can be written 2a is of type R3. The preimage of a in Φ   a1 , . . . ,  a m,  am}. The root  ai is orthogonal to every other root in this set as { a1 ,  bi =  ai +  ai is also a root. The image of  bi in Φ equals 2a. except for  ai, and  Let Ua be the root subgroup corresponding to the relative root a. Recall that by convention U2a ⊂ Ua . The structure of Ua is now slightly more complicated.  be the unique other root mapping  mapping to a and let  Choose  a∈Φ a ∈ Φ   and maps to 2a. The to a and not orthogonal to  a. Then b =  a+ a  lies in Φ extension Fa/Fb is Galois of degree 2, with automorphism given by the unique element of Θ that switches  a and  a . The product U[ a] := Ua  · Ua  · Ub  ⊂ G is a 3-dimensional connected unipotent Fb-subgroup of G. It is non-canonically isomorphic to the group UFa /Fb of (2.7.2). We have the isomorphism algebraic F-groups RFb /F U[ a ] → Ua

(2.6.2)

and hence the isomorphism of abstract groups U[ a] (Fb ) → Ua (F). In particular, these groups are non-commutative. But the non-commutativity comes only from the non-commutativity of U[ a]  UFa /Fb . In other words, over the algebraic closure the individual copies of U[ a] in the group [F :F] Ua ×F Fs = (RFb /F U[ a] ) ×F Fs  (U[ a] ) b

commute. The isomorphism (2.6.2) sends u ∈ U[ a] (Fb ) to



σ ∈Θ/Θ b

σ(u).

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Algebraic Groups

(c) The Relative Root System of a Tits Index We continue with an arbitrary field F with a separable closure Fs . In [Tit66], Tits gives the classification of semi-simple algebraic F-groups. A lot of information about such a group is encoded in its Tits index, which consists of the absolute root system decorated by further invariants, such as the F-split rank, the absolute root system of the anisotropic kernel, and the size of the quotient through which Gal(Fs /F) acts on the absolute Dynkin diagram via the ∗-action. In [Tit66, §2.5.2], Tits gives a recipe for computing the relative root system of an absolutely simple adjoint F-group from its Tits index. We will provide here an alternative recipe, which may be easier to use in some situations. It is based on the coefficients of the expression of the highest root of an irreducible root system as a linear combination of simple roots, cf. Table 2.6.1. This recipe will be used implicitly in the classification of connected reductive groups over a discretely valued Henselian field k with perfect residue field f satisfying dim(f)  1 given in §10.7. Thus we consider a set of absolute simple roots Δ, a subset Δ0 consisting of the simple roots of the anisotropic kernel, and an action of a finite quotient Θ of Gal(Fs /F) on Δ that preserves Δ0 (the so called *-action). This data constitutes the Tits index. Let DFs and DF be the absolute and relative Dynkin diagrams. The recipe for computing DF is as follows. (1) The vertices of DF are the Θ-orbits in Δ − Δ0 . (2) Two vertices of DF are joined by an edge if and only if there exist representatives α, α  ∈ Δ of the corresponding Θ-orbits, as well as distinct α1, . . . , αk ∈ Δ0 , such that {α, α1, . . . , αk , α  } is a connected subdiagram of DFs . (3) The coefficient with which a given vertex of DF enters into the expression of the highest root for DF is the sum over all members of the Θ-orbit in Δ − Δ0 of the coefficients with which these members enter the expression of the highest root for DFs . (4) Given two vertices a, b ∈ DF that are joined by an edge, the multiplicity and direction of this edge can be determined by considering the union D  = O a ∪ Ob ∪ Δ0 , where O a, Ob ⊂ Δ − Δ0 are the two Θ-orbits corresponding to a and b. There exists a connected component D  of D  that intersects both O a and Ob (in fact, Θ acts transitively on the set of such connected components). Let na be the sum of the positive integers with which the elements of O a ∩ D  enter in the expression of the highest root for D , and let nb be the corresponding sum for Ob ∩ D . Assume without loss of generality that na  nb . The edge between a and b has

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higher multiplicity if and only if na < nb , in which case na = 1, the multiplicity is nb , and the edge points towards b. (5) Given a vertex a ∈ DF , to check whether the corresponding relative simple root is multipliable one considers the union D  = O a ∪ Δ0 . Then a is multipliable if and only if there exists a connected component D  of D  such that the sum of the integers with which the elements of O a ∩ D  enter in the expression of the highest root for D  is larger than 1. We note that once the vertices and edges (without orientation and multiplicity) are known by (1) and (2), the information in (3) is enough to distinguish all possible Dynkin types for DF except B and C, in particular to determine whether DF is reduced. The information in (4) is only necessary to distinguish between types B and C. The information in (5) is not necessary to determine the Dynkin type, but it can be used to determine exactly where the multipliable simple root in a system of type BC occurs. Example 2.6.12 We give an example of how multipliable vertices of DF can occur. In the following example of the index 2 A(2) 9,2 , DF has two vertices.

The union of the left Θ-orbit with Δ0 is a diagram of type A3 × A3 × A1 and each element of the orbit is contained in a different connected component, so the vertex is not multipliable. The union of the right Θ-orbit with Δ0 is a diagram of type A5 × A1 × A1 and each member of the Θ-orbit is contained in the component A5 and enters the expression of the highest root of A5 with the coefficient 1; thus the sum of these coefficients over the Θ-orbit is 2, making this vertex multipliable. (2) , DF has again two vertices. In the following example of the index 1 D6,2

The union of the left Θ-orbit with Δ0 is a diagram of type A3 × A1 × A1 and the unique member of the Θ-orbit enters the expression of the highest root of A3 with the coefficient 1, so the corresponding vertex of DF is not multipliable. The union of the right Θ-orbit with Δ0 is a diagram of type D4 × A1 and the unique member of the Θ-orbit enters the expression of the highest root of D4 with coefficient 2, so that vertex of DF is multipliable. Example 2.6.13 We give an example of how bonds with higher multiplicity (2) , DF has three vertices. can occur. In the following example of the index 2 A11,3

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To compute the multiplicity of the bond between the left and middle edges, the union D  of the two Θ-orbits and Δ0 is of type A5 × A5 and the contribution of each orbit to the highest root of A5 is 1, so the bond is simple. For the bond between the middle and the right edge, the type of D  is A7 × A1 × A1 , with the elements of the middle Θ-orbit contributing a total of 2 to the highest root of A7 , and the element of the right (singleton) Θ-orbit contributing 1, so the bond is double and pointing towards the middle edge. (2) , DF has three vertices. In the following example of the index 1 D7,3

For the bond between the left and the middle vertex we consider the union of these singleton Θ-orbits with Δ0 , which is a system of type A6 , so both vertices contribute the coefficient 1 to its highest root, hence the bond between the two vertices is single. For the bond between the middle and the right vertex the corresponding union is a system of type D5 × A1 , with the middle vertex contributing a coefficient of 2 and the right vertex a coefficient of 1 to the highest root of D5 . Therefore the bond between the two vertices is double, with an arrow pointing towards the middle vertex. In the reminder of this section we prove that this recipe is correct. We accept as given that (1) is correct, cf. [Bor91, §21.8]. To prove the remaining points we will use the following lemma. Lemma 2.6.14 Let Φ be a reduced irreducible root system and let Δ be a basis. Let D be the Dynkin diagram. (1) Let α ∈ Φ and let Δ ⊂ Δ be the subset of those simple roots that occur in the expression of α. The subdiagram of D corresponding to Δ is connected. (2) Conversely, let Δ ⊂ Δ be such that the corresponding subdiagram of D is connected. Then α∈Δ α ∈ Φ. Proof (1) Fix a Weyl-invariant scalar product. By contradiction we assume that α ∈ Φ is positive of minimal height with the property that the subdiagram for Δ is not connected. According to [Bou02, Chapter VI, §1, no. 6, Proposition 19], the minimality of α implies that this subdiagram has two connected components, one of which is a point, that is, Δ = Δ ∪ {β}, and moreover γ := α − β ∈ Φ is a sum of elements of Δ. The disconnectedness of {β} and Δ implies (β, γ) = 0. Since β + γ ∈ Φ we conclude that both β and γ must be

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short, and hence that Δ consists of short roots. Therefore α lies in the span of the set of short simple roots. This span is a simply laced subroot system. But then β + γ ∈ Φ implies, (γ, β) < 0, a contradiction. (2) We induct on the size of Δ, the case |Δ | = 1 being trivial. We can decompose Δ = Δ ∪ {β} such that the subdiagram for Δ is still connected. By induction γ = α∈Δ α ∈ Φ. Since Dynkin diagrams do not have loops, there exists a unique α ∈ Δ with (α, β)  0. Thus (γ, β) = (α, β) < 0, hence γ + β ∈ Φ by [Bou02, Chapter VI, §1, no. 3, Theorem 1].  Let ΦFs and ΦF be the root systems corresponding to DFs and DF . We interpret these root systems as formal linear combinations of the elements of Δ and ΔF , respectively, where ΔF is the set of simple roots of ΦF corresponding to Δ. Recall that ΦF is the set of non-zero restrictions of the elements of ΦFs , where in the current abstract setting restriction means mapping α∈Δ Fs nα α to

 a ∈Δ F α∈Oa nα a. (2) If we assume that a, a  ∈ ΔF are joined by an edge, then (a, a ) < 0, so a + a  ∈ ΦF . Choose a root in Φ whose restriction equals a + a . This root is positive and we can write it as a non-negative integral linear combination of elements of Δ. This linear combination contains elements α, α  ∈ Δ whose restrictions are the elements a, a  ∈ ΔF , respectively. All the other elements {α1, . . . , αk } must have trivial restrictions and thus lie in Δ0 , and part (1) of the above lemma proves that {α, α1, . . . , αk , α  } is connected. Conversely, let a, a  ∈ ΔF be distinct, α, α  ∈ Δ lifts of a, a , and α1, . . . , αk ∈ Δ0 be such that {α, α1, . . . , αk , α  } is connected. Then part (2) of the above lemma implies that α + α1 + · · · + αk + α  ∈ Φ, hence a + a  ∈ ΦF , so part (1) of the above lemma applied to the reduced subsystem of ΦF implies that {a, a  } ⊂ ΔF is connected. This completes the proof of (2). (3) This follows at once from [Bou02, Chapter VI, §1, no. 8, Proposition 25]. (4) Assume without loss of generality that either a and b have equal length, or a is longer. Consider the rank 2 subdiagram of DF consisting of a, b, and the edge between them. Let ma and mb be the coefficients with which a and b enter the expression of the highest root for that subdiagram. Then a is longer than b if and only if ma < mb . But this highest root is the restriction of the highest root of the subdiagram D  of DFs . (5) The root a ∈ ΔF is multipliable if and only if 2a ∈ ΦF , which is equivalent to the existence of β ∈ ΦFs that is supported on O a ∪ Δ0 and to which the elements of O a have a combined contribution of 2, equivalently > 1. The element β lies in an irreducible component of the root system spanned by O a ∪ Δ0 .

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Table 2.6.1 The coefficients for the highest root in the irreducible root systems Label

Diagram

An

(n  1)

Bn

(n  3)

Cn

(n  2)

BCn

(n  1)

Dn

(n  4)

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

1

2

2

2

2

2

2 1

1

2

2

2

2

2 1

2

E6

1

2

3

2

1

4

3

2

1

4

3

2

E7

2

3 3

E8 F4 G2

2

4

6

5

2

3

4

2

2

3

2

In the rest of this section k will denote a discretely valued Henselian field with perfect residue field.

(d) The Valuation Homomorphism and the Subgroup G(k)1 Notation 2.6.15 Let G be a connected reductive k-group. Recall the notation Θ = Gal(k s /k). Generalizing the discussion of §2.5(b) we define a homomorphism ωG : G(k) → HomZ (X∗ (G)Θ, Z),

g → ( χ → −ω( χ(g)).

The kernel of ωG will be denoted by G(k)1 .

(2.6.3)

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Lemma 2.6.16 (1) The homomorphism ωG is functorial in G. (2) A homomorphism G → H of connected reductive groups maps G(k)1 to H(k)1 . (3) The group G(k)1 is the preimage of Gab (k)b , where Gab = G/Gder . (4) G(k)1 = {g ∈ G(k) | ω( χ(g)) = 0 for all χ ∈ X∗ (G)Θ } = {g ∈ G(k) | ω( χ(g)) = 0 for all χ ∈ X∗ (G)}. Proof The projection G → Gab induces an isomorphism X∗ (G) → X∗ (Gab ). This isomorphism is functorial in G, since a homomorphism G → H maps Gder to Hder and therefore induces a homomorphism Gab → Hab . The functoriality of ωG is now evident from the definition. This proves (1) and (2), while (3) follows from Proposition 2.5.8 and (4) follows from (3) and Proposition 2.5.9.  Lemma 2.6.17 Let /k be a Galois extension of finite ramification degree, equipped with the unique valuation that extends ω. Then ωG is the restriction to G(k) of ωG . Proof

Same as for Lemma 2.5.6.



Remark 2.6.18 Note that for a k-torus T the maximal bounded subgroup T(k)b coincides with the subgroup T(k)1 . For a connected reductive group G, every bounded subgroup of G(k) is contained in G(k)1 , but G(k)1 itself need not be bounded; it is bounded if and only if Gder is anisotropic, in which case it equals the maximal bounded subgroup of G(k), cf. Proposition 2.2.12. Let AG be the maximal k-split torus in the center of G. The restriction map X∗ (G)Θ → X∗ (AG ) is injective and its image is of finite index. Therefore HomZ (X∗ (G)Θ, Q) = HomZ (X∗ (AG ), Q). Composing the homomorphism ωG : G(k) → HomZ (X∗ (G)Θ, Z) of (2.6.3) with the inclusion HomZ (X∗ (G)Θ, Z) → HomZ (X∗ (G)Θ, Q) = HomZ (X∗ (AG ), Q)  Q ⊗Z X∗ (AG ), we obtain the following homomorphism, which we will denote again by ωG : ωG : G(k) → Q ⊗Z X∗ (AG ).

(2.6.4)

Consider now a Levi subgroup M ⊂ G. Let AG ⊂ G and AM ⊂ M be the maximal split tori in the centers of the corresponding group. Let AM be the maximal torus in AM ∩ Gder . The natural map AM → AM → AM /AG is an isogeny and thus induces an isomorphism of vector spaces Q ⊗Z X∗ (AM ) → Q ⊗Z X∗ (AM /AG ).

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Algebraic Groups

This gives a canonical section of the natural inclusion Q ⊗Z X∗ (AM ) → Q ⊗Z X∗ (AM ), and hence a decomposition Q ⊗Z X∗ (AM ) = Q ⊗Z X∗ (AG ) ⊕ Q ⊗Z X∗ (AM ). Lemma 2.6.19 (1) The restriction of ωG to M(k) equals the composition of ω M with the homomorphism Q ⊗Z X∗ (AM ) → Q ⊗Z X∗ (AG ). (2) M(k)1 = {m ∈ G(k)1 ∩ M(k) | ω M (m) = 1 in Q ⊗Z X∗ (AM )}. Proof The first statement is immediate from the definitions, and the second statement follows from the first. 

(e) The Subgroups G(k) and G(k)0 Definition 2.6.20 Let G(k) be the image of the natural map Gsc (k) → G(k). Fact 2.6.21 A morphism G → H carries G(k) into H(k) . Proof This follows from the fact that the morphism lifts (in fact uniquely) to  a morphism Gsc → Hsc . Fact 2.6.22 If M ⊂ G is a Levi subgroup, then G(k) = G(k) · M(k). Proof We may assume without loss of generality that M is minimal, thus M = ZG (S) for a maximal split torus S ⊂ G. The claim is then immediate from the Bruhat decomposition [Bor91, IV.21.15] and the fact that the Weyl group  Wk = NG (S)(k)/ZG (S)(k) is the same for G and Gsc . Definition 2.6.23 Let G be a connected reductive group. Denoting the center of G by Z(G), we define the subgroup G(k)0 ⊂ G(k) as follows. (1) If G is quasi-split, choose a maximally split maximal torus T ⊂ G and define G(k)0 = G(k) · T(k)0 , with T(k)0 as in Definition 2.5.13. (2) If G/Z(G) is anisotropic, define G(k)0 = G(K)0 ∩ G(k), where G(K)0 is defined by (1) since G K is quasi-split by Corollary 2.3.8. (3) For general G, choose a minimal Levi subgroup M0 ⊂ G and define G(k)0 = G(k) · M0 (k)0 . Lemma 2.6.24 The above definition does not depend on the choices of T or M0 . The group G(k)0 is normal and open.

2.6 Reductive Groups

105

Proof The independence of T and M0 follows from the fact that any two minimal Levi subgroups are conjugate under Gsc (k). Note that the isomorphism Ad(g) : M0 (k) → gM0 g −1 (k) sends M0 (k)0 to gM0 g −1 (k)0 . Normality follows from the normality of G(k) and the independence of T and M0 . Openness  follows from the openness of T(k)0 in T(k) and Fact 2.6.22. Remark 2.6.25 We note that the three parts of Definition 2.6.23 agree with each other in cases to which they apply simultaneously, and moreover agree with Definition 2.5.13 when G = T is a torus: (1) The case that G is a torus is precisely the case in which (1) and (2) apply, because this is equivalent to G/Z(G) being quasi-split and anisotropic, and hence trivial (being semi-simple). In that case (1) is vacuous, and (2) is consistent with Definition 2.5.13. (2) When both (1) and (3) apply, then G is quasi-split, so M0 is a maximally split maximal torus, hence (1) and (3) coincide. (3) When both (2) and (3) apply, then G/Z(G) is anisotropic, so M0 = G, hence (3) is vacuous. Remark 2.6.26 The definition of G(k)0 given here is rather ad hoc, but has the advantage of being elementary. We will give a more natural interpretation in Chapter 11, cf. Remark 11.5.5, which will in particular imply that G(k)0 is functorial, just like G(k) and G(k)1 . We will also be able to describe the quotients G(k)/G(k)0 , G(k)1 /G(k)0 , and G(k)/G(k)1 , cf. Corollary 11.6.2. The inclusion G(k)0 ⊂ G(K)0 ∩ G(k) is obvious from Definition 2.6.23, but the converse inclusion is not. We will prove that G(k)0 = G(K)0 ∩ G(k), cf. Proposition 9.3.24; this will be a consequence of Bruhat–Tits theory. Lemma 2.6.27 Let G be simply connected and let M0 ⊂ G be a minimal Levi subgroup. Then M0 (K)0 = M0 (K)1 . In particular, M0 (k)0 = M0 (k)1 . Proof Since M0 is anisotropic modulo center, it is enough by Definition 2.6.23 to show M0 (K)0 = M0 (K)1 . The inclusion M0 (K)0 ⊂ M0 (K)1 is immediate; we argue the opposite inclusion. Let T ⊂ M0 be a maximally split maximal K-torus. Then it is also a maximally split maximal K-torus in the quasi-split simply connected group G. Therefore T is induced. Let Tsc be the intersection of T with the derived subgroup of M0 . The latter is simply connected, so Tsc is also an induced torus, as is the quotient D = T/Tsc . The image of an element m ∈ M0 (K)1 in D(K) lies in D(K)1 , which equals D(K)0 by Lemma 2.5.18. By Lemma 2.5.20 this element lifts to an element t ∈ T(K)0 ⊂ M0 (K)0 . The  element t −1 m thus lies in M0,der (K) ⊂ M0 (K)0 . Corollary 2.6.28

Let G be a connected reductive group, M0 ⊂ G a minimal

106

Algebraic Groups

Levi subgroup. Then G(K)0 ∩ M0 (k)1 = M0 (k)0 . Hence, G(k)0 ∩ M0 (k)1 = M0 (k)0 . Proof The second statement follows from the first and the inclusion G(k)0 ⊂ G(K)0 . For the first, we use the identities M0 (k)1 = M0 (K)1 ∩ M0 (k) and M0 (k)0 = M0 (K)0 ∩ M0 (k), the latter by Definition 2.6.23. It is thus enough to show G(K)0 ∩ M0 (K)1 = M0 (K)0 , which in turn reduces to showing that G(K) ∩ M0 (K)1 ⊂ M0 (K)0 . To see this, let M0 be the preimage of M0 in Gsc . Then the image of M0 (K) is G(K) ∩ M0 (K). The kernel of the map M0 → M0 being finite, the preimage of M0 (K)1 in M0 (K) equals M0 (K)1 , which by Lemma 2.6.27 equals M0 (K)0 . Since M0 → M0 maps M0 (K)0 to M0 (K)0 , we conclude  that G(K) ∩ M0 (K)1 ⊂ M0 (K)0 . The above corollary holds true for an arbitrary Levi subgroup of G, not just a minimal Levi subgroup, cf. Lemma 11.5.6. Lemma 2.6.29

The index of G(k)0 in G(k)1 is finite.

Proof Let S ⊂ G be a maximal split torus, Z = ZG (S). We will reduce the problem for G to the problem for Z. Using the Bruhat decomposition and the fact that every element of NG (S)(k)/ZG (S)(k) is realized in Gsc (k), we see G(k)1 /G(k)0 = (Z(k) ∩ G(k)1 )/[(Z(k) ∩ G(k) ) · Z(k)0 ]. Assuming that the problem for Z has been solved, it is enough to show that the image of Z(k) ∩ G(k) in (Z(k) ∩ G(k)1 )/Z(k)1 is of finite index. The latter embeds via the valuation homomorphism into Q ⊗Z X∗ (S ), cf. Lemma 2.6.19. This image is a lattice containing X∗ (S ) (since the image of S (k) ⊂ Z(k) is precisely X∗ (S )) and contained in e−1 X∗ (S ) for some e ∈ Z. On the other hand, the image of Z(k) ∩ G(k) contains the lattice X∗ (Ssc ), where Ssc is the maximal torus in the preimage of S in Gsc . Since Ssc → S  is an isogeny, X∗ (Ssc ) is of finite index in X∗ (S ), hence also in e−1 X∗ (S ). The reduction from G to Z is complete. It allows us to assume that G = Z. In that case, G(k)0 = G(K)0 ∩ G(k). Since also G(k)1 = G(K)1 ∩ G(k), we can now assume k = K. Then G is quasi-split by Corollary 2.3.8, which implies that now Z is a torus. We apply again the reduction from G to Z and then Lemma 2.5.14.  Definition 2.6.30

Define G(k)b = G(k) · Z(k)1 = G(k)0 · Z(k)1 .

It is clear that G(k)0 ⊂ G(k)b ⊂ G(k)1 . In particular, G(k)b is open. It is also normal. Both of these inclusions can be strict, as the following example shows. Moreover, unlike the relations G(k)1 = G(K)1 ∩ G(k) and G(k)0 = G(K)0 ∩ G(k), it is generally not true that G(k)b = G(K)b ∩ G(k).

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107

Example 2.6.31 To see the strictness of the second inclusion, consider G = PGL2 and T = Gm the diagonal torus. Then T(k)0 = T(k)1 and G(k)1 = G(k), but G(k)0 = G(k) · T(k)0 = G(k)b is the subgroup of index 2 consisting of those matrices modulo central translation whose determinant, well-defined as an element of k × /k ×,2 , lies in o× /o×,2 . To see the strictness of the first inclusion, consider a ramified quadratic ex 2 , with μ2 embedded diagonally  = R/k SL2 , and let G = G/μ tension /k, let G  2 . Then one checks that  Then T  = R/k Gm and T = T/μ into the center of G. 0 1 T(k) is of index 2 in T(k) . This can be done either by hand, or by using the Kottwitz homomorphism that we will introduce in §11.1. Corollary 2.6.28 implies G(k)b /G(k)0 = T(k)1 /T(k)0 , showing that G(k)0 is also of index 2 in G(k)b . To see the failure of G(k)b = G(K)b ∩ G(k), let k = Q p and let G be the non-trivial inner form of PGL2 /k. Then G(k) is anisotropic, hence G = Z and G(k)b = G(k)1 = G(k) = D× /k × , where D/k is the unique quaternion algebra. On the other hand G(K) = PGL2 (K) and we have just seen that G(K)b  G(K)1 . In fact, we see that G(K)b ∩ G(k) is the subgroup of D× /k × represented by elements of D× whose reduced norm has even valuation. Lemma 2.6.32 Assume that G is quasi-split and let T ⊂ G be a maximally split maximal torus of G. Let Gsc be the simply connected cover of the derived subgroup of G and let Tsc be the preimage of T. Let N be the image of NGsc (Tsc )(k) in G(k). Then (1) WG (T)(k) = NG (T)(k)/T(k). (2) NG (T)(k) = N · T(k). (3) NG (T)(k) ∩ G(k)0 = N · T(k)0 . Proof For the first and second points, we have the commutative diagram with exact rows. 1

/ Tsc (k)

/ NG (Tsc )(k) sc

/ WG (Tsc )(k) sc

/1

1

 / T(k)

 / NG (T)(k)

/ WG (T)(k)

/1

Exactness on the top right follows from the vanishing of H1 (Θ,Tsc ) due to the fact that Tsc is induced, while exactness on the bottom right follows from exactness on the top right. For the third point the inclusion N ⊂ G(k)0 implies NG (T)(k) ∩ G(k)0 = (N ·T(k))∩G(k)0 = N ·(T(k)∩G(k)0 ), so it is enough to show T(k)∩G(k)0 =

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Algebraic Groups

T  · T(k)0 , where T  is the image of Tsc (k) in G(k). But T(k) ∩ G(k)0 = T(k) ∩ (G(k) · T(k)0 ) = (T(k) ∩ G(k) ) · T(k)0 = T  · T(k)0, 

as claimed.

2.7 The Group SU3 In this section we will review the structure of the unique quasi-split but nonsplit reductive group of rank 1, namely SU3 . The relative root groups of this group occur as building blocks of relative root groups of more general quasisplit connected reductive groups, as will be discussed in Construction 2.9.16. Moreover, the building of SU3 provides an instructive example of Bruhat–Tits theory, and will be described in §3.2. Let k be a field and let /k be a separable quadratic extension with Galois automorphism denoted by σ or x → x. The quasi-split group G := SU3 associated to /k is defined as the outer twisted form of the group SL3 by the pinned automorphism σ defined by g → Ad(J)g −t ,

⎡ ⎢ J = ⎢⎢ ⎢1 ⎣

−1

1⎤⎥ ⎥, ⎥ ⎥ ⎦

where g −t denotes the inverse of the transpose of the matrix g. Thus, the group SU3 (k) is given by {g ∈ SL3 () | g = Ad(J)g −t }, where g denotes the matrix obtained from g by applying the Galois automorphism of /k to each individual entry. The standard Borel pair (T, B) of SL3 , consisting of the group of diagonal matrices T and the group of upper triangular matrices B, is invariant under this twisted Galois action and is therefore a Borel pair of SU3 defined over k. Let S be the maximal k-split torus contained in T and N be its normalizer in G. Let NG (T) be the normalizer of T in G. Then NG (T)(k) = N(k). Using the diagonal entries as coordinates we obtain X∗ (T) = Z30 , the group of triples of integers that sum to zero, and X∗ (T) = Z3 /Z, the quotient of Z3 by the diagonal copy of Z. The Galois group Γ operates through its quotient Γ/k , with σ(x, y, z) = (−z, −y, −x) on both Z30 and Z3 /Z. Let S be the k-split subtorus of T. Thus S = Gm and the natural inclusion S → T sends x to the diagonal matrix with entries (x, 1, x −1 ). The inclusion X∗ (S) = Z → Z30 = X∗ (T) is given by

2.7 The Group SU3

109

x → (x, 0, −x), and the projection X∗ (T) = Z3 /Z → Z = X∗ (S) is given by (x, y, z) → x − z. We have the absolute root system Φ(T) = Φ(T, G) ⊂ X∗ (T), consisting of the weights for the action of T on the Lie algebra of G over the field , over which the group G becomes the split group SL3 . The system Φ(T, B) of positive b}, while the subsystem of simple roots is { a,  a  }, where absolute roots is { a,  a ,     a+ a = (1, 0, −1). We also have the  a = (1, −1, 0),  a = (0, 1, −1), and b =  relative root system Φ(S) = Φ(S, G) ⊂ X∗ (S), consisting of the weights for the action of the split torus S on the Lie algebra of G. The system Φ(S, B) of positive relative roots is {a, 2a} ⊂ X∗ (S), where a = 1 ∈ Z = X∗ (S). Under the a → a,  a  → a, and  b → 2a. natural projection X∗ (T) → X∗ (S) we have  The absolute root groups Ua and Ua are each defined only over , not over k. They are both isomorphic to Ga / via the standard coordinates of SL3 . The absolute root group Ub is defined over k, but its isomorphism to Ga is not Γ-equivariant. If we transport the Γ-action on Ub to Ga via this isomorphism we obtain the algebraic group 0 Ga := ker(Tr/k : R/k Ga → Ga ) R/k

(2.7.1)

whose functor of points sends any k-algebra R to {x ∈ R ⊗k  | x + x = 0}. Here we have denoted by “bar” the automorphism idR ⊗ σ of R ⊗k . We also have the relative root subgroups U2a and Ua , both of which are defined over k. The inclusion Ub → U2a is an isomorphism of algebraic groups over . In coordinates we have ⎧ ⎡ ⎪ ⎨ ⎢1 ⎪ U2a () = ⎢⎢0 ⎪ ⎪ ⎢0 ⎩⎣

0 1 0

⎫ v ⎤⎥  ⎪ ⎬ ⎪  ⎥ 0⎥ v ∈   , ⎪ ⎪ 1⎥⎦ ⎭

0 G , hence U (k) = and taking into account the Galois action we see U2a = R/k a 2a 0 0  , where  denotes the kernel of the trace map  → k. The group Ua , which by definition contains U2a , is the entire unipotent radical of the chosen Borel subgroup. Thus

⎧ ⎡ ⎪ ⎨ ⎢1 u ⎪ Ua () = ⎢⎢0 1 ⎪ ⎪ ⎢0 0 ⎩⎣

v ⎤⎥ w ⎥⎥ 1 ⎥⎦

 ⎫ ⎪  ⎬ ⎪   u, v, w ∈  . ⎪  ⎪ ⎭

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Algebraic Groups

We see ⎡1 u ⎢ σ ⎢⎢0 1 ⎢0 0 ⎣

v ⎤⎥ ⎡⎢1 w ⎥⎥ = ⎢⎢0 1 ⎥⎦ ⎢⎣0

w 1 0

uw − v ⎤⎥ u ⎥⎥ . 1 ⎥⎦

Setting ⎡1 u ⎢ ua (u, v) = ⎢⎢0 1 ⎢0 0 ⎣

v ⎤⎥ u⎥⎥ , 1⎥⎦

⎡1 ⎢ u−a (u, v) = ⎢⎢u ⎢v ⎣

0 1 u

0⎤⎥ 0⎥⎥ , 1⎥⎦

we obtain U±a (k) = {u±a (u, v) | u, v ∈ , v + v = uu}. Thus U±a (k) = U/k (k), where U/k is the connected unipotent algebraic group defined over k with functor of points sending a k-algebra R to U/k (R) = {u, v ∈ R ⊗k  | v + v = uu}.

(2.7.2)

The group structure is given by (u, v) · (u , v ) = (u + u , v + v  + uu ), and the inverse of (u, v) is (−u, v). Note that this group is non-commutative. Let 0 ⊂U U/k /k denote the subgroup consisting of {(0, v)}. 0 is central in U Fact 2.7.1 The subgroup U/k /k . The map (0, v) → v induces 0 0 an isomorphism U/k → R/k Ga . The map (u, v) → u induces an isomorphism 0 (k) → . The fiber over u ∈  is in natural bijection with U/k (k)/U/k

 tr=1 = {λ ∈  | tr/k (λ) = 1} by λ → (u, λuu). Proof Immediate.



2.8 Separable Quadratic Extensions of Local Fields Let k be a non-archimedean Henselian valued field with valuation ω. We assume that ω(k × ) = Z. There is a unique extension of ω to any algebraic extension of k and we will denote this extension again by ω. In this section we will collect some facts about the structure of separable quadratic extensions /k. Their study is motivated by the structure of the root group U/k of SU3 that was defined in (2.7.2), and will be used in the description of the building of SU3 in §3.2, as well as in the construction of integral models of U/k in Appendix C.

2.8 Separable Quadratic Extensions of Local Fields

111

Let /k be a separable quadratic extension. For x ∈  write x for its Galois conjugate. As we just saw the sets  0 = {x ∈  | x + x = 0}

and  tr=1 = {x ∈  | x + x = 1}

play a role in the description of the root subgroups of SU3 for reduced relative roots. The number μ = sup{ω(x) | x ∈  tr=1 }

(2.8.1)

will play a role in describing certain natural filtrations of these groups. Note that all x ∈  tr=1 satisfy ω(x)  0, so the supremum μ exists and is a non-positive element of e1 Z, where e ∈ {1, 2} is the ramification degree of /k. Since ω is discrete, the supremum is in fact a maximum, so we can consider the set  tr=1,max = {x ∈  tr=1 | ω(x) = μ}.

(2.8.2)

We will collect here some results about the set  0 and the number μ. Their behavior depends on the ramification of /k. If /k is ramified, the lower ramification filtration of the Galois group of /k has a unique jump s ∈ Z0 . It is defined as   s = 2ω(σ() − ) − 1 = 2ω (σ()/) − 1 (2.8.3) for any uniformizer  ∈ . The number s has the property that Gal(/k)s+1 = {1} while Gal(/k)s  {1}. If δ/k is the discriminant of /k, then ω(δ/k ) = s + 1. The following are equivalent. (1) The extension /k is tamely ramified. (2) p  2. (3) s = 0. When p = 2 so that s > 0 and /k is wildly ramified, there are two cases. The special case is when 2 | s. Then the characteristic of k is zero, s = 2ek where ek = ω(2) is the absolute ramification index of k, and  = k() for a uniformizer  ∈  with  = −. Note that this case is very analogous to the case p  2. The generic case is when 2  s. If k has characteristic zero then s < 2ek . For more information cf. [FV02, III.2]. Lemma 2.8.1 (1) If /k is unramified, then μ = 0 and ω( 0 ) = Z. (2) If /k is ramified, then μ = −s/2 and ω( 0 ) = Z + 21 + μ. Remark 2.8.2 Before we prove the lemma, let us explicate the case of /k ramified a little more. First, the lemma states that ω( 0 ) is the unique Z-coset in 12 Z not containing μ. Moreover, μ = 0 is equivalent to p  2. When p = 2 we have the following distinction.

112

Algebraic Groups

(1) If 2 | s then μ = −ek ∈ Z and ω( 0 ) = Z + 12 . (2) If 2  s then μ = −s/2 ∈ Z + 12 and ω( 0 ) = Z. Proof of Lemma 2.8.1 Since  0 is a 1-dimensional k-vector subspace of , ω( 0 ) is a Z-coset in ω( × ). The situation when p  2 is much simpler. In that case 12 ∈  tr=1 and ω( 12 ) = 0, so μ = 0. When /k is unramified then ω( × ) = Z and hence also ω( 0 ) = Z. When /k is ramified there exists a uniformizer  ∈  with  2 ∈ k, thus  ∈  0 showing ω( 0 ) = Z + 12 . For the rest of this proof assume p = 2. Assume first that /k is unramified. Then ω( × ) = Z and hence also ω( 0 ) = Z. The trace map induces an isomorphism of f-vector spaces f /f → f. Let ζ ∈ f  f have trace equal to 1. Then ζ is a root of the polynomial X 2 − X + γ, where γ ∈ f is the norm of ζ. Let g ∈ o be a lift of γ. By Hensel’s lemma the polynomial X 2 − X + g has a root z ∈ o that lifts ζ. The trace of z is again equal to 1 and ω(z) = 0, thus μ = 0. Assume now that /k is ramified. We first show that μ  ω( 0 ). Assume the contrary and let x ∈  tr=1,max and y ∈  0 satisfy ω(x) = ω(y). Then x y −1 ∈ o× and we may multiply y by a unit in k to achieve that the image of x y −1 in f× = f× equals −1. Then ω(x + y) > μ, but at the same time x + y ∈  tr=1 , a contradiction. In particular we see that ω( 0 ) is a single Z-coset in ω( × ). We now compute μ as follows. Let x ∈  tr=1,max . Since μ  ω( 0 ) there exists a uniformizer  ∈  such that x ∈  0 . This implies  = −x x −1 and therefore 1 + s = 2ω( − ) = 1 + 2ω(1 + x x −1 ) = 1 + 2ω(x −1 ) = 1 − 2μ.



2.9 Chevalley Systems When working with classical matrix groups, such as SL2 (k), one can use the natural coordinates given by matrix entries to study their structure. For a general quasi-split reductive group the role of coordinates is played by the notion of a Chevalley system, which we will review here. In this section k will denote an arbitrary field and G a connected reductive quasi-split k-group.

(a) Pinnings Definition 2.9.1 A k-pinning of G is a triple (B,T, {Xa}) consisting of a Borel k-subgroup B, a maximal k-torus T ⊂ B, and for each  a in the set of simple

2.9 Chevalley Systems

113

absolute roots  Δ a non-zero vector Xa ∈ ga(k s ) such that the set {Xa} is stable under Θ. Note that a k-pinning is more than just a pinning, since we are requiring Θ-invariance. When G is split, then a k-pinning is just a pinning. That a k-pinning always exists can be seen as follows. Recall the stabilizer  and its fixed field k a in k s . Galois descent for vector spaces implies a∈Φ Θa of  that the 1-dimensional k s -subspace ga of g has a k a-structure. One can choose a representative  a for each Θ-orbit in Δ(T, B), choose 0  Xa ∈ ga(k a), and = σ(Xa). then set Xσ( a) Fact 2.9.2 Giving a k-pinning is equivalent to giving the following for each simple relative root a ∈ Δ(S, B). (1) If a is of type R1 and  a ∈ Δ(T, B) is a lift, an isomorphism Rk a /k Ga → Ua . (2) If a is of type R2 and  a,  a  are lifts whose sum  b is also a root, an isomorphism Rkb /k Uk a /kb → Ua (see the group Uk a /kb in (2.7.2)). We may refer to the tuple (G, B,T, {Xa}) as a pinned group. It gives rise to a based root datum. The following theorem is a cornerstone in the theory of reductive groups. Theorem 2.9.3 Let (G, B,T, {Xa}) and (G , B ,T , {Xa }) be two pinned groups. Every morphism between their based root data arises from a unique morphism G → G  respecting the two pinnings. We recall here from [SGA3, Exp. XXI, §6, Definition 6.1] and [SGA3, Exp. XXIII, §1, 1.5] that a morphism (X, Δ,Y, Δ∨ ) → (X , Δ,Y , Δ∨ ) of based root data is a morphism f : X → X  of abelian groups that restricts to a bijection Δ → Δ as well as a bijection between the root systems generated by Δ and Δ, and such that its dual f ∨ : Y  → Y restricts to a bijection Δ∨ → Δ∨ as well as a bijection between the root systems generated by Δ∨ and Δ∨ . The proof of the above theorem is given in [SGA3, Exp. XXIII, §4, Theorem 4.1] assuming G is split. However, the uniqueness statement of the theorem and Galois descent extend the result to the quasi-split case. Another proof, in the special case when f is an isomorphism, can be found in [Spr09, Theorem 16.3.2]. Corollary 2.9.4 Let (G, B,T, {Xa}) be a pinned group. There exists a unique automorphism of G that preserves T, acts by inversion on T, and sends Xa to −X− a , where X− a ∈ g− a (k s ) is the unique element satisfying [Xa , X− a] = ∨ d a (1) in Lie(Gsc ). This automorphism is called the Chevalley involution of the pinning.

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Algebraic Groups

(b) The Split Case Let G be a connected, semi-simple, simply connected, split k-group and let g  = Φ(T). be its Lie algebra. Let T ⊂ G be a split maximal torus and let Φ = Φ For a ∈ Φ and a non-zero Xa ∈ ga (k) there is a unique homomorphism ηXa : SL2 → G defined over k whose differential satisfies " " # # 0 1 1 0 → Xa, → Ha, 0 0 0 −1 where Ha = da∨ (1) ∈ t(k) is the coroot associated to a. Define " # 0 1 w(Xa ) = ηXa . −1 0

(2.9.1)

Remark 2.9.5 Given Xa there exists a unique Xa∨ ∈ g−a (k) such that Ha = [Xa, Xa∨ ]. It is given by " # 0 0 Xa∨ = ηXa . 1 0 The element w(Xa ) can be described alternatively as w(Xa ) = ua (1)u−a (−1)ua (1), where u±a : Ga → U±a is the unique isomorphism whose differential sends 1 to X±a . When k has characteristic zero, we can take u±a (t) = exp(t X±a ), where exp is the usual exponential map. Remark 2.9.6 Using the notation w(Xa )Xb := Ad(w(Xa ))Xb , we see that w(Xa )Xa = −Xa∨ and w(−Xa )Xa = −Xa∨ . Fact 2.9.7 (1) w(Xa ) normalizes T and lifts the reflection sa ∈ W(T). (2) One has w(Xa )2 = a∨ (−1), w(Xa )3 = w(−Xa ), and [Xa, w(Xa )Xa ] = −Ha . (3) If σ is a k-automorphism of G preserving T, then ηdσ(Xa ) = σ ◦ ηXa . In particular, w(dσ(Xa )) = σ(w(Xa )). (4) w(w(Xa )Xb ) = w(Xa )w(Xb )w(Xa )−1 . Definition 2.9.8 (1) A weak Chevalley system is a set Xweak = {±Xa | a ∈ Φ}, containing for each a ∈ Φ a pair ±Xa of non-zero elements of ga (k s ), such that for each a ∈ Φ the set Xweak is preserved by w(Xa ). (2) A Chevalley system (or Chevalley basis) is a set X = {Xa | a ∈ Φ} such that [Xa, X−a ] = Ha and such that Xweak = {±Xa | a ∈ Φ} is a weak Chevalley system. We say that X refines Xweak .

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(3) Given a weak Chevalley system, the subgroup W ⊂ NG (T)(k) generated by w(Xa ) for all a ∈ Φ is called the Tits group. Remark 2.9.9 A weak Chevalley system Xweak satisfies Chevalley’s rule: for a, b ∈ Φ with c = a + b ∈ Φ we have [Xa, Xb ] = ±(r + 1)Xc , where r is the largest integer such that b − ra ∈ Φ, cf. [SGA3, Exp. XXIII, Corollary 6.5]. It also satisfies the following identity: for a, b ∈ Φ with c = a + b ∈ Φ we have [Xa, Xb ] [w(Xa )Xa, w(Xb )Xb ] = . Xc w(Xc )Xc When the numbers r + 1 are invertible in k one can show that a set X = {±Xa | a ∈ Φ} satisfying [Xa, X−a ] = ±Ha and either of these identities must be a weak Chevalley system. Lemma 2.9.10

Let Xweak be a weak Chevalley system.

(1) Given a, b ∈ Φ then w(Xa )w(Xb )w(Xa )−1 = w(Xsa b )±1 . (2) If Δ is a set of simple roots, the Tits group is generated by w(Xa ) for all a ∈ Δ. Proof The first claim is implied by Fact 2.9.7 and w(Xa )Xb = ±Xsa b . We turn to the second claim. If b ∈ Φ is a positive root there exists a sequence of simple roots a1, . . . , an ∈ Δ such that b = ra1 · · · ran−1 (an ) by [Bou02, Chapter VI, §1, no. 6, Corollary 2]. By the first claim we know w(Xb )±1 = w(Xa1 ) · · · w(Xan−1 )w(Xan )w(Xan−1 )−1 · · · w(Xa1 )−1 .



Proposition 2.9.11 Let (B,T, {Xa }) be a k-pinning of the split group G and Δ ⊂ Φ the set of B-simple roots. (1) There exists a unique weak Chevalley system extending it, and every weak Chevalley system arises this way. (2) The Tits group is an extension of W by the 2-torsion subgroup of T(k). (3) If w = ra1 · · · rak ∈ W(T) is a reduced expression, then the product n(w) = w(Xa1 ) · · · w(Xak ) ∈ NG (T)(k) depends only on w and not on the reduced expression. The assignment w → n(w) is a set-theoretic splitting of the extension 1 → T(k)[2] → W → W(T) → 1. Proof We begin by considering the subgroup W ⊂ NG (T)(k) generated by w(Xa ) for a ∈ Δ. The relation w(Xa )2 = a∨ (−1) and the fact that the set {a∨ | a ∈ Δ} is a basis for X∗ (T) shows that T(k)[2] is a normal subgroup   The surjection NG (T)(k) → W induces a surjection W/T(k)[2] → W. of W.  The group W/T(k)[2] is generated by w(Xa ) and subject to the braid relation [Spr09, Proposition 9.3.2] and w(Xa )2 = 1. But these are exactly the relations

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among the generators {sa | a ∈ Δ} according to [Bou02, Chapter VI, §1, no. 5,  Theorem 2]. Therefore the map W/T(k)[2] → W is an isomorphism. As discussed in [Spr09, 9.3.3], for each w ∈ W the product n(w) depends only on w and not on the reduced expression w = ra1 · · · rak . The fact that w → n(w) is a set-theoretic splitting is immediate. We now extend the pinning to a weak Chevalley system. For each a ∈ Φ choose w ∈ W(T) such that b = w −1 a ∈ Δ(T, B). Choose any w ∈ W mapping to  b ∈ ga (k). We claim that Xa is up to sign independent w. Define Xa = Ad(w)X  Fixing w, the choice of w is unique up to an element of the choices of w and w. of T(k)[2], which acts on ga by multiplication by ±1. Given w1, w2 ∈ W(T) with b = w1−1 a = w2−1 a we have b = w2−1 w1 b and [Spr09, 9.3.5] implies n(w2−1 w1 )Xb = Xb . But n(w2−1 w1 ) = t · n(w2 )−1 n(w1 ) for some t ∈ T(k)[2]. This shows b(t) ∈ {±1} and Ad(t)Xb = ±Xb , and the claim is proved. Define Xweak = {±Xa | a ∈ Φ}. This set is invariant under W by construction, and hence a weak Chevalley system. The fact that every weak Chevalley system arises this way is immediate. 

(c) The Quasi-split Case Let G be a semi-simple, simply connected, quasi-split k-group. Let T ⊂ G be  = Φ(T) and Φ = Φ(S) be the absolute a maximally split maximal torus. Let Φ and relative root systems. Definition 2.9.12 A weak Chevalley–Steinberg system for G is a Θ-invariant weak Chevalley system for G ks . Δ the set of B-simple absolute Fact 2.9.13 Let (B,T, {Xa}) be a k-pinning and  roots. The associated weak Chevalley system for G ks is a weak Chevalley– Steinberg system. The splitting W → W ⊂ NG (T)(k s ) is Θ-invariant. Lemma 2.9.14 Every weak Chevalley–Steinberg system Xweak can be refined  to a Chevalley system X with the following Θ-invariance property. Let  a∈Φ restrict to a ∈ Φ and σ ∈ Θ. (1) If a is of type R1 or R2, then σ(Xa) = Xσ a.   (2) If a is of type R2 and  a ∈ Θ· a is the unique root such that  b= a + a  ∈ Φ, and σ ∈ Θb, then σ(Xb) = (σ) · Xb, where  : Θb/Θa → {±1} is the unique non-trivial character. a Proof This follows from the fact that Θa acts trivially on ga(k a) when   restricts to a of type R1 or R2. When  a restricts to a of type R2, then b restricts a and  a  and acts trivially on gb(k b), while Θb/Θa switches to 2a, and Θa fixes 

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 a and  a  and acts by the character  on gb(k b). This can be read off from the Dynkin diagram.  Definition 2.9.15 A Chevalley system with the properties enumerated in Lemma 2.9.14 will be called a Chevalley–Steinberg system. Construction 2.9.16 Let X be a Chevalley–Steinberg system. If a ∈ Φ is of  is a lift we obtain the isomorphism of k-groups type R1 and  a∈Φ Rk a /k Ga → Ua

(2.9.2)

by composing (2.6.1) with the isomorphism of k a-groups Ga → Ua whose differential sends 1 to Xa. The isomorphism (2.9.2) is independent of the choice of  a in the following sense. If  a  is another lift of a, there is a canonical a= a , and isomorphism Rk a /k Ga → Rk a /k Ga , effected by any σ ∈ Θ with σ this isomorphism identifies the two versions of (2.9.2).  are two lifts such that  b= a+ a  also If a ∈ Φ is of type R2 and  a,  a ∈ Φ  we obtain the isomorphisms of k-groups lies in Φ, Rkb /k Uk a /kb → Ua,

Rkb /k Uk0a /k  → U2a

(2.9.3)

b

as follows (recall here (2.7.2)). The isomorphisms of k a-groups Ga → Ua, Ga → Ua and Ga → Ub determined by Xa, Xa , and Xb, splice together to an isomorphism of k b-groups Uk a /kb → U[ a] , where we recall that U[ a] = Ua · Ua · Ub. We compose this isomorphism with (2.6.2). The isomorphisms (2.9.3) are again independent of the choice of  a in the same way as in the case of (2.9.2).

(d) Commutation Relations We continue with a semi-simple, simply connected, quasi-split k-group G, a maximally split maximal torus T ⊂ G and a Chevalley–Steinberg system X.  = Φ(T) and Φ = Φ(S) for the absolute and relative root As before we write Φ  we have the isomorphism ua : Ga → Ua defined over systems. For each  a∈Φ k a. Consider two non-proportional relative roots a, b ∈ Φ. We have the commutator map Ua × Ub → G,

−1 (ua, ub ) → ua ub u−1 a ub .

We would like to record here the explicit formulas for this map that arise from the parameterizations of Ua and Ub given by the Chevalley–Steinberg system.

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In the case that G is split and hence T = S, these formulas were obtained by Chevalley, and are given by  p q u pa+qb (±cp, q xa xb ), (2.9.4) ua (xa )ub (xb )ua (xa )−1 ub (xb )−1 = p, q ∈Z >0  pa+qb ∈ Φ

where cp, q are integers equal to 1, 2, or 3, cf. [SGA3, Exp. XXIII, Prop 6.4]. In the case that G is not split, but quasi-split, the formulas are more complicated. We reproduce the summary given in [BT84a, Appendix A]. These formulas were independently verified in [PR84, §§1.11–1.14]. We may assume without loss of generality that G is simply connected and k-simple. Then it is the restriction of scalars R/k of a simply connected and absolutely simple group. The commutator maps are then also obtained by applying R/k to the commutator maps over . Therefore we can reduce to the case that G is simply connected and absolutely simple. We may further reduce, upon replacing G by the subgroup generated by U±a and U±b , that G is of k-rank at most 2. If after these reductions G has become split, the commutator formula is again (2.9.4). The remaining cases are 2 A2n , 2 A2n+1 , 2 Dn , 3 D4 , 6 D4 , or 2 E6 . Since after the reductions our base field is again k, we will reuse the letter  for the minimal splitting field of G. Thus [ : k] equals the top left superscript. The root group Ua associated to a non-divisible relative root a ∈ Φ is then identified by the Chevalley–Steinberg system with a group Va that is, according to the discussion in §(b), one of the following: (1) Ga, k , (2) R  /k Ga,  for / /k with [  : k] equal to 2 or 3, (3) the group U/k and [ : k] = 2; cf. Equation (2.7.2). We are interested in the map γa, b : Va × Vb → Ua × Ub →

 c

Uc →



Vc ,

c

where the product runs over all non-divisible roots c ∈ Φ that are rational linear  combinations of a and b, the morphism Ua × Ub → c Uc is the commutator map, and the isomorphisms Va → Ua , Vb → Ub , and Uc → Vc are given by c for the c-coordinate of this the Chevalley–Steinberg system. We will write γa, b c map. We will give the formula for each γa, b , but we will allow the following ambiguities, which stems from the sign ambiguity in (2.9.4). (1) The inversion automorphism on Vc .

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(2) The automorphism on Vc coming from the unique element of Gal( /k) when Vc = R  /k Ga,  and [  : k] = 2, or when Vc = U  /k . (3) The composition of these two automorphisms. We will write ∼ for equality up to such ambiguity. Consider the cases 2 A2n+1 , 2 Dn , or 2 E6 . Then Φ is reduced and [ : k] = 2. Let a, b ∈ Φ. We have the following cases. (1) ∠(a, b) = 90◦ . Then a, b are short and the only possibility for c is c = a+b, which is long. We have Va = Vb = R/k Ga, , Vc = Ga, k , and c γa, b (u, v) ∼ Tr/k (uv).

(2) ∠(a, b) = 120◦ . The only possibility for c is c = a + b and a, b, c are either all short or all long. We have Va = Vb = Vc = Ga, k and c γa, b (u, v) ∼ uv.

(3) ∠(a, b) = 135◦ . Then a, b are of different length. Say a is short and b is long. Then Va = R/k Ga, and Vb = Ga, k . The possibilities for c are either c = a + b short and Vc = R/k Ga, , or c = 2a + b long and Vc = Ga, k . Then a+b γa, b (u, v) ∼ uv,

2a+b γa, b (u, v) ∼ N/k (u) · v.

Next consider the case 2 A2n . Then Φ is non-reduced and [ : k] = 2. Let a, b ∈ Φ be non-divisible. We have the following cases. (1) ∠(a, b) = 90◦ and a, b are of type R1. Then a + b is of type R3. Set c = (a + b)/2. Then Va = Vb = R/k Ga, and Vc = U/k , and c γa, b (u, v) ∼ (0, uv − uv).

(2) ∠(a, b) = 90◦ and a, b are of type R2. Then c = a + b is of type R1, Va = Vb = U/k , Vc = R/k Ga, , and c    γa, b ((u, v), (u , v )) ∼ uu .

(3) ∠(a, b) = 120◦ . Then a, b are of type R1 and so is c = a + b, Va = Vb = Vc = R/k Ga , and c γa, b (u, v) = uv.

(4) ∠(a, b) = 135◦ . Then one of a, b is of type R1 and the other is of type R2. Say a is of type R1 and b of type R2. Then Va = R/k Ga, and Vb = U/k . We have two subcases.

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Algebraic Groups (i) ∠(a, c) = 45◦ . Then c = a + b is of type R2, Vc = U/k , and  c   c    γa, b (u, (u , v )) ∼ γb, a ((u , v ), u) ∼ (uu , uuv ).

(ii) ∠(a, c) = 90◦ . Then c = a + 2b is of type R1, Vc = R/k Ga, , and c   c    γa, b (u, (u , v )) ∼ γb, a ((u , v ), u) ∼ uv .

Finally consider the cases 3 D4 and 6 D4 . Let  /k be a cubic subextension of /k. There exists a unique endomorphism θ of the scheme R  /k Ga,  such that N  /k (u) = uθ(u). Let a, b ∈ Φ be non-divisible. We have the following cases. (1) ∠(a, b) = 60◦ . Then c = a + b, Va = Vb = R  /k Ga,  , Vc = Ga , and c  γa, b (u, v) ∼ tr /k (uv).

(2) ∠(a, b) = 120◦ and a, b are short. Then Va = Vb = R  /k Ga,  . The options for c are c = a + b with Vc = R  /k Ga,  , c = 2a + b with Vc = Ga, k , and c = a + 2b with Vc = Ga, k . Then a+b γa, b (u, v) ∼ θ(u + v) − θ(u) − θ(v) 2a+b  γa, b (u, v) ∼ tr /k (θ(u)v) a+2b  γa, b (u, v) ∼ tr /k (uθ(v)).

(3) ∠(a, b) = 120◦ and a, b are long. Then c = a + b is long, Va = Vb = Vc = Ga, k , and c γa, b (u, v) ∼ uv.

(4) ∠(a, b) = 150◦ . Then a, b are of different length. Say a is short and b is long. The options for c are c = a + b with Vc = R  /k Ga,  , c = 2a + b with Vc = R  /k Ga,  , c = 3a + b with Vc = Ga, k , and c = 3a + 2b with Vc = Ga, k . This is the only case in which the root groups Uc do not commute with each other, so we choose arbitrarily an order on the set c . They are {a + b, 2a + b, 3a + b, 3a + 2b}. This pins down the maps γa, b given by a+b γa, b (u, v) ∼ uv 2a+b γa, b (u, v) ∼ θ(u)v 3a+b  γa, b (u, v) ∼ N /k (u)v 3a+2b 2  γa, b (u, v) ∼ C · N /k (u)v .

Here C is either 1 or 2 and the value depends on the chosen order of the c (v, u) are given by set {a + b, 2a + b, 3a + b, 3a + 2b}. The values of γb, a the same list as above, with the only difference that the constants C in the

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3a+2b (u, v) and γ 3a+2b (v, u), that is, if one fourth case are not equal for γa, b b, a is 1 then the other must be 2 and vice versa.

(e) Simply Connected Cover Let G be a connected reductive quasi-split k-group. We can apply the discussion of the previous paragraphs to the simply connected cover Gsc of the derived subgroup Gder of G. When we speak of a (weak) Chevalley system, or a (weak) Chevalley–Steinberg system for G, we will mean one for Gsc . As discussed in the beginning of §2.6, passing from G to Gsc does not change the root groups, so all results concerning their structure, as well as the commutation relations, remain valid for G.

2.10 Integral Models In this section, k will denote a field equipped with a discrete valuation ω, o its ring of integers that is assumed to be Henselian. We denote by K a fixed maximal unramified extension of k and the ring of integers of K by O. The residue field of k will be denoted by f. The residue field of K is the separable closure fs of f. Let X be an affine variety over k, by which we mean a reduced affine scheme of finite type over k. Let k[X] be its coordinate ring, so that X = Spec(k[X]). Definition 2.10.1 (1) An integral model of X is an affine scheme X over o, flat and of finite type, together with an isomorphism X ×Spec(o) Spec(k) → X. (2) If Y is another affine variety over k and θ : X → Y is a morphism of affine varieties, and X and Y are integral models, an extension of θ is a morphism of o-schemes X → Y making the diagram / Y ×Spec(o) Spec(k)

X ×Spec(o) Spec(k)  X

θ

 /Y

commute. (3) Given two integral models X1 and X2 of X, a morphism of integral models of X is an extension of the identity automorphism of X. Remark 2.10.2 Given an integral model X of X we obtain from the isomorphism X ×Spec(o) Spec(k) → X a map X (R) → X(R ⊗o k) for any o-algebra R. This map is injective if R is flat.

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Remark 2.10.3 According to the previous remark we can view X (o) as a subset of X(k). We can endow X(k) with the analytic topology induced by k as discussed in Remark 2.2.2. Then X (o) is a closed, open, bounded subset of X(k). Indeed, we may choose the closed immersion X → An to be the base change of a closed immersion X → An and then X (o) = X(k) ∩ o n . If k is locally compact, then X (o) is open and compact. More generally, we can apply the above observation to any algebraic extension  of k. Let o[X ] be the coordinate ring of X , that is, the ring of global sections of the structure sheaf of X . Then giving an isomorphism X ×Spec(o) Spec(k) → X is the same as giving an isomorphism o[X ] ⊗o k → k[X] of k-algebras. Composing this map with the map o[X ] → o[X ] ⊗o k,

f → f ⊗ 1

we obtain a homomorphism o[X ] → k[X] of o-algebras. Since X is flat, this homomorphism is injective and realizes o[X ] as an o-subalgebra of k[X]. We shall henceforth view o[X ] as an o-subalgebra of k[X]. In addition, we see that any morphism θ : X → Y has at most one extension X → Y . Indeed, this extension is given on the level of coordinate rings by the restriction of θ ∗ : k[Y ] → k[X] to the o-subalgebra o[Y ], and it exists if and only if θ ∗ (o[Y ]) ⊂ o[X ]. Fact 2.10.4 Let π ∈ k be a uniformizer. The functor sending an integral model X to the subalgebra o[X ] ⊂ k[X] is inverse to the functor sending a finite type o-subalgebra A ⊂ k[X] satisfying A[π −1 ] = k[X] to the affine scheme Spec(A). Remark 2.10.5 Among all integral models of X there is the “trivial” integral model X , obtained by composing the structure morphism X → Spec(k) with the morphism Spec(k) → Spec(o) induced by the inclusion o → k. This is an integral model because k is an o-algebra of finite type and the multiplication map k ⊗o k → k is an isomorphism of algebras. It is the unique integral model that is not faithfully flat over o. We have X (o) = ∅. Lemma 2.10.6 Assume that X is irreducible. Let X be a faithfully flat integral model of X. Then all irreducible components of the special fiber of X have dimension equal to the dimension of X. Proof This is [EGAIV3 , Lemma 14.3.10]. Indeed, the structure morphism X → Spec(o) is surjective since X is assumed faithfully flat. In particular the special fiber of X is non-empty. Moreover, the irreducibility of X and the flatness of X imply the irreducibility of X , so the above reference applies. 

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Remark 2.10.7 If X is an affine scheme over o, there exists a maximal closed subscheme X flat that is flat over o, namely the zero locus of the torsion ideal of o[X ]. The closed immersion X flat → X induces an isomorphism on generic fibers. Remark 2.10.8 Assume that X is smooth has the structure of a group scheme, and that the integral model X has a compatible structure of a group scheme. The tangent space at 1 (that is, the Lie algebra Lie(X), but without the bracket) is canonically identified with the k-vector space Derk (k[X], k), where k is considered as a k[X]-module via the identity section e : k[X] → k. The Lie algebra Lie(X ) is the o-module Dero (o[X ],o). The inclusion o[X ] ⊂ k[X] realizes Lie(X ) as an o-lattice in Lie(X), as follows. The tangent space Lie(X ) is the o-dual of the cotangent space ωX ,e . If Ie ⊂ o[X ] is the kernel of the identity section e : o[X ] → o and Ω1X /o is the module of differentials of the o-algebra o[X ], then f → df ⊗ 1 is an isomorphism of o-modules Ie /Ie2 → Ω1X /o ⊗o[X ] o, and these canonically isomorphic o-modules are the cotangent space to X at e, and will be denoted by ωX , e . Then ωX ,e is a finite rank o-module that may have m-torsion. We have Dero (o[X ],o) = Homo (ωX ,e,o) and Derk (k[X], k) = Homo (ωX ,e, k). Thus Lie(X ) ⊂ Lie(X) and the natural map Lie(X ) ⊗o k → Lie(X) is an isomorphism. Furthermore, the Lie algebra of the special fiber of X is Derf (f[X ], f) = Homo (ωX , e, f). The following statements are equivalent. (1) (2) (3) (4)

The model X is smooth. The o-module ωX , e is m-torsion-free, hence free. dimk (Lie(X)) = dimf (Lie(Xf )), where Xf := X ×o f. The homomorphism Lie(X ) ⊗o f → Lie(Xf ) is surjective, hence an isomorphism.

Notice that the m-torsion submodule of ωX , e is not detected by Lie(X ), since o is m-torsion-free; but it is detected by Lie(Xf ). Thus whether or not X is smooth cannot be read off directly from the tangent space Lie(X ), but it can be read off from the cotangent space ωX , e , or from the relationship between the tangent spaces Lie(X ) and Lie(Xf ). We will be particularly interested in integral models that are smooth. Then we can describe the o-subalgebra o[X ] ⊂ k[X] in terms of the subset X (o) ⊂ X(k) as follows. Lemma 2.10.9

If X is a smooth integral model of X then o[X ] = { f ∈ k[X] | f (X (O)) ⊂ O},

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where f ∈ k[X] is considered as a function X(K) → A1 (K) = K. Proof The inclusion ⊂ is immediate. To argue the opposite inclusion, assume by way of contradiction that f ∈ k[X] satisfies f (X (O)) ⊂ O but f  o[X ].  Since k[X] = n0 π −no[X ] there exists a unique smallest n > 0 such that π n f ∈ o[X ]. Since π n f is not divisible by π in o[X ] its image in o[X ] ⊗o f = o[X ]/(π) is non-zero. Call this image g. At the same time we have [π n f ](X (O)) ⊂ Mn ⊂ M since n > 0, so the regular function g of Xfs vanishes on the image of the reduction map X (O) → X (fs ). The smoothness of X implies that this reduction map is surjective (this is a standard fact, reviewed in Lemma 8.1.3), so the function g vanishes on all of X (fs ). The latter is schematically dense in the special fiber of X , implying g = 0, a contradiction.  An O-scheme X satisfying O[X ] = { f ∈ K[X] | f (X (O)) ⊂ O} is called étoffé in [BT84a, 1.7.2]. Corollary 2.10.10 Let X and Y be affine and of finite type over k. Assume X is smooth. Let X be a smooth o-model of X, and Y any model of Y . A morphism θ : X → Y extends (necessarily uniquely) to a morphism X → Y if and only if θ(X (O)) ⊂ Y (O). Proof A given θ : X → Y is the base change of a morphism X → Y if and only if the k-algebra map θ ∗ : k[Y ] → k[X] arises via extension of scalars from an o-algebra map o[Y ] → o[X ], which is equivalent to f ∗ (o[Y ]) ⊂ o[X ]. By Lemma 2.10.9 this is equivalent to θ(X (O)) ⊂ Y (O).  Corollary 2.10.11 The functor X → X (O) ⊂ X(K) is a fully faithful functor from the category of smooth integral models of X to the category of bounded open subsets of X(K) and inclusions. Remark 2.10.12 The affineness of Y in Corollary 2.10.10, and hence the affineness of X in Corollary 2.10.11, is essential. Indeed, regular semi-stable curve singularities provide examples of smooth affine X and smooth non-affine Y , where a morphism between generic fibers θ : X → Y satisfies θ(X (O)) ⊂ Y (O) but does not extend to a morphism X → Y . In brief, one can take X to be the affine line over O, and Y to be the maximal smooth subscheme of the blow-up of X at an arbitrary point of the special fiber. The natural morphism Y → X is an isomorphism between generic fibers, and a bijection on O-points. But the inverse isomorphism XK → YK does not extend to a morphism X → Y , for if it did, then Y → X would be an isomorphism, which is impossible considering the special fibers. The details of such an example are spelled out in Example 2.10.14. On the other hand, in the case

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when Y is affine, but X may not be, Corollary 2.10.10 continues to hold, as we now show. Lemma 2.10.13 Let X be a smooth o-scheme and Y a smooth affine oscheme. A morphism θ : Xk → Yk extends (necessarily uniquely) to a morphism X → Y if and only if θ(X (O)) ⊂ Y (O). Proof Write X and Y for the generic fibers of X and Y . Choose an affine open   cover X = i Ui and let Ui be the generic fiber of Ui , so that X = Ui is an affine open cover. If an extension θo : X → Y exists, then clearly, θ(X (O)) ⊂ Y (O). Moreover, θo |Ui is uniquely determined by θ|Ui by Corollary 2.10.10. This implies that θo is uniquely determined by θ. Assuming that θ(X (O)) ⊂ Y (O), the restriction θ to any Ui is a morphism Ui → Y that sends Ui (O) to Y (O), and hence has an extension θ i : Ui → Y by Corollary 2.10.10. Since θ i |Ui ∩U j and θ j |Ui ∩U j both extend θ|Ui ∩U j , the uniqueness statement implies that the restrictions of θ i and θ j to Ui ∩ U j agree. Therefore the collection of morphisms (θ i ) glues to a unique morphism  θo : X → Y . The preceding lemma stated that given a smooth O-scheme X and a smooth affine O-scheme Y , a morphism f : XK → YK extends to a morphism X → Y if and only if f (X (O)) ⊂ Y (O). We now give an example showing that this criterion fails when Y is not assumed to be affine. This example was communicated to us by Johan de Jong and Brian Conrad. Example 2.10.14 Let O be a discrete valuation ring, K be its field of fractions. Let π ∈ O be a uniformizing element. Let X = A1O = Spec(O[y]) be the affine line over O and let X  be the blow-up of X at the origin of its special fiber. One can describe X  explicitly as follows. Let U1 = Spec(O[x1 ]) be the affine X -scheme defined by y → πx1 and let U2 be the affine open X -scheme Spec(O[y, x2 ]/(x2 y − π)). In U1 and U2 we have the open pieces where x1 respectively x2 are invertible: thus Spec(O[x1, x1−1 ])  Gm and Spec(O[x2, x2−1, y]/(x2 y − π))  Spec(O[x2, x2−1 ])  Gm . We glue U1 and U2 along these isomorphic open pieces via the isomorphism x1 → x2−1 . Note that U2 has generic fiber Gm and the gluing produces A1K = XK as generic fiber of X , as it should. Consider the special fiber of X  and its map to the special fiber of X . This is the gluing of the line (U1 )f = Spec(f[x1 ]) = A1f and the coordinate cross (U2 )f = Spec(f[y, x2 ]/(x2 y)) over the open subsets {x1  0} = Gm and {x2  0} = Gm via inversion. The line (U1 )f maps to the origin of Xf = A1f , and so does the line {x2 = 0} in (U2 )f . Thus, Xf  has two irreducible

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components: a copy of P1f over the origin of Xf = A1f , and a copy of A1f that maps isomorphically to Xf ; it is the closure of the copy of Xf − {0} that naturally lies in Xf . In particular, the O-scheme X  is flat, of finite type, and O-smooth away from the one point in its special fiber where the two lines cross. The complement of that single non-smooth f-point in Xf  is the maximal O-smooth open subscheme in X . Let us denote it by Y . The natural map q : X  → X is an isomorphism away from 0 in Xf but the fiber over 0 in Xf is a copy of P1f . Since q is proper and qK is an isomorphism, the valuative criterion for properness implies X (O) = X (O). We claim further that Y (O) = X (O). In other words, there is no O-point of X  that passes through the singularity in Xf . Indeed, such a point ξ existed, then from the construction of X  we see that ξf is the origin in the special fiber of U2 , so ξ factors through the open subscheme U2 ⊂ X  by Lemma 8.1.1. The corresponding O-algebra map ξ ∗ : O[x2, y]/(x2 y − π) → O must carry x2 and y to 0 mod π, i.e. it carries each of x2 and y into the maximal ideal πO. But then the product x2 y would have to land in π 2 O, yet x2 y = π in the source ring for ξ ∗ , and ξ ∗ is an O-algebra map. Since π does not belong to π 2 O, this is a contradiction. We have thus seen X (O) = X (O) = Y (O). The natural morphism Y → X is an isomorphism between the generic fibers. Its inverse XK → YK maps X (O) to Y (O). However, it cannot extend to a morphism X → Y , for if it did, this would be an inverse of the morphism Y → X . However, we have seen that the special fibers of Y and X are not isomorphic. Definition 2.10.15 An open bounded subset U ⊂ X(K) is called schematic, if there exists a (necessarily unique) smooth integral model X of X with X (O) = U. It is furthermore called connected, if X has connected fibers. Note that we are demanding the existence of an o-model, not just O-model, even though we are working with K-points. In particular, a schematic subset U ⊂ X(K) is necessarily Gal(K/k)-invariant. In fact, we have the following very useful reduction. Fact 2.10.16 Let U ⊂ X(K) be a closed, open, bounded subset. Assume that U is Gal(K/k)-invariant and that there exists a smooth model X  of XK such that X (O) = U. Then X  descends to o, that is, there exists a smooth model X of X and an isomorphism X ×o O → X . In particular, U is schematic. Proof

This is unramified Galois descent over discrete valuation rings, see

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for example [BLR90, §6.2, Ex.B], together with Corollary 2.10.11. Note that  o[X ] = O[X ] ∩ k[X]. Remark 2.10.17 The process of smoothening, which we shall review later in the setting of group schemes, shows that in the definition of schematic we can drop the assumption that the model X is smooth. However, in the definition of connected this assumption cannot be dropped, because the smoothening of a connected model can be disconnected. Remark 2.10.18 In terms of coordinate rings, Corollary 2.10.11 implies the following. If U ⊂ X(K) is schematic then the coordinate ring of the corresponding integral model is { f ∈ k[X] | f (U) ⊂ O}. If A ⊂ k[X] is a o-subalgebra that is the coordinate ring of a smooth integral model, then the corresponding schematic subset is {x ∈ X(K) = Homk (k[X], K) | x(A) ⊂ O}. In principle, describing the smooth integral models of X is now reduced to answering the question: which open, closed, and bounded subsets of X(K) are schematic? For such a subset U the integral model is then given by the above formula. Additional work may be required if that formula is not sufficiently explicit for applications. Remark 2.10.19 If X is a group scheme and U ⊂ X(K) is an open, closed, and bounded subgroup that is schematic, then Corollary 2.10.10 implies that the smooth integral model X , whose set of O-points is the set U, is a group scheme. Furthermore, if Y is a group scheme over o with generic fiber Y and θ : X → Y is a morphism of group schemes sending X (O) to Y (O), then its canonical extension X → Y is also a morphism of group schemes. 2.10.20 Vector groups Let R be a commutative unital ring (for us, the main examples are o, k, and f). Let M be a finite rank free R-module. The tensor ∞ ⊗n equipped with the algebra is defined as the R-module T(M) = n=0 M multiplication map coming from the canonical isomorphism M ⊗n1 ⊗ M ⊗n2 → M ⊗(n1 +n2 ) . The symmetric algebra Sym(M) is defined as the quotient of T(M) by the ideal generated by the set {m1 ⊗ m2 − m2 ⊗ m1 |m1, m2 ∈ M }. Since this ideal is homogeneous, the grading of T(M) endows Sym(M) with a grading. ∞ n n Thus Sym(M) = n=0 Sym (M), with Sym (M) being the R-module quotient ⊗n of M by the operation of the group of permutations of n letters. The algebra Sym(M) satisfies the following universal property. Given an Ralgebra A and an R-linear map M → A satisfying f (m1 ) f (m2 ) = f (m2 ) f (m1 )

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for all m1, m2 ∈ M, there is a unique R-algebra homomorphism Sym( f ) : Sym(M) → A whose restriction to M = Sym1 (M) equals f . Moreover, if A is graded and f takes image in A1 , then Sym( f ) is a homomorphism of graded algebras. In particular, if f : M → N is a homomorphism of free R-modules of finite rank, there is a unique homomorphism Sym( f ) : Sym(M) → Sym(N) of graded Ralgebras whose restriction to M = Sym1 (M) equals the composition of f with the inclusion N = Sym1 (N) → Sym(N). The algebra homomorphism Sym(R) → R[T], induced by the linear map R → R[T] given by multiplication by T, is an isomorphism between Sym(R) and the polynomial algebra R[T]. For two finite rank free R-modules M1 and M2 there is a natural isomorphism Sym(M1 ⊕ M2 ) = Sym(M1 ) ⊗ Sym(M2 ). In particular, any choice of basis (m1, . . . , mn ) of M provides an isomorphism Sym(M) → R[T1, . . . ,Tn ] of graded R-algebras. The algebra Sym(M) has a natural structure of a Hopf algebra, with comultiplication induced by m → m ⊗ 1 + 1 ⊗ m for all m ∈ M, counit given ∞ n 0 by the projection of n=0 Sym (M) onto the direct factor Sym (M) = R, and antipode induced by m → −m. It is cocommutative. Given a homomorphism f : M → N of free R-modules of finite rank, the homomorphism Sym( f ) : Sym(M) → Sym(N) of graded R-algebras preserves the Hopf algebra structures. The module dual to M is defined by M ∨ = HomR (M, R). It is a free R-module of the same rank as M. The functor G M assigning to any R-algebra A the abelian group underlying the A-module M ⊗R A is a commutative affine algebraic group, represented by the scheme associated to the algebra Sym(M ∨ ), and with group structure reflected by the Hopf algebra structure of Sym(M ∨ ), cf. [DG70, Chapter II, §1, 2.1]. We call such an affine algebraic group a vector group. Given two free R-modules M1 and M2 of finite rank, the natural isomorphism (M1 ⊕ M2 )∨ → M1∨ ⊕ M2∨ of R-modules and the natural isomorphism of Hopf algebras Sym(M1 ⊕M2 ) = Sym(M1 )⊗Sym(M2 ) lead to the natural isomorphism G M1 ⊕M2 = G M1 × G M2 . In particular, a choice of a basis (m1, . . . , mn ) of M induces an isomorphism G M → (Ga )n , showing that a vector group is a smooth group scheme, and characterizing the vector groups as those affine group schemes that are (non-canonically) isomorphic to products of copies of the additive group. If R is an algebraically closed field, then a vector group can be alternatively characterized as a smooth connected affine commutative unipotent group, subject to the additional condition that its elements have order equal to the char-

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acteristic of R whenever that characteristic is positive, cf. [CGP15, Lemma B.1.10]. A homomorphism f : M → N of free R-modules of finite rank induces a homomorphism G M → G N of group schemes, which on the level of coordinate rings is given by Sym( f ∨ ). Given a ring extension R → S and an S-module N, the Weil restriction of scalars RS/R G N exists and is naturally identified with G N R , where NR is the R-module obtained from N by “forgetting” the Sstructure. On the other hand, the base change G M ×Spec(R) Spec(S) is naturally identified with G M ⊗ R S . In particular, if R = o, then G M is a smooth model of the affine k-group GV , where V is the k-vector space M ⊗o k. The Lie algebra of G M is naturally identified with M, so that the homomorphism Lie(G M → G N ) is identified with f : M → N, and, when R = o, the lattice Lie(G M ) inside of Lie(GV ) is naturally identified with M inside of V. Given two free R-modules M and N of finite rank and an R-linear map f : M ⊗ M → N, one obtains a morphism of schemes G M × G M → G N that, for any R-algebra A, is given on A-points by f ⊗id A

(M ⊗R A)×(M ⊗R A) → (M ⊗R A) ⊗ A (M ⊗R A) = (M ⊗R M) ⊗R A → N ⊗R A. On coordinate rings this homomorphism can be described as follows. Pointwise product of linear functionals induces an isomorphism M ∨ ⊗R M ∨ → (M ⊗R M)∨ . Using this isomorphism, we can view f ∨ as an R-linear map N ∨ → M ∨ ⊗R M ∨ . The identity M ∨ = Sym1 (M ∨ ) leads to the embedding M ∨ ⊗ M ∨ → Sym(M ∨ )⊗R Sym(M ∨ ), whose image lies in a commutative subalgebra. Composing f ∨ with this embedding we obtain from the universal property the desired homomorphism Sym(N ∨ ) → Sym(M ∨ ) ⊗R Sym(M ∨ ) of R-algebras. Th morphism f : G M × G M → G N is a 2-cocycle for the trivial action of G M on G N and thus induces a new group structure G M × f G N given by (x1, y1 ) · (x2, y2 ) = (x1 + x2, y1 + y2 + f (x1, x2 )). The map G N → G M × f G N sending y to (0, y) is a closed immersion whose image is a normal subgroup scheme. The quotient is isomorphic to G M via the projection map (x, y) → x. The group scheme G M × f G N is commutative if and only if f is a symmetric bilinear mapping. We refer to Appendix A for a discussion of various operations that can be performed on integral models, such as

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(1) schematic closure, which propagates a closed subscheme of the generic fiber into the integral model; (2) dilatation, which propagates a closed subscheme of the special fiber into the integral model; (3) Weil restriction, which reduces the construction of integral models to a few simple cases; (4) Greenberg functor, which generalizes certain aspects of Weil restriction to mixed characteristic; (5) smoothening, which replaces a possibly non-smooth integral model by a smooth one via a sequence of dilatations. In Appendix B we discuss integral models of tori in detail, and in Appendix C we discuss integral models of relative root groups of quasi-split reductive groups.

2.11 Group Scheme Actions and the Dynamic Method We give here a brief overview of the dynamic method as well as various results about actions of group schemes on schemes. The details of these results are contained in [CGP15], and we confine ourselves to stating the results that we will need (not always in their full generality) and referring to [CGP15] for proofs. The dynamic method associates to a 1-parameter group λ : Gm → G closed subgroups U (λ) ⊂ P(λ) and describes their properties. This method works in the generality of an affine group scheme G over any (commutative unital) base ring A. When A is a field and G is connected and reductive, P(λ) is a parabolic subgroup and U (λ) is its unipotent radical. We will be interested in the three cases A = k, A = f, and A = o, and the affine group scheme G will usually be smooth, but not necessarily reductive. More generally, this method works for any action λ : Gm × G → G , which we write as (t, g) → t · g. The case of a 1-parameter subgroup is the special case in which the action is given by the conjugation action. Given a commutative A-algebra R and g ∈ G (R) we say that the limit limt→0 t · g exists if the orbit map (Gm )R → GR defined by t → t · g extends (necessarily uniquely) to an R-morphism A1R → GR . In that case, the value of this limit is the element of G (R) that is the value at 0 ∈ A1R (R) of this R-morphism. The closed subgroup PG (λ) is defined via its functor of points by PG (λ)(R) = {g ∈ G (R) | lim t · g exists} t→0

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while UG (λ) is defined via UG (λ)(R) = {g ∈ G (R) | lim t · g = 1}. t→0

Proposition 2.11.1 ([CGP15, Lemmas 2.1.4 and 2.1.5, Corollary 2.1.9]) (1) PG (λ) is represented by a closed subgroup scheme of G and UG (λ) is represented by a closed normal subgroup scheme of PG (λ). (2) ZG (λ) = PG (λ) ∩ PG (−λ) is the scheme-theoretic centralizer of λ in G . (3) The formation of PG , UG , and ZG , is compatible with any extension of the base ring. (4) If H is a closed subgroup scheme of G that is stable under the action of Gm , then PH (λ) = PG (λ) ∩ H , and the same holds for UH and ZH . (5) Assume A is a field and G , H are connected and of finite type. If G → H is a quotient map (a faithfully flat surjective homomorphism), then so are the maps PG (λ) → PH (λ), UG (λ) → UH (λ), and ZG (λ) → ZH (λ). (6) If G is of finite type over A, then the fibers of UG are unipotent. For n ∈ Z let Lie(G )n be the n-weight space for the action of Gm via λ. Define Lie(G )+ and Lie(G )− to be the direct sum of the weight spaces for all positive respectivelyall negative n. Proposition 2.11.2 ([CGP15, Lemma 2.1.5, Proposition2.1.8]) G is of finite presentation.

Assume that

(1) The subgroups PG (λ), UG (λ), ZG (λ) are of finite presentation. (2) The fibers of UG (λ) are connected, and if the fibers of G are connected, then the same holds for PG (λ) and ZG (λ). (3) Lie(ZG (λ)) = Lie(G )0 , Lie(UG (λ)) = Lie(G )+ . (4) Lie(PG (λ)) = Lie(G )0 ⊕ Lie(G )+ . (5) The multiplication map induces an isomorphism of A-group schemes ZG (λ)  UG (λ) → PG (λ). Proposition 2.11.3 ([CGP15, Propositions 2.1.8, 2.1.10, 2.1.12, 2.2.9, Corollary 2.2.5]) Assume that G is of finite presentation and smooth. (1) The subgroups ZG (λ), PG (λ), UG (λ) are smooth. (2) The multiplication map UG (−λ) × PG (λ) → G is an open immersion. This open immersion is an isomorphism if A is a field and G is connected and solvable.

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(3) If G is a reductive group scheme, then PG (λ) is a parabolic subgroup scheme and UG (λ) is the unipotent radical of this parabolic subgroup scheme. (4) If A is a field the unipotent group UG (λ) is split. Finally we consider more general actions of group schemes. Let A again be any (commutative unital) base ring. Let Λ be a finitely generated abelian group and let M be the corresponding A-split multiplicative group, characterized by the property X∗ (M) = Λ. A functorial action of M on an A-module V is the datum of an R-linear action of the abstract group M(R) on the R-module VR = V ⊗ A R, for each A-algebra R, functorial in R. In other words, it is a scheme-theoretic action M × V → V. Proposition 2.11.4 ([CGP15, Lemma A.8.8] or [SGA3, Exp. I, 4.7.3]) There is a natural bijection between the set of functorial actions ρ of M on an A module V and the set of Λ-gradings V = λ∈Λ Vλ . This bijection is given by   λ(m)vλ . ρ(m) vλ = In particular, we see that the geometric fibers of M are linearly reductive. Given an action of M on an A-scheme Y we define the fixed point locus Y M to be the subfunctor that assigns to each A-algebra R the subset of points y ∈ Y (R) whose image in Y (R ) is fixed by M(R ) for any R-algebra R . Proposition 2.11.5 ([CGP15, Propositions A.8.10, A.8.12, and A.8.14]) Let Y be a separated scheme, locally of finite type over A. Let M be a split multiplicative A-group scheme (or, more generally, an affine A-group scheme of finite type whose geometric fibers are linearly reductive) equipped with a left action on Y . (1) The fixed point locus Y M is represented by a closed subscheme of Y locally of finite type. If Y is locally of finite presentation, then Tany (Y H ) = Tany (Y ) H for any y ∈ Y M (A). (2) Assume that A is noetherian. If Y  is another separated scheme locally of finite type over A and f : Y → Y  is smooth, then Y M → Y M is smooth. In particular, if Y is smooth, then Y M is smooth. (3) If A is a field and G is connected reductive, then the identity component of G M is connected reductive. (4) If A is a field and G is a smooth connected affine algebraic A-group whose A-unipotent radical Ru, A(G) is M-stable, then Ru, A((G M )0 ) = (Ru, A(G) M )0 . Moreover, Ru, A(G) M is connected if M is a closed subgroup of a torus acting on G.

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2.11.6 Let S be a base scheme (for us this will be the spectrum of a discretely valued Henselian field, or its ring of integers, or its residue field). Given an Sscheme X and an S-group scheme G , an left action of G on X is a morphism f : G ×S X → X of S-schemes satisfying f ◦ (idG × f ) = f ◦ (mG × idX ) and f ◦ (eG × idX ) = idX , where id denotes the identity automorphism of G or X , m : G ×S G → G denotes the multiplication morphism, and eG : S → G is the unit section. As an example, the group scheme G acts on itself in three different ways: via left multiplication, right multiplication by inverses, and conjugation. For any g ∈ G (S) the morphism f (g, −) : X → X obtained by X

S ×S X

g,idX

/ G ×S X

a

/X

is an automorphism of the S-scheme X with inverse given by f (g −1, −). We will denote the automorphism f (g, −) simply by g when there is no danger of confusion. If U ⊂ X is an open (or closed) subscheme, so is gU . 2.11.7 Let S be a non-trivial o-split torus and G be a smooth affine ogroup scheme equipped with a left action of S by automorphisms. This action induces an action on the Lie algebra Lie(G ) of G , under which this Lie algebra  decomposes as a direct sum of weight spaces a ∈X∗ (S ) Lie(G ) a . A character a of S will be called an S -root (of G ) if a  0 and Lie(G )a  {0}, and Lie(G )a will be called the a-root space. Let G be the generic fiber of G . Then Lie(G ) is an o-lattice in the k-vector space Lie(G) (cf. Remark 2.10.8) and tensoring with k we obtain Lie(G) =  a (Lie(G ) a ⊗o k). This shows that the S -roots of G coincide with the Sroots of G, and the a-root space of G is a lattice in the a-root space of G. 2.11.8 We now formulate an integral version of 2.6.3. Consider the closed subgroup scheme Ca := ker a of S . It is o-flat, because it is isomorphic to Hom(X∗ (S )/Za, Gm ) and is therefore the direct product copies of Gm and μn for various n. As noted in 2.6.3, Ca may not be smooth. According to [SGA3, Exp. VIII, Theorem 3.1], a descends to an isomorphism S /Ca → Gm . Proposition 2.11.9 Given an S -root a of G , there is a unique closed o-smooth subgroup scheme Ua of G with the following properties: (1) Ua is stable under the action of S and Ca acts trivially on it, (2) its fibers are connected and unipotent and hence it is contained in the relative identity component G 0 of G ,

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(3) the Lie algebra of Ua is the sum of S -root spaces for S -roots that are positive integral multiples of a. Furthermore, Ua has the following properties: (1) (2) (3) (4)

Ua is the schematic closure of Ua in G , Ua is contained in G 0 , Ua (o) = Ua (k) ∩ G (o) = Ua (k) ∩ G 0 (o), the special fiber U a of Ua is the a-root group for the action of S on G .

Proof Let Ga be the centralizer of Ca in G . This centralizer is smooth and its Lie algebra is the sum of root spaces for S -roots that are integral multiples of a, cf. Proposition 2.11.5(1). The action of S on G induces an action of the quotient S /Ca on Ga . Consider the isomorphism λa : Gm → S /Ca that is the inverse of a. Let Ua denote the closed o-subgroup scheme UG a (λa ) defined in §2.11. This subgroup scheme is clearly stable under the action of S ; by Propositions 2.11.2 and 2.11.3 it is o-smooth, its fibers are connected and its Lie algebra is the sum of all root spaces for S -roots that are positive integral multiples of a. According to Proposition 2.11.1 the fibers of Ua are unipotent. To prove the uniqueness of Ua , let V be an o-subgroup scheme of G with the three properties listed in the proposition. Property (1) implies that V is contained in Ga . Now we consider UV (λa ); it is contained in Ua = UG a (λa ) by Proposition 2.11.1, but it has the same Lie algebra as Ua and V , so V = UV (λa ) = Ua . We now come to the additional properties. Since the a-root group Ua with respect to Sk of Gk can be constructed in the same way as Ua , we conclude that the generic fiber of Ua is the a-root group Ua of Gk . Hence, the schematic closure of Ua in G is the root group Ua , and Ua (k) ∩ G (o) = Ua (o). As Ua is contained in the relative identity component G 0 of G , we conclude that Ua (k) ∩ G 0 (o) = Ua (o) = Ua (k) ∩ G (o). Let S be the special fiber of S . It acts on the special fiber G of G by f-group automorphisms. It is obvious from the above construction of the a-root group of G that the special fiber U a of Ua is the a-root group of the special fiber G  with respect to the torus S . Definition 2.11.10 Root groups The subgroup scheme Ua provided by the preceding proposition will be called the a-root group of G with respect to S . As Ua is contained in G 0 , it is also the a-root group of the latter.

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Remark 2.11.11 Propositions 2.11.1(4) and 2.11.3(1) imply that if G  is a closed o-smooth subgroup scheme of G that is stable under the action of S , then Ua := Ua ∩ G  is the a-root group of G . Hence, Ua is smooth and has connected split unipotent fibers. We now consider a base field k and a smooth affine group G over k equipped with an action of a split torus S. Given a smooth connected subgroup H ⊂ G  H) for the set of weights of S in Lie(H), stable under this action, we write Φ(S, possibly including 0, and wt(S, H) for the subsemigroup of X∗ (S) generated by  H). Then Φ(S,  H) = Φ(S,  G) ∩ wt(S, H). Φ(S, Proposition 2.11.12 ([CGP15, Proposition 3.3.5]) Let H1, H2 be smooth connected k-subgroups stable under the S-action. Let H is the subgroup generated by them and let [H1, H2 ] be their commutator subgroup. Then wt(S, H) is the subsemigroup of X∗ (S) generated by wt(S, H1 ) and wt(S, H2 ), while wt(S, [H1, H2 ]) ⊂ wt(S, H1 ) + wt(S, H2 ). Proposition 2.11.13 ([CGP15, Proposition 3.3.6]) For any subsemigroup A ⊂ X∗ (S) there exists a unique smooth connected S-invariant k-subgroup H A of G such that Lie(H A) is the span of the a-weight spaces in Lie(G) for  G). Any smooth connected S-invariant k-subgroup H  with all a ∈ A ∩ Φ(S,   H ) ⊂ A is a subgroup of H A. If 0  A then H A is unipotent and we write Φ(S, U A instead of H A to emphasize that. Consider 0  a ∈ X∗ (S) and let A = a . Let A j = {na ∈ A|n  j} for any positive integer j. Let U j a be the smooth connected unipotent group associated to A j . We obtain the descending filtration of Ua ⊃ U2a ⊃ U3a · · ·. Proposition 2.11.14 ([CGP15, Lemma 3.3.8]) Each U ja is normal in Ua . The successive quotient U j a /U(j+1)a is a vector group. If it is non-trivial,  U j a /U(j+1)a ) = { ja}. In particular, U j a /U(j+1)a has a unique Sthen Φ(S, equivariant F-linear structure. Theorem 2.11.15 ([CGP15, Thmeorem 3.3.11]) Let G be a smooth connected solvable k-group equipped with an action of a split k-torus S. Given a disjoint  G) = $n Ψi such that the subsemigroups Ai ⊂ union decomposition Φ(S, i=1 X∗ (S) generated by Ψi are pairwise disjoint, the k-scheme map H A1 × · · · × H A n → G defined by multiplication is an isomorphism. Definition 2.11.16 Quasi-reductive Algebraic Groups Let k be a field. For a connected smooth affine k-group H, the k-unipotent radical of H is defined

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to be the maximal smooth connected normal unipotent k-subgroup. The kunipotent radical of H is denoted by Ru,k (H). The k-split unipotent radical of H is defined to be the maximal smooth connected normal k-split unipotent k-subgroup of H; it is denoted by Rus, k (H). It is obvious that Rus, k (H) is the maximal connected k-split subgroup of Ru, k (H). The group H is said to be quasi-reductive if Rus,k (H) is trivial. It is easily seen that for any H as above, the k-group H/Rus, k (H) is quasi-reductive. This quotient is denoted by H qred ; it is the maximal quasi-reductive quotient of H. The k-unipotent radical of H qred is Ru, k (H)/Rus,k (H); this quotient is a kwound unipotent group (that is, there is no non-trivial homomorphism of Ga into this group). Every toral action on such a group is trivial (see [CGP15, B.2 and Proposition B.4.4]). If H is quasi-reductive, then its k-unipotent radical Ru, k (H) is k-wound, and hence every k-torus of H commutes with it. Since Rus, k (H) is k-split, the Galois cohomology H1 (k, Rus,k (H)) is trivial, and hence the natural homomorphism H(k) → H qred (k) is surjective. If k is perfect, then for any H as in this definition, Rus, k (H) = Ru (H), hence the maximal quasi-reductive quotient of H is reductive, so it is the maximal reductive quotient of H. Proposition 2.11.17 ([CGP15, Proposition C.2.24]) Let G be a connected quasi-reductive k-group, S ⊂ G a maximal k-split torus, a ∈ Φ(S, G), and U±a be the root groups associated to ±a. Given u ∈ Ua (k)∗ = Ua (k) − {1}, there exist unique u , u  ∈ U−a (k) such that the element m(u) = u uu  ∈ G(k) normalizes S. The elements u , u  are different from 1, and the element m(u) acts as the reflection along the root a. If a is non-multipliable, u  = u  = m(u)−1 · u · m(u) and m(u)2 ∈ S(k). Remark 2.11.18 We refer the reader to the proofs of Lemmas 3.1.3 and 3.2.8 for explicit formulas for u , u , and m(u) for the groups SL2 and SU3 . The elements u  and u  are not equal to each other in general if a is multipliable (cf. Lemma 3.2.8). The uniqueness of u  and u  has a number of consequences. First, m(u) = m(u ) = m(u ): this follows from m(u) = (m(u)u  m(u)−1 )u u = uu (m(u)−1 u  m(u)), and the fact that m(u)u  m(u)−1 and m(u)−1 u  m(u) belong to Ua (k). In fact, these identities give expressions for the elements (u ) and (u ) of Ua (k) associated to u , as well as the elements (u ) and (u ) of Ua (k) associated to u . Second mu−1 = m(u)−1 . Third, if 2a is also a root and u ∈ U2a (k)∗ ⊂ Ua (k)∗ , then u , u  ∈ U−2a (k).

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137

It is clear that m(u) lies in G a (k), where G a is the connected quasi-semisimple subgroup of G generated by U±a .

P AR T T WO BRUHAT–TITS THEORY

3 Examples: Quasi-split Simple Groups of Rank 1

In this chapter we will discuss the two most basic examples in Bruhat–Tits theory: those of the split group SL2 and the quasi-split group SU3 . These examples will provide us with basic intuition for the theory without requiring much notation. The examples of higher-dimensional classical groups, such as SLn , Sp2n , or SUn , will be discussed in Chapter 15. Let k be a field equipped with a surjective non-archimedean discrete valuation ω : k × → Z, o ⊂ k the ring of integers, m ⊂ o the maximal ideal, and f = o/m the residue field, which is assumed perfect.

3.1 The Example of SL2 We begin by considering the simplest example, that of the group G(k) = SL2 (k) of 2 × 2 matrices with entries in k and determinant 1. This group has two conjugacy classes of maximal bounded subgroups, with representatives given by %" #& o o P0 := o o

%" and

P1/2 :=

o m

m−1 o

#& .

Here boundedness means that the valuations of the four matrix entries are bounded below. If the Bruhat–Tits building is supposed to parameterize the maximal bounded subgroups, then as a first approximation we could take it to be the union of these two conjugacy classes. Indeed, it will turn out that this is the set of vertices (simplices of minimal dimension) in the actual building. Note that P0 and P1/2 are conjugate under PGL2 (k). 141

142

Examples: Quasi-split Simple Groups of Rank 1

(a) The Standard Apartment To give a more systematic definition of the building, we first describe a piece of it, called the standard apartment. Let T be the standard maximal torus of G, that is, the subgroup of diagonal matrices and N be its normalizer in G = SL2 . We have the standard positive root a ∈ X∗ (T) and its coroot a∨ ∈ X∗ (T) defined by " " # # x 0 x 0 2 ∨ a : Gm → T, x → → x , . a : T → Gm, 0 x −1 0 x −1 The morphism a∨ is an isomorphism and its inverse ωa ∈ X∗ (T) is the fundamental weight. We use these isomorphisms to identify X∗ (T) = Z. There is a unique maximal bounded subgroup in T(k), called . It comes equipped with a descending filtration T(k)n indexed by non-negative integers n. These are defined as T(k)0 = a∨ (o× ),

T(k)n = a∨ (1 + m n ),

n > 0.

The root system of T is the set Φ = {a, −a} ⊂ X∗ (T). We have the 1-parameter subgroups ua, u−a : Ga → G given by " " # # 1 u 1 0 ua (u) = , u−a (v) = . 0 1 v 1 Each root is a linear form on the real line A := X∗ (T) ⊗Z R = R, which we call the standard apartment. To any x ∈ A we associate the bounded open subgroup Px of G(k) generated by T(k)0 and ub (m−  b,x  ) for b ∈ {a, −a}. Here we have used the following notation. Notation 3.1.1 For x ∈ R we denote by x the largest integer less than or equal to x, and by x the smallest integer greater than or equal to x. The reader can check that for x = 0 and x = 1/2 we recover the bounded open subgroups P0 and P1/2 introduced earlier. For 0 < x < 1/2 we obtain a new bounded open subgroup, which turns out to be the intersection of P0 and P1/2 and is given by " # o o . m o It is called an Iwahori subgroup. In contrast to P0 and P1/2 , it is clearly not a maximal bounded subgroup.

3.1 The Example of SL2

143

(b) The Affine Roots We have thus obtained a partial parameterization of the bounded open subgroups of G(k) by the real line A. It would be useful to record those points x for which the corresponding bounded open subgroup Px is maximal. These are precisely the points for which a, x ∈ Z. One way to record them is to introduce the concept of affine roots, which is the set of affine functionals on A given by Ψ = {b + j | b ∈ Φ, j ∈ Z}. Then x ∈ A corresponds to a maximal bounded open subgroup if and only if some affine root takes the value 0 at x. The set of affine roots endows the line A with a simplicial structure – the 0-dimensional simplices (also called vertices) are the zeros of the affine roots, that is, the set 12 Z ⊂ R, and the 1-dimensional simplices (also called edges) are the intervals between two consecutive 0-dimensional simplices. The group Px depends only on the simplex F containing x and we may thus denote it by PF . To each affine root ψ = b + j we may introduce the open bounded subgroup Uψ of Ub (k) defined by ub (m j ). Then PF is the subgroup of G(k) generated by T(k)0 and Uψ for all affine roots ψ with ψ(F)  0. Note that we now have two systems for indexing the filtration on Ub (k) obtained by transporting the natural filtration of k via the isomorphism ub : k → Ub (k). The first indexing system is via the points of A, each point x ∈ A corresponding to ub (ω−1 ([− b, x , ∞])). The second indexing system is via affine roots. The two systems are compatible in the following sense. Given a point x ∈ A and a root b there is a unique affine functional that vanishes at x and has derivative b. This functional may not be an affine root, but there is a smallest affine root ψ greater than or equal to it. Then Uψ = ub (ω−1 ([− b, x , ∞])). While the system using points of A is more immediate, the system using affine roots will eventually turn out to be more flexible.

(c) The affine Weyl group Next we notice that, while for two distinct simplices F1 and F2 the groups PF1 and PF2 are distinct, they may happen to be G(k)-conjugate. This behavior is captured by the action of the affine Weyl group. First, the group Z = X∗ (T) acts on A by translations. This action preserves the simplicial structure, and PF+1 is conjugate to PF via the diagonal matrix a∨ (π −1 ), where π ∈ k is any uniformizer. Second, the Weyl group of the root system Φ, which is Z/2Z, acts on A = R by inversion, and P−F is conjugate to PF by the matrix " w0 =

# 0 1 . −1 0

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Examples: Quasi-split Simple Groups of Rank 1

These two actions together make up an action of the group Z  Z/2Z. This group is identified with the subgroup of transformations of A generated by reflections along the zeros of affine roots. There is another way to realize the affine Weyl group, which ties in with the way the various bounded open subgroups are conjugate to each other. Namely, we apply ⊗Z X∗ (T) to the exact sequence ω

1 → o× → k × → Z → 0 to obtain a homomorphism ω : T(k) → X∗ (T) and introduce an action of T(k) on A by t · x = x − ω(t). This action extends to an action of NG (T)(k) by letting w0 act as inversion. In this way we obtain an action of N(k) on A by simplicial automorphisms that has the property PnF = nPF n−1 for n ∈ N(k). The kernel of this action is T(k)0 , and N(k)/T(k)0 is canonically isomorphic to Z  Z/2Z. The following statement can be proved easily and is left as an exercise. Fact 3.1.2

Given x ∈ A, the stabilizer of x in N(k) is N(k) ∩ Px .

The two realizations of the affine Weyl group, as the group generated by reflections along affine roots, and as the quotient N(k)/T(k)0 , can be called the abstract and concrete affine Weyl group. They can be related more precisely as follows. For any u ∈ Ua (k) there exist unique u , u  ∈ U−a (k) such that m(u) := u uu  normalizes T.  For an affine functional ψ, let Uψ+ denote Uψ over all affine functionals ψ  > ψ. Lemma 3.1.3 Let u ∈ Uψ − Uψ+ . (1) u , u  ∈ U−ψ − U−ψ+ . (2) The element m(u) realizes the affine reflection along the affine root ψ. (3) U−ψ ⊂ U−ψ+ ∪ Uψ · m(u) · T(k)0 · Uψ . The set {m(u) | u ∈ Ua (k)} generates N(k). Proof We consider the case where ψ has derivative a. The case of derivative −a is analogous. Write ψ = a+ j. Then −ψ = −a− j. We have u = ua (x) for some x ∈ m j − m j+1 . The elements u , u  are determined by u  = u  = u−a (−x −1 ), which one sees via the matrix computation " # 0 x   = a∨ (x)w0 . u uu = −x −1 0 This shows that u , u  ∈ U−ψ − U−ψ+ and that m(u) = a∨ (x)w0 acts on A by multiplication by −1 followed by translation by − ja∨ . Thus it realizes the

3.1 The Example of SL2

145

reflection that fixes the point − ja∨ /2 ∈ A, which is exactly the point where ψ = a + j vanishes. Let v ∈ U−ψ − U−ψ+ . Then mv ∈ m(u) · T(k)0 . There are v  = v  ∈ Uψ − Uψ+ with mv = v  vv , hence v = (v )−1 mv (v )−1 ∈ Uψ · m(u) · T(k)0 · Uψ . The group generated by the set {m(u) | u ∈ Ua (k)} contains both m1 = w0 and m(x)m1−1 = a∨ (x), and therefore equals N(k). 

(d) The Building Let us summarize what we have done so far. We have introduced on the real line A the structure of a simplicial complex. To each simplex F we have assigned a bounded open subgroup PF of G(k). We have moreover introduced an action of N(k)/T(k)0 on A by simplicial automorphisms, having the property PnF = nPF n−1 . The maximal bounded subgroups among the PF do not exhaust all maximal bounded subgroups of G(k), but every maximal bounded subgroup of G(k) is G(k)-conjugate to one of the PF . Moreover, two PF are G(k)-conjugate if and only if they are N(k)-conjugate, and this is detected by the action of N(k)/T(k)0 on A. More precisely, any two 1-dimensional simplices of A are in the same N(k)-orbit, so the corresponding groups PF are N(k)-conjugate. There are precisely two orbits of 0-dimensional simplices, and any two adjacent simplices are representatives for these orbits. We are now ready to construct the building of G(k). It will be the solution to the following problem. We consider a 1-dimensional simplex F0 in A together with its two endpoints x0 and y0 and the bounded open subgroups Px0 and Py0 . We would like to have a simplicial complex, endowed with a G(k)-action by simplicial automorphisms, which contains F0 , and such that the stabilizers of x0 and y0 for the G(k)-action are precisely Px0 and Py0 . This space is constructed explicitly by taking the set G(k)×A and dividing out by the equivalence relation (g, x) ∼ (h, y) ⇔ ∃n ∈ N(k) : y = nx, g −1 hn ∈ Px . The resulting set is the Bruhat–Tits building B for G(k). It has an action of G(k) by g(h, y) = (gh, y). It is clear that for every z ∈ B there exists g ∈ G(k) such that gz ∈ A. Fact 3.1.2 allows us to conclude that x → (1, x) is an injection A → B and that the stabilizer of x in G(k) is Px for all x ∈ A. The subset A ⊂ B is called the standard apartment. More generally, an apartment of B is a subset of the form gA for any g ∈ G(k). The space B is a simplicial complex of dimension 1, in fact a tree. The underlying abstract simplicial complex (that is, the set of vertices and their incidence) is the set of maximal bounded subgroups with two such forming an

146

Examples: Quasi-split Simple Groups of Rank 1

edge if and only if their intersection contains (equivalently, equals) an Iwahori subgroup, which by definition is any G(k)-conjugate of I = P0 ∩ P1/2 . A maximal bounded subgroup is also called a maximal parahoric subgroup. We warn the reader that for a general reductive group the notions of maximal parahoric subgroups and maximal bounded subgroups need not coincide (cf. Example 4.2.16); they do if the group is semi-simple and simply connected. When the residue field f is finite, the tree B is locally finite, that is, there are finitely many edges emanating from each vertex. Thus, it is a special case of Example 1.5.10. The number of such edges is #P1 (f) = #f + 1. The occurrence of P1 is not coincidental: this is the variety of Borel subgroups of SL2 , whose relation to the Bruhat–Tits building will be explained in Theorem 8.4.19.

(e) The Moy–Prasad Filtration Subgroups We can use the same construction that defined Px for any x ∈ A to define a descending filtration Px,r for r > 0. We could have done this at the same time as defining Px , but we wanted to emphasize that Px is all that is needed for the definition of the building. We let Px,r be the subgroup of G(k) generated by the three subgroups T(k) r  and ub (m r− b,x  ) for b ∈ {a, −a}. As is  customary, we set Px,r+ = s>r Px, s . The latter can also be given explicitly as the subgroup generated by T(k) r +1 and ub (m r− b,x +1 ). We also set Px,0 = Px . Just like Px,0 , the subgroup Px,0+ depends only on the facet containing x. This is no longer true for general r > 0. For any x = (g, y) ∈ B we define Px,r = gPy,r g −1 .

(f) The Apartment as an Affine Space The discussion so far has crucially used the natural coordinates on the group G(k) = SL2 (k), that is, the fact that we can speak of “matrix entries” of an element of G(k). This will not be the case for a general reductive group. To see what this might entail, consider now instead of SL2 (k) the group G(k) = SL(V), where V is a 2-dimensional k-vector space. If we choose a basis ψ = (v1, v2 ) of V, then we obtain an isomorphism V → k 2 and an isomorphism SL(V) → SL2 (k) and the previous discussion applies. But the resulting objects – apartment, building, and bounded subgroups – are not intrinsic to G, since they also depend on the basis. Let us see how they change if we change the basis. The basic objects we used were the homomorphism a∨ : Gm → G whose image was the maximal torus T, and the homomorphisms ua, u−a : Ga → G whose images were the two root subgroups. The simplest way to change the basis is to replace ψ = (v1, v2 ) by ψ  = (x1 v1, x2 v2 ) with some x1, x2 ∈ k × .

3.2 The Example of SU3

147

Then a∨ is unchanged, but ua is multiplied by x1 /x2 , while u−a is multiplied by x2 /x1 . For x ∈ A = X∗ (T) ⊗Z R, the bounded subgroup Px constructed with respect to the basis ψ  will be equal to Px−ω(x1 /x2 )/2 constructed with respect to the basis ψ. In other words, while the collection of open bounded subgroups {Px | x ∈ A} remains unchanged, their indexing shifts. We would like the apartment A to be an object associated to the maximal torus T with the property that a point x ∈ A provides an open bounded subgroup Px of G(k), without the need for additional choices. As we have just seen, the definition A = R ⊗Z X∗ (T) does not have this property, because the group Px depends in addition on the choice of basis ψ. In the situation at hand we can resolve the problem as follows. Fix a basis ψ and consider the set {Px | x ∈ 12 Z}. This set receives a simply-transitive action of the abelian group 1 1 2 Z. This 2 Z-torsor is unchanged if we rescale the basis vectors, or switch their order. Therefore it depends only on the torus T. We define A to be the R-torsor obtained from this 21 Z-torsor by extending scalars. Thus A is a 1-dimensional affine space, but it has no natural identification with R. We review affine spaces in §1.2. A generalization of this approach will work for any quasi-split connected reductive group.

3.2 The Example of SU3 We shall now study the simplest quasi-split but non-split reductive group, namely SU3 . A number of new phenomena occur when we go beyond the split case. This example will serve both to introduce them and to supply the basic calculations that will be needed in the case of general quasi-split groups. The structure of the group was reviewed in §2.7. As in the case of SL2 , it turns out that there are two SU3 (k)-conjugacy classes of maximal bounded (parahoric) subgroups, namely ⎡o ⎢ ⎢o ⎢ ⎢o ⎣ 

o o o

o ⎤⎥ o ⎥⎥ o ⎥⎦

and

⎡ o ⎢ ⎢ o ⎢ ⎢m−1 ⎣ 

m o o

m ⎤⎥ m ⎥⎥ . o ⎥⎦

Unlike the case of SL2 , however, these two are not conjugate even under the adjoint group PU3 (k).

(a) The Filtration of the Maximal Torus We endow the group T(k) with a decreasing filtration. The isomorphism Z2 → Z30,

(x, y) → (x, y − x, −y)

148

Examples: Quasi-split Simple Groups of Rank 1

identifies the action of σ on the right with the action σ(x, y) = (y, x) on the left and therefore corresponds to the isomorphism R/k Gm → T, which gives the identification T(k) = . In other words, ⎫ ⎧ ⎡x ⎤  ⎪ ⎪ ⎬ ⎪ ⎨⎢ ⎪ ⎥ ⎥  x ∈ × . T(k) = ⎢⎢ x/x ⎥ ⎪ ⎪  ⎪ ⎪⎢ x −1 ⎥⎦ ⎭ ⎩⎣ The group T(k) inherits the filtration  × ⊃ o× ⊃ 1 + m ⊃ 1 + m2 ⊃ · · ·, which we write as T(k)0 = o×,

T(k)r = {x ∈ T(k)0 | ω(x − 1)  r }, r > 0.

The jumps of this filtration are the elements of e1 Z0 . We have T(k)0 /T(k)0+ = f× = T(f), where T = Rf /f Gm . The torus T is the special fiber of the finite type Néron model of T that will be discussed in Chapter 8.

(b) The Filtrations of the Root Subgroups We endow the root groups Ua (k) and U2a (k) with decreasing filtrations. The case of U2a (k)   0 is simpler. We can restrict the natural filtration of , given by powers of the prime ideal m , to  0 . The rth filtration subgroup would then be {ua (0, v) ∈ U2a (k) | ω(v)  r }. This would give a perfectly good theory, with the slight blemish that the jumps of this filtration will depend not only on the ramification index e of /k, but also on the jump s of the lower ramification filtration of Gal(/k). Indeed, Lemma 2.8.1 states that when /k is ramified the set of jumps of this filtration is Z + 12 + μ. Bookkeeping becomes easier if we absorb μ into the definition of the filtration and set 0 (k) | ω(v)  r + μ}. U2a (k)r = {ua (0, v) ∈ U/k

(3.2.1)

We now turn to Ua (k) = {ua (u, v) | u, v ∈ , uu = v + v}. This is a 3-dimensional k-subvariety of  2  k 4 . If we restrict the natural filtration of  ×  to U/k (k) we obtain a filtration indexed by two parameters. More precisely the filtration piece indexed by the pair of real numbers (r1, r2 ) is {ua (u, v) ∈ Ua (k) | ω(u)  r1, ω(v)  r2 }. This two-parameter filtration is certainly useful and we will use it in the finer analysis of open bounded subgroups. But for our immediate purposes we would

3.2 The Example of SU3

149

like to work with a one-parameter filtration. We can obtain such a filtration by composing with any function R → R × R. The relation uu = v + v implies 2ω(u) = ω(v + v)  ω(v). This suggests that a good choice of a function R → R × R would be r → (r, 2r). We are led to define the filtration on Ua (k) so that the piece indexed by the real number r is {ua (u, v) ∈ Ua (k) | ω(u)  r, ω(v)  2r }, which is then equal to {ua (u, v) ∈ Ua (k) | ω(v)/2  r }. As in the case of U2a we shift this definition by μ to ease bookkeeping, and set Ua (k)r := {ua (u, v) ∈ Ua (k) | ω(v)/2  r + μ/2}.

(3.2.2)

Fact 3.2.1 For each r ∈ R, the subsets Ua (k)r ⊂ Ua (k) and U2a (k)r ⊂ U2a (k) are subgroups. Proof



This is a direct computation.

The filtration of Ua (k) induces a filtration on the quotient Ua (k)/U2a (k). The rth filtration piece is the image of Ua (k)r in this quotient, namely Ua (k)r · U2a (k)/U2a (k). A first indication that the built-in shift by μ is helpful is the following observation. Lemma 3.2.2 The isomorphism Ua (k)/U2a (k) →  of Fact 2.7.1 identifies the filtrations on both sides. Proof Fix λ ∈  tr=1,max . Given u ∈ , Fact 2.7.1 describes the preimage of u in Ua (k)/U2a (k) as the set {ua (u, λuu + v ) | v  ∈  0 }. This set is in the rth filtration piece for Ua (k)/U2a (k) if and only if some element of it lies in Ua (k)r , which is equivalent to ω(λuu + v )/2  r + μ/2. This in turn is equivalent to  ω(λuu)/2  r + μ/2; that is, ω(u)  r. Remark 3.2.3 Let X be a set endowed with a decreasing separated filtration {X } indexed by r ∈ R, which we recall means Xr ⊂ Xs when r > s and

r r ∈R Xr = {1}. The information contained in the filtration {Xr } can be encoded in a function ϕ : X → R ∪ {∞} defined by ϕ(x) = sup{r ∈ R ∪ {∞} | x ∈ Xr }. The filtration subset Xr can then be recovered as {x ∈ X | ϕ(x)  r }. The set of values of ϕ is exactly the set of jumps of the filtration Xr , that is, the set of r  for which Xr  Xr+ , where Xr+ = s>r Xs . Let ϕa , ϕ2a , and ϕa/2a , be the functions associated by Remark 3.2.3 to the filtrations on Ua (k), U2a (k), and Ua (k)/U2a (k). These are then given by ϕ2a (ua (0, v)) = ω(v) − μ,

ϕa (ua (u, v)) =

1 (ω(v) − μ) 2

(3.2.3)

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Examples: Quasi-split Simple Groups of Rank 1

and ϕa/2a (ua (u, v) · U2a (k)) = sup{ϕa (ua (u, v )) | ua (u, v ) ∈ ua (u, v) · U2a (k)} = ω(u),

(3.2.4)

the last equality by Lemma 3.2.2. The corresponding jump sets are Ja = ϕa (Ua (k) − {1}),

J2a = ϕ2a (U2a (k) − {1}),

(3.2.5)

and Ja/2a = ϕa/2a (Ua (k)/U2a (k) − {1}).

(3.2.6)

For the sake of uniformity we set J2a/4a = J2a . From Lemma 2.8.1 and Lemma 3.2.2 we obtain the following result. Corollary 3.2.4 (1) If /k is unramified, then Ja = 12 Z, Ja/2a = Z, J2a = Z. (2) If /k is ramified, then Ja = 14 Z, Ja/2a = 12 Z, J2a = Z + 12 . By definition, for any relative root b the quotient Ub (k)r /Ub (k)r+ is nontrivial precisely when r ∈ Jb . Let us examine these quotients more precisely. Lemma 3.2.5 (1) Let r ∈ J2a . Then U2a (k)r /U2a (k)r+ is non-canonically isomorphic to (i) f0 if /k is unramified; (ii) f = f if /k is ramified. (2) Let r ∈ 12 J2a ⊂ Ja and r  Ja/2a . Then the inclusion U2a (k)2r /U2a (k)2r+ → Ua (k)r /Ua (k)r+ is an isomorphism, so Ua (k)r /Ua (k)r+ is a 1-dimensional f-vector space. (3) Let r ∈ Ja/2a and r  12 J2a ⊂ Ja . Then /k is ramified and the projection Ua (k)r /Ua (k)r+ → [Ua (k)/U2a (k)]r /[Ua (k)/U2a (k)]r+ is an isomorphism, so Ua (k)r /Ua (k)r+ is a 1-dimensional f-vector space. (4) Let r ∈ Ja/2a ∩ 12 J2a . Then /k is unramified, Ua (k)r /Ua (k)r+ is noncanonically isomorphic to Ua (k)0 /Ua (k)0+ , and the reduction map Ua (k)0 /Ua (k)0+ → Uf /f (f) is an isomorphism.

3.2 The Example of SU3

151

0 which is an o-module, in fact an Proof We have U2a (k)r /U2a (k)r+ = r0 /r+ f-vector space, contained in the f-vector space lr /lr+  f , the last isomorphism 0 is depending on a choice of uniformizer of . If /k is unramified then r0 /r+ 0 contained in the 1-dimensional f-subspace f and must therefore be equal to it. If /k is ramified, then f = f is already 1-dimensional. We now consider Ua (k)r /Ua (k)r+ . We use the exact sequence

1 → U2a (k) → Ua (k) → Ua (k)/U2a (k) → 1, which, by the definitions of the various filtrations, gives 1 → U2a (k)2r → Ua (k)r → [Ua (k)/U2a (k)]r → 1 and leads to 1→

Ua (k)r [Ua (k)/U2a (k)]r U2a (k)2r → → → 1, U2a (k)2r+ Ua (k)r+ [Ua (k)/U2a (k)]r+

where we have used the fraction notation to denote right coset spaces. If r  Ja/2a then the third term vanishes and the first map is an isomorphism. If r  12 Ja then the first term vanishes and the second map is an isomorphism. By Fact 2.7.1 and Lemma 3.2.2 we have the filtration-preserving isomorphism Ua (k)/U2a (k) → , therefore [Ua (k)/U2a ]r /[Ua (k)/U2a ]r+  lr /lr+  f . By Corollary 3.2.4 the extension /k is ramified and hence f = f. If r ∈ Ja/2a ∩ 21 J2a then by Corollary 3.2.4 the extension /k is unramified and hence r ∈ Z. If π ∈ k is a uniformizer, the map (u, v) → (π·u, π 2 ·v) is an automorphism of Ua (k) identifying Ua (k)r with Ua (k)r+1 . Now (u, v) ∈ U/k (k)0 implies u, v ∈ o and the reduction map o → f induces a group homomorphism U/k (k)0 → Uf /f (f) whose kernel is U/k (k)0+ by definition of the filtration on U/k (k). The surjectivity of this group homomorphism follows from the 0 (k) =  0 ∩ o → f 0 = surjectivity of the reduction homomorphisms U/k 0   0 0 (k)] = l → f = Uf /f (f) and of the reduction homomorphism [U/k (k)/U/k 0 0   [Uf /f (f)/Uf0 /f (f)]. We have thus far discussed only the positive roots a and 2a. For the negative roots the discussion is analogous, but we must use the opposite shift by μ. That is, we have ϕ−2a (u−a (0, v)) = ω(v) + μ, The jump sets remain unchanged.

ϕ−a (u−a (u, v)) =

1 (ω(v) + μ). 2

(3.2.7)

152

Examples: Quasi-split Simple Groups of Rank 1

(c) The Standard Apartment and the Affine Roots We define the standard apartment of SU3 to be A = R ⊗Z X∗ (S) = R ⊗Z X∗ (T)Γ = [R ⊗Z X∗ (T)]Γ . The isomorphism X∗ (S) = Z gives the isomorphism A = R. For every v ∈ A we have the filtration subgroups Ua (k) a,v and U2a (k) 2a,v defined in the previous subsection. We declare v to be special, if at least one of these filtrations jumps at v, by which we mean that at least one of the inequalities Ua (k) a,v  Ua (k) a,v + and U2a (k) 2a,v  U2a (k) 2a,v + holds. By definition, the first filtration jumps if and only if a, v ∈ Ja , while the second jumps if and only if 2a, v ∈ J2a . But Corollary 3.2.4 shows that 2a, v ∈ J2a implies a, v ∈ Ja . Therefore v is special if and only if a, v ∈ Ja . Under the isomorphism A = R the linear form a : A → R becomes the identity. With this identification we see that v ∈ R is special if and only if v ∈ Ja . For every v ∈ A we define the bounded open subgroup Pv ⊂ G(k) to be the group generated by T(k)0 and Ub (k) −b,v for all b ∈ {a, −a, 2a, −2a}, and P+v ⊂ G(k) the group generated by T(k)0+ and Ub (k) −b,v + for all b ∈ {a, −a, 2a, −2a}. If v is not special, then Pv /P+v = T(k)0 /T(k)0+ = T(f). If v is special then Pv /P+v is the group of f-points of a connected reductive algebraic group Gv defined over f with maximal torus T. This group is the reductive quotient of the special fiber of a certain o-model of G constructed in Chapter 8. For now, we will accept its existence and use it as a motivation. The group Gv has f-rank 1 and root subgroups Ua (k) −a,v /Ua (k) −a,v + and U−a (k) a,v /U−a (k) a,v + . From Lemma 3.2.5 we obtain the following information. If a, v ∈ 12 J2a − Ja/2a , then Ua (k) −a,v /Ua (k) −a,v + is a 1dimensional f-vector space on which T acts by the character 2a, thus the root system of Gv with respect to T is {2a, −2a} and therefore the derived subgroup of Gv is SL2 /f. If a, v ∈ Ja/2a − 12 J2a then Ua (k) −a,v /Ua (k) −a,v + is a 1-dimensional f-vector space on which T acts by the character a, so the root system of Gv is {a, −a} and therefore the derived subgroup of Gv is PGL2 /f. If v ∈ Ja/2a ∩ 12 J2a then Ua (k) −a,v /Ua (k) −a,v + is a 3-dimensional unipotent group over f, so the root system of Gv is {a, −a, 2a, −2a} and the derived subgroup of Gv is SU3 for the quadratic extension f /f (in this case the extension /k is necessarily unramified). This discussion can be made systematic by introducing the set of affine roots. An affine linear function ψ : A → R is called an affine root if (1) its derivative b is a root; thus b ∈ {a, −a, 2a, −2a} and ψ = b + r with r ∈ R;

3.2 The Example of SU3

153

(2) r is a jump of the filtration of Ub (k) if b ∈ {2a, −2a}, and r is a jump of the filtration of Ub (k)/U2b (k) if b ∈ {a, −a}; in other words r ∈ Jb/2b . This definition is made precisely so that for v ∈ A the root system of Gv with respect to T is the set of derivatives of those affine roots that vanish at v. Let Ψ be the set of affine roots. From Corollary 3.2.4 we obtain the following description of Ψ. We have added the description of the groups Gv without proof, and refer to Theorem 8.4.10 for the proof of that description. Corollary 3.2.6

(1) If /k is unramified the affine root system is {a + Z, −a + Z, 2a + Z, −2a + Z}.

We have G0  SU3 and (G1/2 )der  SL2 . (2) If /k is ramified the affine root system is ' a + 12 Z, −a + 12 Z, 2a + 12 + Z, −2a +

1 2

( +Z .

We have G0  PGL2 and G1/4  SL2 . Remark 3.2.7 Recall from Definition 1.3.39 the notions of a “special” and 1 Z, and “extra special” point x ∈ A. In this case, the set of special points is 2e 1 the set of extra special points is e Z.

(d) The Affine Weyl Group We have again the abstract and concrete realizations of the affine Weyl group. The abstract realization is the subgroup of affine automorphisms of A generated by reflections along the affine roots. The concrete realization is the group N(k)/T(k)0 . The latter is an extension 1 → T(k)/T(k)0 → N(k)/T(k)0 → W(T)(k) → 1. The element ⎡ ⎢ w0 = ⎢⎢ −1 ⎢1 ⎣

1⎤⎥ ⎥ ⎥ ⎥ ⎦

of N(k) has order 2. Its image in W(T)(k) generates that group. Therefore the above extension is split. We let T(k) act on A by t·x = x−ω(t), where ω(t) ∈ A is the element defined by ω(t), a = ω(a(t)) (in terms of the isomorphism A = R induced by X∗ (S) = Z the element ω(t) is identified with the valuation of the upper left corner of the matrix t). This action factors through T(k)/T(k)0 . We let w0 act by w0 · a = −a + δμ , where δμ ∈ A is the element defined by δμ , a = μ (again in terms of A = R we have δμ = μ). The shift by δμ reflects the shift by

154

Examples: Quasi-split Simple Groups of Rank 1

μ of the filtration of Ua (k). In this way we obtain an action of N(k)/T(k)0 on A. It is clear that this action identifies N(k)/T(k)0 with a subgroup of the group of affine automorphisms of A. The following fact implies that this subgroup is none other than the abstract affine Weyl group. Lemma 3.2.8 Let ψ : A → R be an affine functional and let x ∈ Uψ − Uψ+ . (1) There exist x , x  ∈ U−ψ − U−ψ+ such that the element m(x) = x  x x  lies in N(k) and realizes the affine reflection along Hψ . (2) We have U−ψ ⊂ U−ψ+ ∪ Uψ · m(x) · T(k)0 · Uψ . The set {m(x) | x ∈ Ua (k)} generates the quotient N(k)/T(k)0 . Proof We consider the cases where ψ has derivative either a or 2a. The cases where the derivative is −a or −2a are analogous. We denote by (u, v) and (u , v ) the elements ⎡ 1 0 0⎤ ⎡1 u v ⎤ ⎢  ⎢ ⎥ ⎥ ⎢u 1 0⎥ ⎢0 1 u ⎥ and ⎢ ⎢ ⎥ ⎥ ⎢ v  u  1⎥ ⎢0 0 1 ⎥ ⎣ ⎣ ⎦ ⎦ of Ua (k) and U−a (k), respectively. Consider first the case ψ = 2a+ j with j ∈ J2a . We have x = (0, v) with v ∈  0 and ω(v) = j + μ. Note that then −v −1 ∈  0 . We have x  = x  = (0, −v −1 ). Then ⎡ ⎢ m(x) = x x x = ⎢⎢ ⎢−v −1 ⎣ 



1

v ⎤⎥ ⎥ = t v · w0 , ⎥ ⎥ ⎦

⎡v ⎢ tv = ⎢⎢ ⎢ ⎣

−1

⎤ ⎥ ⎥. ⎥ −1 −v ⎥⎦

We have a, ω(tv ) = ω(v) = j + μ. Thus for y ∈ A we have a, m(x)y = − a, y + a, δμ − a, ω(tv ) = − a, y − j. The condition y ∈ Hψ is equivalent to 2a, y = − j and we see that m(x) preserves Hψ . Since its image in W(T)(k) is the non-trivial element we conclude that m(x) realizes the affine reflection along Hψ . Consider now the case ψ = a + j with j ∈ Ja . Then x = (u, v) with ω(v) = 2 j + μ. Recall the relation v + v = uu. The elements x  and x  of U−a (k) are given by x  = (−u/v, 1/v) and x  = (−u/v, 1/v). One then computes that ⎡ ⎢ m(x) = x  x x  = ⎢⎢ ⎢1/v ⎣

−v/v

v ⎤⎥ ⎥ = t v · w0 , ⎥ ⎥ ⎦

⎡v ⎢ tv = ⎢⎢ ⎢ ⎣

v/v

⎤ ⎥ ⎥. ⎥ 1/v ⎥⎦

We have a, ω(tv ) = ω(a(tv )) = ω(v) = 2 j + μ. Thus for y ∈ A we have a, m(x) · y = − a, y + a, δμ − a, ω(tv ) = − a, y − 2 j. We have y ∈ Hψ if and only if a, y = − j and we see that m(x) fixes Hψ . Since its image in

3.2 The Example of SU3

155

W(T)(k) is the non-trivial element we conclude that m(x) is the affine reflection along Hψ . In the cases the derivative of ψ is a or 2a, let y ∈ U−ψ − U−ψ+ . Then m(y) ∈ m(x) · T(k)0 . There are y , y  ∈ Uψ − Uψ+ with m(y) = y  yy , hence y = (y )−1 m(y)(y )−1 ∈ Uψ · m(x) · T(k)0 · Uψ . Finally, since we already know that for x  1 the element m(x) maps to the unique non-trivial element of N(k)/T(k), it is enough to show that every element of T(k)/T(k)0 is a product of elements of the form m(x). The map v → tv is an isomorphism  × → T(k) identifying o with T(k)0 . It is therefore enough to show that there are x1 = (u1, v1 ) and x2 = (u2, v2 ) such that v1 v2−1 is a uniformizer in  × . That this is always possible follows from Lemma 2.8.1 when /k is ramified. When /k is unramified we can take u1 = u2 = 0, v2 ∈  0 ∩ o× ,  and v1 = π · v2 with π ∈ k × a uniformizer. For any affine functional ψ = b + j : A → R we define Uψ to be the group Ub (k) j . The action of N(k) on A induces an action of N(k) on the set of affine functionals. Fact 3.2.9 For any affine functional ψ : A → R and any n ∈ N(k) we have nUψ n−1 = Unψ . Proof It is enough to check n = w0 and n = t ∈ T(k). In both cases the computation is immediate. 

(e) Unshifted Filtrations Remark 3.2.10 In order to ease bookkeeping we have chosen to incorporate the arithmetic quantity μ of (2.8.1), which in turn depends on the quantity s of (2.8.3), into the definition of the filtrations of Ua and U2a and hence of the functions ϕa and ϕ2a . This is of course not strictly necessary. We could have instead made the more straightforward definitions Ua (k)r = {ua (u, v) ∈ Ua (k) | ω(v)/2  r } and 0 (k) | ω(v)  r }, U2a (k)r = {ua (0, v) ∈ U/k

which would have resulted in the functions 1 ω(v). 2 The price we would then pay is that the shift by μ would show up at various other places, most notably the jump sets Ja and Ja/2a , as well as the formulas ϕ2a (ua (0, v)) = ω(v),

ϕa (ua (u, v)) =

156

Examples: Quasi-split Simple Groups of Rank 1

for the affine root systems. For example, when /k is ramified the affine root system will have the form ( ' a + 12 Z, −a + 12 Z, 2a + 12 + Z, −2a + 12 + Z when 2 | s, and the form ( ' a + 12 Z + 14 , −a + 12 Z + 14 , 2a + Z, −2a + Z when 2  s. Of course these two affine root systems are isomorphic.

(f) The Building We summarize again what we have done so far. We recall that T is the diagonal torus of the quasi-split G = SU3 ; S is the maximal k-split torus of G contained in T. Then the centralizer Z of S in G equals T. Moreover, as we mentioned above, NG (T)(k) = N(k), where N is the normalizer of S in G. We have introduced on the real line A the structure of a simplicial complex. To each simplex F we have assigned a bounded open subgroup PF of G(k). We have moreover introduced an action of N(k)/Z(k)0 = N(k)/T(k)0 on A by simplicial automorphisms, having the property PnF = nPF n−1 . The building B for SU3 is constructed in the same way as that of SL2 , namely as the quotient of G(k) × A by the equivalence relation (g, x) ∼ (h, y) ⇔ ∃ n ∈ N(k) : y = nx, g −1 hn ∈ Px . Again, x → (1, x) is an injection A → B and the stabilizer of x in G(k) is Px for all x ∈ A. The subset A ⊂ B is called the standard apartment. More generally, an apartment of B is a subset of the form gA for any g ∈ G(k). The space B is a simplicial complex of dimension 1, in fact a tree (cf. [Ser03]). When the residue field f is finite, this tree is locally finite, that is, there are finitely many edges emanating from each vertex. The structure of the tree B depends on the extension /k. When /k is ramified, then B is a regular tree of degree #P1 (f) = #f + 1. In other words, the building of ramified SU3 is exactly the same as the building of SL2 , despite the two groups being rather different. On the other hand, when /k is unramified, then B is not a regular tree. Instead, given any two vertices joined by an edge, the number of edges emanating from one of them is #f + 1, while the number of edges emanating from the other is (#f)3 + 1. The vertices of the second type are the hyperspecial vertices, while the vertices of the first type are special, but not hyperspecial.

4 Overview and Summary of Bruhat–Tits Theory

Let k be a field given with a discrete valuation ω : k → Z ∪ {∞}. Let o = {x ∈ k | ω(x)  0} be the ring of integers, m = {x ∈ k | ω(x) > 0} the maximal ideal of o and f = o/m the residue field. We assume that o is Henselian and the residue field f is perfect. Let K be a maximal unramified extension of k. We denote again by ω the unique extension of the valuation of k to K. Let O be the ring of integers of K and M the maximal ideal of O. Then the residue O/M =: f of K is an algebraic closure of f. Let G be a connected reductive k-group.

4.1 Axiomatization of Bruhat–Tits Theory In this section we will introduce a list of axioms that encapsulate the essence of Bruhat–Tits theory. In Chapters 6, 7, 8, and 9, we will establish that these axioms do hold, first for quasi-split groups, and then for general reductive groups. The core axioms are 4.1.1, 4.1.4, and 4.1.9. We will see in §4.4 that they characterize the building up to unique isomorphism. The remaining axioms however are essential in most applications of Bruhat–Tits theory, as well as in the development of the theory itself (for example, they are used in deriving the theory for general reductive groups from the case of quasi-split reductive groups). Axiom 4.1.1 Existence of a building There exists a building B = B(G) in the sense of Definition 1.5.5, equipped with an action of G(k) by polysimplicial automorphisms. The apartments of this building are in bijective G(k)-equivariant correspondence with the maximal split tori of G. In particular, the stabilizer in G(k) of an apartment A = A(S) corresponding to a maximal k-split torus S is N(k), where N denotes the normalizer of S in G. A surjective homo157

158

Overview and Summary of Bruhat–Tits Theory

morphism G → G  with central kernel induces an equivariant isomorphism B(G) → B(G ). In particular, the buildings of G, Gad , Gder , and Gsc , are canonically identified, and the action of G(k) on its building factors through Gad (k). Recall the partial order “≺” on the set of non-empty subsets of B(G) introduced in Notation 1.3.34: given two non-empty subsets Ω and Ω of B(G), Ω ≺ Ω if the closure Ω  of Ω contains Ω. For facets F and F  of B(G), if F ≺ F , we say that F is a face of F . In a collection C of facets, a facet is maximal if it is not a proper face of any facet belonging to C , and a facet is minimal if no proper face of it belongs to C . In B(G), the maximal facets are called chambers, and the minimal facets are called vertices. Axiom 4.1.2 Bounded subgroups The stabilizer G(k)1x of a point x ∈ B(G) for the action of G(k)1 is a bounded open subgroup of G(k). Definition 4.1.3

Let F be a facet of B(G).

(1) The stabilizer G(k)0F of F in G(k)0 is called the parahoric subgroup of G(k) associated to F. (2) The minimal parahoric subgroups, that is, those associated to chambers, are called Iwahori subgroups. The reader should bear in mind that the group G(k)0x may be a proper subgroup of G(k)1x , cf. Example 4.2.16. Our definition of parahoric subgroup follows Haines–Rapoport [PR08, Appendix] and is different (yet equivalent) to that of Bruhat–Tits. It has the advantage of being elementary and independent of the construction of integral models. In the case of a non-archimedean local field k, there is an alternative characterization of Iwahori subgroups, cf. Proposition 13.5.2. Axiom 4.1.4 Affine structure on the apartment Let S be a maximal k-split torus of G. Let S  be the unique maximal subtorus of S that is contained in the derived subgroup Gder of G. Let V(S) = R⊗Z X∗ (S) and V(S ) = R⊗Z X∗ (S ). If AG is the maximal k-split torus in the center of G, then S  → S → S/AG is an isogeny, hence V(S ) → V(S) → V(S/AG ) is an isomorphism. This provides a canonical splitting V(S) → V(S ) of the inclusion V(S ) → V(S), and hence a canonical decomposition V(S) = V(S ) ⊕ V(AG ). Let Z and N be the centralizer and normalizer of S in G. The apartment A = A(S) of B(G) corresponding to S satisfies the following axioms. A 0 A is equipped with the structure of an affine space under V  = V(S ).

4.1 Axiomatization of Bruhat–Tits Theory

159

A 1 The action of N(k) on A is by affine transformations. Let f : N(k) → Aff(A) be the action map, where Aff(A) is the group of affine automorphisms of A. For n ∈ N(k), the derivative df (n) ∈ Aut(V ) is the automorphism of V  induced by the action of N(k) on X∗ (S ), i.e. the Weyl group action. In particular, the group Z(k) acts on A by translations, where the translation ν(z) ∈ V  for z ∈ Z(k) is the image under V(S) → V(S ) of −ω Z (z), where ω Z is the valuation homomorphism (2.6.4) for the group Z(k). In particular, the maximal bounded subgroup Z(k)b of Z(k), and the center ZG (k), act trivially. For convenience, we note here that a(ν(s)) = −ω(a(s)) for all a ∈ Φ(S) and s ∈ S(k). A 2 For any g ∈ G(k), the isomorphism g : A(S) → A(gSg −1 ) is affine and its derivative is the natural isomorphism g : V(S ) → V(gS  g −1 ). We remark that these three conditions uniquely determine the affine structure on A, see Proposition 4.4.3. In fact, we will give in that proposition a simple and direct construction of this affine space, which will however turn out to be too abstract to be useful. Remark 4.1.5 The minus sign in the formula ν(z) = −ω Z (z) in Axiom 4.1.4(A 1) might appear puzzling at first sight. Of course, it is just a convention, but the reason for this convention is the following. In the field k, elements close to zero have positive valuation, while elements far from zero, thus close to +∞, have negative valuation. In an affine space we would prefer to have the opposite convention, where elements close to +∞ result in translations of large (hence positive) magnitude. From now on we will not distinguish between the polysimplicial complex B(G) and its geometric realization for which each apartment has the affine space structure described in 4.1.4. Axiom 4.1.6 Affine root system Let S be a maximal k-split torus, A = A(S) the affine space of Axiom 4.1.4, and Φ = Φ(S, G) the relative root system. Given a ∈ Φ and u ∈ Ua (k)∗ the elements u , u  ∈ U−a (k)∗ and m(u) = u  · u · u  ∈ N(k) are as in Proposition 2.11.17. AS 1 The action of m(u) on A fixes a hyperplane Hu that is an affine subspace of A under the vector subspace ker(a). AS 2 Let ψau ∈ A∗ be the unique functional with derivative a that vanishes on Hu . Then Ψ  = {ψau | a ∈ Φ, u ∈ Ua (k)∗ } is an affine root system with derivative Φ.

160

Overview and Summary of Bruhat–Tits Theory

AS 3 For ψ ∈ A∗ with a = ψ ∈ Φ define Uψ = {u ∈ Ua (k)∗ | ψau  ψ} and Uψ+ = {u ∈ Ua (k)∗ | ψau > ψ}. Then Ψ = {ψ ∈ Ψ  | Uψ  Uψ+ · U2a (k)} is an affine root system with derivative Φ, where we set U2a = {1} if 2a  Φ. AS 4 For ψ ∈ Ψ , at least one of ψ, 2ψ belongs to Ψ. AS 5 The image of the action map f : N(k) → Aff(A) contains W(Ψ) and is contained in W(Ψ)ext . If G is simply connected, this image equals W(Ψ). If G is quasi-split and adjoint, this image equals W(Ψ)ext . AS 6 The polysimplicial structure on A endowed by the affine root hyperplanes of Ψ (equivalently Ψ ) as in Remark 1.3.16 coincides with the polysimplicial structure coming from B. Remark 4.1.7 Note that as ψ ∈ A∗ varies with a fixed derivative a ∈ Φ, Uψ forms a decreasing filtration of Ua (k). An affine functional ψ is a “jump” of this filtration; that is, Uψ+  Uψ , if and only if ψ = ψau for some u ∈ Ua (k)∗ . In other words, we recover Ψ  as {ψ ∈ A∗ | a = ψ ∈ Φ, Uψ+  Uψ }. If we use overline to denote the natural map Ua (k) → Ua (k)/U2a (k), we can write analogously Ψ = {ψ ∈ A∗ | a = ψ ∈ Φ, U ψ+  U ψ }. According to AS 4 the hyperplanes coming from Ψ are the same as those coming from Ψ , while by AS 2 these are the same as the hyperplanes Hu for a ∈ Φ and u ∈ Ua (k)∗ . In particular, the Weyl groups of Ψ and Ψ  are the same. The reflection along ψau is realized by m(u) according to AS 1. Since the action of N(k) by conjugation preserves the set {m(u) | a ∈ Φ, u ∈ Ua (k)∗ }, it preserves the set of hyperplanes {Hu | a ∈ Φ, u ∈ Ua (k)∗ }, hence the set Ψ , hence the set of groups {Uψ | ψ ∈ Ψ  }, hence also the subset Ψ ⊂ Ψ . In other words, the image of the action map f : N(k) → Aff(A) is contained in W(Ψ)ext and contains W(Ψ). By Remark 2.11.18 we have m(u) = m(u ) = m(u ). This   implies Hu = Hu = Hu , and hence ψau = −ψau = −ψau . Given a non-empty bounded subset Ω of an apartment A of B(G), let G(k)1Ω denote the subgroup of G(k)1 consisting of elements that fix Ω pointwise. Let G(k)0Ω denote the subgroup of G(k)0 consisting of elements that fix Ω pointwise. Axiom 4.1.8 Action of Uψ Let S ⊂ G be a maximal k-split torus, A ⊂ B the corresponding apartment, and ψ ∈ Ψ ⊂ A∗ as in Axiom 4.1.6. (1) The action of Uψ on B fixes the half-apartment Aψ0 . (2) Given Ω ⊂ A, the group G(k)0Ω is generated by Z(k)0 and Uψ for all ψ ∈ Ψ with ψ(Ω) ⊂ R0 . (3) For a ∈ Φ, Ua (k) ∩ G(k)0Ω = Uψ , where ψ ∈ Ψ  is the smallest element with derivative a that only takes non-negative values on Ω.

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Axiom 4.1.9 Tits system. Let A be an apartment in B(G) corresponding to a maximal k-split torus S and let C ⊂ A be a chamber. Let G = G(k)0 , I = G(k)0C , and N = N(k) ∩ G. Then (I, N) is a saturated Tits system in G whose associated restricted building is B(G). In other words, the stabilizer G(k)0F of a facet F of B(G) for the action of G = G(k)0 is precisely the parabolic subgroup of this Tits system, cf. Definition 1.4.1, Remark 1.4.2, Theorem 1.5.6 and Definition 1.5.20. Remark 4.1.10 In view of Tits’ Propositions 1.5.6 and 1.5.28, Axioms 4.1.1 and 4.1.9 are complementary. If Axiom 4.1.1 holds and one can prove that G(k)0 acts on B(G) by special automorphisms and transitively on pairs consisting of an apartment and a chamber, then Axiom 4.1.9 follows from Proposition 1.5.28. Conversely, if one is given an abstract apartment with a polysimplicial structure and Axiom 4.1.9 holds, then Axiom 4.1.1 follows from Proposition 1.5.6, provided one can match up the a priori given polysimplicial structure of A with that of the resulting building. We will use the second approach in Chapter 7, and the first approach in Chapter 9. Remark 4.1.11 The subgroups G(k)0F , associated to the facets F of B(G), that is, the parabolic subgroups of the Tits system of Axiom 4.1.9, are thus the parahoric subgroups of G(k) as defined in Definition 4.1.3. 4.1.12 Let F be a facet in B(G). From Axiom 4.1.9 and Proposition 1.5.13 we have the following. (1) The group G(k)0F is the stabilizer, as well as the pointwise stabilizer, of F in G(k)0 . (2) The subset of points of B(G) fixed under G(k)0F is precisely the closure F of F. (3) The action of G(k)0F on the set of apartments containing F is transitive. (4) The action of G(k)0 , hence also of G(k), on the set of ordered pairs consisting of an apartment of B(G) and a chamber in it, is transitive. Note this implies, in particular, that given a maximal k-split torus S ⊂ G, its normalizer, N(k) in G(k), acts transitively on the set of chambers in the apartment A corresponding to S. (5) Two points x, y ∈ B(G) lie in the same facet if and only if G(k)0x = G(k)0y . Remark 4.1.13 Axiom 4.1.9 and the identities Gsc (k)0 = Gsc (k) and B(G) = B(Gsc ) imply the following. If Ω is a subset of a facet of B(G), the stabilizer of Ω in G(k) fixes Ω pointwise. Moreover, if an element g of G(k) belongs to G(k)1Ω , then it is actually contained in G(k)0Ω .

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Definition 4.1.14 For a subset Ω ⊂ B(G) let cl(Ω) denote the union of the closures of all facets of B(G) that intersect Ω non-trivially. Remark 4.1.15 It is clear that G(k)0Ω = G(k)0cl(Ω) and G(k)1Ω = G(k)1cl(Ω) . If Ω is contained in an apartment A, then so is cl(Ω). Axiom 4.1.16 Iwahori decomposition. Let S ⊂ G be a maximal k-split torus with associated apartment A ⊂ B(G). Let C ⊂ A be a chamber and I = G(k)0C the corresponding Iwahori subgroup. For any positive system of roots Φ+ ⊂ Φ = Φ(S, G) the product map (U + (k) ∩ I) × Z(k)0 × (U − (k) ∩ I) → I is bijective, where U + is the unipotent radical of the minimal parabolic subgroup containing S corresponding to Φ+ , and U − is the unipotent radical of the Sopposite parabolic subgroup. Moreover, the product map  Uψa → U ± (k) ∩ I a ∈Φ±, nd

is a bijection, where the product is taken in any order, and for any non-divisible root a, ψa is the smallest affine functional on A with derivative a that is non-negative on C. Note that the three factors U + (k) ∩ I, Z(k)0 , and U − (k) ∩ I, can be taken in any order, since U + and U − can be switched by replacing Φ+ by Φ− = −Φ+ , while Z(k)0 normalizes the other two factors. Axiom 4.1.17 Compatibility with isomorphisms Consider two Henselian valued fields k 1 and k2 and an isomorphism fk : k 1 → k2 . Let Gi be a connected reductive ki -group and let fG : G1 → G2 be an isomorphism covering fk ; that is, fG : G1 ×k1 k2 → G2 is an isomorphism of k2 -groups. The pair ( fk , fG ) induces an isomorphism of buildings fB : B(G1, k1 ) → B(G2, k2 ). It satisfies fB (g1 x1 ) = fG (g1 ) fB (x1 ) for g1 ∈ G1 (k1 ) and x1 ∈ B(G1, k1 ). Before we state the next axiom, let us introduce the following notation. Definition 4.1.18 Let G be a smooth o-group scheme. We denote by G 0 the largest subgroup scheme with connected fibers. We refer to G 0 as the relative identity component. It is shown in [SGA3, Exp. VIB , Theorem 3.10] that G 0 exists and is an open subscheme of G . In fact, it is the union of the neutral components of the generic and special fibers. If G is affine, then so is G 0 [PY06, Lemma in §3.5].

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Remark 4.1.19 We will often deal with a smooth affine group scheme G with connected generic fiber. Then G is a connected scheme, but the special fiber of G may still be disconnected, so G 0 may be a proper subscheme of G . Axiom 4.1.20 Bruhat–Tits group schemes Given a non-empty bounded subset Ω of an apartment A of B(G), there is a smooth affine o-group scheme GΩ1 , with generic fiber G, such that GΩ1 (o) = G(k)1Ω . Letting GΩ0 be the relative identity component of GΩ1 we have GΩ0 (o) = G(k)0Ω . Note that if k is strictly Henselian this condition determines GΩ1 up to isomorphism, cf. Corollary 2.10.11. Note further that G(k)0Ω and G(k)1Ω are open and bounded. The group scheme GΩ1 has the following properties. (1) If Ω is a subset of a facet F, then GΩ0 = GF0 . (2) If Ω is the closure of Ω, then GΩ0 = G 0 . Ω (3) If the above apartment A corresponds to the maximal k-split torus S of G, then there is a closed o-split torus S in GΩ0 with generic fiber S. The special fiber S of S is a maximal f-split torus in the special fiber G Ω0 of GΩ0 . (4) If G is semi-simple and simply connected, then GΩ0 = GΩ1 . Since the o-group schemes GΩ1 and GΩ0 are smooth, their coordinate rings have the following description by Lemma 2.10.9: O[GΩ1 ] = { f ∈ K[G] | f (GΩ1 (O)) ⊂ O}; O[GΩ0 ] = { f ∈ K[G] | f (GΩ0 (O)) ⊂ O}. Definition 4.1.21 Let F be a facet of B(G). The group scheme GF0 is called the Bruhat–Tits parahoric group scheme. Axiom 4.1.22 Structure of the reductive quotient Given non-empty and bounded subsets Ω and Ω  of an apartment of B(G), with Ω ≺ Ω , there is an o-group scheme homomorphism ρΩ,Ω : GΩ1 → GΩ1 that is the identity homomorphism on the generic fiber G, and which on o-points recovers the inclusion G(k)1Ω ⊂ G(k)1Ω . Recall that when k is strictly Henselian the existence and uniqueness of such a homomorphism follows from Corollary 2.10.10. The above homomorphism restricts to an o-group scheme homomorphism 0 GΩ → GΩ0 and induces an f-homomorphism ρΩ,Ω : G Ω0  → G Ω0 . The restriction of ρΩ,Ω to any torus of G Ω0  is an isomorphism onto a torus of G Ω0 . The homomorphism ρΩ,Ω has the following properties. (1) The kernel of ρΩ,Ω is a smooth connected unipotent f-subgroup of G Ω0  . In addition, the following properties hold in the special case of facets F ≺ F  of B(G).

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(2) The image pF (F ) := ρF,F  (G F0  ) is a parabolic f-subgroup of G F0 . (3) Let S be a maximal k-split torus of G such that the apartment of B(G) corresponding to S contains F . Let S be the closed o-split torus of GF0  with generic fiber S, and let S be the special fiber of S . Then S is a maximal f-split torus of G F0 , as well as of G F0  , and its isomorphic image in the maximal reductive quotient G F := G F0 /Ru (G F0 ) of G F0 is a maximal f-split torus of G F . We will denote the image of pF (F ) in G F by ℘F (F ). Let x be a point of F and v be a vector in V(S) such that v + x is a point of F . Then the nonzero weights of S in the Lie algebra of the parabolic f-subgroup ℘F (F ) of G F are the roots a of G F (with respect to S ) such that a(v)  0. (4) The inverse image of the subgroup pF (F )(f) of G F0 (f), under the natural homomorphism πF : GF0 (o) → G F0 (f) is GF0  (o). (5) F  → ℘F (F ) is an order-preserving bijective map of the partially ordered set {F  | F ≺ F  } onto the set of parabolic f-subgroups of G F partially ordered by the opposite of inclusion. (6) The root datum of G F with respect to the maximal f-split torus S is given by (X∗ (S), ΦF , X∗ (S), Φ∨F ), where ΦF ⊂ Φ ⊂ X∗ (S) is the image under the derivative map Ψ → Φ of the subset ΨF of affine roots that vanish on F, and Φ∨F = {a∨ ∈ Φ∨ | a ∈ ΦF }. (7) Let G(k)0+ (⊂ G(k)0F ) be the preimage of Ru (G F0 )(f) under the surjective F ⊂ G(k)0+ . homomorphism πF : GF0 (o) → G F0 (f). Then G(k)0+ F F Remark 4.1.23 Note that (4) implies that the inverse image P+F under πF of the normal subgroup Ru (G F0 )(f)(⊂ pF (F )(f)) of G F0 (f) is contained in GF0  (o). So P+F fixes every facet F , F ≺ F , pointwise. (5) implies that a strictly “increasing” sequence F ≺ F1 ≺ F2 ≺ · · · ≺ Fn corresponds to a strictly decreasing sequence ℘F (F1 ) ⊃ ℘F (F2 ) ⊃ · · · ⊃ ℘F (Fn ) of parabolic f-subgroups of G F . Therefore, the codimension of F is the f-rank of the derived subgroup of G F . Hence, a facet F of B(G) is a chamber (i.e., it is a maximal facet) if and only if G F does not contain a proper parabolic f-subgroup, or, equivalently, in G F every f-split torus is central, cf. Proposition 2.4.7. (6) gives a combinatorial recipe to compute the Dynkin diagram of G F from the affine Dynkin diagram of Ψ. Indeed, if C is a chamber that contains F in its closure and Δ is the basis of Ψ determined by C, then ΔF = ∇(Δ ∩ ΨF ) is a basis of ΦF according to Proposition 1.3.35. Therefore, the Dynkin diagram of

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G F is obtained by removing from the affine Dynkin diagram of Ψ all vertices

of the facet F, and all edges connected to them, cf. Fact 1.3.66.

Remark 4.1.24 Let G(k)†Ω denote the stabilizer of Ω in G(k)1 . We claim that the subgroup G(k)1Ω is of finite index in G(k)†Ω . Indeed, any element of G(k)1 which stabilizes Ω permutes the facets of the building that meet Ω. Every such facet is contained in the apartment that contains Ω. Therefore, there are only finitely many such facets, and hence a subgroup of finite index of G(k)†Ω keeps each facet that meets Ω stable and fixes every vertex of such a facet, hence it fixes pointwise every facet that meets Ω. Thus a subgroup of finite index of G(k)†Ω fixes Ω pointwise and therefore lies in G(k)1Ω . The claim is thus proved. Note that it implies that G(k)†Ω is bounded, since G(k)1Ω is by Axiom 4.1.2. Fact 4.1.25 Let C ⊂ A ⊂ B be a chamber and an apartment in B. Let I and S be the corresponding Iwahori subgroup and maximal k-split torus. Then G(k) = I · N(k) · I. Proof Set G = G(k)0 and N = N(k) ∩ G. Then (G, I, N) is a Tits system by Axiom 4.1.9. The Bruhat decomposition (cf. Proposition 1.4.5) for this Tits system implies G = I · N · I, which together with G(k) = Z(k) · G (Fact 2.6.22) implies the statement.  Definition 4.1.26 We say that Bruhat–Tits theory is available for G over k if Axioms 4.1.1, 4.1.4, 4.1.9, 4.1.6, 4.1.17, 4.1.20, and 4.1.22 hold. Axiom 4.1.27 Unramified descent Let K be a maximal unramified extension of k. In view of Axiom 4.1.17 the Galois group Gal(K/k) acts on B(G K ). UR 1 There is a G(k)-equivariant embedding B(G) → B(G K ) whose image is precisely the set of Gal(K/k)-fixed points in B(G K ). UR 2 If S ⊂ G is a maximal k-split torus, there exists a k-torus T ⊂ G containing S such that TK is a maximal K-split torus of G K . For any such T, A(S) = A(TK )Gal(K/k) . UR 3 Given an affine root ψ ∈ Ψ(S) ⊂ A(S)∗ and a root b ∈ Φ(TK , G K ) whose restriction to S equals the derivative of ψ, there exists an affine root ψb ∈ Ψ(TK ) ⊂ A(TK )∗ with derivative b whose restriction to A(S) = A(TK )Gal(K/k) equals ψ. Conversely, if an affine root in Ψ(TK ) has nonconstant restriction to A(S), then this restriction is an affine root in Ψ(S). UR 4 For a subset Ω of an apartment of B(G), the base change to O of the o-models GΩ0 and GΩ1 recovers the O-models GΩ0 and GΩ1 associated to

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Overview and Summary of Bruhat–Tits Theory the image of Ω in B(G K ). In particular, G(k)0Ω = G(K)0Ω ∩ G(k) and G(k)1Ω = G(K)1Ω ∩ G(k).

4.1.28 Where are these axioms proved in this book? Assume first that G is quasi-split. The apartment associated to a maximal k-split torus S is constructed in §6.1, in particular Definition 6.1.27. In Axiom 4.1.4, A 0 and A 2 are true by construction and A 1 is verified in Lemma 6.1.26. Axiom 4.1.6 is verified in §6.2 and §6.3, with AS 1 being Proposition 6.2.1, AS 2 and AS 3 in Proposition 6.3.13, AS 4 in Proposition 6.3.8, AS 5 in Proposition 6.6.2, and AS 6 due to the construction of B(G). Theorem 7.5.3 constructs the Iwahori–Tits system and verifies the first part of Axiom 4.1.9. That B(G) is the building of that Tits system holds by definition, cf. Definition 7.6.1. The latter also establishes Axiom 4.1.1, except for the bijection between maximal split tori and apartments, which is Proposition 7.6.3. The groups G(k)0x and G(k)1x are first constructed by hand, independently of the action of G(k) on B(G), in §7.4 and §7.7. They are open and bounded by Corollary 7.3.14 and Proposition 7.7.1. That these groups coincide with the stabilizer of x for the action of G(k)0 respectively G(k)1 on B(G) is proved in Propositions 7.6.4 and 7.7.5. This establishes Axiom 4.1.2 and the second part of Axiom 4.1.9. For Axiom 4.1.8, (1) is proved in Fact 7.6.7, (2) follows from Proposition 7.7.5 and the construction of G(k)0Ω “by hand” in §7.4, and (3) follows from Lemma 7.7.3 and Definitions 6.3.1 and 6.3.4. Axiom 4.1.16 is proved in Corollary 7.4.9, and Axiom 4.1.17 is proved in Proposition 7.9.5. Axiom 4.1.20 is proved mostly in Chapter 8. The construction of GΩ1 is given in Theorem 8.3.2, and (1) is proved in Theorem 8.3.13 together with the following remark. Part (2) follows from G(K)0Ω = G(K)0 , which is due to the Ω fact that G(k) acts on B(G) by polysimplicial automorphisms, and Corollary 2.10.11. Part (3) is proved in Propositions 8.2.4 and 8.2.2. Part (4) follows from G(K)0 = G(K)1 = G(K) and Corollary 2.10.11. Axiom 4.1.22 is proved in §8.4. Part (1) is Proposition 8.4.15, parts (2)–(5) are Theorem 8.4.19, part (6) is Theorem 8.4.10, part (7) is Corollary 8.4.12. Axiom 4.1.27 is handled in the general case of G not necessarily quasi-split. Assume now that G is general. The proofs of the axioms for G rely on the validity of these axioms for the quasi-split group G K , cf. Corollary 2.3.8. A summary of where each axiom for G is derived from the validity of all axioms for G K is given in §9.5. Note that once the availability of valuations of root data is established in §9.6, the material of Chapters 6 and 7 becomes available for non-quasi-split groups as well.

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4.1.29 Analogy with Borel–Tits theory for loop groups Consider the special case of the function field k = f((t)) for some algebraically closed field f. Some results of Bruhat–Tits theory for a connected reductive k-group G can be interpreted as analogs of statements of the classical structure theory for reductive groups due to Borel–Tits applied to the loop group associated to G. We illustrate this principle following [Ric16b] and [PR08]. The functor R → G(R((t))) from the category of f-algebras to the category of groups is representable by a strict ind-affine ind-group scheme LG over f, called the (twisted) loop group. In general LG need not be reduced or connected. An example when LG is reduced is when G is semi-simple, split over a tamely ramified extension of k, and the order of the fundamental group π1 (Gder ) is prime to p, cf. [PR08, Theorem 0.2]. To a smooth integral model G of G we can associate the positive (twisted) loop group L + G , which is the functor R → G (R[[t]]) from the category of f-algebras to the category of groups; this functor is representable by a reduced affine subgroup scheme of LG, which is connected if the special fiber of G is connected. Recall that a subgroup P of G is parabolic if and only if the variety G/P is proper, in which case it is projective (this variety is called a (partial) flag variety). The analog in Bruhat–Tits theory is the statement that a smooth integral model G with connected special fiber is parahoric if and only if the fpqc-quotient LG/L + G is representable by an ind-proper ind-scheme, in which case it is ind-projective (this scheme is called an affine flag variety, denoted by FlG ), cf. [Ric16b, Theorem A]. An interpretation of the subgroup G(k)0 , or more generally of the Kottwitz homomorphism κ : G(k) → π1 (G)I discussed in Chapter 11, is that κG induces a bijection π0 (LG) → π1 (G)I , and thus that G(k)0 = LG0 (f), cf. [PR08, Theorem 0.1]. In particular, when G is semi-simple, then LG is connected if and only if G is simply connected. Other analogies come from the fact that the parabolic subgroups of G(k), as well as the parahoric subgroups of G(k) = LG(f), form a building. But one needs to be mindful to work with the identity component of LG, thus with G(k)0 = LG0 (f). For example, the statement that any two parabolic subgroups contain a common maximal k-split torus is analogous to the statement that any two parahoric subgroups contain the maximal bounded subgroup of a maximal k-split torus. The statement that a parabolic subgroup is any subgroup that contains a minimal parabolic subgroup is analogous to the statement that a parahoric subgroup is any bounded subgroup of G(k)0 that contains an Iwahori subgroup. The statement that all minimal parabolic subgroups are conjugate under G(k) is analogous to the statement that all Iwahori subgroups are conju-

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gate under G(k)0 . The statement that a parabolic subgroup is equal to its own normalizer in G(k) is analogous to the statement that a parahoric subgroup is equal to its own normalizer in G(k)0 .

4.2 Metric We assume given a building B satisfying Axioms 4.1.1, 4.1.2, 4.1.4, and 4.1.9. In this section we will see that these axioms allow us to endow B with a metric; we will study properties of this metric and draw some implications. Recall from Proposition 1.5.24 that, given an apartment A of B and a chamber C of A we have the retraction ρ = ρA,C : B → A. Given a maximal k-split torus S ⊂ G, let S  be the maximal torus in S ∩ Gder . We have the real vector space V(S ) := R ⊗Z X∗ (S ). Given two such tori S1, S2 there exists g ∈ G(k) such that Int(g)S1 = S2 . The element g then induces an isomorphism V1 := V(S1 ) → V(S2 ) =: V2 of real vector spaces. The set of all such isomorphisms is a torsor for the Weyl group WG (S)(k). Therefore, if we choose a Euclidean metric on V(S ) invariant under the Weyl group WG (S)(k) for a fixed S ⊂ G, we obtain a Euclidean metric on V(S ) for every S ⊂ G. We shall call such a choice of metric on each V(S ) a compatible system of metrics. We assume henceforth that such a choice has been made. Note that when G is k-simple, there is a unique choice up to rescaling. In fact, there is a canonical choice for a Weyl-invariant metric, see [Bou02, Chapter VI, §1, no. 1, Proposition 3]. Since the apartment A := A(S) of B(G) is an affine space under V(S ), the Euclidean metric on V(S ) endows A(S) with a Euclidean metric. We will write dA for this metric. Proposition 4.2.1 There exists a unique function d : B(G) × B(G) → R0 whose restriction to each apartment A coincides with the Euclidean metric dA . Proof Given two apartments A1, A2 and g ∈ G(k) such that gA1 = A2 , the isomorphism g : A1 → A2 is an isometry, because it translates the action of v ∈ V1 on A1 to the action of gv ∈ V2 on A2 , and the isomorphism g : V1 → V2 is an isometry. Given x, y ∈ B there exists an apartment A containing both x, y, due to Definition 1.5.5. We define d(x, y) = dA (x, y) and claim that this does not depend on the choice of A. If A  is another apartment containing x, y, there exists g ∈ G(k)0 such that gA = A  and gx = x and gy = y, by Proposition  1.5.13. Therefore dA (x, y) = dA (x, y).

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Given points x, y in an apartment A we shall denote by [x, y] the line segment in A connecting x and y. Lemma 4.2.2 Let C ⊂ A ⊂ B be a chamber and let ρ = ρA,C : B → A. Let x, y ∈ B be the retraction. (1) The restriction of ρ to any apartment containing C is an isometry. (2) If either x ∈ C or there exists a facet F whose closure contains x and y, then d(ρ(x), ρ(y)) = d(x, y). (3) For arbitrary x, y ∈ B one has d(ρ(x), ρ(y))  d(x, y). Proof For the first point, ρ| A is by construction given by an element g ∈ G(k)0 sending A to A and fixing C, hence it is an isometry. This implies the second point, since C, x, and y, are contained in a common apartment A. To see the third point, choose an apartment A containing both x and y. Let x = x0, x1, . . . , xn = y be a sequence of points lying on the line segment [x, y] ⊂ A such that each pair (xi , xi+1 ) lies in the closure of a facet Fi+1 of A. Then n n   d(ρ(x j−1 ), ρ(x j )) = d(x j−1, x j ) = d(x, y).  d(ρ(x), ρ(y))  j=1

j=1

Proposition 4.2.3 (1) The function d is a metric. (2) Given x, y ∈ B(G), the set of points D = {z ∈ B(G) | d(x, y) = d(x, z) + d(z, y)} lies in every apartment A that contains both x, y and equals the line segment [x, y] there. (3) Any isometry of B(G), in particular every element of G(k), that fixes both x and y, also fixes every point of D. Proof Let x, y, z ∈ B(G). Clearly d(x, y) = d(y, x) and d(x, y) = 0 if and only if x = y. Choose an apartment A containing x, y and let C be a chamber of A. Let ρ = ρA,C . Recall that ρ : B → A is the identity on A and that d |A is a metric. Thus d(x, y) = d(ρ(x), ρ(y))  d(ρ(x), ρ(z)) + d(ρ(z), ρ(y))  d(x, z) + d(z, y). This proves that d is a metric. Now assume that z ∈ D and let t ∈ [x, y] ⊂ A be such that d(x, z) = d(x, t). Choose a chamber C of A whose closure contains t. Since z ∈ D, the above chain of inequalities becomes a chain of equalities. This implies that the inequalities d(x, ρ(z)) = d(ρ(x), ρ(z))  d(x, z) and d(y, ρ(z)) = d(ρ(y), ρ(z))  d(y, z)

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become equalities as well. Therefore ρ(z) = t. Since t ∈ C, Proposition 1.5.24 implies z = t. The third point follows immediately.  Definition 4.2.4 (1) We will write [x, y] for the set D of Proposition 4.2.3(2). (2) For t ∈ [0, 1] we will write t x + (1 − t) y for the unique point z ∈ [x, y] with the property d(x, z) = (1 − t)d(x, y). Corollary 4.2.5 Let x, y ∈ B(G) and let c : [0, 1] → B be the curve defined by c(t) = t x + (1 − t) y. Then c is the unique geodesic joining x and y in the sense of Definition 1.1.2. In particular, B is a uniquely geodesic space in the sense of Definition 1.1.4. Proof Let A be an apartment containing x, y. By Proposition 4.2.3 the image of c is contained in A. It is immediate that c : I → A is continuous and geodesic with respect to the Euclidean metric. Since the metric of B restricts to the Euclidean metric on A, the same holds for c : I → B. If c  is a geodesic joining x and y, then for any t ∈ [0, 1] we have d(x, c (t)) = t d(x, y) and d(c (t), y) = (1 − t)d(x, y). Thus d(x, c (t)) + d(c (t), y) = d(x, y). It follows from Proposition 4.2.3 that the image of c  equals [x, y] and that  c (t) = c(t). Remark 4.2.6 Recall from Definition 1.1.12 the notion of a convex subset of a uniquely geodesic space. With the current notation, a subset Ω ⊂ B(G) is called convex if x, y ∈ Ω implies [x, y] ⊂ Ω. Every apartment, and more generally every subset of an apartment that is convex for the Euclidean metric, is a convex subset of B(G). In particular, any facet is convex. Proposition 4.2.7 The metric space B is non-positively curved in the sense of Definition 1.1.8. Proof Let x, y ∈ B and let A be an apartment containing x, y. Let m be the midpoint of [x, y]. It also lies on A by Proposition 4.2.3. The parallelogram law holds in A; that is, we have d(x, z )2 + d(y, z )2 = 2d(m, z )2 + (1/2)d(x, y)2 for any z  ∈ A. Choose a chamber of A whose closure contains m and let ρ : B → A be the retraction centered at that chamber. For any z ∈ B, let z  = ρ(z). Lemma 4.2.2 implies d(m, z ) = d(m, z), d(x, z )  d(x, z), d(y, z )  d(y, z). So d(x, z)2 + d(y, z)2  2d(m, z)2 + (1/2)d(x, y)2 . Lemma 4.2.8



Let x, y, z, z  ∈ B. Assume that z ∈ [x, y] and that there exists

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 > 0 such that d(x, z )  d(x, z) +  d(x, y) and d(y, z )  d(y, z) +  d(x, y). Then d(z, z )2  4 d(x, z)d(y, z) +  2 d(x, y)2 . Proof Let A be an apartment containing x, y. According to Proposition 4.2.3 the geodesic [x, y] is contained in A. Let C ⊂ A be a chamber whose closure contains z. Let z  = ρA,C (z ). Lemma 4.2.2 implies d(z, z ) = d(z, z ) and further that the points x, y, z, z  satisfy the same property as the points x, y, z, z . But now x, y, z, z  lie in A and the claim follows from basic Euclidean geometry.  Corollary 4.2.9

The map

[0, 1] × B × B → B,

(t, x, y) → t x + (1 − t) y

is continuous. In particular, B is a contractible topological space. Proof

This is immediate from Lemma 4.2.8.



Theorem 4.2.10 The metric space B is complete. If X is a set of facets, then  F ∈X F is closed. Proof The proof is more transparent when B is locally finite, so we will first give it in this special case. The second statement is then immediate, so we prove the first. It will be enough to show that there exists  > 0 such that for every x ∈ B, the ball B(x, ) is contained in a compact subset of B. Indeed, once this is shown, if we are given a Cauchy sequence (xn ) in B, we can choose N ∈ N such that for m, n  N, d(xm, xn ) < . In particular, (xn )nN is a Cauchy sequence contained in B(x N , ), hence in a compact set, so it converges. To show the claim, it is enough to consider points of the closure of a fixed chamber C, since every point in B is G(k)-conjugate to a point of C and G(k) operates by isometries. But C is the union of finitely many facets, and each is contained in the closure of finitely many chambers. The union of the closures of all chambers obtained in this way is a compact subset of B and C is contained in its interior. Since C is compact, the claim is proved. Now we give the proof without assuming that B is locally finite. We will use the following lemma. Lemma 4.2.11 Let x ∈ B. There exists an open ball D with center x such that if a facet F intersects D, then x ∈ F. Moreover, if g ∈ G(k)0 is such that gx ∈ D, then gx = x. Proof Let C be a chamber containing x in its closure and A an apartment containing C. According to Axiom 4.1.6 and Proposition 1.3.12 there exists δ > 0 such that if α1, α2 are two affine roots with the same derivative, and

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H1, H2 ⊂ A are their vanishing hyperplanes, then d(H1, H2 )  δ (Definition 1.1.1). This implies that there exists  > 0 such that if D  is the ball with center x and radius  in A, then D  meets only those vanishing hyperplanes of affine roots that already contain x. Therefore, if F is a facet of A meeting D , then x ∈ F. Let D be the ball with center x and radius  in B. Let F be a facet of B that meets D. Let A be an apartment containing C and F by Proposition 1.5.14. Proposition 1.5.13 implies the existence of g ∈ G(k)0 that fixes pointwise the closure of C and such that gA = A. Then g fixes x and therefore leaves invariant D. Thus gF meets D ∩ A = D , so x lies in the closure of gF, so x = g −1 x lies in the closure of F, as claimed. Finally, if g ∈ G(k)0 is such that gx ∈ D, then the facet F containing gx clearly meets D, so x ∈ F. This implies gF = F. Proposition 1.5.13 implies  that G(k)0F fixes pointwise the closure of F, hence gx = x. Returning to the proof of the above theorem, it is enough to show that  M = F ∈X F is complete. Let {xn } be a Cauchy sequence in M. We will show that a subsequence of {xn } converges to a point of M. Let C be any chamber of B. For each n there exists a gn ∈ G(k)0 such that yn = gn−1 xn ∈ C. Since C is compact, we can pass to a subsequence of yn (hence also of xn ) and assume that yn converges to a point y ∈ C. Let D be the ball centered at y provided by Lemma 4.2.11. We estimate d(gm y, gn y)  d(gm y, gm ym ) + d(gm ym, gn yn ) + d(gn yn, gn y) = d(y, ym ) + d(xm, xn ) + d(yn, y). Therefore there exists a natural number N such that for all m, n  N we have gn−1 gm y ∈ D and hence gm y = gn y =: x. For n  N we have d(xn, x) = d(gn yn, gn y) = d(yn, y). So the sequence {xn } converges to x. Applying Lemma 4.2.11 again we obtain an open ball D with center x; for all large n, xn lies in D. So if F ∈ X is a facet containing xn in its closure, then D intersects F, which implies that  x ∈ F ⊂ M. Corollary 4.2.12 The Bruhat–Tits fixed point theorem (Theorem 1.1.15) holds for B. Proof This follows from Theorem 4.2.10 and Proposition 4.2.7. Proposition 4.2.13 equivalent.



For a subset X of G(k)1 the following statements are

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(1) X is a bounded subset of G(k) in the sense of Definition 2.2.1. (2) For one, hence any, pair (C, A) of an apartment A of B and chamber C of A, there exists a finite subset {n1, . . . , nr } ⊂ N(k) such that X ⊂  I · ni · I, where I and S are the Iwahori subgroup and maximal k-split torus corresponding to C and A. (3) The action of X on B is bounded; that is, for every x ∈ B, X · x is of bounded diameter. Proof The group I is bounded by Axiom 4.1.2. Lemma 2.2.6 implies that X is bounded if and only if I · X · I is bounded, which in turn is equivalent, by Fact 4.1.25, to the existence of a bounded subset Y ⊂ N(k) such that X ⊂ I · Y · I. The quotient N(k)/Z(k) is finite. Choosing a finite set S of representatives for the finite quotient N(k)/Z(k), we see that the boundedness of X is equivalent to the existence of a bounded subset Y  ⊂ Z(k) such that X ⊂ I · Y  · S · I. The injection Z(k)/Z(k)1 → HomZ (X∗k (Z), Z) shows that a subset Y  ⊂ Z(k) is bounded if and only if its image under this map is bounded; that is, if and only if it is contained in a finite union of translates of Z(k)1 . Since Z(k)0 is of finite index in Z(k)1 and is contained in I, we finally conclude that (1) and (2) are equivalent. For the equivalence of (2) and (3) let x0 ∈ C be a point. For every g ∈ Ini I, d(gx0, x0 ) = d(ni x0, x0 ). This implies that Ini I · x0 is a subset of bounded diameter for each i. Thus (2) implies (3). If, conversely, X cannot be covered by finitely many I-double cosets, let {ni } be an infinite subset of N(k) such that the corresponding I-double cosets are all disjoint, intersect X non-trivially, and their union contains X. Then in the collection {ni C} of chambers, there are infinitely many distinct chambers, so the subset {ni x} of A has infinite diameter.  Corollary 4.2.14 A bounded subgroup G of G(k) has a fixed point in B(G). If G preserves a non-empty closed convex (not necessarily bounded) subset of B(G), then it has a fixed point in that subset. Proof This follows from the above Proposition and the Bruhat–Tits fixed point theorem, Theorem 1.1.15.  Theorem 4.2.15 (1) Let x ∈ B(G) be a vertex. The group G(k)1x is a maximal open and bounded subgroup of G(k). (2) Every maximal bounded subgroup of G(k) is of the form G(k)1x for a point x ∈ B(G). If G(k)0 = G(k)1 (this holds for example when G is semi-simple and simply connected) then x is a vertex. (3) For a subgroup K of G(k) the following statements are equivalent.

174

Overview and Summary of Bruhat–Tits Theory (i) K is a maximal parahoric subgroup (Definition 4.1.3). (ii) K is a maximal bounded subgroup of G(k)0 .

Proof Let K be a bounded subgroup of G(k). By Corollary 4.2.14, K fixes a point y ∈ B. Since K is bounded, it is contained in G(k)1 . Therefore K ⊂ G(k)1y . By Axiom 4.1.2, the group G(k)1y is open and bounded. If we assume that K is maximal bounded, then the boundedness of G(k)1y implies that K = G(k)1y . Assume in addition that G(k)1 = G(k)0 . Let F be the facet containing y and let x be a vertex contained in F. Then G(k)1y = G(k)0y = G(k)0F ⊂ G(k)0x = G(k)1x by Axiom 4.1.9 and the maximality of K implies G(k)0y = G(k)0x . But according to Proposition 1.5.13(4), x is the only point fixed by G(k)0x . Hence x = y; that is, F is a vertex. This proves (2). To prove (1) assume that x is a vertex, and that K is a bounded subgroup of G(k) containing G(k)1x . As was just argued, K ⊂ G(k)1y for a point y ∈ B. Thus G(k)1x fixes y. But x is the only point fixed by G(k)0x , hence also by G(k)1x . Therefore x = y, hence K = G(k)1x . The maximality of G(k)1x follows. The observations made so far also prove (3).  Example 4.2.16 When G(k)0 = G(k)1 does not hold, there can exist maximal bounded subgroups G(k)1x for x not a vertex of B(G). For example, the subgroup of PGL2 (k) generated by the standard Iwahori subgroup, that is, the preimage in PGL2 (o) of the upper triangular Borel subgroup in PGL2 (f), and the element " # 0 1 , π 0 is a maximal bounded subgroup and the unique point in B that it stabilizes is the barycenter of the standard chamber. The following simple lemma is sometimes useful. We will use it in Chapter 14. Lemma 4.2.17 Let (xn ) be a sequence of points of B converging to a point x. For any subgroup G of G(k) we have Gxn ⊂ Gx .

Proof Let S be the set of all xn . Then Gxn is the pointwise stabilizer of the set S for the action of G on B. Since this action is continuous, this pointwise stabilizer fixes every point of the closure of S.  Proposition 4.2.18

The action map G(k) × B → B is continuous.

Proof It is enough to prove sequential continuity, since G(k) is second countable and B is a metric space. Given a sequence (gn ) in G(k) converging to

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g ∈ G(k) and a sequence (xn ) in B converging to x ∈ B, we want to prove that the sequence (gn xn ) in B converges to gx. But d(gn xn, gx)  d(gn xn, gn x) + d(gn x, gx). Now d(gn xn, gn x) = d(xn, x) converges to 0 by assumption. On the other hand, d(gn x, gx) = d(g −1 gn x, x). Since the stabilizer of x in G(k) is open (Axiom 4.1.2) and g −1 gn converges to 1 ∈ G(k), g −1 gn x = x for all sufficiently large n.  Remark 4.2.19 Recall from Proposition 1.1.20 that if Y ⊂ B is a closed convex subset then for each x ∈ B there exists a unique y ∈ Y that is closest to x. Assume in addition that Y is the union of the closures of a set of chambers of B. If y lies in a chamber, then x = y. Indeed, if x  y then the geodesic [x, y] meets the closure of the chamber containing y in a point z  y, and then d(x, z) < d(x, y), contradicting the assumption that y is the closest point to x contained in Y . This argument is valid for any chamber complex B endowed with a metric that is complete and non-positively curved. Lemma 4.2.20 B.

For any x ∈ B, the convex hull of the G(k)-orbit of x equals

Proof Let A be an apartment of B containing x, let S be the corresponding maximal k-split torus of G, and as before, let N be its normalizer. Since x is contained in the closure of some chamber and N(k) acts transitively on the set of chambers, we conclude that the orbit N(k) · x intersects the closure of every chamber, so the convex hull of N(k) · x is equal to all of A. Since G(k) acts transitively on the set of apartments in B, the lemma follows.  Corollary 4.2.21 The only G(k)-equivariant self-isometry of B is the identity. Proof

Immediate from Proposition 1.5.13(4).



The following two lemmas provide necessary and sufficient condition for the intersection of finitely many parahoric subgroups to be itself a parahoric subgroup. We write G = G(k)0 and denote by (G, I, N, R) the Tits system of Axiom 4.1.9.

Lemma 4.2.22 Let P1, . . . , Pq be parahoric subgroups. Then nq Pn is a parahoric subgroup if and only if Pi ∩ P j is a parahoric subgroup for every 1  i  j  q.

Proof If nq Pn is a parahoric subgroup, then since Pi ∩ P j contains it, the latter is also a parahoric subgroup for all i, j  q. To prove the converse by induction on q, we can assume that Pi  P j for

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i  j  q, and assume that Q := nq−1 Pn is a parahoric subgroup. For n  q, let Fn be the facet corresponding to the parahoric subgroup Pn and F be the facet corresponding to Q. Then Fn ≺ F for all n  q − 1. Let A be an apartment that contains the facets F and Fq . Then for n  q −1, Fn is contained in A. Let Ω(⊂ A) be the convex hull of F and Fq . Then using the fact that for n  q − 1, Pn ∩ Pq is a parahoric subgroup and so it determines a facet in A, we see that Ω is contained in the same side of each root hyperplane. Therefore, Ω is contained in the closure of a facet in A and the corresponding parahoric

 subgroup is contained in nq Pn . Lemma 4.2.23 We assume that the Tits system (G, I, N, R) is irreducible. Let P  P  be two maximal parahoric subgroups of G. Then Q := P ∩ P  is a parahoric subgroup if and only if it is a maximal proper subgroup of P. This assertion is false if the Tits system (G, I, N, R) is not assumed to be irreducible. Proof If P ∩ P  is a parahoric subgroup, then after conjugating both P and P  by an element of G we can (and will) assume that P ∩ P  contains I. Since (G, I, N, R) is irreducible, there exist r, r  ∈ R such that P = IR−{r } and P  = IR−{r  } . Then P ∩ P  = IR−{r ,r  } is a maximal proper subgroup of P. Conversely, suppose P ∩ P  is maximal in P. Let x and x  be the vertices of the affine building given by the Tits system (G, I, N, R) corresponding to the maximal parahoric subgroups P and P , respectively, Then P ∩ P  fixes every point of the geodesic [x, x ]. Lemma 4.2.11 implies that for any y ∈ [x, x ] sufficiently close to and distinct from x, the unique facet F that contains y, contains x in its closure. Hence the parahoric subgroup Q corresponding to F is properly contained in P and it contains P ∩ P . As P ∩ P  has been assumed  to be a maximal proper subgroup of P, Q = P ∩ P . Using the metric we obtain the following strengthening of the first assertion of Proposition 1.5.13. Proposition 4.2.24 Let A1, A2 be two apartments of B. There exists g ∈ G(k)0 such that gA1 = A2 and gx = x for every x ∈ A1 ∩ A2 . Proof Note that A1 ∩ A2 is a closed convex subset of the building B. If y ∈ A1 ∩ A2 is contained in the facet F, then F ⊂ A1 ∩ A2 . Therefore, A1 ∩ A2 is a union of facets. Let F ⊂ A1 ∩ A2 be a facet whose dimension is maximal among all facets contained in A1 ∩ A2 . Then F is an open subset of A1 ∩ A2 . If F is of dimension zero (that is, it is a vertex), then being connected and of dimension zero, A1 ∩ A2 = F, and so there is nothing to prove in this case. We assume now that dim F > 0. According to Proposition 1.5.13(1) there

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exists g ∈ G(k)0 such that gA1 = A2 and gx = x for all x ∈ F. We claim that, in fact, gy = y for all y ∈ A1 ∩ A2 . Let y ∈ A1 ∩ A2 . Fix a x ∈ F and let D = [x, y] be as in Definition 4.2.4(1). Then gD = [x, gy]. The sets D and gD are line segments in A2 of the same length and having x as an end point. We claim that they are equal. To show this, it is enough to find just one point other than x contained in both D and gD. But since F is open in A1 ∩ A2 , the set D ∩ F contains more points than just x, and all these are fixed by g and thus belong to D ∩ gD.  Corollary 4.2.25 Let A be an apartment of B and S ⊂ G the corresponding maximal k-split torus. For g ∈ G(k) there exists n ∈ N(k) such that gx = nx for all x ∈ A ∩ g −1 A. Proof Using Proposition 4.2.24 choose h ∈ G(k) such that hA = g −1 A and hx = x for all x ∈ A ∩ g −1 A. Since n := gh stabilizes A, it belongs to N(k) by Axiom 4.1.1, and has the desired property.  Corollary 4.2.26 Let A1 and A2 be two apartments of B(G) with corresponding maximal k-split tori S1 and S2 . Let C be a subtorus of S1 and assume that for some x ∈ A1 ∩ A2 , C(k) · x is contained in A2 . Then C ⊂ S2 . Proof Let g ∈ G(k)0 be as in Proposition 4.2.24. Then gS1 g −1 = S2 . Fix a k-embedding G → SLn . Via the conjugation action of SLn on the k-vector space Mn (k) of n×n matrices, we obtain a k-rational representation of G, which we restrict to C. The C(k)-orbit of the element g ∈ Mn (k) lies in the bounded subset G(k)0x . Lemma 2.2.8 implies that g is fixed under the conjugation action of C; that is, g commutes with C. But then gCg −1 = C is contained in S2 .  The following result shows that the only apartment contained in a tubular neighborhood of an apartment A is A. Proposition 4.2.27 Let A1, A2 be two apartments of B. Assume there exists r > 0 such that for every x1 ∈ A1 there exists x2 ∈ A2 such that d(x1, x2 )  r. Then A1 = A2 . Proof Let πi : B → Ai be the closest point map, cf. Proposition 1.1.20. Consider f := π1 |A2 ◦π2 |A1 : A1 → A1 . For every x ∈ A1 we have d(x, f (x))  2r. We claim that f is surjective. To see this, choose a point in A1 and use it to identify A1 with Rn . The metric on A1 is identified with the usual Euclidean metric on Rn (up to scaling). For y ∈ Rn consider fy : Rn → Rn defined by fy (x) = f (y − x) − (y − x). This map carries the ball of radius 2r in Rn to itself. By Brauer’s fixed point theorem, there exists x0 in the ball of radius 2r fixed by fy , that is, f (y − x0 ) = y, proving the claim.

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The claim implies that π1 maps A2 onto A1 . This implies, by Remark 4.2.19, that A2 contains every chamber of A1 . Since A2 is closed in B, it contains A1 . For i = 1, 2, let Si be the masimal k-split subgr oup of G corresponding to the apartment Ai . Then Corollary 4.2.26 implies that S1 ⊂ S2 . So from the  maximality of S1 , we see that S1 = S2 , which implies that A1 = A2 .

4.3 The Enlarged Building According to Axiom 4.1.2 the stabilizer G(k)1x of a point x ∈ B(G) in the group G(k)1 is bounded and open. Since the action of G(k) on B(G) factors through Gad (k) according to Axiom 4.1.1, the stabilizer of x in G(k) is the preimage of Gad (k)1x and contains the full center ZG of G. Therefore G(k)x will be bounded if and only if the maximal k-split torus AG in ZG is trivial. It is sometimes convenient to modify the building so that G(k)x is itself bounded. This leads to the enlarged building, which we will now describe. Consider the real vector space R ⊗Z X∗ (AG ). If we let Gab = G/Gder , then 0 → G Gab is a torus and the natural map ZG ab is an isogeny. Therefore, the Θ induced map R ⊗Z X∗ (AG ) → R ⊗Z X∗ (Gab ) is bijective. The group Gab (k) acts on the vector space R ⊗Z X∗ (Gab )Θ as a group of translations, by the rule gλ, χ = λ, χ − ω( χ(g)), for g ∈ Gab (k), λ ∈ R ⊗Z X∗ (Gab )Θ , χ ∈ X∗ (Gab )Θ . In this way we obtain an action of G(k) on R ⊗Z X∗ (AG ). We will write V(AG ) = R ⊗Z X∗ (AG ). Lemma 4.3.1 (1) (2) (3) (4)

For g ∈ G(k) the following are equivalent.

g leaves invariant a non-empty bounded subset of V(AG ). g fixes a point of V(AG ). g acts trivially on V(AG ). g ∈ G(k)1 .

Proof The equivalence of the first three points is immediate from the fact that g acts by a translation on the vector space V(AG ). Let λ ∈ V(AG ) = R ⊗Z X∗ (Gab )Θ and g ∈ G(k). Then gλ = λ implies  ω( χ(g)) = 0 for all χ ∈ X∗ (Gab )Θ = X∗ (G)Θ , therefore g ∈ G(k)1 .  Definition 4.3.2 The enlarged building of G, denoted by B(G), is the cartesian product B(G) × V(AG ). Given a maximal k-split torus S ⊂ G, the enlarged  is the cartesian product A(S) × V(AG ). apartment A(S)

4.3 The Enlarged Building

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 The group G(k) acts on B(G) by the diagonal action on the product  x ∈ B(G) we let G(k)x be the stabilizer of  x for the action B(G) × V(AG ). For   of G(k) on B(G).  Lemma 4.3.3 Let  x ∈ B(G) and let x ∈ B(G) be the image of  x . Then 1 G(k) x = G(k) ∩ G(k)x = G(k)1x . More generally, G(k)1Ω is equal to the  subgroup of G(k) whose action on B(G) fixes Ω × V(AG ). Proof We have  x = (x, λ) for some λ ∈ V(AG ). Thus G(k)x = G(k)x ∩ G(k)λ .  But G(k)λ = G(k)1 by Lemma 4.3.1. 4.3.4 Let S be a maximal k-split torus of G and let S  be the unique maximal k-split torus of Gder contained in S. Then S = AG · S . This gives a bijection between the maximal split tori in G and those in Gder . Recall from Axiom 4.1.4 the canonical decomposition V(S) = V(S ) ⊕ V(AG ). The translation action of V(S ) on A(S) and the translation action of V(AG ) on itself provide a translation   into an affine space over V(S). The action of V(S) on A(S), which turns A(S)   by using the V(S )-affine space A(S) embeds into the V(S)-affine space A(S)  This distinguished point 0 ∈ V(AG ) and mapping x ∈ A(S) to (x, 0) ∈ A(S).  to x ∈ A(S), and this embedding has a retraction, by projecting (x, v) ∈ A(S)  by the action of V(AG ). retraction realizes A(S) as the quotient of A(S)  via its action The normalizer N(k) of S in G(k) acts on the affine space A(S) on A(S) and its action (through the quotient Gab (k)) on V(AG ). In particular, the derivative of this action is the natural action of N(k) on V(S) which factors through the Weyl group N(k)/Z(k). The action of z ∈ Z(k) is given by the translation ν(z) ∈ V(S) that is the negative of the valuation homomorphism Z(k) → V(S), cf. (2.6.4). In particular, the maximal bounded subgroup Z(k)b acts trivially. 4.3.5 As discussed in §4.2, the choice of a compatible system of metrics on V(S ) for each maximal k-split torus S leads to a metric d on B(G) with respect to which B(G) is complete, non-positively curved, and uniquely geodesic. Complementing this with the choice of a Euclidean metric on the real vector   with a metric with respect to which B(G) is comspace V(AG ) endows B(G) plete, non-positively curved, and uniquely geodesic. Note that combining a Weyl group-invariant Euclidean metric on V(S ) with an arbitrary Euclidean metric on V(AG ) is the same as choosing a Weyl group-invariant Euclidean  metric on V(S). Therefore, the metric on B(G) is obtained from a compatible system of metrics on each V(S), in the obvious sense generalizing this concept from the case of V(S ).  sending x to (x, v). Every v ∈ V(AG ) induces an embedding B(G) → B(G)

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This embedding is a Gder (k)-equivariant isometry. Projecting onto the first  factor gives a G(k)-equivariant surjective map B(G) → B(G). Its restriction to each subset B(G) × {v} is a Gder (k)-equivariant isometric isomorphism. We record the direct analog of Corollary 4.2.14.  Corollary 4.3.6 A bounded subgroup G of G(k) has a fixed point in B(G). If G preserves a non-empty closed convex (not necessarily bounded) subset of  B(G), then it has a fixed point in that subset. Proof Since G is bounded, it is contained in G(k)1 and therefore acts trivially  in particular the on V(AG ). Thus G preserves each subset B(G) × {v} of B(G),   distinguished subset B(G) = B(G) × {0} ⊂ B(G). If X ⊂ B(G) is a closed convex subset preserved by G, then its intersection with B(G) × {0} is a closed convex subset of B(G) stable under G. Corollary 4.2.14 implies that G fixes a point in X ∩ B(G).  The following lemma is stated for completeness. It will not be used in this book.  is of the Lemma 4.3.7 Every Gder (k)-equivariant isometry B(G) → B(G) form x → (x, v) for some v ∈ V(AG ). Therefore the set of Gder (k)-equivariant  isometries B(G) → B(G) is canonically isomorphic to the real vector space V(AG ).  Proof Let ι : B(G) → B(G) be a Gder (k)-equivariant isometry, let x ∈ B(G), and let (y, v) = ι(x). By Lemma 4.2.20 applied to Gder the convex hull of the Gder (k)-orbit of x equals B(G) and therefore the image of ι is the convex hull of the Gder (k)-orbit of (y, v), which equals B(G) × {v}. Composing ι with the  projection B(G) → B(G) gives a Gder (k)-equivariant self-isometry of B(G), which equals the identity by Corollary 4.2.21.  One often uses the embedding corresponding to 0 ∈ V(AG ).

4.4 Uniqueness of the Apartment and the Building In this section we will discuss the uniqueness properties of the Bruhat–Tits building of G. Let S be a maximal k-split torus of G. We will first show that Axiom 4.1.4(A 1) characterizes the apartment A associated to S up to unique isomorphism. In fact, we will give a simple and direct construction of A. Let S  be the maximal k-split torus of Gder contained in S and denote by  N (⊂ N) and Z  (⊂ Z) respectively the normalizer and the centralizer of S  in

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Gder . We will denote the maximal bounded subgroup of Z (k) by Z (k)b , and W will denote the k-Weyl group N (k)/Z (k) of Gder , equivalently of G. We have the exact sequence 1 → Z (k)/Z (k)b → N (k)/Z (k)b → W → 1. Let V  := R ⊗Z X∗ (S ). The homomorphism ν : Z(k) → V  described in Axiom 4.1.4(A 1) is W-equivariant. The push-out of the above extension along ν| Z  (k) gives the following extension: R → W → 1. 1 → V → W

(4.4.1)

Lemma 4.4.1 The action of V  by conjugation on the set of splittings of the extension (4.4.1) realizes that set as an affine space over V . This affine space satisfies Axiom 4.1.4(A 1), and the collection of these affine spaces as S varies satisfies Axiom 4.1.4(A 2). Proof We have the cohomology groups Hi (W,V ) for i  0. Since multiplication by |W | is an automorphism of V , it is also an automorphism of Hi (W,V ) for any i by functoriality. On the other hand, multiplication by |W | annihilates Hi (W,V ) for i > 0 due to the inflation-restriction formula. Therefore Hi (W,V ) = {0} for all i > 0. The class of this extension is an element of H2 (W,V ), whose vanishing shows that this extension is split. The set of V -orbits of splittings is a torsor under the group H1 (W,V ), whose vanishing shows that there is a unique V orbit of splittings. The action of V  on that orbit is transitive by definition. We R is a splitting, the conjugation claim that it is also simple. Indeed, if s : W → W  action of an element v ∈ V preserves it if and only if for all w ∈ W the identity s(w)vs(w)−1 = v holds. This means v ∈ V W . But X∗ (S )W = {0}, hence v = 0. The fact that Axioms A 1 and A 2 are satisfied by this affine space is immediate from the construction.  4.4.2 Let A be an affine space satisfying Axiom 4.1.4(A 1). The action of N(k)/Z(k)b on A given by this axiom is compatible with the action of V  on A by translation, and hence factors through the push-out and induces an action R on A. of W Proposition 4.4.3 Let A be an affine space satisfying Axiom 4.1.4(A 1). Between each two of the following three affine spaces under V  there is a unique N(k)-equivariant isomorphism that induces the identity on V : (1) the affine space A, (2) the set of splittings of the extension (4.4.1),

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(3) the set of maximal compact (equivalently, maximal finite) subgroups of R . the Lie group W x of W R whose action on More precisely, x ∈ A corresponds to the subgroup W R → W to W x splits A fixes x, and the restriction of the natural projection W this projection. In particular, between any two affine spaces satisfying Axiom 4.1.4(A 1) there exists a unique N(k)-equivariant isomorphism that induces the identity on V . x of x in W R is a maximal Proof Let x ∈ A. We claim that the stabilizer W x  compact, in fact maximal finite, subgroup of WR , and the assignment x → W  is a bijection between A and the set of maximal compact subgroups of WR . x = {1}, which shows that W x is finite. Conversely, let Indeed, we have V  ∩ W R . Since V  has no non-trivial compact 0 be a maximal compact subgroup of W W 0 = {1}. This shows that W 0 is finite. Thus its action on A subgroups, V  ∩ W x . 0 ⊂ W has a fixed point x (take the barycenter of an arbitrary orbit). So W 0 was assumed maximal we x is finite. Since W We have already seen that W x . Thus the map x → W x is a surjective map from A onto 0 = W conclude W R , and all of these subgroups are the set of maximal compact subgroups of W in fact finite. R of the extension (4.4.1). Its image is a Consider now a splitting s : W → W x for some x ∈ A. But then it must R , hence contained in W finite subgroup of W x maps isomorphically  equal Wx . We conclude that for each x ∈ A, the group W  onto W. Since the action of W on V fixes no other vector besides 0, the action x x on A fixes no other point besides x. This shows that the map x → W of W x is a bijection is injective. We have thus established the claim that x → W R . Moreover, we between A and the set of maximal compact subgroups of W have also shown that each such subgroup maps isomorphically onto W, and hence provides a splitting of (4.4.1). Conversely, the image of any splitting of R as we have already argued. (4.4.1) is a maximal compact subgroup of W To see that these isomorphisms are N(k)-equivariant and induce the identity R we have W wx = w · W x · w −1 in on V  it is enough to note that for any w ∈ W  WR . To see the uniqueness statement it is enough to show that the identity on A is the only automorphism of A that is N(k)-equivariant and induces the identity on V . Let f : A → A be such an automorphism. Then f (x) = x + v for some fixed v ∈ V . Equivariance under N(k) is equivalent to wv = v for all w ∈ W,  thus v ∈ V W = {0}. We now turn to the uniqueness properties of the building. We assume given a building B satisfying Axioms 4.1.1, 4.1.4, and 4.1.9. We identify B with its

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183

geometric realization, in which each apartment carries the structure of an affine space from Axiom 4.1.4. Proposition 4.4.4 Let S ⊂ G be a maximal k-split torus with normalizer N and associated apartment A as an affine space. The map G(k) × A → B,

(g, x) → gx

is surjective, G(k)-equivariant, and its fibers are the equivalence classes with respect to the following equivalence relation (g, x) ∼ (h, y) ⇔ ∃n ∈ N(k) : nx = y, g −1 hn ∈ G(k)x .

(4.4.2) 

In the above equivalence relation, one can replace G(k)x by G(k)x , or any intermediate subgroup. Proof Given y ∈ B there exists an apartment A containing y by Definition 1.5.5(3) and an element g ∈ G(k) such that gA = A due to the conjugacy of maximal split tori under G(k) and the bijection between such tori and apartments due to Axiom 4.1.1. This proves surjectivity. Consider now two pairs (g, x) and (h, y) such that gx = hy in B. Applying Corollary 4.2.25 to x = g −1 hy we obtain an element n ∈ N(k) such that nx = y. Therefore g −1 hn ∈ G(k)x , showing that (g, x) and (h, y) are in the same equivalence class. The converse is immediate.  To see that G(k)x can be replaced by G(k)x or any intermediate subgroup it  is enough to show that, for a given x ∈ A, we have G(k)x = G(k)x · N(k)x . This follows by choosing a chamber C of A containing x in its closure and applying Fact 4.1.25 to see G(k)x = I · N(k)x · I, and observing that nIn−1 ⊂ G(k)0x for  all n ∈ N(k)x , and moreover Z(k)0 ⊂ N(k)x . Let B1 and B2 be two buildings for G satisfying Axioms 4.1.1, 4.1.4, and 4.1.9. For a maximal k-split torus S ⊂ G, Proposition 4.4.3 provides a canonical isomorphism between the apartments A1 and A2 in B1 and B2 associated to S. Corollary 4.4.5 If there exists a pair of points x1 ∈ A1 and x2 ∈ A2 that correspond under the canonical isomorphism of Proposition 4.4.3 such that G(k)0x1 ⊂ G(k)0x2 , then the canonical isomorphism of Proposition 4.4.3 extends uniquely to a G(k)-equivariant isomorphism B1 → B2 of buildings. In particular, for any pair of facets F1, F2 that correspond under this isomorphism we have G(k)0F1 = G(k)0F2 . Proof We present each building as in Proposition 4.4.4. The assumption that G(k)0x1 ⊂ G(k)0x2 implies that the isomorphism G(k) × A1 → G(k) × A2 that is the identity on the first factor, and the canonical isomorphism of Proposition

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4.4.3 on the second factor, descends to a surjective map f : B1 → B2 . This map is obviously G(k)-equivariant. Given g ∈ G(k), the restriction of this map to the apartment gA1 is the canonical isomorphism of Proposition 4.4.3 between the apartments in B1 and B2 associated to the maximal k-split torus gSg −1 . To show that f must also be injective, consider x  y ∈ B1 . Choose an apartment A1 containing both x, y. We have just argued that the restriction of f to A1 is bijective, and the injectivity of f follows. We have thus established a canonical bijective map f : B1 → B2 that is G(k)equivariant, and restricts to the canonical isomorphism of Proposition 4.4.4 between the apartments on its source and target corresponding to a maximal ksplit torus S of G. The G(k)-equivariance of this map implies that if f (y1 ) = y2 , then G(k)0y1 = G(k)0y2 . According to 4.1.12 this implies that the map f respects facets and is therefore an isomorphism of polysimplicial complexes.  Assume that B satisfies, in addition to Axioms 4.1.1, 4.1.4, 4.1.9, also Axiom 4.1.8(2). Proposition 4.4.6 Fix a maximal k-split torus S ⊂ G and let A be the associated apartment; that is, the canonical affine space of Proposition 4.4.3. Consider a G(k)-set B equipped with an N(k)-equivariant embedding A → B. Assume that  (1) B = gA, g∈G(k)

(2) Axiom 4.1.8(1) holds for B. The identity on A extends uniquely to a G(k)-equivariant bijection B → B. In this book we will show that a building B satisfying the axioms assumed in this proposition exists. In fact, we will show that B satisfies all axioms listed in §4.1. This proposition then implies that this building, as a G(k)-set, is uniquely determined just by the assumptions placed on B. Proof Let x ∈ A. We claim that the stabilizer of x for the action of G(k) on B equals the stabilizer of x for the action of G(k) on B. In order to distinguish the two actions, we will write x  for x thought of as a point of B. According to Axiom 4.1.8(2), the stabilizer G(k)0x of x in G(k)0 is generated by T(k)0 and Uψ for ψ ∈ Ψ with ψ(x)  0. Since G(k)0x acts transitively on the set of apartments of B containing x by 4.1.12, we see that the stabilizer G(k)x of x in G(k) equals G(k)0x · N(k)x . By assumption (2), the action of G(k) on B, when restricted to Uψ , fixes x . The action of N(k) on A is the same whether A is considered as a subset of B or B. Therefore we see that G(k)x ⊂ G(k)x . To see the converse inclusion,

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we use the affine Bruhat decomposition of Fact 4.1.25 with a chamber whose closure contains x to conclude G(k) = G(k)0x · N(k)x · G(k)0x . By what was just proved, the action of G(k)0x on B fixes x . Therefore the stabilizer of x  is G(k)0x · N(k)x · G(k)0x , and thus is contained in G(k)x . The claim is proved. Define a map f : B → B as follows. Given x ∈ B choose g ∈ G(k) such that y := g −1 x ∈ A and set f (x) = g · y , where y  is the image of y under the inclusion A → B. By the claim that was just proved, this map is well defined. It is surjective by assumption (1). If it failed to be injective, that would imply that for some point x ∈ B the stabilizer of x in G(k) is strictly smaller than the stabilizer of f (x) ∈ B. By assumption (1) and the G(k)-equivariance of f we may assume x ∈ A. Now the claim that was just proved gives a contradiction. Finally, the uniqueness of the bijection is clear: any self-bijection of B that is G(k)-equivariant and fixes A pointwise automatically fixes B pointwise.  Corollary 4.4.7 Let B1 and B2 be two buildings for G that satisfy Axioms 4.1.1, 4.1.4, 4.1.9, and 4.1.8. There is a unique bijection B1 → B2 that restricts, for each maximal k-split torus S ⊂ G, to the canonical identification A1 (S) = A2 (S) of Proposition 4.4.3. Proof Fix a maximal k-split torus S and apply Proposition 4.4.6 to B1 and the map A1 (S) = A2 (S) ⊂ B2 to obtain a bijection of G(k)-sets B1 → B2 . 

5 Bruhat, Cartan, and Iwasawa Decompositions

Bruhat, Cartan, and Iwasawa decompositions to be discussed in this chapter are very important results which are frequently used in the representation theory of, and harmonic analysis on, reductive groups over local fields. In fact, establishing these decompositions for these groups was a major goal of Bruhat–Tits theory. We will use Iwasawa decomposition in the proof of Proposition 13.6.3. Let k be a field given with a discrete valuation ω : k → R ∪ {∞}. Let o := {x ∈ k | ω(x)  0} be the ring of integers, m = {x ∈ k | ω(x) > 0} the maximal ideal of o, and f = o/m the residue field. We assume that o is Henselian and f is perfect. Let G be a connected reductive k-group and S a maximal k-split torus of G. We will denote by N the normalizer of S in G and by Z (⊂ N) its centralizer. We know that Z(k) contains a unique maximal bounded subgroup (Proposition 2.2.12) which we will denote by Z(k)b . Wk will denote the k-Weyl group N(k)/Z(k). Let A be the apartment corresponding to S in the Bruhat–Tits building B(G) of G(k).

5.1 The (affine) Bruhat Decomposition This result was already stated as Fact 4.1.25 and used in the construction of the metric. We state it here again so that it can be located easily. Proposition 5.1.1 (Bruhat decomposition) We fix a chamber C in the apartment A and let I be the Iwahori subgroup of G(k) corresponding to C. Then G(k) = IN(k)I. 186

5.2 The Cartan Decomposition

187

5.2 The Cartan Decomposition We fix a positive system of roots Φ+ in the root system Φ = Φ(S, G) of G with respect to S. Define Z = {z ∈ Z(k) | a(ω Z (z))  0 for all a ∈ Φ+ },

(5.2.1)

where ω Z : Z(k) → V(S) is the valuation homomorphism of (2.6.4) for Z(k). More explicitly, recall that X∗k (Z) is embedded in X∗ (S) as a subgroup of finite index. For every χ ∈ X∗ (S) and z ∈ Z(k) we define ω( χ(z)) to be n1 ω(n χ(z)) for any positive integer n such that n χ ∈ X∗k (Z); it is easily seen that this does not depend on the choice of n. Then Z = {z ∈ Z(k) | ω(a(z))  0 for all a ∈ Φ+ }.

(5.2.2)

The following is a version of the Cartan decomposition [BT72, Proposition 4.4.3] that was first formulated in this form in [HR10, Theorem 1.0.3]. Theorem 5.2.1 (Cartan decomposition) Let x be a special vertex in A (Definition 1.3.39) and Px be the corresponding parahoric subgroup of G(k). (1) G(k) = Px · Z · Px . (2) For z, z  ∈ Z, Px zPx = Px z  Px is equivalent to z  z−1 ∈ Z(k)0 . Proof We fix a chamber C in A that has x as one of its vertices. Let I be the Iwahori subgroup of G(k) corresponding to C. Then I ⊂ Px and as x is a special vertex, the natural homomorphism N(k) ∩ Px → Wk is surjective due to Lemma 1.3.42 and Axiom 4.1.6. Since N(k) ∩ Px maps onto Wk , we see that Px Z(k) contains IN(k), and therefore, the affine Bruhat decomposition (Proposition 5.1.1) implies G(k) = Px · Z(k) · I. The surjectivity of N(k) ∩ Px maps onto Wk further implies that the conjugates nZn−1 , for n ∈ N(k) ∩ Px , cover Z(k). We conclude that G(k) = Px · Z · Px , proving (1). We will now prove (2). Since Z(k)0 ⊂ Px it is clear that z  z −1 ∈ Z(k)0 implies Px zPx = Px z  Px . To prove the converse, we assume given z, z  ∈ Z(k) with Px zPx = Px z  Px . Then the images of z and z  in G(k)/G(k)0 are equal, therefore z  z−1 ∈ G(k)0 . Corollary 2.6.28 reduces the proof to showing that z  z −1 ∈ Z(k)1 . We first claim that z  ∈ Nx zNx , where Nx = Px ∩ N(k). By assumption there exist g, g  ∈ Px such that g z = z  g . Then x lies in A ∩ g · A and z  x = z  g  x = g z x also lies in A ∩ g · A. By Proposition 1.5.13(1) and Axiom 4.1.9 (alternatively Proposition 4.2.24) there exists an element g0 ∈ Px ∩ z  Px z −1

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such that gA = g0 A. This implies that g0−1 g ∈ Nx , and z −1 g0 z  ∈ Px . So h := g0−1 g z = g0−1 z  g  = z (z −1 g0−1 z )g  ∈ Nx z ∩ z  Px . Since both z, z  belong to Z(k) ⊂ N(k), we have Nx z ∩ z  Px = Nx z ∩ z  Nx . Hence h belongs to Nx z ∩ z  Nx and we conclude z  ∈ hNx ⊂ Nx zNx as claimed. The claim just proved implies that there exists n1, n2 ∈ Nx such that z  = n1 zn2 = n1 n2 · n2−1 zn2 . Defining z  = n2−1 zn2 ∈ Z(k), we obtain z0 := z  z −1 = n1 n2 ∈ Z(k) ∩ Nx ⊂ Z(k) ∩ Px , which is contained in Z(k)1 by Axiom 4.1.2. In particular, the images of z  and z  under the valuation homomorphism ω Z : Z(k) → Q ⊗Z X∗ (S) are equal. Therefore, all the three elements z, z , z  map under ω Z to the subset { v ∈ Q ⊗Z X∗ (S) | a(v)  0 for all a ∈ Φ+ }. But that subset is a fundamental domain for the action of the Weyl group Wk and the homomorphism ω Z is N(k)-equivariant, which implies that the images of z and z  (= n2−1 zn2 ) under ω Z must be equal. Therefore, there exists c ∈ ker(ω Z ) = Z(k)1 such that z  = cz. So z0 = z  z −1 = z  z−1 c−1 , and hence  z  z−1 = z0 c ∈ Z(k)1 .

5.3 The Iwasawa Decomposition We fix a positive system of roots Φ+ in the root system Φ = Φ(S, G) and let Δ be the basis of Φ contained in Φ+ . Let ZU = U Z be the minimal parabolic k-subgroup of G containing S, with U its unipotent radical, determined by the positive system of roots Φ+ , and Z := ZG (S). For a ∈ Φ we will denote the corresponding root group by Ua ; it is a connected unipotent k-group normalized by Z. The unipotent radical U is generated by the root groups Ua for nondivisible roots a ∈ Φ+ . Recall from Proposition 2.11.17 that for any u ∈ Ua (k)∗ = Ua (k) − {1} there exist unique u , u  ∈ U−a (k)∗ such that m(u) = u  uu  normalizes S. The image of m(u) in the Weyl group Wk = N(k)/Z(k) equals the reflection ra Lemma 5.3.1 Let A be the apartment of B(G) corresponding to the maximal k-split torus S and x ∈ A. Let a ∈ Φ and u ∈ Ua (k)∗ . If u  Px , then u , u  ∈ Px . Proof According to Axiom 4.1.8, the assumption u  Px implies u  Uψ for any ψ ∈ Ψ  with ψ(x)  0. Since u ∈ Uψau (k) we see ψau (x) < 0. According u (x) > 0 and ψ u (x) > 0, thus u , u  ∈ P by to Remark 4.1.7 this means ψ−a x −a Axiom 4.1.8. 

5.3 The Iwasawa Decomposition

189

For a ∈ Φ, we denote by Ma the subgroup of G generated by Ua , U−a and Z. We will denote the subgroup Px ∩ Ma (k) of Px by Px, a . It is obvious that M−a = Ma and Px,−a = Px, a . For every a ∈ Φ+ , we fix an element na ∈ N(k) that maps onto the reflection in a in the k-Weyl group Wk ; note that na ∈ Ma (k). We set n−a = na . Let Ξa = Ua (k)Z(k){1, na }Px, a . Lemma 5.3.2 (1) na Ua (k) ⊂ na Px, a ∪ Ua (k)Z(k)Px, a ⊂ Ξa . (2) Ma (k) ⊂ Ξa = Ua (k)Z(k){1, na }Px, a . Proof (1) Let u be a non-trivial element of Ua (k). If it lies in Px , then na u ∈ na Px, a . So let us assume that u does not lie in Px . By Lemma 5.3.1 then both u  and u  lie in Px , and hence they lie in Px, a . As u = u −1 m(u)u −1 , na u = na · u −1 m(u)u −1 ∈ na m(u) · m(u)−1 u −1 m(u)Px, a . Since na and m(u) have the same image in Wk and this image is an element of order 2, na m(u) lies in Z(k); also m(u)−1 (u −1 )m(u) belongs to Ua (k). So, if u  Px , then na uPx, a ⊂ Z(k)Ua (k)Px, a = Ua (k)Z(k)Px, a . Thus for all u ∈ Ua (k), na uPx, a is contained in na Px, a ∪ Ua (k) Z(k)Px, a . (2) Recall that by the Bruhat decomposition, Ma (k) = Ua (k)Z(k) ∪ Ua (k)Z(k)na Ua (k). So to show that Ma (k) ⊂ Ξa (= Ua (k)Z(k){1, na }Px, a ), it is enough to observe  that according to (1), na Ua (k) ⊂ Ξa . Theorem 5.3.3 (Iwasawa Decomposition) Let C be a chamber in the apartment A and PC be the corresponding Iwahori subgroup of G(k). Then G(k) = U(k) · N(k) · PC . Proof Let Ξ = U(k)N(k)PC . We will prove that for each a ∈ Δ, Ξ is stable under multiplication on the left by U−a (k), that is, U−a (k) · Ξ = Ξ. On the other hand, Ξ is clearly stable under multiplication on the left by U(k) Z(k). It is well known that the latter together with the subgroups U−a (k) for all a ∈ Δ generate G(k). Then it would follow that Ξ is stable under multiplication on the left by all of G(k), and hence it equals G(k). For a ∈ Δ, let Ua be the k-subgroup of U generated by the root groups Ub , for non-divisible positive roots b different from a. Then Ua is normalized by U±a and U = Ua  Ua . Now U−a (k) · Ξ = U−a (k)U(k)N(k)PC = U−a (k)Ua(k)Ua (k)N(k)PC = Ua(k)U−a (k)Ua (k)N(k)PC ⊂ Ua(k) Ma (k)N(k)PC . Now using (2) of the preceding lemma for any x ∈ C, we obtain that Ma (k) ⊂

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Ua (k)Z(k){1, na }(PC ∩ Ma (k)). Also, we know from Axiom 4.1.16 and the compatibility of the open cell with closed subgroups (cf. Proposition 2.11.1(4)) that PC ∩ Ma (k) is contained in Z(k)Ua (k)U−a (k) ∩ Z(k)U−a (k)Ua (k). Hence, we conclude from the above that U−a (k) · Ξ ⊂ U(k)Z(k){1, na }(PC ∩ Ma (k))N(k)PC ⊂ U(k)Z(k){Ua (k)U−a (k) ∪ na U−a (k)Ua (k)}N(k)PC = U(k)Z(k)U−a (k)N(k)PC, since, na U−a (k)Ua (k)n−1 a = Ua (k)U−a (k) and Z(k) normalizes Ua (k). Now to complete the proof of the theorem, it is enough to show that for every n ∈ N(k) and v ∈ U−a (k), vn lies in Ξ. Set u = n−1 vn ∈ U−b (k), where b = n−1 · a. Then vn = nu ∈ nU−b (k). Using (1) of the preceding lemma for any x ∈ C and −b in place of a, we see that nb u ∈ U−b (k)Z(k){1, nb }PC . Hence, −1 −1 vn = nu = nn−1 b nb u ∈ n · nb U−b (k)nb · Z(k){1, nb }PC

= nUb (k)n−1 · Z(k){n, nn−1 b }PC ⊂ Ua (k)N(k)PC ⊂ Ξ.  The following is a more popular form of the Iwasawa decomposition, in which an arbitrary parabolic subgroup of G is allowed. Theorem 5.3.4 For any special vertex x ∈ B(G) and any parabolic ksubgroup P of G, we have G(k) = P(k)Px = Px P(k). Proof Let C be a chamber in B(G) one of whose vertices is x, and A be an apartment that contains C. Let S be the maximal k-split torus of G corresponding to A. Let Z be the centralizer of S in G, N its normalizer and U Z = ZU be a minimal parabolic k-subgroup of G containing Z. Since x is a special vertex, Px ∩ N(k) maps onto N(k)/Z(k), and hence Z(k)Px = N(k)Px . Therefore, N(k)PC ⊂ N(k)Px = Z(k)Px . Now by Theorem 5.3.3, G(k) = U(k)N(k)PC ⊂ U(k)Z(k)Px ⊂ G(k). By conjugacy of minimal parabolic k-subgroups [Bor91, Theorem 20.9(i)] there exists a g ∈ G(k) such that P contains the minimal parabolic k-subgroup g −1 (U Z)g. As G(k) = U(k)Z(k)Px , there is a h ∈ U(k)Z(k) such that hg lies in Px . Now we see that P(k)Px ⊃ g −1 (U(k)Z(k))gPx = (hg)−1 (U(k)Z(k))(hg)Px = (hg)−1 (U(k)Z(k)Px )(hg) = G(k).

5.4 The Intersection of Cartan and Iwasawa Double Cosets

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Lemma 5.3.5 Let P be a bounded subgroup of G(k)0 containing Z(k)0 . For z, z  ∈ Z(k), U(k)zP = U(k)z  P is equivalent to z −1 z  ∈ Z(k)0 . Proof Since Z normalizes U, the equality of the double cosets U(k)zP and U(k)z  P implies that there exists an element of U(k), say u, such that uz−1 z  belongs to P. Then as uz −1 z  lies in the bounded subgroup P ∩ (U  Z)(k) of (UZ)(k), its image under the natural projection UZ → Z lies in the bounded subgroup Z(k)1 of Z(k). But that image equals z −1 z . Therefore z −1 z  ∈ Z(k)1 and also z −1 z  ∈ u−1 P ⊂ G(k)0 , hence z−1 z  ∈ Z(k)1 ∩ G(k)0 = Z(k)0 by Corollary 2.6.28. 

5.4 The Intersection of Cartan and Iwasawa Double Cosets Let S ⊂ G be a maximal k-split torus with apartment A. Choose a positive system of roots Φ+ ⊂ Φ = Φ(S, G) and let U be the unipotent radical of the corresponding minimal parabolic subgroup. Lemma 5.4.1 Let ρ : B → A be the retraction centered at some chamber of A. Given g ∈ G(k) and x, y ∈ A with x − y ∈ V(S) dominant, we have ρ(gx) − ρ(gy)  x − y. We recall here from [Bou02, Chapter VI, §1, no. 6] that, given v, v  ∈ V(S), we write v   v if every element of Φ+ takes non-negative values on v − v . Proof Let x = x0, x1, . . . , xn = y be a monotonic sequence of points on the straight line connecting x and y in A having the property that each pair xi , xi+1 is contained in the closure of some chamber. Note that for each i = 0, . . . , n − 1 the vector xi − xi+1 is dominant. To prove the lemma, it is enough to show ρ(gxi ) − ρ(gxi+1 )  xi − xi+1 for all i, for then we can add all inequalities together. Thus we may assume that x, y are contained in the closure of a chamber C. By Lemma 1.5.29 there exists h ∈ G(k) such that ρ(gz) = hz for all z ∈ C. This means that h sends the chamber C to another chamber of A. Therefore there exists n ∈ N such that hz = nz for all z ∈ C. Thus ρ(gx) − ρ(gy) = nx − ny = w(x − y), where w ∈ W is the image of n. Now the dominance of x − y implies w(x − y)  x − y by [Bou02, Chapter VI, no. 6, Proposition 18].  Proposition 5.4.2 Let x ∈ A. Let Z be as in §5.2. Let z ∈ Z and z  ∈ Z(k) be such that Px zPx ∩ U(k)z  Px  ∅. Then for every dominant weight ξ relative to the ordering determined by Φ+ , ω(ξ(z−1 z ))  0.

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Proof Since Px zPx and U(k)z  Px intersect non-trivially, there exists a u ∈ U(k) such that Px z and uz  Px intersect non-trivially. Let C be a chamber in A such that the corresponding Iwahori subgroup PC of G(k) contains u. Then Px z ∩ PC z  Px  ∅, and hence z −1 Px z ∩ (z −1 PC z · z−1 z  Px )  ∅. For simplicity, we introduce the notations y = z−1 x ∈ A, C  = z−1 C ⊂ A, and z0 = z −1 z  ∈ Z(k). Then Py ∩ PC z0 Px  ∅. We fix g ∈ Py ∩ PC z0 Px . Then g y = y. Since g = g  z0 g , with g  ∈ PC and g  ∈ Px , we see that g x = g  z0 x. Now we consider the retraction ρ := ρA,C of the Bruhat–Tits building B(G) onto A with center C , Proposition 1.5.24. Then, as g x = g  z0 x and g  lies in PC whereas y and z0 x belong to A, we see that ρ(g y) = ρ(y) = y and ρ(g x) = z0 x. Let f : Z(k) → Aff(A) be the action map. We observe that x − y = x − z −1 x = f (z), and ρ(g x) − ρ(g y) = z0 x − y = z −1 z  x − z−1 x = f (z ). As z ∈ Z, for a ∈ Φ+ , a( f (z))  0, thus f (z) is dominant. So Lemma 5.4.1 can be applied and we obtain f (z )  f (z); that is, 0  − f (z −1 z ) and hence  ω(ξ(z−1 z )) = ξ(− f (z−1 z ))  0 for all dominant ξ. We will now make use of the following result about affine buildings. We will not use this proposition elsewhere in this book, so we will not prove it here. Interested readers can find a proof in [BT72, Proposition 2.8.1] or [AB08, Theorem 11.53]. The proof does not involve Bruhat–Tits theory of reductive groups. Proposition 5.4.3 Let X be an affine building, and Y be a bounded subset of X that is either convex or has a non-empty interior. Then Y is contained in an apartment of X if and only if it is isometric to a subset of an apartment. Proposition 5.4.4 Let Ω, Ω and Ω be three non-empty bounded subsets of the apartment A. We assume that at least one of them has a non-empty interior. Then (PΩ · PΩ ) ∩ (PΩ · PΩ ) = (PΩ ∩ PΩ ) · PΩ . Proof Let g = p p = p q, with p ∈ PΩ , p ∈ PΩ , and p, q ∈ PΩ . Let M = Ω ∪ Ω ∪ g · Ω and f : Ω ∪ Ω ∪ Ω → M be the map that is the identity on Ω ∪ Ω and is translation by g on Ω. Then f is well defined since g fixes every point of Ω ∩ (Ω ∪ Ω). The map f is easily seen to be an isometry. Now Proposition 5.4.3 implies that there is an apartment A that contains M. As Ω ∪ Ω ⊂ A ∩ A, there exists h ∈ PΩ ∩ PΩ such that h · A = A. Hence, hg · Ω ⊂ A. Therefore, by Proposition 4.2.24, there exists n ∈ N(k) such that hg · x = n · x for every x ∈ Ω. For x ∈ Ω and y ∈ Ω ∪ Ω we have d(n−1 · y, x) = d(y, n · x) = d(y, hg · x) = d(y, g · x) = d(y, x). If Ω (respectively, Ω ∪ Ω) contains a non-empty open subset of A, we deduce

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193

that n−1 · y = y and n ∈ PΩ ∩ PΩ (respectively, n · x = x and n ∈ PΩ ). Thus,  in either case, we have g ∈ h−1 nPΩ ⊂ (PΩ ∩ PΩ ) · PΩ . Corollary 5.4.5 Under the hypothesis of the preceding proposition, we have PΩ ∩ (PΩ · PΩ ) = (PΩ ∩ PΩ ) · (PΩ ∩ PΩ ). Proof If g = p p ∈ PΩ , with p ∈ PΩ and p ∈ PΩ , Proposition 5.4.4 implies that p = p−1 g ∈ (PΩ ∩ PΩ ) · (PΩ ∩ PΩ ). We can therefore assume  that p ∈ PΩ and then p ∈ PΩ also. Proposition 5.4.6 Assume that k is locally compact. Let n ∈ N(k) and x, y ∈ A. Assume that y  := n · y lies in the positive cone with vertex at x; that is, in the following subset of A: X := {x + v | v ∈ V(S ), a(v)  0 for all a ∈ Φ+ }. Then (Px nPy ) ∩ (U(k)nPy ) = nPy . Proof It is enough to show that Px ∩ U(k)Py ⊂ Py . This reduces to showing that for y that lies in X, (Px · Py ) ∩ (U(k) · Py ) = Py . So we need to prove that, given u ∈ U(k), if Px ∩ uPy  ∅, equivalently, if u ∈ Px · Py , then u fixes y. Since u ∈ G(k) , if we could show that u fixes y, then this would imply that u lies in Py (see Remark 4.1.13), and hence Px ∩ (U(k)Py ) ⊂ Py . We will now show that u does indeed fix y. Let C be a chamber of A that contains x in its closure. Let π be a uniformizer in k and λ : Gm → S  be any 1-parameter subgroup such that a(λ) > 0 for all a ∈ Φ+ . Then as u lies in U(k), we see that lim n→∞ λ(π n )uλ(π −n ) = 1, and hence for all sufficiently large positive integers n, λ(π n )uλ(π −n ) lies in PC . Therefore, λ(π n )uλ(π −n ) fixes C and hence u fixes λ(π −n ) · C = C + nλ for all sufficiently large positive integers n. Let U be the union of the chambers C+nλ, for λ as in the preceding paragraph and n a positive integer such that C + nλ is fixed by u. The observation in that paragraph is that for any given λ as before, for all sufficiently large positive integers n, C + nλ is contained in U. Now using the fact that R · X∗ (S ) is dense in V(S ), we easily see that the smallest closed convex subset of A containing x and U contains X. In particular, this closed convex set contains y. Now we consider an increasing family {Ci }n1 of subsets of U such that  each Cn is a union of finitely many chambers contained in U and n Cn = U. Then uG(k)1y ∩ (Px ∩ PCn )  ∅ for all n  1. Therefore u ∈ PCi and hence it

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lies in PCi ∩ (Px · Py ). However, by Corollary 5.4.5, PCi ∩ (Px · Py ) = (PCi ∩ Px ) · (PCi ∩ Py ). So there is a sequence {pi } ⊂ PCi ∩ Py (⊂ Py ) such that u pi ∈ PCi ∩ Px . Now finally we use the hypothesis that k is locally compact. Then Py is compact and hence a subsequence of {pi } converges to an element p0 of Py . We see

that up0 lies in ( i PCi ) ∩ Px . Hence it fixes the closed convex set spanned by  U = Ci and x. As this closed convex set contains y, we see that up0 fixes y.  But since p0 also fixes y, we conclude that u fixes y.

6 The Apartment

Let k be a field endowed with a discrete valuation ω : k × → R. We assume that k is Henselian and denote the unique extension of ω to any algebraic field extension of k also by ω. We denote by k s a separable closure of k and set Θ = Gal(k s /k). In this chapter we do not assume that the residue field of k is perfect. Except in §6.5 we will assume that ω(k × ) = Z. We refer to §6.5 for a discussion of how things change once this assumption is dropped. Let G be a connected reductive k-group. For a maximal k-split torus S of G, denote by S  the maximal torus in S ∩ Gder . Recall the vector spaces V(S) = R ⊗Z X∗ (S) and V(S ) = R ⊗Z X∗ (S ). The purpose of this section is to construct, for each maximal k-split torus S, an affine space A(S) over the vector space V(S ) satisfying Axioms 4.1.4, 4.1.6, and 4.1.9, although the last one of these will only be proved in the next chapter. An affine space satisfying Axiom 4.1.4 was already constructed rather easily in §4.4, and we know from Proposition 4.4.3 that it must be the correct one. But this construction is too abstract to allow Axioms 4.1.6 and 4.1.9 to be proved. Therefore, in this chapter, we will give a new and independent construction of the same affine space that does allow Axioms 4.1.6 and 4.1.9 to be proved. We will prove Axiom 4.1.6 in §6.3, while Axiom 4.1.9 will only be proved in §7.5. In what follows S will denote a fixed maximal k-split torus of G, N its the normalizer in G, and Z (⊂ N) its centralizer. The images of S, N and Z under the adjoint map G → Gad will be denoted by Sad , Nad and Zad respectively. In §6.1 we will construct A(S) in the case when G is quasi-split. This construction is based on the concept of “valuation of the root datum” due to Bruhat–Tits, Definition 6.1.2. The construction of such a valuation is in turn based on the concept of a weak Chevalley–Steinberg system, Definition 2.9.12. In §6.2 we will study the action of N(k) on the affine space A(S). We will allow general connected reductive groups in that section, even though at this 195

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point we will not know that valuations of the root datum exist in this generality; this will only be proved in Chapter 9. In §6.3 we will construct the affine root system in A(S)∗ and associated filtration groups of the root groups, again for a general reductive group G subject to the assumption that a valuation of the root datum exists. In §6.4 and §6.6 we will give explicit descriptions of the affine root system and the affine Weyl group, in the case that G is quasi-split.

6.1 The Apartment of a Quasi-split Reductive Group We shall construct the building of a general quasi-split reductive group G in a manner similar to the examples of SL2 and SU3 discussed in Chapter 3. Given a maximal k-split torus S we will first construct an affine space A(S), endow it with simplicial structure, associate to each simplex a bounded open subgroup, introduce an action of N(k), and then form the building to be the solution of the same problem as for SL2 or SU3 . The two essential differences from these basic examples, apart from the obviously necessary systematic use of Lie theory, are the following. First, there is no natural choice of a maximal k-split torus. We will need to make an arbitrary choice and then argue that the result is independent of this choice. Second, and more importantly, even once a maximal k-split torus S is fixed, there is no natural choice for the 1-parameter root subgroups Ga → G associated to each absolute root. The implication of this will be that, unlike in the case of SL2 , for a general split semi-simple group the apartment A(S) is not equal to R ⊗Z X∗ (S), but is rather an affine space under the vector space R ⊗Z X∗ (S). This was already alluded to at the end of §3.1. Each choice of a system of 1-parameter root subgroups will correspond to a choice of an origin of A(S), and thus a choice of identification between A(S) and the real vector space R ⊗Z X∗ (S). For a review of the concept of affine spaces, we refer the reader to §1.2. We will first treat the case of split semi-simple groups, where the ideas are most visible. Then we will turn to quasi-split semi-simple groups. The case of quasi-split reductive groups is then just a matter of notation.

(a) Split Semi-simple Groups We assume first that G is split and semi-simple and describe the construction of the apartment in this special case, where the notational burden is low and the ideas become most clear. Let Φ = Φ(S, G) ⊂ X∗ (S) be the root system of G. For every a ∈ Φ, there is a canonical subgroup Ua ⊂ G, the root subgroup corresponding to

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a. It is determined by the property that it is normalized by S and its Lie algebra is the root space ga . The group Ua is isomorphic to Ga , but there is no canonical choice for this isomorphism. Fixing an isomorphism ua : Ga → Ua is equivalent to fixing a non-zero element Xa ∈ ga , the equivalence being given by dua (1) = Xa . The isomorphism ua is defined over k if and only if Xa ∈ ga (k). Recall from the example of SL2 that we used the isomorphisms ua to construct the parahoric subgroups. We will want to do the same here, but for this we will need certain natural compatibilities between the various Xa with respect to taking commutators. The appropriate compatibilities are captured in the concept of a Chevalley system of Definition 2.9.8. Let us for now fix a Chevalley system and denote it by o. We obtain a system of isomorphisms uo, a : Ga → Ua defined over k. Set V(S) = R ⊗Z X∗ (S). For every v ∈ V(S) we can now define a bounded open subgroup Po,v ⊂ G(k) just as in the case of SL2 : we have S(k)0 = S(k)0 = X∗ (S) ⊗Z o× and define Po, v to be the subgroup generated by S(k)0 and uo, a (m− a(v) ) for all a ∈ Φ, where we are using Notation 3.1.1. We could now in principle proceed just as we did in the case of SL2 : define the set of affine roots as Ψ = {a + k | a ∈ Φ, k ∈ Z} and view them as affine functionals on the real vector space V(S), associate to each affine root a + k its vanishing hyperplane {v ∈ V(S) | a(v) = −k}, define the affine Weyl group to be the subgroup of affine linear transformations of V(S ) generated by the reflections along these hyperplanes, define an action of NG (S)(k)/S(k)1 on V(S) and relate it to the action of the affine Weyl group, and finally define the building by the same formula as for SL2 . The trouble with this approach would be that we made an arbitrary choice of a Chevalley system for the Lie algebra g, as well as an arbitrary choice for the maximal torus S. Especially the choice of Chevalley system is something whose effect on these constructions is not a priori clear. Thus, instead of going this route, we first stop to examine how much of the information inherent in the Chevalley system is really needed for the construction of the groups Po, v , upon which the construction of the building is ultimately based. Say that we chose for each a ∈ Φ a unit za ∈ o× and defined Xa = za Xa . Let us ignore for a moment that the collection (Xa )a may fail to be a Chevalley system, and let us denote it by o. Then uo , a (x) = uo, a (za x) and we see that Po, v = Po , v for any v ∈ V(S). Thus, for our purposes, we may freely modify Xa , or equivalently ua : Ga → Ua , by a unit. This can be reformulated as follows. If we are given an isomorphism ua : Ga → Ua defined over k, then

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we obtain a map ϕa : Ua (k)∗ = Ua (k) − {1} → Z by composing the inverse of ua with the valuation ω : k × → Z. If ua : Ga → Ua is another isomorphism defined over k, with corresponding ϕa , then ϕa = ϕa if and only if ua and ua differ by a unit. So for the purposes of constructing Po, v , it is enough to remember the collection (ϕo, a ) determined by the Chevalley system o. Let us make the convention that ϕo, a (1) = ∞. Then Po, v is the subgroup generated −1 ([−a(v), ∞]) ⊂ U (k). by S(k)0 and the subgroups ϕo, a a In fact, we can push this new language just a little bit further. First, notice −1 ([−a(v), ∞]) = ϕ−1 ([−a(v), ∞]). Define ϕ that ϕo, o+v, a (u) = ϕo, a (u)+a(v). a o, a 0 −1 Then Po, v is the subgroup generated by S(k) and ϕo+v, a ([0, ∞]). Note that ϕo+v, a now takes values in R ∪ {∞} instead of Z ∪ {∞}. The following lemma collects the most important properties of the collection (ϕo+v, a )a . Lemma 6.1.1 (1) The collection (ϕo+v, a )a depends only on the weak Chevalley system underlying the Chevalley system o. (2) The subset ϕo+v, a (Ua (k) − {1}) of R is a Z-torsor. −1 (3) For any r ∈ R, the preimage Ua,r := ϕo+v, a ([r, ∞]) is a bounded open −1 subgroup of Ua (k), and Ua,∞ := ϕo+v, a (∞) = {1}. (4) For t ∈ Sad (k) and u ∈ Ua (k), we have ϕo+v, a (tut −1 ) = ϕo+v, a (u) + ω(a(t)). (5) If n lies in the Tits group W ⊂ NG (S )(k) associated to the Chevalley system o (cf. Definition 2.9.8) and w ∈ W is its image, then ϕo+wv, wa (nun−1 ) = ϕo+v, a (u). Proof Since for each a, the element Xa in the weak Chevalley system is ambiguous only up to multiplication by the unit −1, the first point is clear. The second and third are immediate from the construction of ϕo+v, a and the fact that ua : Ga → Ua is an isomorphism. The fourth and fifth are direct computations, using the fact that a weak Chevalley system is invariant under its Tits group by definition.  These properties and the commutation relations in Construction 2.9.16(d) imply that the collection (ϕo+v, a )a is a valuation of the root datum of (G, S), compatible with ω, in the sense of the following definition due to Bruhat and Tits. Since we will eventually work with arbitrary connected reductive groups, we will discuss the definition in its proper generality. Thus, for a moment, we let G be an arbitrary connected reductive k-group and S ⊂ G a maximal k-split torus. Thus Φ = Φ(S, G) is a root system that may not be reduced. For a ∈ Φ, Ua denotes the a-root group of G with respect to S.

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Definition 6.1.2 A valuation of the root datum of (G, S) is a collection ϕ = (ϕa )a ∈Φ of morphisms ϕa : Ua (k) → R ∪ {∞}, satisfying the following properties. V 0 For each a, ϕa (Ua (k)) has at least 3 elements. V 1 For each a and r ∈ R, Ua,ϕ,r := ϕ−1 a ([r, ∞]) is a subgroup of Ua (k), and −1 Ua,ϕ,∞ := ϕa (∞) = {1}. V 2 For every n ∈ N(k) realizing the reflection along a, the function u → ϕa (u) − ϕ−a (nun−1 ) is constant on Ua (k)∗ = Ua (k) − {1}. V 3 Given a, b ∈ Φ and i, j ∈ R such that b  −R>0 · a, the commutator subgroup (Ua,ϕ,i , Ub,ϕ, j ) is contained in the group generated by Upa+qb,ϕ, pi+q j for all p, q ∈ Z>0 with pa + qb ∈ Φ. V 4 If a, 2a ∈ Φ then ϕ2a = 2ϕa |U2a (k) . In particular Ua, x,r ∩ U2a (k) = U2a, x,2r . V 5 For a ∈ Φ, and u ∈ Ua (k), let u , u  ∈ U−a (k) be as in Proposition 2.11.17. Then ϕ−a (u ) = −ϕa (u). A valuation ϕ of the root datum of (G, S) is said to be compatible with ω, or ω-compatible, if the following holds. V 6 For a ∈ Φ and z ∈ Z(k), ϕa (z−1 uz) = ϕa (u) + a(ν(z)), where ν : Z(k) → V(S ) is the homomorphism described in Axiom 4.1.4(A 1). In particular, ϕa (s−1 us) = ϕa (u) − ω(a(s)) for s ∈ S(k). In addition, a valuation ϕ of the root datum of (G, S) is said to be compatible with the topology on G(k) if the following holds. V 7 For a ∈ Φ, the subgroups Ua,ϕ,r , for r ∈ R, are bounded open subgroups of Ua (k) and they constitute a neighborhood base of 1 in Ua (k). If ϕ is compatible with the topology on G(k), then a sequence {ui } contained in Ua (k) converges to 1 if and only if the sequence {ϕa (ui )} converges to ∞. All the valuations constructed in this book have the properties V 6 and V 7. Moreover, the functions ϕa constructed and used in this book are locally constant on Ua (k)∗ , and for every a ∈ Φ, ϕa (Ua (k)∗ ) is a discrete subset of R. According to the conventions of §1.6, we can also index the filtration {Ua,ϕ,r }  with Ua,ϕ,r+ = s>r Ua,ϕ, s . by R, Remark 6.1.3 The definition of a valuation of the root datum due to Bruhat– Tits is more general, and allows for a variety of scenarios, including functions ϕa with dense image, or finite image.

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Remark 6.1.4 It is clear that except for V 3, each of the Axioms V 0 – V 5 depends only on the subgroup Ma of G generated by Z and the root groups U±a . In fact, V 0, V 1, V 4, V 5 depend only on the simply connected cover of the semi-simple subgroup G a generated by U±a . The same would be true for V 2, if we knew that for each z ∈ ZG (S)(k) and a ∈ Φ, ϕa (u) − ϕa (zuz−1 ) is constant on Ua (k)∗ . Indeed, V 2 implies this property, as we will see in the proof of Proposition 6.2.3. If V 6 is assumed, then this property holds independently of V 2, and then V 2 can be checked using G a . If Φ  ⊂ Φ is a closed subsystem and GΦ is the connected reductive subgroup of G generated by T and Ua for all a ∈ Φ , then a valuation (ϕa )a ∈Φ of the root datum of (G, S) restricts to a valuation (ϕa )a ∈Φ of the root datum of GΦ . Remark 6.1.5 For a ∈ Φ, r ∈ R and u ∈ Ua (k), from the definition of Ua,ϕ,r given in V 1, we see that u lies in this subgroup if and only if r  ϕa (u). From this it is clear that ϕa (u−1 ) = ϕa (u). Now for u ∈ Ua (k)∗ , let u , u  be as in Proposition 2.11.17, and m(u) = u uu . Then m(u)−1 = u −1 u−1 u −1 = m(u−1 ). Using V 5 we see that −ϕa (u) = −ϕa (u−1 ) = ϕ−a (u −1 ) = ϕ−a (u ). Thus V 5 implies that ϕ−a (u ) = −ϕa (u). Given u, v ∈ Ua (k) with ϕa (u) < ϕa (v) we infer that ϕa (uv) = ϕa (u) as follows. Letting r = ϕa (u) we know that uv ∈ Ua,ϕ,r since the latter is a group, and if ϕa (uv) > r then uv ∈ Ua,ϕ,r+ , which would imply u = uv −1 · v ∈ Ua,ϕ,r+ and hence ϕa (u) > r, a contradiction. Next we introduce an action of V(S) on the set of valuations of root data. Given a valuation ϕ = (ϕa ) a ∈Φ and v ∈ V(S) we define the collection ϕv = (ϕv, a ) a ∈Φ of maps ϕv, a : Ua (k) → R ∪ {∞}, where ϕv, a (u) = ϕa (u) + a(v). Lemma 6.1.6 For every v ∈ V(S), ϕv = (ϕv, a ) a ∈Φ is a valuation of the root datum of (G, S). Moreover, Ua,ϕ,r = Ua,ϕv ,r+a(v) . Proof It is easily seen from the definition of ϕv that it has the properties V 0, V 2, V 4, and V 5. To verify the properties V 1 and V 3 we first observe directly from Definition 6.1.2 that for a ∈ Φ and r ∈ R, Ua,ϕv ,r+a(v) = {u ∈ Ua (k) | ϕv, a (u)  r + a(v)} = {u ∈ Ua (k) | ϕa (u)  r } = Ua,ϕ,r . Using this  we easily see that ϕv also has the properties V 1 and V 3. It is clear that (ϕ, v) → ϕv is an action of the abelian group V(S) on the set of valuations of the root datum of (G, S). We will therefore write ϕv = ϕ + v. This action is not always simple: if AG is the maximal split torus in the center of G, then V(AG ) ⊂ V(S) is the kernel of this action, equivalently the stabilizer

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of any point. More importantly, this action is not transitive. Its orbits will play an essential role in what follows, so we make the following definition. Definition 6.1.7 Two valuations ϕ and ϕ  of the root datum of (G, S) are called equipollent if there exists v ∈ V(S) such that ϕa (u) = ϕa (u) + a(v). The equipollence class of ϕ is thus the orbit through ϕ of the action of V(S) on the set of valuations of the root datum of (G, S). In particular, it is clear that equipollence is an equivalence relation. Note also that if ϕ is compatible with ω, then so is any valuation equipollent to ϕ. Thus, compatibility with ω is a property of an equipollence class of valuations of the root datum. 6.1.8 Next we introduce an action of the normalizer N(k) of S in G(k) on the set of valuations of the root datum of (G, S): N(k) acts on the character group X∗ (S) and the kernel of this action is Z(k). So this action factors through the Weyl group W = N(k)/Z(k), and it keeps the set Φ = Φ(S, G) of roots stable. For n ∈ N(k) and a ∈ Φ, the transform of a by n will be denoted by n · a. Let ϕ = (ϕa ) a ∈Φ be a valuation of the root datum of (G, S). For n ∈ N(k), and a ∈ Φ, we define n · ϕa to be the map u → ϕn−1 ·a (n−1 un) for u ∈ Ua (k). It is easily seen that a → n · ϕa , for a ∈ Φ, is a valuation of the root datum of (G, S). This valuation will be denoted by nϕ. The map (n, ϕ) → nϕ is an action of N(k) on the set of valuations of the root datum of (G, S). More generally, consider the natural surjective homomorphism G → Gad . Its kernel is central (but it may not be smooth) and hence the conjugation action of G on itself factors through an action of Gad on G, which we will denote again by (x, g) → xgx −1 for x ∈ Gad (A) and g ∈ G(A) for any commutative k-algebra A. We denote by Nad and Zad the images of N and Z in Gad respectively. Note that Nad is the normalizer of S in Gad and Zad is the centralizer of S. Then Nad (k) acts on Φ since the map G → Gad identifies Φ with the root system for Gad . Now for n ∈ Nad (k), if we define n · ϕa again to be the map u → ϕn−1 ·a (n−1 un) for u ∈ Ua (k), then we easily see that a → n · ϕa , for a ∈ Φ, is a valuation of the root datum of (G, S). As in the preceding paragraph, we will denote this valuation by nϕ. The map (n, ϕ) → nϕ, for n ∈ Nad (k), is an action of Nad (k) on the set of valuations of the root datum of (G, S). The action of N(k) is obtained from the action of Nad (k) via the natural map N(k) → Nad (k). The action of Nad (k) on S induces an action on V(S). We observe that for a ∈ Φ, v ∈ V(S) and n ∈ Nad (k), a(n · v) = (n−1 · a)(v). Therefore, n · (ϕ + v) = n · ϕ + n · v, for all n ∈ Nad (k) and v ∈ V(S). In other words, the actions of Nad (k) and V(S) on the set of valuations of root data are compatible, that is they fuse together to give an action of V(S)  Nad (k). We now assume again that G is split. The concepts of valuations of the root

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datum and their equipollence will in a moment allow us to dispense with the choice of a (weak) Chevalley system. We begin by defining, for each valuation ϕ of the root datum of (G, S), the bounded open group Pϕ to be the subgroup of G(k) generated by S(k)0 and Ua,ϕ,0 for all a ∈ Φ. If ϕ is the valuation ϕo+v obtained from a (weak) Chevalley system o and v ∈ V(S) then Pϕ = Po, v . With the above notation we have ϕo+v = ϕo + v. Since G is semi-simple, AG = {1} and the action of V(S) on the set of valuations of root data is simple. Therefore, the set of valuations of the root datum of (G, S) that are of the form ϕo+v , that is, the equipollence class of ϕo in the sense of Definition 6.1.7, has a natural structure of an affine space under the vector space V(S) in the sense of Definition 1.2.1. The open bounded subgroups Po, v are naturally parameterized by this equipollence class of valuations. The key question now is: how does this affine space depend on the choice of weak Chevalley system o? As we have just discussed, the (weak) Chevalley system o contains more information than is strictly needed to specify the valuation ϕo . It turns out that, instead of choosing elements Xa for all roots a ∈ Φ, it is enough to choose elements Xa for a set of simple roots. That is, we choose a Borel subgroup B containing S, and for each simple root a ∈ Δ(S, B) a non-zero element Xa ∈ ga (k). The tuple (S, B, {Xa }) is called a k-pinning of G, cf. Definition 2.9.1. Unlike in the case of a Chevalley system, there are no further conditions placed on the elements Xa . As is shown in Proposition 2.9.11, there exists a unique weak Chevalley system extending the pinning, that is, containing the set {Xa }a ∈Δ(S,B) . Therefore a pinning leads to a valuation of the root datum. Definition 6.1.9 A valuation of the split root datum coming from a pinning is called a Chevalley valuation. We have not yet removed the choice of a weak Chevalley system, but we have at least reduced it to the choice of a pinning. To summarize, given a pinning (S, B, {Xa }) of G, which we now denote by the same symbol o that we used for the choice of a Chevalley system, we obtain a valuation ϕo of the root datum of (G, S), and for each v ∈ V(S) we have the equipollent valuation ϕo+v and hence the bounded open subgroup Po,v of G(k). There are obvious actions of Nad (k) on the set of all pinnings, as well as on the set of all Chevalley bases. We have also defined above an action of Nad (k) on the set of all valuations of the root datum of (G, S). The reader can readily verify that these three actions are compatible. In particular, we obtain the following result. Fact 6.1.10 If ϕ is a Chevalley valuation, then nϕ is also a Chevalley valuation for every n ∈ Nad (k).

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We have also remarked that the action of Nad (k) on the set of all valuations of the root datum is compatible with the action of V(S) in the sense that the two actions splice to an action of V(S)  Nad (k). But there is a second kind of compatibility between these two actions that is essential. To state it, we need some preparation. Let Ssc be the preimage of S in the simply connected cover Gsc of G. The isogenies Ssc → S → Sad lead to inclusions X∗ (Ssc ) → X∗ (S) → X∗ (Sad ) with finite cokernels, which then become isomorphisms upon tensoring with R. In this way, the coroot lattice Q∨ = X∗ (Ssc ) and the coweight lattice P∨ = X∗ (Sad ) are naturally embedded in the real vector space V(S). Recall the functorial valuation homomorphism (2.5.1) ωS : S(k) → X∗ (S) ⊂ V(S). Fact 6.1.11 Under the valuation homomorphism, the images of Ssc (k), S(k), and Sad (k) in V(S) are Q∨ = X∗ (Ssc ), X∗ (S), and P∨ = X∗ (Sad ), respectively. Lemma 6.1.12

Let ϕ be a valuation of the root datum of (G, S).

(1) For any n ∈ Nad (k) with image w ∈ W we have n[ϕ + v] = nϕ + wv. (2) If ϕ is equipollent to a Chevalley valuation, then for any t ∈ Sad (k) we have t ϕ = ϕ − ω(t). Proof

The first claim is a direct computation: [n(ϕ + v)]a (u) = (ϕ + v)w −1 a (n−1 un) = ϕw −1 a (n−1 un) − w −1 a(v) = [nϕ]a (u) − a(wv) = [nϕ + wv]a (u).

For the second, we reduce using the first to the case that ϕ is itself a Chevalley valuation. Let (Xa ) be a Chevalley system giving rise to ϕ and let (ua ) be the corresponding system of isomorphisms ua : Ga → Ua . Then −1 −1 −1 [t ϕ]a (u) = ϕa (t −1 ut) = ω(u−1 a (t ut)) = ω(a(t) u a (u)) = ϕ a (u) − ω(a(t)).

 The following simple but essential lemma is the key to removing the choice of pinning, and hence of Chevalley system. Lemma 6.1.13 Let ϕ be a Chevalley valuation and ϕ  an arbitrary valuation. Then ϕ  is a Chevalley valuation if and only if there exists a (necessarily unique) v = v(ϕ, ϕ ) ∈ X∗ (Sad ) ⊂ V(S) such that ϕ  = ϕ + v. In particular, any two Chevalley valuations are equipollent.

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Proof Assume first that ϕ  is also a Chevalley valuation. Then both ϕ and ϕ  arise from weak Chevalley systems. Choose a Borel subgroup containing S and let {Xa}a∈Δ(S, B) and {Xa }a∈Δ(S, B) be subsets of the respective weak Chevalley systems. Thus (S, B, {Xa}) and (S, B, {Xa }) are two pinnings of G, giving rise to ϕ and ϕ  respectively. Let t ∈ Sad (k) be the unique element determined by Xa = Ad(t)Xa. Then ϕ  = t ϕ = ϕ − ω(t) and we can take v = −ω(t). Conversely, assume that ϕ  = ϕ + v for some v ∈ X∗ (Sad ) and choose an arbitrary t ∈ Sad (k) with v = −ω(t). If (S, B, {Xa}) is a pinning giving rise to  ϕ, then Ad(t)(S, B, {Xa}) is a pinning giving rise to ϕ  = t ϕ. Remark 6.1.14 If in the above lemma one does not assume that the valuation ω : k × → R is normalized by ω(k × ) = Z, then one must replace the lattice X∗ (Sad ) of V(S) by X∗ (Sad ) ⊗Z ω(k × ). Lemma 6.1.13 implies that a given valuation ϕ is equipollent to one fixed Chevalley valuation if and only if it is equipollent to every Chevalley valuation. Hence, the following object is well defined. Definition 6.1.15 Let A(S) be the set of all valuations of the root datum that are equipollent to one, hence every, Chevalley valuation. This set is called the apartment associated to S. Said differently, the vector space V(S) acts simply on the set of all valuations of the root datum, and there is a distinguished orbit for this action, namely the orbit containing all Chevalley valuations. This orbit is A(S), and the vector space V(S) acts simply transitively on A(S). If o is a weak Chevalley system, then ϕo ∈ A(S). The map ιo : V(S) → A(S),

v → ϕo + v

(6.1.1)

is a bijection. It is clear that this bijection translates the action of V(S) on A(S) to the action of V(S) on itself by addition. In other words, A(S) is an affine space under the vector space V(S) in the sense of Definition 1.2.1. The coweight lattice X∗ (Sad ) ⊂ V(S) acts simply transitively on the subset of A(S) consisting of Chevalley valuations, by Lemma 6.1.13. By Fact 6.1.10 and Lemma 6.1.12, the action of Nad (k) on the set of all valuations preserves A(S) and acts on it by affine automorphisms. The affine action of Nad (k) on A(S) induces, by taking derivative, a linear action on V(S) which factors through the Weyl group W and coincides with the natural action of W on V(S). The points of the apartment A(S) are thus valuations of the root datum. Rather than writing ϕ ∈ A(S), we shall write x ∈ A(S) and denote by ϕx, a the corresponding Ua (k) → R ∪ {∞}. Every point x ∈ A(S) determines a bounded open subgroup Px , namely the group generated by S(k)0 and ϕ−1 x, a ([0, ∞]).

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Remark 6.1.16 We have seen in Lemmas 6.1.12 and 6.1.13 that the action of N(k) on the set of valuations of the root datum of (G, S) preserves A(S) and the restriction of this action to S(k) is given by translations according to the valuation homomorphism ωS : S(k) → V(S). This turns out to be true for any equipollence class of valuations compatible with ω, not just the distinguished one that contains all Chevalley valuations. Moreover, for a general connected reductive group G and a maximal k-split torus S, the valuation homomorphism S(k) → V(S) can be extended naturally to a valuation homomorphism Z(k) → V(S) through which Z(k) acts on a given equipollence class of valuations of the root datum of (G, S). We refer the reader to §6.2, in particular to Proposition 6.2.4.

(b) Quasi-split Semi-simple Groups We now drop the assumption that G is split, and only assume that it is quasi-split. We still assume that G is semi-simple. Let S be a maximal k-split torus of G and let T := Z be its centralizer; T is  = Φ(T, G) and Φ = Φ(S, G) be a maximally split maximal torus of G. Let Φ the absolute and relative root systems. We shall define the apartment A(S) as an affine space under the vector space V(S) := R ⊗Z X∗ (S) = R ⊗Z X∗ (T)Θ = (R ⊗Z X∗ (T))Θ . In the split case, A(S) was defined as the set of valuations of the relative root datum that are equipollent to one, hence any, Chevalley valuation. In the current case of a quasi-split group the construction will be quite analogous, and will use the relative root datum. The definition of a valuation of a root datum remains the same, namely Definition 6.1.2, but we must now use the relative root datum of the maximal split torus S, and not the absolute root datum of the maximal torus T. The definition of equipollence remains the same, namely Definition 6.1.7; since G is semi-simple we still have S  = S. What we must do is define what a Chevalley valuation is and prove that all Chevalley valuations are equipollent to each other. This is based on the notion of a weak Chevalley–Steinberg system, defined in Definition 2.9.12. Until the end of this section, let /k be the splitting extension of G. Construction 6.1.17 Consider a weak Chevalley–Steinberg system. Let a ∈ Φ. We construct a function ϕa : Ua (k)∗ → R as follows. According to Lemma 2.9.14 the weak Chevalley–Steinberg system can be refined to a Chevalley–Steinberg system.  lifting a. The isomorphism (2.9.2) induces If a is of type R1 choose  a∈Φ

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an isomorphism k a → Ua (k), and ϕa is the composition of its inverse with the fixed valuation ω : k a → R ∪ {∞}.  lifting a such that   We If a is of type R2, choose  a,  a ∈ Φ b= a+ a  ∈ Φ. 0 have the isomorphisms Uk a /kb (k b) → Ua (k) and Uk /k (k b) → U2a (k) given a  b  by (2.9.3). Therefore it is enough to define ϕa : Uk a /kb (k b) → R ∪ {∞} and ϕ2a : Uk0 /k (k b) → R ∪ {∞}. This situation was discussed in §3.2. As stated in a  b  Remark 3.2.10 we may or may not incorporate a μ-shift into the definition. For the moment we will use the unshifted version, leaving the shifted version for the next construction. Thus we set ϕa ((u, v)) = 12 ω(v) and ϕ2a ((0, v)) = ω(v), for (u, v) ∈ Uk a /kb (k b). Since different Chevalley–Steinberg systems refining the given weak system differ from one another only by multiplication by signs, the resulting functions ϕa depend only on the weak Chevalley–Steinberg system. As was discussed in §3.2, the description of the structure of root groups for multipliable roots in residual characteristic 2 involves a certain quantity, which was denoted by μ. We saw that incorporating this quantity into the valuation of the root datum made the description of the affine root system uniform. We will now generalize this to the case of an arbitrary quasi-split semi-simple group. This is only relevant when p = 2 and Φ is not reduced. Construction 6.1.18 Let B be a Borel k-subgroup of G containing the chosen maximal k-split torus S. Define an element v ∈ V(S) by requiring that for each relative simple root a the quantity a(v) is zero if a is of type R1, and equals μa /2 if a is of type R2, where μa is the number μ of (2.8.1) for the quadratic  is a lift of a;   is the unique other lift of a a∈Φ a ∈ Φ extension k a/k b, and    that is not orthogonal to  a, and b =  a+ a. We emphasize that μa is to be computed with respect to the valuation on k a that extends the valuation on k. Thus, if we are using on k the valuation normalized so that ω(k × ) = Z, then the value of μa is equal to the inverse of the ramification degree of k a/k multiplied by the value provided by Lemma 2.8.1 for the quadratic extension k a/k b. Proposition 6.1.19 The collection of maps ϕ = (ϕa )a ∈Φ of Construction 6.1.17 is a ω-compatible valuation of the relative root datum of (G, S). The same is true for the shifted collection ϕ − v, where v is as in Construction 6.1.18. Proof According to Lemma 6.1.6, the axioms of Definition 6.1.2 are insensitive to shifting, so we may verify them either for ϕ or for ϕ − v. Axiom V 3 for ϕ follows from the commutation relations §2.9(d). Following Remark 6.1.4, we

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may prove the other axioms for a given root a ∈ Φ by considering the subgroup G a of G generated by T, Ua , U−a . In fact, it is enough to consider non-divisible roots a, for each non-non-divisible root will also be covered this way. If t ∈ S(k), then it is immediate from Construction 6.1.17 that ϕa (tut −1 ) = ϕa (u) + ω(a(t)). Therefore it is enough to consider the simply connected cover of G a . This is the restriction of scalars of either SL2 or SU3 . Since Construction 6.1.17 is compatible with restriction of scalars, it is enough to check SL2 and SU3 . The case of SL2 was already dealt with in Lemma 6.1.1. We consider the case of SU3 and treat the roots a and 2a simultaneously. The images of (ϕ − v)a and (ϕ − v)2a are the sets Ja and J2a computed in Corollary 3.2.4. All these are infinite, hence V 0. Axiom V 1 follows directly from the formulas for (ϕ − v)a , as stated in Fact 3.2.1. Axiom V 4 is again immediate from the definition of (ϕ − v)a and (ϕ − v)2a . Axioms V 2 and V 5 follow from Lemma 3.2.8 and Fact 3.2.9. 

Definition 6.1.20 The valuation ϕ of the relative root datum of Construction 6.1.17 is called a Chevalley valuation. Its shift ϕ − v with v as in Construction 6.1.18 is called an adjusted Chevalley valuation. Construction 6.1.21 Consider the split group G and its maximal split torus T . The Galois group of /k acts on the set of all valuations of the absolute root datum of that split group as follows. If σ is a Galois automorphism and ϕ = (ϕa) is a valuation of the absolute root datum, then (σϕ)a : Ua() → R ∪ {∞} is defined as ϕσ −1 a(σ −1 (u)). The Galois group also acts on the set of weak Chevalley systems for the split group G and its split maximal torus T and one checks immediately that the assignment of a Chevalley valuation to a weak Chevalley system is equivariant. In particular, the Chevalley valuation for G associated to a weak Chevalley–Steinberg system is fixed by the Galois action. Thus a weak Chevalley–Steinberg system gives a Chevalley valuation ϕ ∈ A(S) by Construction 6.1.17 and also a Chevalley valuation ϕ  ∈ A(T ) by the analogous construction in the split case. These two are related as follows.  If a ∈ Φ is a relative root of type R1, choose any absolute root  a ∈ Φ lifting a. Then ϕa : Ua (k) → R ∪ {∞} is the composition of the isomorphism Ua (k) → Ua(k a) ⊂ Ua() described in (2.6.1) with ϕ , a : Ua() → R ∪ {∞}.  lifting If a ∈ Φ is a relative root of type R2, choose any absolute root  a∈Φ  be the unique other such root for which   a and let  a ∈ Φ b =  a+ a  ∈ Φ. Recall the isomorphism Ua (k) → U[ a] (k b ) ⊂ U[ a] () = Ua  () · Ua  () · Ub  () given by (2.6.2). Then ϕa : Ua (k) → R ∪ {∞} equals the composition of the

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isomorphism (2.6.2) with the map   (u a, u a , u b) → min ϕ , a(u a), ϕ , a (u a ), 12 ϕ , b (u b) . In particular, for any non-divisible a ∈ Φ and r ∈ R one has   Ub,ϕ ,r Ua,ϕ,r = Ua (k) ∩

(6.1.2)

b

where b runs over the non-divisible elements of Φ(T ) whose restriction to T is either a or 2a. One expresses this relationship by saying that ϕ is the descent of ϕ  . We will study this relationship in the more general setting of arbitrary connected reductive groups in §9.6. From the case of split groups we obtain the affine space A(T ) under the vector space V(T ) = R ⊗Z X∗ (T ). The action of Θ on the set of valuations of the root datum preserves A(T ). There is a unique injection of affine spaces A(S) → A(T ) that sends every Chevalley valuation ϕ to ϕ  . It is V(S)-equivariant and identifies A(S) with A(T )Θ . Recall the valuation homomorphism (2.5.1) ω = ωT : T(k) → Q ⊗Z X∗ (T)Θ ⊂ V(S). Proposition 6.1.22 Let ϕ be a Chevalley valuation and ϕ  an arbitrary valuation. Then ϕ  is a Chevalley valuation if and only if there exists a (necessarily unique) v = v(ϕ, ϕ ) ∈ ω(Tad (k)) ⊂ V(S) such that ϕ  = ϕ + v. In particular, any two Chevalley valuations are equipollent. Proof The proof is essentially the same as for Lemma 6.1.13. Assume first that ϕ  is a Chevalley valuation. Then both ϕ and ϕ  arise from weak ChevalleySteinberg systems. These can be refined to Chevalley–Steinberg systems by Lemma 2.9.14. Choose a Borel k-subgroup containing T and let {Xa}a∈Δ(T , B) and {Xa }a∈Δ(T , B) be subsets of the respective Chevalley–Steinberg systems. Thus (T, B, {Xa }) and (T, B, {Xa }) are two k-pinnings of G, giving rise to ϕ and ϕ  respectively. Let t ∈ Tad (k) be the unique element determined by X = Ad(t) Xb. Then ϕ  = t ϕ = ϕ − ω(t) and we can take v = −ω(t). b Conversely, assume that ϕ  = ϕ + v for some v ∈ ω(Tad (k)) and choose an arbitrary t ∈ Tad (k) with v = −ω(t). If (T, B, {Xa }) is a pinning giving rise to  ϕ, then Ad(t)(T, B, {Xa }) is a pinning giving rise to ϕ  = t ϕ. Remark 6.1.23 The lattice ω(Tad (k)) ⊂ V(S) can be computed explicitly, see Lemma 6.4.10.

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Definition 6.1.24 The apartment of G relative to S is the affine space A(S) under V(S) whose elements are the valuations of the relative root datum that are equipollent to one, hence any, Chevalley valuation (equivalently, adjusted Chevalley valuation). Remark 6.1.25 If either p  2 or Φ is reduced, then v = 0 so an adjusted Chevalley valuation is the same as a Chevalley valuation. When p = 2 and Φ is non-reduced then adjusted Chevalley valuations are more convenient base points in A than Chevalley valuations. Lemma 6.1.26

Let x ∈ A(S), v ∈ V(S).

(1) For any t ∈ Tad (k) we have t ϕ = ϕ − ω(t). (2) For any n ∈ Nad (k) with image w ∈ WG (T) we have n[ϕ + v] = nϕ + wv. (3) The kernel of the action of T(k) on A(S) is T(k)1 . Proof (1) and (2) follow from Construction 6.1.21 and Lemma 6.1.12, while  (3) follows from the fact that T(k)1 is the kernel of ωT . As in the split case, we observe that the action of Nad (k) on the set of valuations of the root datum preserves the set of Chevalley valuations, so the above Lemma implies in particular that this action preserves A(S).

(c) Quasi-split Reductive Groups Let G be a quasi-split reductive group. Let S  be a maximal k-split torus in the derived subgroup of G, S the unique maximal k-split torus of G containing S , and T (= Z) the unique maximal torus of G containing S , equivalently S. Then S = S  if and only if the center of G is anisotropic, and S = T if and only if G is split. Each of S , S, and T, determines the other two. We will denote the subtorus T ∩ Gder by Tder . The inclusion S  → S induces an inclusion V(S ) → V(S). This inclusion is canonically split. Indeed, if AG is the maximal split torus in the center of G, then composing the inclusion S  → S with the projection S → S/AG induces an isogeny S  → S/AG , hence an isomorphism V(S ) → V(S/AG ). The projection map V(S) → V(S/AG ) composed with the inverse of this isomorphism provides the desired splitting. In particular, we have the identifications V(S ) = R ⊗Z X∗ (S ) = R ⊗Z X∗ (Tsc )Θ = R ⊗Z X∗ (Tad )Θ = R ⊗Z X∗ (S/AG ).

(6.1.3)

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A valuation of the root datum of (G, S) is the same as a valuation of the root datum of S  and Gder . We can apply the constructions of the previous subsections to the derived subgroup Gder to obtain a canonical affine space A(S ) of valuations of the root datum. We can also denote it by A(S) or A(T). This affine space does not change if we replace G by its derived subgroup Gder , or the simply connected cover Gsc of Gder , or the adjoint group Gad . The underlying vector space is V(S ). The action of V(S) on A(S) factors through the quotient map V(S) → V(S/AG ) = V(S ). Definition 6.1.27 The apartment A(S) associated to S is the affine space A(S ) associated to the semi-simple group Gder and its maximal k-split torus S  (⊂ S) by Definitions 6.1.15 respectively 6.1.24. Lemma 6.1.28 The action of Nad (k) on the set of valuations of the root datum preserves the affine space A(S). Proof Fact 6.1.10 shows that this action preserves the set of all Chevalley valuations, first over the splitting field of G, but by Construction 6.1.17 also over k. Lemma 6.1.26 completes the proof.  Note that the group N(k) acts on A via the quotient map NG (T)(k) → Nad (k). The kernel of this action is no longer just T(k)1 , because the center ZG (k) acts trivially, but it may not lie in T(k)1 . Lemma 6.1.29 if n ∈ T(k)1 .

An element n ∈ N(k) ∩ G(k)1 acts trivially on A if and only

Proof If n acts trivially on the affine space A, it also acts trivially on its space of translations V(S ). Therefore the image of n in the Weyl group WG (T)(k) := NG (T)(k)/T(k) is trivial and we conclude n ∈ T(k). The group T(k)1 is the kernel of the map ω : T(k) → X∗ (T)Θ obtained by Q ×

0 ×T applying ⊗Z X∗ (T) to the valuation ω : k → Q. The isogeny ZG der → T 0 induces a Θ-equivariant inclusion X∗ (ZG ) ⊕ X∗ (Tder ) → X∗ (T) with finite co0 )Θ ⊕X (T )Θ . → X∗ (ZG kernel, which in turn induces an isomorphism X∗ (T)Θ ∗ der Q Q Q 0 )Θ and X (T )Θ are Therefore n ∈ T(k)1 if and only if its images in X∗ (ZG ∗ der Q Q trivial. The first image being trivial is by definition equivalent to n ∈ G(k)1 , and the second image being trivial is by definition equivalent to the image of n in Tad (k) belonging to Tad (k)1 , which in turns is equivalent to n acting trivially on A. 

Remark 6.1.30 Let a ∈ Φ be a non-divisible relative root. Let G a be the subgroup of G generated by T and the root groups U±a and U±2a . If a is of type

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R1, then (G a )sc is isomorphic to Rk a /k SL2 . If a is of type R2, then (G a )sc is isomorphic to Rk b /k SU3 (k a/k b). Note that G a is a Levi subgroup of G. We will denote by A(S, G) and A(S, G a ) the apartments associated to the maximal split torus S in the two groups G and G a , and by V  and Va the underlying vector spaces. The vector space Va is canonically a direct summand of V , as explained in §2.6(d). We will now construct a map A(S, G) → A(S, G a ) whose vector part is the projection V  → Va . Let ϕ = (ϕb ) b ∈Φ be a valuation lying in A(T, G). We claim that ϕ  = (ϕb ) b ∈ {±a,±2a } is a valuation lying in A(T, G a ). All the axioms in Definition 6.1.2 carry over from ϕ to ϕ , so ϕ  is indeed a valuation of the root datum of G a relative to T, but we need to show that ϕ  is equipollent to a Chevalley valuation. This follows from the following two observations. First, if ϕ is a Chevalley valuation, then so is ϕ , because if ϕ arises from a Chevalley–Steinberg system {X b} b ∈ Φ(T  ,G) for G,  then {X b} b ∈ Φ(T  ,G a ) is a Chevalley–Steinberg system for G a and ϕ arises from       it. Second, if v ∈ V has image v ∈ Va , then (ϕ + v) = ϕ + v . This completes the construction of the map A(T, G) → A(T, G a ). We just argued that this map respects Chevalley valuations. It also respects adjusted Chevalley valuations, because the element v ∈ V(T) corresponding to a Borel k-subgroup of G maps under V  → Va to the element v ∈ Va corresponding to B ∩ G a . Since the jump sets of Definition 6.4.3 depend only on the root subgroups, we see that the elements of Ψ(S, G) with derivative in the set {±a, ±2a} are precisely the pull-backs of the elements of Ψ(S, G a ). We collect our observations as follows. The map A(T, G) → A(T, G a ) (1) is a surjective morphism of affine spaces; (2) identifies A(T, G a ) with the quotient of A(T, G) by the subspace ker(a) ⊂ V ; (3) has as its vector part the natural projection V  → Va ; (4) maps Chevalley valuations to Chevalley valuations, and adjusted Chevalley valuations to adjusted Chevalley valuations; (5) identifies the affine roots on A(T, G) with derivative in {±a, ±2a} with the affine roots on A(T, G a ). We remark that there is no natural affine map A(T, G a ) → A(T, G) whose vector part is the natural inclusion Va → V . One can obtain such a map by choosing a point x ∈ A(T, G), taking its image x  ∈ A(T, G a ), and using A(T, G a )

ι−1 x

/ V a

/ V

ιx

/ A(T, G).

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6.2 Affine Reflections and Uniqueness of Valuations In §6.1 we constructed a canonical affine space A(S) associated to a maximal k-split torus S of a quasi-split reductive k-group G, cf. Definition 6.1.27. Its points were valuations of the root datum of (G, S) in the sense of Definition 6.1.2. More precisely, A(S) is a single equipollence class (Definition 6.1.7) of valuations of root data, and in fact the unique equipollence class that contains all Chevalley valuations (Definition 6.1.9). In this section we consider a general connected reductive k-group G that may not be quasi-split. We let again S  be a maximal k-split torus in the derived subgroup of G, S the unique maximal k-split torus of G containing S . The normalizer (respectively, the centralizer) of S  coincides with the normalizer N (respectively, the centralizer Z) of S in G. The centralizer Z is a torus if and only if G is quasi-split. In the generality being considered in this section, there do not exist Chevalley valuations. After much work we will be able to construct, in Chapter 9, a canonical affine space A(S), and then prove in hindsight that its points are equipollent valuations of the root datum of (G, S), compatible with the valuation ω of k and topology on G(k). At the moment we do not know that. Instead, in this section we will collect basic results about an arbitrary equipollence class of valuations of the root datum. These results will be most relevant either to the canonical A(S) of Definition 6.1.27 when G is quasi-split, or to the canonical A(S) constructed in Chapter 9 for general G. In Proposition 6.2.6 we will show that these affine spaces are unique: a valuation of the root datum of S compatible with ω is unique up to equipollence. Let Φ = Φ(S, G) = Φ(S , G) be the relative root system of G with respect to S; this root system may not be reduced. For any root a ∈ Φ, Ua will denote the corresponding root group, and Ua (k)∗ will denote the subset Ua (k) − {1} of Ua (k). We will denote by G a the connected semi-simple subgroup of G generated by the root groups U±a . Consider an equipollence class A of valuations of the root datum of (G, S). Recall that A is an affine space under V(S ) = R ⊗Z X∗ (S ). For u ∈ Ua (k)∗ , the subset Hu = {ϕ ∈ A | ϕa (u) = 0} ⊂ A is immediately seen to be a hyperplane with derivative a⊥ ⊂ V(S ). There exists a unique affine endomorphism of A that fixes Hu pointwise and whose derivative is the reflection ra of V(S ). It is given explicitly by ϕ → ϕ − ϕa (u)a∨ . We will call this reflection the affine reflection along Hu . Proposition 6.2.1 For u ∈ Ua (k)∗ , let m(u) = u uu  be as in Proposition 2.11.17. Then for ϕ ∈ A, ϕ − m(u)ϕ = ϕa (u)a∨ . Thus the action of m(u) on the

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213

set of valuations of the root datum of (G, S) preserves A and acts on it as the affine reflection in Hu . Proof for the canonical affine space of Definition 6.1.27 We already know that the derivative of the action of m(u) is the reflection ra , so it suffices to show that m(u) fixes Hu . For any point x ∈ A we have Hu = a⊥ + x − 12 xa (u)a∨ . We choose x to be a Chevalley valuation. The element m(u) lies in the simply connected cover of the semi-simple group of k-rank 1 that is generated by U±a and U±2a . Up to restriction of scalars, this group is either SL2 or SU3 . In Lemmas 3.1.3 and 3.2.8 we computed that m(u) is the product of the element w0 belonging to the Tits group, and the translation by −xa (u)a∨ . The element w0 fixes the hyperplane a⊥ + x and maps − 12 xa (u)a∨ to 12 xa (u)a∨ , and the claim follows.  Proof in the general case Let ϕ ∈ A and ϕa (u) = s. For b ∈ Φ, as m(u)−1 · b = m(u) · b = −b(a∨ )a + b, to prove the proposition, it would suffice to show that for t ∈ R, m(u)−1 · Ub, ϕ, t · m(u) = U−b(a∨ )a+b, ϕ, −b(a∨ )s+t . We first consider the case where b is not a rational multiple of a. Then using V 3 we see that in the unipotent group U generated by Uma+nb, ϕ, ms+nt , for m, n ∈ Z, n  1, such that ma + nb ∈ Φ, the subgroup V generated by Uma+nb, ϕ, ms+nt , for m, n ∈ Z, n > 1, with ma + nb ∈ Φ, is a normal subgroup. Both U and V are stable under the conjugation action of u , u, u , and hence also of m(u), on G(k). Moreover, U/V is a commutative group, and the product (in any order) of Uma+b, ϕ, ms+t , m ∈ Z such that ma + b ∈ Φ, maps bijectively onto U/V under the natural projection π : U → U/V. Thus, U/V is isomorphic to the direct prod

uct of the groups π Uma+b, ϕ, ms+t , for m ∈ Z such that ma+b ∈ Φ. It is obvious

 that under the induced action of m(u)−1 on U/V, the factor π Uma+b, ϕ, ms+t of U/V corresponding to the root ma + b is mapped isomorphically onto the factor corresponding to the root m(u)−1 · (ma + b) = m(u) · (ma + b). In particular, 

as m(u) · b = −b(a∨ )a + b, the factor π Ub, ϕ, t corresponding to b is mapped

 isomorphically onto the factor π U−b(a∨ )a+b, ϕ, −b(a∨ )s+t . From this, we infer that m(u)−1 · Ub, ϕ, t · m(u) = U−b(a∨ )a+b, ϕ, −b(a∨ )s+t . It remains to treat the case where b is a rational multiple of a. In view of the Axioms V 4 and V 5, it would suffice to treat the case b = −a. We need to show in this case that for t ∈ R, m(u)−1 · U−a,ϕ,t · m(u) = Ua, ϕ, 2s+t . For this purpose, we observe that m(u) = uu  (m(u)−1 u  m(u)), and as m(u)−1 u  m(u) belongs to Ua (k), we conclude that m(u ) = m(u) and from Axiom V 5 we see that −s = ϕ−a (u ) = −ϕa (m(u)−1 u  m(u)). Thus ϕa (m(u)−1 u  m(u)) = s =

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−ϕ−a (u ). However, according to V 2, x → ϕ−a (x) − ϕa (m(u)−1 x m(u)) is constant on U−a (k)∗ . As the value of this function at x = u  is ϕ−a (u ) − ϕa (m(u)−1 u  m(u)) = −2s, we see that ϕ−a (x) − ϕa (m(u)−1 x m(u)) = −2s. This is equivalent to 2s + ϕ−a (x) = ϕa (m(u)−1 x m(u)) for all x ∈ U−a (k)∗ and implies that m(u)−1 · U−a,ϕ,t · m(u) = Ua, ϕ, 2s+t .



The next proposition will describe a homomorphism ν : Z(k) → V(S ) of groups via which Z(k) acts on the affine space A. If the equipollence class consists of valuations that are compatible with ω in the sense of Definition 6.1.2, this is simply the negative of the valuation homomorphism of (2.6.4) for the group Z(k), composed with the natural projection V(S) → V(S ). Since these will be the valuations we are primarily interested in, the reader may safely skip ahead to Proposition 6.2.4. Lemma 6.2.2 Let Π be a basis of the root system Φ, and ϕ = (ϕa ) a ∈Φ and ϕ  = (ϕa ) a ∈Φ be two valuations of the root datum of (G, S). Then ϕ  = ϕ if and only if ϕa = ϕa for all a ∈ Π. Proof In view of Axiom V 4, it is enough to show that if ϕa = ϕa for all a ∈ Π, then ϕb = ϕb for every non-divisible root b ∈ Φ. Given a non-divisible root b, there is a root a ∈ Π and an element w in the Weyl group W := N(k)/Z(k) such that w · a = b. We express w as a product ra1 · · · ran of reflections, with ai ∈ Π for i  n. We fix u i ∈ Uai (k)∗ , then the image of m(u1 ) · · · m(un ) in the Weyl group is w. Now repeated application of  Proposition 6.2.1 implies that ϕb = ϕb . Proposition 6.2.3 There exists a unique homomorphism μ : Z(k) → V(S ) such that for all a ∈ Φ, z ∈ Z(k), ϕ ∈ A, and u ∈ Ua (k)∗ , the following identity holds ϕa (z −1 uz) = ϕa (u) + a(μ(z)). Moreover, if the valuation ϕ is compatible with ω, then μ = ν, where ν is as in Axiom V 6. Proof The uniqueness is clear. It is also clear that if μ satisfies the above identity for one ϕ ∈ A, it does so for all ϕ ∈ A. To prove existence, fix ϕ ∈ A and a ∈ Φ. Let n be an element of N(k) that maps onto the reflection ra in the Weyl group W, and let z be an arbitrary element of Z(k). Then from Axiom V 2 we see that the functions u → ϕ−a (u) − ϕa (nun−1 )

and u → ϕ−a (u) − ϕa (z−1 nun−1 z)

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215

are constant on U−a (k)∗ . Their difference −ϕa (nun−1 )+ϕa (z−1 nun−1 z) is therefore also constant on U−a (k)∗ . Replacing nun−1 (∈ Ua (k)∗ ) with x, we see that the function −ϕa (x) + ϕa (z −1 xz) is constant on Ua (k)∗ ; denote this constant by μa (z)(∈ R). Then ϕa (z−1 xz) = ϕa (x) + μa (z). Since ϕa (z −1 z −1 xzz ) − ϕa (x) 



= ϕa (z −1 xz) − ϕa (x) + ϕa (z −1 z−1 xzz ) − ϕa (z−1 xz) , we see that μa (zz ) = μa (z) + μa (z ). Thus μa is a homomorphism. For each a ∈ Φ, we have thus constructed a homomorphism μa : Z(k) → R. Fix now a basis Π of Φ. Since Π is a basis of V ∗ (S ) = R ⊗Z X∗ (S ), there exists a unique vector μ(z) ∈ V(S ) such that μa (z) = a(μ(z)) for all a ∈ Π. We denote the function z → μ(z) by μ. Consider the valuation ϕμ(z) = ϕ + μ(z) of the given root datum, it is clear that for a ∈ Π and z ∈ Z(k), (ϕμ(z) )a = (zϕ)a , and hence according to Lemma 6.2.2, zϕ = ϕμ(z) . The construction of μ is now complete. In the case that ϕ is compatible with ω, so that, Axiom V 6 holds, then uniqueness of μ implies that it equals ν.  The next proposition was already proved for the canonical affine space A(S) of Definition 6.1.27 in the case that G is quasi-split, namely in Lemma 6.1.28. We now state and prove it for an arbitrary equipollence class of valuations of the root datum of an arbitrary connected reductive group. Let μ : Z(k) → V(S ) be as in the preceding Proposition. Proposition 6.2.4 The action of N(k) on the set of valuations of the root datum of (G, S) preserves A and acts on it by affine transformations. The derivative of this action is the natural Weyl group action of N(k) on V(S ). The restriction of this action to Z(k)(⊂ N(k)) is given by z ϕ = ϕ + μ(z). In case A consists of valuations compatible with ω, then as μ = ν, the kernel of the action of N(k) ∩ G(k)1 on A equals Z(k)1 . Proof Let Π be a basis of Φ. For each a ∈ Π, we fix an element ua ∈ Ua (k)∗ . Then as the reflections ra , a ∈ Π, generate W = N(k)/Z(k), we see that N(k) is generated by Z(k) and the elements m(ua ), a ∈ Π. Now the proposition follows from Propositions 6.2.1 and 6.2.3.  Proposition 6.2.5 Let ϕ be a ω-compatible valuation of the root datum of (G, S) and G. The equipollence class A(S) of ϕ satisfies parts A 0 and A 1 of Axiom 4.1.4. If, for every g ∈ G(k), we define A(gSg −1 ) to be the equipollence class of the valuation of the root datum of gSg −1 obtained by transporting ϕ by

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The Apartment

Int(g), then A(gSg −1 ) depends only on the torus gSg −1 , and not on g, and the resulting collection of affine spaces satisfies part A 2 of Axiom 4.1.4. Proof Part A 0 holds by definition of equipollence. Proposition 6.2.4 implies part A 1, as well as the claim that A(gSg −1 ) depends only on the torus gSg −1 and not on the choice of g. Note that, since ϕ has been assumed to be compatible with ω, Proposition 6.2.3 implies that μ = ν, where μ and ν are as in that proposition. It is obvious that part A 2 holds.  Proposition 6.2.6 Any two ω-compatible valuations of the root datum of (G, S) are equipollent. Proof Let ϕ1, ϕ2 be two ω-compatible valuations of the root datum of (G, S). Let A1 and A2 be the equipollence classes of ϕ1 and ϕ2 , respectively. According to Propositions 6.2.5 and 4.4.3 there exists an N(k)-equivariant isomorphism of affine spaces A1 → A2 . After replacing ϕ2 with an equipollent valuation, we assume that this isomorphism maps ϕ1 onto ϕ2 . Then for any n ∈ N(k) we have ϕ1 − nϕ1 = ϕ2 − nϕ2 . Let a ∈ Φ and u ∈ Ua (k)∗ . According to Proposition 6.2.1, ϕi − m(u)ϕi = ϕi, a (u)a∨ . From this we conclude ϕ1, a (u) = ϕ2, a (u).  The following lemmas give useful descriptions of the sets of “jumps” of the filtration of Ua (k)∗ given by a valuation ϕ of the root datum of (G, S). Lemma 6.2.7 Let a, b ∈ Φ, and ra (b) = b − b(a∨ )a =: c. We assume that ϕc (Uc (k)∗ ) contains 0. Then b(a∨ ) · ϕa (Ua (k)∗ ) ⊂ ϕb (Ub (k)∗ ). Proof Proposition 6.2.1 implies that for u ∈ Ua (k)∗ , ϕ − ϕa (u)a∨ = m(u)ϕ. Now we fix an element x ∈ Uc (k)∗ such that ϕc (x) = 0. Then ϕb (m(u) x m(u)−1 ) − b(a∨ )ϕa (u) = (m(u)ϕ)b (m(u) x m(u)−1 ) = ϕc (x) = 0, for all u ∈ Ua (k)∗ . So we conclude that b(a∨ ) · ϕa (Ua (k)∗ ) ⊂ ϕb (Ub (k)∗ ).



Lemma 6.2.8 Let Π ⊂ Φ be a set of simple roots. For v ∈ V(S ), the following statements are equivalent. (1) a(v) ∈ ϕa (Ua (k)∗ ) for all a ∈ Π. (2) b(v) ∈ ϕb (Ub (k)∗ ) for all non-divisible roots b ∈ Φ. Proof As (2) ⇒ (1) is trivial, we must show (1) ⇒ (2). For this purpose, we fix for each a ∈ Π, an element ua ∈ Ua (k)∗ such that ϕa (ua ) = a(v). Let b ∈ Φ be a non-divisible root such that b(v) ∈ ϕb (ub ) for some ub ∈ Ub (k)∗ . Using

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217

Proposition 6.2.1 we see that for any a ∈ Π, 

ϕra (b) m(ua )−1 ub m(ua ) = (m(ua )ϕ)b (ub ) = ϕb (ub ) − b(a∨ )ϕa (ua ) = b(v) − b(a∨ )a(v) = (b − b(a∨ )a)(v) = ra (b)(v). Now let ura (b) = m(ua )−1 ub m(ua ) ∈ Ura (b) (k)∗ , then from the above we see that ra (b)(v) = ϕra (b) (ura (b) ) ∈ ϕra (b) (Ura (b) (k)∗ ). Thus the set of non-divisible b ∈ Φ such that b(v) ∈ ϕb (Ub (k)∗ ) contains Π and it is closed under reflection ra for a ∈ Π. As these reflections generate the Weyl group W of the root system Φ, and W · Π is the set of all non-divisible roots, we conclude that (1) ⇒ (2) holds. 

6.3 Affine Roots and Affine Root Groups In this section we continue with an arbitrary connected reductive group G and a maximal split torus S  of its derived subgroup, S the unique maximal split torus of G containing S . Let A be an equipollence class of valuations of the root datum of (G, S). For example, it could be the canonical equipollence class of Definition 6.1.27 when G is quasi-split. For a ∈ Φ and x ∈ A we have the filtration group Ua, x,r of Definition 6.1.2. Lemma 6.1.6 implies that Ua, y,r = Ua, x,r+ a,x−y for any two x, y ∈ A, where we have written x − y for the unique v ∈ V(S ) such that x = y + v. This property allows us to make the following definition: Definition 6.3.1 Given a real valued affine functional ψ on A with derivative a ∈ Φ, define the subgroup Uψ ⊂ Ua (k) by for any x ∈ A. We define Uψ+

Uψ = Ua, x,ψ(x)  = ψ  >ψ Uψ  .

We now have two systems of indexing the filtration subgroups of the root groups: by triples (a, x, r) consisting of a relative root a ∈ Φ, a point x ∈ A, and r ∈ R; or by affine functionals ψ ∈ A∗ . The translation in one direction is given by the above definition. For the translation in the other direction, note that for any triple (a, x, r) there exists a unique affine functional ψ with derivative a satisfying ψ(x) = r. Remark 6.3.2

One sees at once that Uψ = {u ∈ Ua (k) | xa (u)  ψ(x)}

for one, hence every, x = (xa ) a ∈Φ ∈ A. If 2a is also a root, then Axiom V 4 of Definition 6.1.2 implies that U2ψ = Uψ ∩ U2a (k).

218 Fact 6.3.3

The Apartment For n ∈ NG (S)(k) and ψ ∈ A(S)∗ we have Unψ = nUψ n−1 .

Recall our convention that for a ∈ Φ, U2a = {1} if 2a  Φ. Definition 6.3.4

Define the subsets Ψ ⊂ Ψ  ⊂ A∗ by Ψ  = {ψ ∈ A∗ | ψ ∈ Φ, Uψ+  Uψ }

and Ψ = {ψ ∈ A∗ | ψ ∈ Φ, Uψ  Uψ+ · U2a (k)}. The set Ψ will be called the affine root system of G (with respect to S), and its elements will be called affine roots. For a real valued affine functional ψ on A, the vanishing hyperplane consisting of points of A where ψ vanishes will be denoted by Hψ . For u ∈ Ua (k)∗ the condition u ∈ Uψ − Uψ+ is equivalent to Hψ = Hu . Remark 6.3.5 Let ψ be an affine functional with derivative a. By the above definition, ψ is an affine root if and only if Uψ is not contained in Uψ+ · U2a (k). This can be reformulated by saying that the image of Uψ in Ua (k)/U2a (k) is not contained in the image of Uψ+ . Writing overline to denote these images, we obtain the alternative expression Ψ = {ψ ∈ A∗ | U ψ+  U ψ }, which emphasizes the close relationship between Ψ and Ψ . We will show in Proposition 6.3.13 that both Ψ and Ψ  are affine root systems in A∗ in the sense of Definition 1.3.1, and their derivative root system is Φ. For quasi-split groups a simpler argument will be given in Propositions 6.4.1 and 6.4.8, which will also give an explicit description of Ψ and Ψ . It is clear that Ψ ⊂ Ψ . The converse will be discussed in Proposition 6.3.8 below. We will see that the affine root system Ψ will play the more important role in what follows, while Ψ  will be of a more auxiliary nature. The reason is that the local root systems Ψx , discussed in §1.3, will turn out to describe the structure of the reductive quotient of the special fiber of certain integral models associated to G and x. This will be discussed in Chapter 8. As we will see in Proposition 6.3.8, the affine hyperplanes coming from Ψ  are the same as those coming from Ψ, so the polysimplicial structure of A induced by Ψ  is the same as that induced by Ψ. Notation 6.3.6 For a ∈ Φ, Ψa will denote the set of affine roots with derivative a. If 2a is also a root, 12 Ψ2a will denote the set { 12 ψ | ψ ∈ Ψ2a }. If 2a is not

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219

a root, both Ψ2a and 12 Ψ2a will denote the empty set. Similar notation will be used for the set Ψ . In what follows, we will assume that for a ∈ Φ, a(ν(Z(k))) is a cyclic subgroup of R. This is automatically the case when A(S) consists of valuations compatible with ω. In particular, this is the case for the apartment A of Definition 6.1.27. We will denote by ca the positive generator of the cyclic group a(ν(Z(k))). Proposition 6.3.7 n ∈ Z.

Let a ∈ Φ and ψ, ψ  ∈ Ψa . Then ψ − ψ  = 12 nca for an

Proof As both ψ and ψ  have derivative a, there exists a c ∈ R, such that ψ  = ψ − c. Choose u ∈ Uψ −Uψ+ , u  ∈ Uψ  −Uψ  + . We know from Proposition 6.2.1 that m(u) and m(u ) act on A by reflections rψ and rψ  in the hyperplanes Hψ and Hψ  respectively. Then rψ · rψ  maps onto the trivial element of the Weyl group, and hence z := m(u) · m(u ) lies in Z(k). Now (rψ · rψ  ) · ψ  = −rψ · ψ  = −rψ · (ψ − c) = ψ + c. On the other hand, (rψ · rψ  ) · ψ  = z · ψ  = z · (ψ − c) = z · ψ − c. But Proposition 6.2.3 implies that z · ψ = ψ + a(ν(z)), so ψ + c = ψ + a(ν(z))− c,  therefore, c = 12 a(ν(z)). Hence, c = 12 nca for an n ∈ Z. Proposition 6.3.8 We assume that A consists of valuations of root datum of (G, S) that are compatible with the topology on G(k). In other words, for  Then the following valuations in A, Axiom V 7 holds. Let ψ ∈ Ψ  and let a = ψ. assertions are true. (1) (2) (3) (4)

If 2a is not a root, then ψ ∈ Ψ. If ψ  Ψ, then 2ψ ∈ Ψ. Let ψ  ∈ Ψ be the smallest element with ψ  ψ . Then Uψ = Uψ  · U2ψ . The sets {Hψ | ψ ∈ Ψ  } and {Hψ | ψ ∈ Ψ} are the same.

Proof (1) If 2a is not a root, the conditions ψ ∈ Ψ  and ψ ∈ Ψ are equivalent. (4) follows at once from (2). To prove (2) and (3) we will use the following lemma. Lemma 6.3.9 Let ψ ∈ Ψ  and let u ∈ Uψ − Uψ+ belong to Uψ+ · U2a (k) but not to U2a (k). Then there exist u  ∈ Ua (k)∗ , v ∈ U2a (k)∗ , and ψ  ∈ Ψ, such that u = u  · v, u  ∈ Uψ − Uψ + , v ∈ U2ψ − U2ψ+ , 2ψ ∈ Ψ, and ψ < ψ . Note that if ψ  Ψ, then Uψ ⊂ Uψ+ · U2a (k).

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Proof Since u is contained in Uψ+ · U2a (k), u = u1 v1 with u1 ∈ Uψ+ and v1 ∈ U2a (k). Note that for any x ∈ A, xa (u) = ψ(x) < xa (u1 ), hence xa (v) = xa (u1−1 u) = xa (u) by Remark 6.1.5. Therefore, v ∈ U2ψ − U2ψ+ and hence 2ψ is an affine root. Let ψ1 be the unique affine functional with derivative a such that u1 ∈ Uψ1 − Uψ1 + . Clearly ψ1 > ψ. If ψ1 is an affine root, we are done. Otherwise, Uψ1 ⊂ Uψ1 + · U2a (k). In particular, u1 ∈ Uψ1 + · U2a (k), so we can apply the same reasoning to u1 . Assume by way of contradiction that none of the members of the resulting sequence ψ1, ψ2, . . . is an affine root. Using Proposition 6.3.7 we see that ψi = ψ + 12 ni ca for an increasing sequence {ni } of positive integers, with positive real number ca as in that proposition. Let ui ∈ Uψi − Uψi + be the elements produced in this process. For x ∈ A we have 1 xa (ui ) = ψi (x) = ψ(x) + ni ca → ∞. 2 Hence, the sequence {ui } converges to 1. We infer from this that the sequence {vi }(⊂ U2a (k)) converges to u. Now since U2a (k) is a closed subgroup of  Ua (k), u must lie in U2a (k). This is a contradiction. We will now prove assertion (2) of the preceding proposition. The subgroup Uψ of Ua (k) is open, and so is Uψ+ . Thus Uψ − Uψ+ is a non-empty open subset of Ua (k). It cannot be contained in U2a (k), since the latter is the group of k-points of the Zariski-closed subgroup U2a of Ua . Therefore there exists u ∈ Uψ − Uψ+ that does not lie in U2a (k). We apply the above lemma to this u and obtain ψ  ∈ Ψ, ψ  > ψ, and 2ψ ∈ Ψ . According to (1), 2ψ ∈ Ψ. Assertion (3) follows at once from the preceding lemma.  Corollary 6.3.10

 = Ψ . For a ∈ Φ, Ψa = Ψa ∪ 12 Ψ2a , and Ψ2a 2a

Proof Immediate from Proposition 6.3.8. Corollary 6.3.11



Let x ∈ A, a ∈ Φ.

(1) If 2a  Φ, then Ua, x,r = Uψ , where ψ is the smallest element of Ψ with derivative a such that ψ(x)  r. (2) If 2a ∈ Φ, then Ua, x,r = Uψ1 · Uψ2 , where ψ1, ψ2 ∈ Ψ are the smallest elements with derivative a and 2a respectively, such that ψi (x)  i · r. Proof Let ψ ∈ A∗ be the unique element with derivative a and ψ(x) = r. Then Ua, x,r = Uψ by definition of Uψ . Let ψ  ∈ Ψ  be the smallest element such that ψ   ψ. Then Uψ = Uψ by definition of Ψ . The claim now follows from Proposition 6.3.8.  Lemma 6.3.12 The natural action of N(k)/Z(k)b on A∗ preserves the subsets Ψ and Ψ .

6.3 Affine Roots and Affine Root Groups

221

Proof Let a ∈ Φ, n ∈ N(k), and b = n · a. For a real valued affine functional ψ on A, with derivative a, n · ψ is a real valued affine functional with derivative b. From Fact 6.3.3 we infer that the action of n preserves Ψ , and furthermore Uψ  Uψ+ · U2a (k)(= Uψ+ if 2a  Φ) if and only if Un·ψ = nUψ n−1  nUψ+ · U2a (k)n−1 = Un·ψ+ · U2b (k). We conclude that ψ is an affine root if and only if n · ψ is an affine root for all n ∈ N(k).  Proposition 6.3.13 The sets Ψ and Ψ  constitute affine root systems in A∗ with derivative root system equal to Φ. They have the same Weyl group. The image of N(k)/Z(k)b in Aff(A) contains W(Ψ) and is contained in W(Ψ)ext . Proof To see that Ψ and Ψ  are affine root systems we apply Proposition 1.3.12, whose assumptions are verified as follows. According to Proposition 6.3.8(1)(2), for any a ∈ Φ there are infinitely many ψ ∈ Ψ with ψ = a. Proposition 6.3.7 shows that the set Ψa does not have an accumulation point. Proposition 6.3.8 shows that the hyperplanes, and hence the reflections, coming from Ψ  are the same as those coming from Ψ. According to Proposition 6.2.1, these reflections are realized by elements of N(k), which according to Lemma 6.3.12 preserves Ψ and Ψ . This verifies all assumptions of Proposition 1.3.12, as well as the claim about the image of N(k)/Z(k)b in Aff(A). That the affine  Weyl groups of Ψ and Ψ  are equal follows from Proposition 6.3.8(3). More precise information about the image of N(k)/Z(k)b in Aff(A) will be obtained in §6.6 when G is quasi-split, and in §9.4 (in particular Proposition 9.4.35) for general G. 6.3.14 Affine roots when Φ is reduced Assume that the set Φ of roots of G is reduced (this is the case, if for example, G is quasi-split and it does not admit a quotient of type 2 A2n ). Fix ϕ ∈ A. For a ∈ Φ and r ∈ R, we denote by ψa,r the real valued affine functional on A, with derivative a, defined by ψa,r (ϕ + v) = a(v) + r. Then Ψ  = Ψ = {ψa,r | a ∈ Φ, r ∈ ϕa (Ua (k)∗ )}. Definition 6.3.15 The hyperplanes in A of the form Hψ for ψ ∈ Ψ will be called affine root hyperplanes. Remark 6.3.16 According to the last assertion of Proposition 6.3.8, these are the same as the hyperplanes Hu for a ∈ Φ, and u ∈ Ua (k)∗ . As the natural action of N(k)/Z(k)b on A∗ preserves the set Ψ, the action of this group on A keeps the set of affine root hyperplanes stable. Remark 6.3.17

Let Ω ⊂ A. Using Proposition 6.3.7 we see that among the

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affine roots ψ with derivative a(∈ Φ), such that ψ | Ω  r, there is a unique smallest one. 6.3.18 Polysimplicial structure. Proposition 6.3.13 and Remark 1.3.16 endow A with a polysimplicial structure. We recall that the chambers (facets of  largest dimension) are the connected components of A − ψ ∈Ψ Hψ , the facets   of codimension 1 are the connected components of ψ Hψ − ψη Hψ ∩ H η , etc. Lemma 6.3.19 Let C ⊂ A be a chamber and let NC be the stabilizer of C for the action of N(k) on A, cf. Proposition 6.3.13. Then G(k) = G(k) · NC . Proof The conjugacy of maximal split tori of G under G(k) implies G(k) = G(k) · N(k). Furthermore, Proposition 6.3.13 and Lemma 1.3.17 show that  N(k) = NG(k) (S) · NC , and the claim follows. Lemma 6.3.20 The facets of A are the largest subsets on which the filtration subgroup Ua, x,0 does not change for any a ∈ Φ. In particular, we may write Ua,F,0 in place of Ua, x,0 , where F is the facet containing x. Proof Let a ∈ Φ and x, y ∈ A be such that Ua, x,0  Ua, y,0 . Interchanging x and y we may (and do) assume that there is a u ∈ Ua, x,0 that is not contained in Ua, y,0 . Thus x and y are separated by the hyperplane Hu . By Proposition 6.3.8 this is an affine root hyperplane. To prove the converse, we assume that x, y are on the different sides of an affine root hyperplane, say of Hψ , for ψ ∈ Ψ. Let a = ψ ∈ Φ. Then one of ψ(x) and ψ(y) is positive and the other one is negative, say ψ(x) > 0. For u ∈ Uψ − Uψ+ we have xa (u) = ψ(x) > 0, hence u ∈ Ua, x,0 , while u  Ua, y,0 ,  thus Ua, x,0  Ua, y,0 . 6.3.21 Let F1 and F2 be facets of A such that F1 ≺ F2 (that is, F1 is contained in the closure of F2 ). Let ψ ∈ Ψ and a = ψ ∈ Φ. Then as ψ is continuous, if it is non-negative on F2 , then it is non-negative on its closure, so in particular, on F1 . This implies that Ua,F2 ,0 ⊂ Ua,F1 ,0 . On the other hand, if ψ takes strictly positive values on F1 , then it takes strictly positive vales on F2 also, so Ua,F1 ,0+ ⊂ Ua,F2 ,0+ . Thus we have the chain of inclusions Ua,F1 ,0+ ⊆ Ua,F2 ,0+ ⊆ Ua,F2 ,0 ⊆ Ua,F1 ,0 . At most one of these inclusions is strict. The precise relationship depends on whether or not there exists ψ ∈ Ψa such that F1 ⊂ Hψ . If such ψ does not exist, then all inclusions are in fact equalities. If such ψ does exist, then it is necessarily unique. We have Ua,F1 ,0+ = Uψ+ and Ua,F1 ,0 = Uψ . According to Lemma 6.1.6 we have the following cases.

6.4 The Affine Root System of a Quasi-split Group

223

(1) ψ(F2 ) = 0: Ua,F1 ,0+ = Ua,F2 ,0+  Ua,F2 ,0 = Ua,F1 ,0 . (2) ψ(F2 ) > 0: Ua,F1 ,0+  Ua,F2 ,0+ = Ua,F2 ,0 = Ua,F1 ,0 . (3) ψ(F2 ) < 0: Ua,F1 ,0+ = Ua,F2 ,0+ = Ua,F2 ,0  Ua,F1 ,0 .

6.4 The Affine Root System of a Quasi-split Group Let G be a quasi-split reductive group defined over k, S  a maximal split torus in the derived subgroup of G, S the unique maximal split torus of G containing S , and T the centralizer of S in G; T is a maximally split maximal torus of G. Let A = A(S) be the apartment of Definition 6.1.27. We have defined affine roots in Definition 6.3.4 and affine root hyperplanes in Definition 6.3.15. We recall that affine root hyperplanes are the vanishing hyperplanes of affine roots. In this section we will give an explicit description of the set Ψ of affine roots, and of the auxiliary set Ψ , and provide alternative proofs that these sets form affine root systems in the sense of Definition 1.3.1 with derivative root system Φ.

(a) Split Groups We assume here that G is split. Fix a Chevalley valuation o ∈ A(S). Proposition 6.4.1 The set of affine roots Ψ of Definition 6.3.4 is a reduced affine root system in the sense of Definition 1.3.1. More precisely, the set of affine roots Ψ equals the affine root system ΨΦ = Φ × Z of Construction 1.3.27. Furthermore, Ψ  = Ψ. Proof Then root system Φ is reduced and for every Chevalley valuation ϕ, ϕa (Ua (k)∗ ) = Z for all a ∈ Φ. Example 6.3.14 shows that Ψ  = Ψ = Φ × Z, where the affine root corresponding to (a, n) ∈ Φ × Z, is the affine functional  ψa, n on A such that ψa, n (o + v) = a(v) + n, for v ∈ V(S ). Recall the concepts of special and extra special vertex from Definition 1.3.39. Lemma 6.4.2

Let x ∈ A. The following are equivalent.

(1) x is special. (2) x is extra special. (3) x is a Chevalley valuation. Proof Since Φ is reduced, a vertex is special if and only if it is extra special (cf. Proposition 1.3.43), and this in turn if and only if for every a ∈ Φ there exists ψ ∈ Ψ with derivative a such that ψ(x) = 0.

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The definition of affine roots makes it clear that o is special. Let x = o + v also be special, for some v ∈ V(T). Thus we have a(v) ∈ Z for all a ∈ Φ. This means that v lies in the coweight lattice P∨ ⊂ V(T). Let t ∈ Tad (k) be such that ω(t) = v. Then t · o = x, implying that x is also a Chevalley valuation. 

(b) Quasi-split Groups Consider now an adjusted Chevalley valuation o ∈ A. Following the discussion of §3.2 we introduce the jump sets Ja and Ja/2a for a ∈ Φ. Definition 6.4.3 Let a ∈ Φ and consider U2a (k) ⊂ Ua (k), with the convention that U2a = {1} if 2a  Φ. Define Ja to be the set of jumps of the filtration of Ua (k) given by any adjusted Chevalley valuation, and Ja/2a to be the set of jumps of the induced filtration on Ua (k)/U2a (k). In other words, we have Ja = ϕa (Ua (k)∗ ) and Ja/2a = ϕa/2a ((Ua (k)/U2a (k))∗ ), where ϕa/2a : Ua (k)/U2a (k) → R is the valuation function for the filtration of Ua (k)/U2a (k) induced by the filtration of Ua (k). More explicitly, for u ∈ Ua (k) ¯ = sup ϕa (u · U2a (k)). with image u¯ ∈ Ua (k)/U2a (k), we have ϕa/2a (u) For a ∈ Φ, recall the definition of the sets Ψa and Ψ!a given in Notation 6.3.6. Identifying a with the element of A∗ whose derivative is a and which takes the value 0 at the point o, we have Ψa = a + Ja,

Ψa = a + Ja/2a .

In order to describe the sets Ψ  and Ψ explicitly it is therefore enough to describe the sets Ja and Ja/2a explicitly. 6.4.4 It is immediate that Ja/2a ⊂ Ja and that, if 2a is not a root, then Ja/2a = Ja . In particular, if 2a is a root, then J2a = J2a/4a . From Corollary 6.3.10 we see that Ja = Ja/2a ∪ 12 J2a . The discussion of §3.2 gives the following information about the sets Ja and Ja/2a , showing in particular that they do not depend on the choice of adjusted  is a lift of a, let e a be the ramification degree Chevalley valuation. If  a ∈Φ of k a/k. The value of e a depends only on a and we can denote it also by ea . b= a+ a , When a is of type R2, let  a,  a  be two non-orthogonal lifts of a and  let ea/2a = ea /e2a , which is the ramification degree of k a/k b. Fact 6.4.5 (1) If a is of type R1, Ja = Ja/2a = e−1 a Z. 1 −1 −1 (2) If a is of type R2 and ea/2a = 1, Ja = 2 ea Z, Ja/2a = e−1 a Z, J2a = e a Z.

6.4 The Affine Root System of a Quasi-split Group

225

(3) If a is of type R2 and ea/2a = 2, Ja =

1 −1 −1 e Z, Ja/2a = e−1 a Z, J2a = e a (2Z + 1). 2 a

Remark 6.4.6 The sets Ja and Ja/2a were denoted by Γa and Γa , respectively, in [BT72] and [BT84a]. They can be defined for any valuation of the root datum, and without the assumption that G is quasi-split, but Fact 6.4.5 is valid only for quasi-split groups. We refer the interested reader to [BT72, §6.2.15–§6.2.24]. Remark 6.4.7 Consider G absolutely simple and not of type 2 A2n . Then all roots are of type R1. If G is unramified then ea = 1 for all a ∈ Φ. If G is ramified, let e be the ramification index of /k, where  is the splitting field of G except in the case that G is of type 6 D4 , in which case we take  to be an arbitrary cubic extension in the sextic extension of k splitting G. Then ea = 1 when a ∈ Φ is long, and ea = e when a ∈ Φ is short. Explicitly, e = 2 when G is of type 2 D2n (then Φ is of type Bn ) or 2 E6 (then Φ is of type F4 ) and e = 3 when G is of type 3 D4 and 6 D4 (then Φ is of type G2 ). Note that in each case e is equal to the weight of the multiple bond in the Dynkin diagram of Φ. Consider now G absolutely simple and of type 2 A2n . Then all roots of type R3 have ea = 1, while all roots of type R1 and R2 have ea = e, where e is the ramification degree of the quadratic extension splitting G. Recall that in Construction 1.3.27 we gave a construction of a reduced affine root system ΨΦ starting from any (possibly non-reduced) finite root system Φ. Recall also that in Example 1.3.28 we introduced a non-reduced affine root system. Proposition 6.4.8 The sets Ψ  and Ψ are affine root systems in the sense of §1.3, with derivative root system Φ. The affine root system Ψ is reduced unless Φ has a component of type BCn that splits over an unramified extension. If G is absolutely almost simple, then Ψ is given as follows. (1) (2) (3) (4)

ΨΦ , if G is unramified and not of type 2 A2n . The system (BCn, Cn ) of Example 1.3.28, if G is unramified of type 2 A2n . (ΨΦ∨ )∨ , if G is ramified and not of type 2 A2n . ΨΦ , if G is ramified of type 2 A2n .

Moreover, Ψ  = Ψ unless Φ is non-reduced. In the latter case, and assuming G is absolutely almost simple, Ψ  is of type (Cn∨, BCn ) when G is ramified of type 2 A , and of type (C ∨, C ) when G is unramified of type 2 A . 2n n 2n n

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Proof The sets Ψ  and Ψ do not change if we pass to the adjoint group. They are the direct sum, in the sense of Construction 1.3.15, of the corresponding sets for the k-simple factors. Assuming that G is k-simple, it is the restriction of scalars of an absolutely simple group. Up to multiplication by a real number the construction of Ψ  and Ψ is unaffected by restriction of scalars. We may thus reduce to the case that G is absolutely simple. We now look at the sets Ja and Ja/2a as described in Remark 6.4.7. If G is unramified and not of type 2 A2n , then Ja = Ja/2a = Z for all a ∈ Φ. Thus Ψ  = Ψ is obtained from Construction 1.3.27 applied to Φ. If Φ is of type BCn that splits over a ramified extension, then Ja/2a = 12 Z when a is of type R1 or R2, and Ja = Z + 12 when a is of type R3. Thus again Ψ is obtained from Φ via Construction 1.3.27 and is of type BCn . On the other hand Ja = 14 Z when a is of type R2 and equal to Ja/2a when a is of type R1 or R3. Therefore Ψ  is of type (Cn∨, BCn ). If G is ramified and not of type 2 A2n , then Ja = Ja/2a = Z when a is long and Ja = Ja/2a = 1 Z when a is short, where  is the ratio of the squares of the two lengths in Φ. Clearly Ψ  = Ψ. According to Remark 1.3.32 we have Ψ = (ΨΦ∨ )∨ . In all of these cases, Ψ is a reduced affine root system. If Φ is of type BCn splitting over an unramified extension, then Ja/2a = Z for all a ∈ Φ, so Ψ = {a + n | a ∈ Φ, n ∈ Z} is the non-reduced affine root system of type (BCn, Cn ) of Example 1.3.28. At the same time, Ja equals 12 Z for roots of type R2, and equal to Ja/2a for all other roots, hence Ψ  is of type (Cn∨, Cn ).  Remark 6.4.9 Assume that G is absolutely almost simple. Associated to G is a Tits label, which is one of An , 2 An , Bn , Cn , Dn , 2 Dn , 3 Dn , 6 Dn , E6 , E7 , E8 , F4 , G2 . It records the type of the absolute root system of G together with the size of the image of the Galois group in the symmetry group of the Dynkin diagram. The type of the corresponding affine root system Ψ, in the sense of Tables 1.3.4 and 1.3.3, can be determined from the Tits labels of G K and G via Proposition 6.4.8. The results are collected in Table 6.4.1.

6.4 The Affine Root System of a Quasi-split Group

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Table 6.4.1 The affine root systems Ψ and Ψ  for an absolutely simple quasi-split group G GK

G

1A n 1A 2n

1A n 2A 2n

1A 2n−1 2A 2n 2A 2n−1 Bn Cn 1D n 1D n 2D n 1D 4 1D 4 2D 4 3D 4 3D 4 6D 4 1E 6 1E 6 2E 6 E7 E8 F4 G2

2A 2n−1 2A 2n 2A 2n−1 Bn Cn 1D n 2D n 2D n 3D 4 6D 4 6D 4 3D 4 6D 4 6D 4 1E 6 2E 6 2E 6 E7 E8 F4 G2

 Φ

Φ

Ψ

Ψ

An A2n A2n−1 A2n A2n−1 Bn Cn Dn Dn Dn D4 D4 D4 D4 D4 D4 E6 E6 E6 E7 E8 F4 G2

An BCn Cn BCn Cn Bn Cn Dn Bn−1 Bn−1 G2 G2 G2 G2 G2 G2 E6 F4 F4 E7 E8 F4 G2

An (BCn, Cn ) Cn BCn Bn∨ Bn Cn Dn Bn−1 ∨ Cn−1 G2 G2 G∨ 2 G∨ 2 G∨ 2 G∨ 2 E6 F4 F4∨ E7 E8 F4 G2

An (Cn∨, Cn ) Cn (Cn∨, BCn ) Bn∨ Bn Cn Dn Bn−1 ∨ Cn−1 G2 G2 G∨ 2 G∨ 2 G∨ 2 G∨ 2 E6 F4 F4∨ E7 E8 F4 G2

Consider again the valuation homomorphism (2.5.1) ωT : T(k) → Q ⊗Z X∗ (T)Θ ⊂ V(S). The following lemma will be used to analyze the relationship between extra special points and adjusted Chevalley valuations, as well as the subgroup of translations in the various incarnations of the affine Weyl group in §6.6. Lemma 6.4.10 The image of ω : Tsc (k) → V(S ) is the subgroup generated by {na∨ | n ∈ Ja }. The image of ω : Tad (k) → V(S ) is the subgroup of elements nd (equivalently all a in a set of simple satisfying a(v) ∈ e−1 a Z for all a ∈ Φ roots Δ). Proof Recall that Tsc and Tad are induced tori. We first note that if  //k is a tower of finite extensions, with  /k Galois, the exact sequence × Θ  × Θ −1 1 → (IndΘ Θ Z) ⊗Z o  → (IndΘ Z) ⊗Z ( ) → (IndΘ Z) ⊗Z (e  Z) → 0

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induces on Θ-fixed points the map ω :  × → e−1 Z ⊂ e−1 Z. Therefore the image of Tsc (k) → V(S ) is generated by all the elements ∨ ∨  We have a∨ = δ  a ∈ Φ. e−1 ∈Θ a b  b , for  b ∈Θ a  b , where δ = 1 when a is of a  type R1 and δ = 2 when a is of type R2. So the image of Tsc (k) → V(S ) is gen−1 ∨ ∨ erated by the elements e−1 a δ a , or equivalently by the elements {na | n ∈ Ja }.  of simple absolute roots. The above argument with Choose a set  Δ ⊂ Φ induced tori shows that the image of Tad (k) → V(S ) is the lattice spanned by ˇ a, as a runs over Δ,  a runs over the set of lifts of a (these the elements e−1 a ω a form a single Θ-orbit), and ωˇ a is the fundamental coweight corresponding to  a. Such an element pairs to 0 with every a  b ∈ Δ, and to e−1 a with a ∈ Δ. Thus the image of Tad (k) in V(S ) is precisely the lattice of elements satisfying  a(v) ∈ e−1 a Z for all a ∈ Δ. We now turn to the analysis of special and extra special points. Recall from Definition 1.3.39 that a point x ∈ A is called extra special if the derivatives of the affine roots that vanish at x contain a basis of Φ, and that x is called special if the same is true after possibly dividing any divisible derivative by 2. The two notions obviously agree when Φ is reduced. On the other hand we have introduced here the notions of a Chevalley valuation and adjusted Chevalley valuation. Propositions 6.4.11 and 6.4.12 compare these notions. Proposition 6.4.11 Let o ∈ A be a Chevalley valuation or an adjusted Chevalley valuation. Let Δ ⊂ Φ be a set of simple roots. For a point x = o + v ∈ A the following are equivalent. (1) x is special. (2) a(v) ∈ Ja for all non-divisible relative roots a ∈ Φ. (3) a(v) ∈ Ja for all a ∈ Δ. In particular, o itself is a special point. Proof Assume first that o is an adjusted Chevalley valuation. The equivalence of the first and second point is immediate from 6.4.4 and the definition of affine roots, while the equivalence of the second and third points is Lemma 6.2.8. To derive from this the result when o is a Chevalley valuation it is enough to show that a(v) ∈ Ja for all non-divisible relative roots a, where v is as in Construction 6.1.18. But a(v) = 0 when a is of type R1 and a(v) = μa /2 when a is of type R2, where μa is the quantity (2.8.1) for the quadratic field extension  are non-orthogonal lifts of a and  a,  a ∈ Φ b =  a+ a . But k a/k b, where   μa ∈ ω(k ×a) = e1a Z = 2Ja .

6.5 Change of Valuation

229

Proposition 6.4.12 A point in A is extra special if and only if it is an adjusted Chevalley valuation. Proof Choose a k-pinning of G and let o be a the associated adjusted Chevalley valuation of Constructions 6.1.17 and 6.1.18. If Δ is the set of simple roots, then Fact 6.4.5 shows that 0 ∈ Ja/2a for each a ∈ Δ, therefore there exists ψ ∈ Ψo with ψ = a, showing that o is extra special. Conversely let x ∈ A be extra special and let ψ1, . . . , ψn ∈ Ψx be such that Δ = {ψ 1, . . . , ψ n } is a set of simple roots of Φ = ∇Ψ. The set Δ determines a Borel k-subgroup of G. Extend (T, B) to a k-pinning and let o be the corresponding adjusted Chevalley valuation. Then x = o + v for a unique v ∈ V(S ). The  assumption ψi ∈ Ψx implies ai (v) ∈ Ja/2a = e−1 a Z for ai = ψi . Lemma 6.4.10  implies that v is in the image of Tad (k) → V(S ), thus x = t · o for some  t ∈ Tad (k), showing that x is an adjusted Chevalley valuation. Remark 6.4.13 In contrast to the case of split groups, cf. Lemma 6.4.2, a special point need not be a Chevalley valuation. For example, when G = SU3 for a ramified quadratic extension /k, o is a Chevalley valuation, a ∈ Φ is a non-divisible root, and v ∈ V(S ) is specified by a(v) = 14 , then o + v is also a special point, but it is not a Chevalley valuation. When p  2, or when p = 2 and 2 | s, then the concepts of Chevalley valuation, adjusted Chevalley valuation, and extra special point coincide, while the concept of a special point is more general. When p = 2 and 2  s, then the concepts of a extra special point and adjusted Chevalley valuation coincide. A Chevalley valuation on the other hand is special, but not extra special.

6.5 Change of Valuation We have been assuming that the valuation ω : k × → R has image equal to Z. This can become inconvenient when one uses arguments involving restriction of scalars along ramified extensions of k. The construction of a valuation of the root datum from a given k-pinning, equivalently a given weak Chevalley–Steinberg system, may be applied with any choice of discrete valuation ω : k × → R. Let us discuss briefly how things change in that case. Consider first the split case. Recall that on the set of all valuations of the root datum we have the actions of N(k) = NG (T)(k) and V(T), which together give an action of V(T)  N(k). To see how things change when the valuation ω : k × → R is changed, we now introduce a third action on the set of all valuations of the root datum, this time by the group R× . We define r ϕ by (r ϕ) a ∈Φ (u) = r · ϕa (u) for r ∈ R× , a ∈ Φ and u ∈ Ua (k). The action of R×

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commutes with the action of N(k) and respects the action of V(T) in the sense that the two actions splice to an action of V(T)  R× , where R× acts on V(T) by multiplication. Recall further that the equipollence classes of valuations of the root datum are the orbits for the action of V(T). Therefore, both groups N(k) and R× respect the grouping of valuations of the root datum into equipollence classes. It can be shown that in fact N(k) preserves each equipollence class. We will not prove this, as we will not need it. On the other hand, it is easy to see that the action of R× on the set of equipollence classes is free. Indeed, all this means is that no valuation ϕ is equipollent to r ϕ if 1  r ∈ R× , which is clear since ϕa − (r ϕ) a ∈Φ = (1 − r)ϕa is not a constant function of Ua (k). Consider now two valuations ωi : k × → R for i = 1, 2. These lead to two equipollence classes Ai of valuations of the root datum. These are the two versions of the apartment of T, one for each valuation ωi . But ω2 = r · ω1 for a unique r ∈ R× and hence A2 = r · A1 for the action of r by multiplication on the set of all valuations of root data, and hence on the set of subsets of that set. Thus A2  A1 if ω2  ω1 . Nonetheless, multiplication by r is a bijective map A1 → A2 which we can use to identify the two versions of the apartments. This map is N(k)-equivariant and translates the action of v ∈ V(T) on A1 to the action of rv on A2 . This discussion generalizes to the quasi-split case as well, the only change that is needed is to replace X∗ (T) by X∗ (S). In addition, if we use a general valuation ω : k × → R, the sets Ja and Ja/2a associated to a relative root a ∈ Φ change as follows. Fact 6.5.1 For a relative root a ∈ Φ let va ∈ R be the common value of a  lifting a. Then a∈Φ generator of the lattice ω(k a) ⊂ R for all absolute roots  (1) a of type R1: Ja = Ja/2a = va Z. (2) a of type R2, ea/b = 1: Ja = 12 va Z, Ja/2a = va Z, J2a = va Z. (3) a of type R2, ea/b = 2: Ja = 12 va Z, Ja/2a = va Z, J2a = va (2Z + 1).

6.6 The Affine Weyl Group We return to a general connected reductive k-group. We maintain the notations S  and S used so far; thus S ⊂ G is a maximal split torus and S  is the maximal torus in S ∩ Gder . Let N and Z be as above. We write Zsc for the preimage of Z in Gsc and Zad for the image of Z in Gad . Let W aff be the group of affine automorphisms of A generated by the reflections along the vanishing hyperplanes of all affine roots. Thus W aff is the affine

6.6 The Affine Weyl Group

231

Weyl group of the affine root system Ψ, equivalently of Ψ , cf. Proposition 6.3.13. In particular, Lemma 1.3.17 holds. Let W ext be the extended affine Weyl group of Ψ of Definition 1.3.71.  1 = N(k)/Z(k)1 . Let  0 = N(k)/Z(k)0 and W Define the discrete groups W  1 with respect to Gsc , and let W ad be the group W  1 with sc be the group W W 0 1 respect to Gad . If G is simply connected then Z(k) = Z(k) according to 1 = W sc . If G is adjoint and quasi-split, 0 = W Lemma 2.6.27 and hence W 1 then Z is an induced torus, hence Z(k) = Z(k)0 by Lemma 2.5.18, thus 1 = W ad . 0 = W W Definition 6.6.1

 0 is called the Iwahori–Weyl group. The group W

The Iwahori–Weyl group was introduced by Haines–Rapoport in [PR08, Appendix] and studied in [Ric16a]. We refer to §7.8 for some of its properties. sc , We can think of W aff and W ext as “abstract affine Weyl groups” and of W  0 and W  1 as “concrete affine Weyl groups.” This is similar to the case of ad , W W a connected reductive group over an algebraically closed field, where one has the “abstract Weyl group” of the root system defined as the group generated by reflections along root hyperplanes, as well as the “concrete Weyl group” defined as NG (T)/T for a maximal torus T. Our goal here is to compare the abstract and concrete realizations of the affine Weyl groups, and in particular obtain more information about the action of N(k) on A.  1 that fits into the chain of maps 0 → W We have the surjective map W 0 1  →W  →W ad . The action of Nad (k) on A induces a map W ad → sc → W W ext W , which is injective by Proposition 6.2.4, and whose image contains W aff according to Proposition 6.3.13. The action of N(k) on A factors through the ad . Since the preimage of Zad (k)1 can be larger than Z(k)1 , this 1 → W map W  1 is also usually not injective, 0 → W map is usually not injective. The map W as Z(k)1 /Z(k)0 is in general non-trivial. The preimages of Z(k)1 and Z(k)0 in sc → W  0, W sc → W  1 , and W sc → W ad , Zsc equal Zsc (k)0 . Therefore the maps W are injective. sc → W aff and Proposition 6.6.2 Assume that G is quasi-split. The maps W ad → W ext are isomorphisms. W sc → W aff is an isomorphism remains valid without the The claim that W assumption that G is quasi-split, and is part of Theorem 7.5.1. Proof The assumption that G is quasi-split implies that Z is a maximal torus of G. We write T = Z to emphasize that. We may also assume without loss of generality that G is semi-simple, as we are only interested in that case. Thus S = S .

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The Apartment

ad → W ext are injective, sc → W We have already remarked that the maps W so it remains to determine their image. Proposition 6.2.1 shows that the image sc and W ad with their images in W ext , we sc contains W aff . Identifying W of W aff sc ⊂ W ad ⊂ W ext . obtain the chain of subgroups W ⊂ W Fix a special vertex x ∈ A, which exists by Proposition 1.3.43, or alternatively Proposition 6.4.11. Using Lemma 1.3.42 we see that it is enough to show the ad , and all translations in W sc lie in following: all translations in W ext lie in W aff W . We first discuss the case of split groups, which is more transparent. Thus S = T. Then [Bou02, Chapter VI, §2, no. 1, Proposition 1] shows that the group of translations in W aff is the coroot lattice Q∨ . This is also the image of Tsc (k) → V(T). At the same time, it is shown in [Bou02, Chapter VI, §2, no. 3] that the subgroup of translations in W ext is the coweight lattice P∨ , which is the image of Tad (k) → V(S ). In the general quasi-split case the argument is essentially the same. Fix a extra special vertex o, that is, an adjusted Chevalley valuation according to Proposition 6.4.12. Let a ∈ Φnd . For u ∈ Ua (k)∗ , the action of m(u) on the apartment A is described in Proposition 6.2.1. It is the composite of reflection ra in a hyperplane through o = ϕ followed by translation by ϕa (u)a∨ . Therefore the group of translations in W aff is given by {sa∨ | a ∈ Φnd, s ∈ Ja }, which is sc . also the image of Tsc (k) in V(S ) by Lemma 6.4.10. This proves W aff = W Consider now W ext . If the translation Tv for v ∈ V(S ) belongs to W ext , then it must preserve the affine root system. This means that a(v) ∈ e−1 a Z. This is also the image of Tad (k) → V(S ) according to Lemma 6.4.10. This proves that ad .  W ext = W  0 be the stabilizer of C for the action 6.6.3 Fix a chamber C of A and let W C  0 on A. Since W aff acts simply transitively on the set of chambers in A by of W Lemma 1.3.17, we obtain the decomposition 0 .  0 = W aff  W W C  0.  1 in place of W The analogous decomposition holds for W

6.7 Projection to a Levi Subgroup

233

6.7 Projection to a Levi Subgroup Let M ⊂ G be a Levi subgroup. Let S ⊂ M be a maximal split torus. Then S is also a maximal split torus of G and we have the natural inclusion Φ(M, S) ⊂ Φ(S, G) of relative root systems.  respectively Let Gder and Mder be the derived subgroups of G and M. Let S G  S M be the maximal torus in S ∩ Gder respectively S ∩ Mder . Let A be the  ∩ Z(M). The composition S  → S  → S  /A of the maximal torus in S G M G G natural inclusion and projection is an isogeny, which leads to the direct sum  ) = V(S  ) ⊕ V(A). decomposition V(S G M Assume given an equipollence class A(S, G) of valuations of the root datum of (G, S). Each element ϕ ∈ A(S, G) is thus a valuation (ϕa )a ∈Φ(S,G) of the root datum of (G, S). We write ϕ M for the collection (ϕa )a ∈Φ(M ,S) obtained from ϕ by forgetting the functions ϕa for all a ∈ Φ(S, G) − Φ(M, S). Define A(M, S) = {ϕ M | ϕ ∈ A(S, G)}. Lemma 6.7.1 (1) Each ϕ M is a valuation of the root datum of (M, S). (2) If ϕ is compatible with ω, then so is ϕ M . (3) The set A(M, S) is an equipollence class. (4) The map π : A(S, G) → A(M, S) descends to an isomorphism π = πS, M ,G : A(S, G)/V(A) → A(M, S)

(6.7.1)

 )/V(A) = V(S  ). of affine spaces over V(S G M (5) Assume that G is quasi-split. If ϕ ∈ A(S, G) is a Chevalley valuation, then so is ϕ M .

Proof That Axioms V 1 – V 6 for ϕ imply the corresponding axioms for ϕ M  ), then ϕ  − ϕ is clear. If ϕ, ϕ  ∈ A(S, G) and v = ϕ  − ϕ ∈ V(S G M = π(v) ∈ M   ) → V(S M ), where π(v) is the image of v under the natural projection V(S G  ) induced by the direct product decomposition V(S  ) = V(S  ) ⊕ V(A). V(S M G M The map π : A(S, G) → A(M, S) sending ϕ to ϕ M is evidently equivariant for  ), in particular invariant under the action of V(A), and hence the action of V(S G descends to an affine map A(S, G)/V(A) → A(M, S). It is surjective by definition and injective by dimension count. Finally, if G is quasi-split, T is the centralizer of S in G, and ϕ corresponds to a weak Chevalley system {±Xα }α∈Φ(T ,G) ,  then ϕ M corresponds to the weak Chevalley system {±Xα }α∈Φ(T ,M) . 6.7.2 Hyperplanes and special points The projection A(S, G) → A(M, S) induces an inclusion of dual spaces A(M, S)∗ → A(S, G)∗ . It is clear from Definition 6.3.4 that under this inclusion we have Ψ(M, S) ⊂ Ψ(S, G) and Ψ (M, S) ⊂ Ψ (S, G).  ) contains the subspace For a ∈ Φ(M, S) the root hyperplane Ha ⊂ V(S G

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The Apartment

V(A). Therefore for α ∈ Ψ(M, S) the image of the affine root hyperplane Hα ⊂ A(S, G) in A(S, G)/V(A) is still a hyperplane, and (6.7.1) identifies it with the affine root hyperplane Hα ⊂ A(M, S). In other words, Hα ⊂ A(S, G) is the preimage of Hα ⊂ A(M, S) under (6.7.1). This implies that if x ∈ A(S, G) is a special point, then its image under (6.7.1) is also a special point. In particular, it is a vertex. The same is also true for extra special points. Note however that in general, the image of a vertex in A(S, G) under (6.7.1) need not be a vertex in A(M, S).

7 The Bruhat–Tits Building for a Valuation of the Root Datum

Throughout this chapter k will denote a field given with a discrete valuation ω : k → R ∪ {∞}. We assume that k is Henselian, and for convenience we assume, except in 7.10, that ω(k × ) = Z. The ring of integers of k will be denoted by o and the maximal ideal of o by m. We do not assume in this chapter that the residue field f := o/m of k is perfect. Let G be a connected reductive k-group and S a fixed maximal k-split torus of G. Let N (respectively, Z) be the normalizer (respectively, centralizer) of S in G. We write S  for the maximal torus in S ∩ Gder . We assume that to each maximal k-split torus S ⊂ G there is assigned an affine space A(S) under the vector space V(S ), so that Axiom 4.1.4 holds. Thus, N(k) acts on A(S) in the form prescribed in A 1 of Axiom 4.1.4, and for every g ∈ G(k) there is an affine isomorphism g : A(S) → A(gSg −1 ) such that the isomorphism for gh is the composition of the isomorphisms for g and h, and the automorphism for n ∈ N(k) is given by the action of N(k) on A(S). In addition, we assume that each A(S) is an equipollence class of valuations of the root datum of (G, S), in the sense of Definitions 6.1.2 and 6.1.7, and the action of N(k) on A(S) is as defined in §6.1.8. The purpose of this chapter is to construct the Bruhat–Tits building for G, which will be denoted by B(G), or simply by B if the group G is understood. Even though this chapter is written for general reductive groups, our primary concern will be the case when G is quasi-split. In that case, a collection of affine spaces A(S) as above is canonically associated to G in Definition 6.1.27. In the case of general G, we proved in Propositions 6.2.5 and 6.2.6 that there exists at most one such collection consisting of valuations of root data that are compatible with ω, but we do not yet know that such a collection actually does exist. In Chapter 9 we will construct the building of G from that of the quasisplit group G K via a descent method. One of the byproducts of this method 235

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The Bruhat–Tits Building for a Valuation of the Root Datum

will be the existence of a valuation of the root datum of (G, S), compatible with ω. Since at this point the building of G will already have been constructed, the material of this chapter, applied to this collection, will provide a different interpretation of that building and of the various open bounded subgroups associated to it. For S as above, write A = A(S) and Φ = Φ(S, G). Recall the affine root system Ψ ⊂ A∗ of Definition 6.3.4 and Proposition 6.3.13. We will write X∗ (S) = X∗k (S) = Hom(S, Gm ) and V = V(S) = R ⊗Z X∗ (S). We also introduce the notations N = NG (S) and Z = ZG (S). The subgroup Z is a connected reductive k-subgroup of G. It is a maximal torus if and only if G is quasi-split. Every point x of A is by definition a valuation x = (xa )a ∈Φ . For a ∈ Φ, r ∈ R ∪ {∞}, we recall the subgroup Ua, x,r = xa−1 ([r, ∞]) of Ua (k) from Definition 6.1.2. For each x ∈ A, this defines a decreasing filtration of the root group Ua (k) indexed by r ∈ R. We will also need a filtration of the group Z(k) by open bounded subgroups, cf. §7.2. To construct the building B(G), we will first construct for each x ∈ A an open bounded subgroup G(k)0x of G(k)0 “by hand,”, that is, as the group generated by the subgroups Z(k)0 of Definition 2.6.23 and Ua, x,0 of Definition 6.1.2 for each a ∈ Φ. These will be the so-called parahoric subgroups. Using them, we will construct a Tits system in G(k)0 and invoke Tits’ Proposition 1.5.6. The resulting restricted building will be the building B(G). A posteriori the group G(k)0x will be recognized as the stabilizer of x for the action of G(k)0 on B(G), and thus as the “admissible parabolic” subgroups of the Tits system. We will also construct various related subgroups of G(k), denoted by  ⊂ G(k)0Ω ⊂ G(k)1Ω ⊂ G(k)†Ω ⊂ G(k)Ω , for a bounded subset Ω ⊂ A, as G(k)Ω  well as subgroups G(k)0x, f for x ∈ A and a concave function f : Φ ∪ {0} → R. Some of these play a technical role in the development of the theory, while others are not needed for the development, but are very useful in applications.

7.1 Commutator Computations In order to prove the necessary properties of the soon to be constructed parahoric subgroups and their variants, we study here the behavior of forming commutators of unipotent elements with respect to valuations of the root datum. Proposition 2.11.17 associates to any element u ∈ Ua (k)∗ , an element m(u) ∈ N(k) − Z(k). This element m(u) will be used often in this chapter.

7.1 Commutator Computations

237

Proposition 7.1.1 Let ϕ ∈ A. Let a ∈ Φ, and u ∈ Ua (k)∗ , u  ∈ U−a (k)∗ , with ϕa (u) = i, and ϕ−a (u ) = j. We assume that i + j = ϕa (u) + ϕ−a (u ) > 0. (1) There exist unique x ∈ Ua (k)∗ , x  ∈ U−a (k)∗ , and z ∈ Z(k), such that uu  = x  z x, ϕa (x) = ϕa (u), ϕ−a (x ) = ϕ−a (u ); z lies in the Iwahori subgroup Z(k)0 of Z(k). (2) If u ∈ U2a (k)∗ , then the commutator (u, u ) = uu u−1 u −1 lies in U−a,ϕ,i+2j · Z(k)0 · Ua,ϕ,2i+j . Moreover, ϕa (xu−1 )  2i + j, so x ∈ Ua,ϕ,2i+j · u. (3) If u  ∈ U−2a (k)∗ , then the commutator (u−1, u −1 ) lies in U−a,ϕ,i+2j · Z(k)0 · Ua,ϕ,2i+j Moreover, ϕ−a (x u −1 )  i + 2 j, so x  ∈ U−a,ϕ,i+2j · u . In view of the importance of this Proposition, we give two proofs: one general and abstract, and one specific to the case of quasi-split groups, which is more concrete. Proof of Proposition 7.1.1 for general G To prove the proposition, we may (and do) replace G by the semi-simple subgroup of k-rank 1 generated by U±a . The roots of G are then either ±a or ±a and ±2a. Then by the Bruhat decomposition ([Bor91]) of G(k), we know that G(k) = N(k) · Ua (k) ∪ U−a (k) · Z(k) · Ua (k). We observe that uu  cannot belong to N(k)·Ua (k), for otherwise, uu  = nv −1 for some n ∈ N(k) and v ∈ Ua (k), and hence uu  v ∈ N(k). But then m(u ) = uu  v, and from Axiom V 5 of Definition 6.1.2 we conclude that ϕ−a (u ) = −ϕa (u), that is, ϕa (u) + ϕ−a (u ) = 0, contrary to the hypothesis. Therefore, uu  belongs to U−a (k) · Z(k) · Ua (k), so we can write uu  = x  z x, with x ∈ Ua (k), x  ∈ U−a (k) and z ∈ Z(k). Since the subsets 



and Z(k) · U−a (k) ∩ Ua (k)∗ Z(k) · Ua (k) ∩ U−a (k)∗ are empty, we see that neither x nor x  is equal to 1. Uniqueness of x, x , and

 hence also of z, follows from the fact that U−a (k) ∩ Z(k) · Ua (k) = {1}. We will now prove the claim about the valuations of x and x . There exist unique y1, y2 ∈ Ua (k)∗ such that x  = y1 m(x ) y2 , and as m(x ) ∈ N(k). Axiom V 5 of Definition 6.1.2 implies that ϕ−a (x ) = −ϕa (y1−1 ). On the other hand,

 u  = u−1 x  z x = u−1 y1 · m(x )z · z−1 y2 z x ∈ Ua (k) · Na (k) · Ua (k) so m(u ) = m(x )z. Moreover, from Axiom V 5 we conclude that ϕ−a (u ) =

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The Bruhat–Tits Building for a Valuation of the Root Datum

−ϕa (y1−1 u). Since ϕa (u) + ϕ−a (u ) > 0, we infer that ϕa (u) > ϕa (y1−1 u). Using Axiom V 1, we see that ϕa (y1−1 ) = ϕa ((y1−1 u) · u−1 ) = ϕa (y1−1 u). Now ϕ−a (x ) = −ϕa (y1−1 ) = −ϕa (y1−1 u) = ϕ−a (u ) We can show that ϕa (x) = ϕa (u) using a similar argument. We will now prove that z lies in the subgroup Z(k)0 of Z(k). Since ϕ−a (x ) = ϕ−a (u ), we see from Proposition 6.2.1 that the actions of m(u ) and m(x ) on the apartment A are the same. But as we observed in the preceding paragraph, m(u ) = m(x )z, so the action of z on A is trivial. Now Proposition 6.2.3 implies that z lies in the maximal bounded subgroup Z(k)b of Z(k). As z lies in the subgroup generated by U±a (k), Corollary 2.6.28 shows that it actually lies in the Iwahori subgroup Z(k)0 of Z(k). This completes the proof of (1). Now we will prove (2). The proof of (3) is entirely analogous (note that uu  = u u(u−1, u −1 )), so it will not be given. We assume that u ∈ U2a (k)∗ and write u = u1 mu2 , with u1 , u2 ∈ U−2a (k)∗ , and m := m(u); u  = u1 m u2 , with u1, u2 ∈ Ua (k)∗ , and m  := m(u ). Using Proposition 6.2.1, we find that ϕ−a (m um −1 ) = ϕa (u) + 2 j = i + 2 j and ϕa (mu  m−1 ) = 2i + ϕ−a (u ) = 2i + j. Now (u, u ) = uu u−1 u −1 = uu1 · m u2 u−1 u2−1 m −1 · u1−1 = uu1 · m u−1 m −1 · u1−1, since u(∈ U2a (k)) and u2 (∈ Ua (k)) commute. By 6.1.5 and Axiom V 5, ϕa (u1 ) = ϕa (u1−1 ) = −ϕ−a (u ) = − j < i = ϕa (u). Hence, ϕa (uu1 ) = ϕa (u1 ) = − j (see Remark 6.1.5). Now as ϕa (uu1 ) + ϕ−a (m u−1 m −1 ) = − j + i + 2 j = i + j > 0, using assertion (1) we conclude that uu1 · m u−1 m −1 ∈ U−a,ϕ,i+2j · Z(k)0 · Ua,ϕ,−j . Now as u1 ∈ Ua,ϕ,−j , we see that (u, u ) ∈ U−a,ϕ,i+2j · Z(k)0 · Ua,ϕ,−j . On the other hand, (u, u ) = uu u−1 u −1 = u1 · mu2 u u2−1 m−1 · u1−1 u −1 = u1 · mu  m−1 · u1−1 u −1, since u (∈ U−a (k)) and u2 (∈ U−2a (k)) commute. Using ϕa (mu  m−1 ) = 2i + j and ϕ−a (u1−1 ) = ϕ−a (u1 ) = −ϕa (u) = −i < j = ϕ−a (u ) = ϕ−a (u −1 ), we obtain ϕ−a (u1−1 u −1 ) = ϕ−a (u1−1 ) = −i (see 6.1.5). Using assertion (1) we see that mu  m−1 · u1−1 u −1 ∈ U−a,ϕ,−i · Z(k)0 · Ua,ϕ,2i+j . This implies that (u, u ) = u1 · mu  m−1 · u1−1 u −1 ∈ U−a,ϕ,−i · Z(k)0 · Ua,ϕ,2i+j .

7.1 Commutator Computations

239

Combining the results of the preceding two paragraphs, we see that 



(u, u ) ∈ U−a,ϕ,i+2j · Z(k)0 · Ua,ϕ,−j ∩ U−a,ϕ,−i · Z(k)0 · Ua,ϕ,2i+j . Since i + 2 j > −i and 2i + j > − j, the above intersection is equal to U−a,ϕ,i+2j · Z(k)0 · Ua,ϕ,2i+j ; thus, as asserted in (2), the commutator (u, u ) belongs to this subset. Now x  z(xu−1 ) = uu u−1 = (u, u )u  ∈ U−a,ϕ,i+2j · Z(k)0 · Ua,ϕ,2i+j · u , and using assertion (1) we see that Ua,ϕ,2i+j · u  ⊂ U−a,ϕ, j · Z(k)0 · Ua,ϕ,2i+j , and from this we conclude that x  z(xu−1 ) lies in U−a,ϕ, j · Z(k)0 · Ua,ϕ,2i+j , and  therefore, xu−1 ∈ Ua,ϕ,2i+j , that is, ϕa (xu−1 )  2i + j. Proof of Proposition 7.1.1 for quasi-split G We will perform a series of reductions. First, we may replace G by the group generated by T and the root groups for all rational multiples of a. Next we may replace G by Gsc . Now G is the restriction of scalars of either SL2 or SU3 and we may assume that G is SL2 or SU3 . Finally, we may replace ϕ by any equipollent valuation, so we take it to be the standard Chevalley valuation for both SL2 and SU3 . It is readily checked that these reductions do not affect the statement. In SL2 we have the computation #" " # " # " # # " 1 γ −1 α 1 α 1 0 γ 0 1 0 = −1  · , ·  1 0 1 γ α 1 0 γ −1 0 α 1 where γ = 1 + αα . In SU3 we compute that ⎡1 ⎢ ⎢0 ⎢ ⎢0 ⎣

α 1 0

β ⎤⎥ ⎡⎢ 1 α⎥⎥ · ⎢⎢α  1 ⎥⎦ ⎢⎣ β 

0 1 α

0⎤⎥ ⎡⎢ 1 0⎥⎥ = ⎢⎢α1 1⎥⎦ ⎢⎣ β1

0 1 α1

0⎤⎥ ⎡⎢γ 0⎥⎥ · ⎢⎢ 1⎥⎦ ⎢⎣

γγ −1

⎤ ⎡1 ⎥ ⎢ ⎥ · ⎢0 ⎥ ⎢ γ −1 ⎥⎦ ⎢⎣0

α1 1 0

β1 ⎤⎥ α1 ⎥⎥ , 1 ⎥⎦

where γ = 1 + αα  + ββ , α1 = (α + α  β)γ −1, and β1 = βγ −1 , α1 = (α  + α β )γ −1 , β1 = β  γ −1 . Notice that in both cases the assumption i + j > 0 implies that γ is a unit. These two equations are the decomposition uu  = x  z x in the two cases of SL2 and SU3 . The first point now follows by inspecting the formulas for β1 and β1 . For the second point, the assumption u ∈ U2a (k)∗ implies α = 0, i.e., u = (0, β), and the values simplify to γ = 1 + β β , α1 = α  βγ −1 , β1 = βγ −1 , α1 = α  γ −1 , β1 = β  γ −1 . A direct computation shows that xu−1 is ⎡1 ⎢ ⎢0 ⎢ ⎢0 ⎣

α  βγ −1 1 0

−β2 β  γ −1 ⎤⎥ α  βγ −1 ⎥⎥ . ⎥ 1 ⎦

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It is then clear that ϕ−a ((α1 , β 1 )) = ϕ−a ((α , β )). Also, ϕa (xu−1 ) = 12 ω(−β2 β  γ −1 ) = ω(β) + 12 ω(β ) = 2i + j. This completes the proof of the second point of (2). We will now establish the assertion in (2) about the commutator (u, u ). We know from (1) that uu  = x  z x, so (u, u ) = x  z(xu−1 )u −1 and as ϕa (xu−1 ) = 2i + j and ϕ−a (u −1 ) = j, we conclude from (1) that (xu−1 )u −1 ∈ U−a,ϕ, j · Z(k)0 · Ua,ϕ,2i+j , so (u, u ) ∈ x  z · U−a,ϕ, j · Z(k)0 · Ua,ϕ,2i+j = U−a,ϕ, j · Z(k)0 · Ua,ϕ,2i+j , since x  ∈ U−a,ϕ, j and z normalizes U−a,ϕ, j . We write u  = u1 m u2 , with u1, u2 ∈ Ua (k)∗ , ϕa (u1 ) = − j = ϕa (u2 ), and  m := m(u ). Then (u, u ) = uu1 · m (u2 u−1 u2−1 )m −1 · u1−1 = uu1 · m u−1 m −1 · u1−1, since u(∈ U2a (k)) commutes with u2 (∈ Ua (k)). Now we recall that Proposition 6.2.1 implies that ϕ−a (m u−1 m −1 ) = 2 j + ϕa (u−1 ) = i + 2 j, and as ϕa (u1−1 ) = ϕa (u1 ) = − j < i = ϕa (u), ϕa (uu1 ) = − j. Therefore, using (1) we find that (u, u ) = uu1 · m u−1 m −1 · u1−1 ∈ U−a,ϕ,i+2j · Z(k)0 · Ua,ϕ,−j . Combining the results of the preceding two paragraphs, we see that (u, u ) lies in the intersection: 



U−a,ϕ,i+2j · Z(k)0 · Ua,ϕ,−j ∩ U−a,ϕ, j · Z(k)0 · Ua,ϕ,2i+j . But since j  i + 2 j and − j  2i + j, the above intersection equals U−a,ϕ,i+2j · Z(k)0 · Ua,ϕ,2i+j . This proves (2). The proof of (3) is analogous.



Lemma 7.1.2 Let i, j ∈ R. We assume that i + j > 0. Let u ∈ Ua,ϕ,i and u  ∈ U−a,ϕ, j . Then (1) The commutator (u, u ) = uu u−1 u −1 lies in U−2a,ϕ,i+3j · U−a,ϕ,i+2j · Z(k)0 · Ua,ϕ,2i+j · U2a,ϕ,3i+j . (2) If u ∈ U2a,ϕ,2i , then the commutator (u, u ) lies in U−2a,ϕ,2i+4j · U−a,ϕ,2i+3j · Z(k)0 · Ua,ϕ,2i+j .

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Proof We assume (as we may) that u  1  u . Write u = u1 mu2 with u1 , u2 ∈ U−a (k)∗ , m := m(u); and u  = u1 m u2 , with u1, u2 ∈ Ua (k)∗ , m  := m(u ). Then uu u−1 u −1 = u1 · mu2 u u2−1 m−1 · u1−1 u −1 = u1 · m(u2 , u )m−1 · mu  m−1 · u1−1 u −1 . Since ϕ−a (u2 ) = −ϕa (u) = −i and ϕ−a (u ) = j, we find that ϕ−2a ((u2 , u ))  −i + j (see Remark 9.6.4). Proposition 6.2.1 implies ϕ2a (m(u2 , u )m−1 )  4i + (−i + j) = 3i + j and ϕa (mu  m−1 ) = 2i + j. We note that ϕ−a (u1−1 ) = ϕ−a (u1 ) = −i and ϕ−a (u −1 ) = ϕ−a (u ) = j > −i, hence, ϕ−a (u1−1 u −1 ) = −i. Now using Proposition 7.1.1(1), (2), and the fact that Ua commutes with U2a , we see that (u, u ) ∈ U−a (k) · Z(k)0 · Ua,ϕ,2i+j · U2a,ϕ,3i+j . Using u  = u1 m u2 , we see that (u, u ) = uu1 · m u2 u−1 u2−1 m −1 · u1−1 = uu1 · m (u2, u−1 )m −1 · m u−1 m −1 · u1−1 . As above, we find that ϕa (uu1 ) = − j, and ϕ−2a (m (u2, u−1 )m −1 )  (i − j) + 4 j = i + 3 j, and ϕ−a (m u−1 m −1 ) = i + 2 j. Now using Proposition 7.1.1(1), (2), we find that (u, u ) ∈ U−2a,ϕ,i+3j · U−a,ϕ,i+2j · Z(k)0 · Ua (k). Therefore, (u, u ) is contained in the following intersection: 

 U−2a,ϕ,i+3j ·U−a,ϕ,i+2j · Z(k)0 ·Ua (k) ∩ U−a (k)· Z(k)0 ·Ua,ϕ,2i+j ·U2a,ϕ,3i+j . As the above intersection is equal to U−2a,ϕ,i+3j · U−a,ϕ,i+2j · Z(k)0 · Ua,ϕ,2i+j · U2a,ϕ,3i+j , we see that (u, u ) belongs to this set; this proves assertion (1). To prove assertion (2), we assume that u ∈ U2a (k)∗ . Then, as before, using  u = u1 m u2 , and the fact that u commutes with Ua (k), we find that (u, u ) = uu u−1 u −1 = uu1 · m u−1 m −1 · u1−1 . Moreover, as ϕa (uu1 )  − j, ϕ−2a (m u−1 m −1 ) = 2i + 4 j, and ϕa (u1−1 ) = − j from Proposition 7.1.1(1)(2), we see that (u, u ) ∈ U−2a,ϕ,2i+4j · U−a,ϕ,2i+3j · Z(k)0 · Ua (k). On the other hand, using u = u1 mu2 , with u1 , u2 ∈ U−2a (k)∗ and m = m(u), we obtain (u, u ) = u1 · mu  m−1 · u1−1 u −1 . Note that ϕ−2a (u1 ) = 2ϕ−a (u1 ) = −2i,

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ϕa (mu  m−1 ) = 2i + j and ϕ−a (u1−1 u −1 ) = −i. Now using Proposition 7.1.1(1) we see that (u, u ) ∈ U−a (k) · Z(k)0 · Ua,ϕ,2i+j . Therefore, the commutator (u, u ) lies in the intersection

 

U−2a,ϕ,2i+4j · U−a,ϕ,2i+3j · Z(k)0 · Ua (k) ∩ U−a (k) · Z(k)0 · Ua,ϕ,2i+j . But this intersection equals U−2a,ϕ,2i+4j · U−a,ϕ,2i+3j · Z(k)0 · Ua,ϕ,2i+j . Thus we have proved (2).  Definition 7.1.3 Consider the commutator of the elements u −1 and u. In the notation of Proposition 7.1.1 it is given by u −1 uu u−1 = (u −1 x )z(xu−1 ) ∈ U−a (k)zUa (k). We call the element z the Z-component of the commutator of u −1 and u. Lemma 7.1.4 Let ψ : A → R be an affine functional with derivative a ∈ Φ,

 and u ∈ Uψ − Uψ+ . Then Uψ ⊂ Uψ+ ∪ U−ψ · m(u) · Z(k)0 · U−ψ . Moreover, if

 Uψ  Uψ+ , then Uψ+ ∩ U−ψ · m(u) · Z(k)0 · U−ψ = ∅. Proof Let ψ(ϕ) = s. Then ϕa (u) = s. As in the proof of the preceding proposition, we may (and do) assume that G equals the semi-simple subgroup generated by U±a . When G is quasi-split, one can easily reduce to the case of either SL2 or SU3 , which has been handled in Lemmas 3.1.3 and 3.2.8. We now discuss the general case and give a uniform proof that does not depend on these two lemmas. To prove the first assertion of the lemma, It will suffice to show that every x ∈ Ua (k)∗ , such that ϕa (x) = −s, lies in U−ψ · m(u) · Z(k)0 · U−ψ . Since ϕa (u) = ϕa (x), we see from Proposition 6.2.1 that the actions of m(u) and m(x) on the apartment A are equal. Therefore, z := m(u)−1 m(x)(∈ Z(k) acts trivially on A, and hence according to Proposition 6.2.3, z lies in the maximal bounded subgroup Z(k)1 of Z(k). But then as z is contained in the subgroup generated by U±a (k), we infer from Corollary 2.6.28 that z ∈ Z(k)0 . We know that there exist x , x  ∈ U−a (k) such that x  m(x) x  = x, and Axiom V 5 of Definition 6.1.2 implies that ϕ−a (x −1 ) = −ϕa (x) = s = ψ(ϕ) = ϕ−a (x −1 ). Axiom V 1 of Definition 6.1.2 implies that ϕ−a (x −1 ) = ϕ−a (x ) and ϕ−a (x −1 ) = ϕ−a (x ). Hence, we see that x , x  lie in U−ψ . Now x = x  m(x) x  = x  · m(u) · z · x  (∈ U−ψ · m(u) · Z(k)0 · U−ψ ). To prove the second assertion by contradiction, we assume that there exist x ∈ Uψ+ ( Uψ ) and x , x  ∈ U−ψ such that x ∈ x −1 m(u)Z(k)0 x −1 . Then m := x  x x  ∈ m(u)Z(k)0 , hence m normalizes S. This implies that m = m(x).

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As ϕa (x) > s, Axiom V 5 of Definition 6.1.2 implies that ϕ−a (x ) < −s =  −ψ(ϕ). Thus x   U−ψ , a contradiction.  r + s > 0. Let Z ⊂ Z(k) be the subgroup Lemma 7.1.5 (1) Let r, s ∈ R, generated by the Z-components of the commutators of all elements of U−a, x,r and Ua, x, s . Then Z ⊂ Z(k)0 and U−a, x,r · Z · Ua, x, s is a group.  r2  2r1 , s2  2s1 , r1  r2 + s1 , s1  s2 + r1 , (2) Let r1, r2, s1, s2 ∈ R, r2 + s2 > 0. Let Z ⊂ Z(k) be the subgroup generated by all Z-components of commutators of U−2a, x,r2 ∪ U−a, x,r1 and Ua, x, s1 ∪ U2a, x, s2 . Then Z ⊂ Z(k)0 and U−2a, x,r2 · U−a, x,r1 · Z · Ua, x, s1 · U2a, x, s2 is a group. (3) Let j ∈ R be such that Ua, x, j+  Ua, x, j . The group generated by U−a, x,−j , Ua, x, j , and Z(k)0 , contains an element m that normalizes S and acts on A as the reflection along the affine hyperplane {z | a(x − z) = j}. Furthermore, this group equals (U−a, x,−j+ · Z(k)0 · Ua, x, j ) ∪ (Ua, x, j · m · Z(k)0 · Ua, x, j ). (4) Let j1, j2 ∈ R be such that U2a, x, j2 +  U2a, x, j2 and j2 < 2 j1 . The group generated by U−2a, x,−j2 , Ua, x, j1 , U2a, x, j2 , and Z(k)0 , contains an element m that normalizes S and acts on it as the reflection along the affine hyperplane {z | a(x − z) = j2 /2}. Furthermore, this group equals the union of U−2a, x,−j2 + · U−a, x, j1 −j2 · Z(k)0 · Ua, x, j1 · U2a, x, j2 and U2a, x, j2 · Ua, x, j1 · m · Z(k)0 · Ua, x, j1 · U2a, x, j2 . Proof The first two points follow immediately from Proposition 7.1.1. Note that in the second point the inequalities that are being assumed imply the inequality r1 + s1 > 0. For the third point consider the affine functional ψ(z) = a(z − x) + j. Proposition 6.2.1 implies the existence of m with mψ = −ψ. In particular, this m realizes ra . Write L1 = U−ψ+ · Z(k)0 · Uψ ,

L2 = Uψ · m · Z(k)0 · Uψ ,

L = L1 ∪ L2 .

Then L is contained in the group generated by Uψ , U−ψ , and Z(k)0 . Conversely,

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it is enough to check that L is invariant under multiplication on the right by Uψ and U−ψ . Since mUψ m−1 = U−ψ this is equivalent to invariance under multiplication on the right by Uψ and m. It is clear from construction that each of L1 and L2 is invariant under multiplication on the right by Uψ . For invariance under m we use that by the first point of this lemma, L1 is a group. Thus we have L1 = Uψ · Z(k)0 · U−ψ+ . Then L1 · m = Uψ · m · Z(k)0 · Uψ+ ⊂ L2 . On the other hand, the relation m2 ∈ Z(k)0 implies L2 · m = Uψ · Z(k)0 · U−ψ . Lemma 7.1.4 shows U−ψ ⊂ U−ψ+ ∪Uψ ·m · Z(k)0 ·Uψ and the proof is complete. The proof of the fourth point is similar to that of the third point. We consider the affine functionals ψ1 (z) = a(z − x) + j1 , ψ2 (z) = 2a(z − x) + j2 , and ψ3 (z) = a(z − x) + j2 − j1 . We write L1 = U−ψ2 + · U−ψ3 · Z(k)0 · Uψ1 · Uψ2 ;

L2 = Uψ2 · Uψ1 · m · Z(k)0 · Uψ1 · Uψ2 .

Let L  be the group generated by Uψ1 , U±ψ2 , and Z(k)0 . Proposition 6.2.1 applied to U±ψ2 shows that L  contains an element m with mψ2 = −ψ2 . In particular, m realizes r2a = ra . From the above expressions for ψ1 , ψ2 and ψ3 , we find that ψ1 = 12 (ψ2 − j2 ) + j1 and ψ3 = 12 (ψ2 + j2 ) − j1 ; these show that mψ1 = −ψ3 . It is therefore clear that L = L1 ∪ L2 is contained in L . To show that L  = L it is enough to show that L is invariant under multiplication on the right by Uψ1 , Uψ2 , and m. Since Uψ1 and Uψ2 commute, each of L1 and L2 is obviously invariant under them by construction, so it remains to check invariance under m. The second point and the assumption j2 < 2 j1 imply that L1 is a group. We can therefore write it as L1 = Uψ2 Uψ1 Z(k)0U−ψ3 U−ψ2 + and obtain L1 · m = Uψ2 · Uψ1 · m · Z(k)0 · Uψ1 · Uψ2 + ⊂ L2 . At the same time, L2 · m = Uψ2 · Uψ1 · Z(k)0 · U−ψ3 · U−ψ2 . By Lemma 7.1.4 we have U−ψ2 ⊂ U−ψ2 + ∪ Uψ2 · m · Z(k)0 · Uψ2 and therefore L2 ·m ⊂ Uψ2 ·Uψ1 ·Z(k)0 ·U−ψ3 ·U−ψ2 + ∪Uψ2 ·Uψ1 ·Z(k)0 ·U−ψ3 ·Uψ2 ·m·Z(k)0 ·Uψ2 . The first set of the union on the right is L1 . To reorganize the second member we note that Uψ2 commutes with U−ψ3 and Uψ1 and is normalized by Z(k)0 ,  and use again mψ3 = −ψ1 to see that this member equals L2 .

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7.2 A Filtration of Z(k) The group Z = ZG (S) has anisotropic derived subgroup. By Proposition 2.2.12 the topological group Z(k) has a unique maximal bounded subgroup Z(k)b . For the discussion of open bounded subgroups of G(k) it would be useful to have a filtration of Z(k)b . Since the filtration subgroups will all be contained in Z(k)b , the filtration will be indexed by R0 rather than R. The essential property of this filtration involves commutators. Definition 7.2.1 A decreasing separated filtration of Z(k)b will be called commutator friendly if Z(k)0 = Z(k)0 and for any a ∈ Φ, ϕ ∈ A, u ∈ Ua (k)∗ , u  ∈ U−a (k)∗ with r := ϕa (u) + ϕ−a (u ) > 0, the Z-component of the commutator u −1 uu u−1 of u −1 and u lies in Z(k)r , cf. Definition 7.1.3. For now we assume that such a filtration of Z(k) is given. For the construction of the building of G we will only need Z(k)0 = Z(k)0 . The filtration subgroups Z(k)r for r > 0 will be needed for the discussion of Moy–Prasad filtration groups, and more generally groups associated to concave functions f with f (0) > 0. So a reader may safely ignore this issue on a first reading. When G is quasi-split, then Z = T is a maximal torus, and we can obtain such a filtration as follows. Definition 7.2.2 The assignment of a decreasing separated filtration T(k)r∗ of T(k)1 to any torus T/k is called functorial if for every morphism f : T1 → T2 of tori over k, the map f : T1 (k)1 → T2 (k)1 satisfies f (T1 (k)r∗ ) ⊂ T2 (k)r∗ . In AppendixB, a functorial filtration, called the minimal congruent filtration, is constructed, independently of the material in the main body of this book. It has the following properties. (1) T(k)0 is the Iwahori group T(k)0 of Definition 2.5.13. (2) If T is induced and r > 0 then T(k)r = {t ∈ T(k) | for all χ ∈ X∗ (T), ω( χ(t) − 1)  r }. As we will discuss in AppendixB, we can take the above formula for T(k)r for a general torus T, provided its ramification is not too wild. If G is either simply connected or adjoint, then T is induced and the filtration of T(k) is determined by (1) and (2) above. For a general k-torus T one must be more careful. We refer the reader to AppendixB for more detail, in particular Definition B.10.8. Lemma 7.2.3 Every functorial filtration satisfying (1) and (2) above is commutator friendly.

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Proof This follows from the argument of the proof of Proposition 7.1.1 in the case of quasi-split G. Indeed, since commutators of elements u ∈ Ua (k) and U−a (k) are computed in the simply connected cover of the semi-simple group of k-rank equal to 1 that is generated by U±a and U±2a , one can argue as in that proof (using the functoriality of the filtration) and reduce to computations in SL2 and SU3 . Those were carried out in that proof, and the element γ computed there is the Z-component of the commutator and has the desired property.  Our construction of the filtration on Z(k) when G is not quasi-split will be given in §9.8.

7.3 Concave Functions In this section we introduce the concept of a concave function, define the open bounded subgroups of G(k) associated to such functions, and study their grouptheoretic properties. The parahoric subgroups of G(k) will be a special case of this construction, as will be the Moy–Prasad filtration subgroups.  is called concave if  = Φ ∪ {0}. A function f : Φ →R Definition 7.3.1 Let Φ f (a + b)  f (a) + f (b)  with a + b ∈ Φ.  for any a, b ∈ Φ  be concave. →R Lemma 7.3.2 Let f : Φ (1) f (0)  0.  are such that a1 + · · · + an ∈ Φ,  then (2) If a1, . . . , an ∈ Φ f (a1 + · · · + an )  f (a1 ) + · · · + f (an ).  and t1, . . . , tn ∈ R0 are such that ti ai = 0, then (3) If a1, . . . , an ∈ Φ ti f (ai )  0. Proof The first point follows from the trivial inequality f (0) = f (0 + 0)  f (0) + f (0). The second point is proved by induction on n. The case n  2 holds by assumption. Assume the statement for n and consider a1, . . . , an+1 . Using the first point and the inductive hypothesis we may assume without loss of generality that ai  0 for all i = 1, . . . , n + 1. Let a = a1 + · · · + an+1 .  If a = 0 we can We claim there exists 1  j  n + 1 such that a − a j ∈ Φ. simply take j = 1. If a  0 let (−, −) be a Weyl group-invariant scalar product on V(T). We have (a, a) > 0 and hence there exists 1  j  n + 1 such that (a, a j ) > 0. According to [Bou02, Chapter VI, §1, Corollary of Theorem 1] we

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 The claim is thus proved. Rearranging so that j = 1 we see have a − a j ∈ Φ. f (a)  f (a1 ) + f (a2 + · · · + an+1 ). The inductive hypothesis finishes the proof of the second point. For the third point we may reduce from the case of ti ∈ R0 to the case of ti ∈ Q0 by observing that the subspace of Rn consisting of those (t1, . . . , tn ) such that ti ai = 0 is defined over Q and therefore its set of Q-points is dense in its set of R-points. We can further reduce from the case ti ∈ Q0 to the case ti ∈ Z0 by multiplying all ti by a sufficiently large positive integer. The claim then follows from the second point.  Definition 7.3.3 Let f be a concave function and x ∈ A. Define the following subgroups of G(k). (1) Ua, x, f = Ua, x, f (a) · U2a, x, f (2a) for a ∈ Φ. (2) G(k)x, f is generated by Ua, x, f for all a ∈ Φ.

(3) Px, f := G(k)x, f := G(k)x, f · Z(k) f (0) .

(4) For a ∈ Φ, G(k)(a) x, f is generated by U±a, x, f .

(a) (a) (a) (a) (5) For a ∈ Φ, Nx, f = N(k) ∩ G(k) x, f and Z x, f = Z(k) ∩ G(k) x, f .

Remark 7.3.4 We have used the usual convention U2a = {1} if 2a  Φ. Assume 2a ∈ Φ. If f (2a) = 2 f (a) then Ua, x, f = Ua, x, f (a) . It is possible however that f (2a) < 2 f (a) and Ua, x, f (a)  Ua, x, f . Remark 7.3.5 While the filtration subgroup Ua, x, f of Ua (k) is unambiguously defined in terms of the valuation of root datum x, the filtration subgroup Z(k) f (0) of Z(k) depends on which filtration of Z(k) we have chosen to work with, as explained in §7.2.  are concave and g  f , then Ua, x,g ⊂ Ua, x, f →R Remark 7.3.6 If f , g : Φ for all a ∈ Φ and x ∈ A; therefore, G(k)x, g ⊂ G(k)x, f and G(k)x,g ⊂ G(k)x, f . Example 7.3.7 (1) The constant function f (a) = 0 is concave. The corresponding group Px, f will play a fundamental role in the theory, and will be studied in §7.4.  by →R (2) If Φ+ ⊂ Φ is a positive system of roots, define fΦ+ : Φ ) 0, a ∈ Φ+ ∪ {0} fΦ+ (a) = 0+, a ∈ Φ− . Then fΦ+ is concave. Let F ⊂ A be a facet and let C ⊂ A be a chamber that contains F in its closure. Let ΨF (C)+ be the positive system of roots of Proposition 1.3.35 and let Φ+ be a positive system of roots that contains

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The Bruhat–Tits Building for a Valuation of the Root Datum the set ∇ΨF (C)+ . For any x ∈ F we have by Lemma 6.3.20 and the observations in 6.3.21, Ua,C,0 = Ua, x, fΦ+ .

(3) Given v ∈ V, the function f (a) = a(v) is linear, in particular concave. (4) More generally, if Ω ⊂ A is a non-empty subset and x ∈ A, denote by  →R x − Ω the subset {v ∈ V | x − v ∈ Ω}. Then the function fΩ, x : Φ defined by fΩ, x (0) = 0, and for a ∈ Φ, fΩ, x (a) = sup a(x − Ω) = inf{ψ(x) | ψ = a, ψ(Ω)  0} is concave. Here the infimum is taken over all affine functionals with derivative a. For f = fΩ, x we have  Ua, x, f = Ua, x, f (a) = Ua, x, = Ua, y,0 = Uψ .  ∈a(x−Ω)

y ∈Ω

 ψ=a,ψ(Ω)0

If Ω is bounded, then f takes values in R.  are concave, then so is f1 + f2 . In particular,  → R Fact 7.3.8 If f1, f2 : Φ if f is concave and v ∈ V, then the function f + v defined by ( f + v)(a) = f (a) + a(v) is also concave. Moreover Ua, x+v, f +v = Ua, x, f for all x ∈ A, hence G(k)x+v, f +v = G(k)x, f .  − {∞} be a concave function. There exists  → R Lemma 7.3.9 Let f : Φ  we have f (a) + a(v)  0. v ∈ V(T) such that for all a ∈ Φ  → R be Proof We first reduce to the case that f takes real values. Let f  : Φ  the function defined by f (a) = ra whenever f (a) = ra or f (a) = ra + with  it is ra ∈ R. Note that f  is still concave. Since f  (a)  f (a) for all a ∈ Φ,  enough to prove the statement with f replaced by f . This allows us to assume that f takes real values.  ⊂ V∗ × R Consider the graph of f , that is the set Γ f = {(a, f (a))| a ∈ Φ} ∗ and let X be the convex cone spanned by it in the vector space V × R. Lemma 7.3.2 states that X does not contain any point of the form (0, t) with t < 0. Farkas’ lemma implies that there exists a linear form λ : V ∗ × R → R such  Under the identification that λ((0, −1)) < 0 and λ(a, f (a))  0 for all a ∈ Φ. (V ∗ × R)∗ = V × R we have λ = (w, b). Then λ((0, −1)) < 0 implies b > 0 and λ(a, f (a))  0 is equivalent to a(w)+b f (a)  0. We may now set v = b−1 w.   − {∞} be a concave function and x ∈ A.  → R Lemma 7.3.10 Let f : Φ  → R such that for every a ∈ Φ,  There exists a concave function f  : Φ Ua, x, f (a) = Ua, x, f  (a) .

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Proof This relies on the fact that the jump sets Ja are discrete. This implies  and r ∈ R there exists  > 0 such that for all 0 <     that for each for a ∈ Φ we have Ua, x,r+ = Ua, x,r+  . Therefore there exists  > 0 such that if we define  with f  so that f (a) = f (a) if f (a) ∈ R and f (a) = ra +  if f (a) = ra + ∈ R  Making  smaller if necessary ra ∈ R, then Ua, x, f (a) = Ua, x, f  (a) for all a ∈ Φ.  we arrange that f  is concave. Lemma 7.3.11 Let f be a concave function and x ∈ A. (1) If a, b ∈ Φ are non-proportional, then the commutator (Ua, x, f , Ub, x, f ) is contained in the subgroup generated by Upa+qb, x, f for all integers p, q > 0 with pa + qb ∈ Φ. (2) Let Φ+ be a positive system of roots and Φ + ⊂ Φ+ a closed subset.  The product a ∈Φ+ Ua, x, f , taken in any order, is a subgroup of G(k) independent of the order. (a) (3) G(k)(a) x, f = U−a, x, f · Ua, x, f · Nx, f . (4) Assume that f (a) + f (−a) > 0 and, when a is multipliable also f (2a) + (a) (a) f (−2a) > 0. Then Nx, f = Z x, f . Proof The first point follows from V3 of Definition 6.1.2. The second point follows from the first. For the third and fourth points we apply Lemma 7.1.5. The concavity of f implies f (2a)  2 f (a), f (−2a)  2 f (−a), f (a)  f (2a) + f (−a), f (−a)  f (−2a) + f (a), f (0)  f (2a) + f (−2a)  2( f (a) + f (−a)), where we disregard the inequalities involving 2a, −2a if a is not multipliable. If a is multipliable and f (2a) + f (−2a) > 0 (hence also f (a) + f (−a) > 0), or if a is not multipliable and f (a) + f (−a) > 0, we can apply the first two points of Lemma 7.1.5. Assume that f (a) + f (−a) = 0. If a is multipliable, then 0  f (0)  f (2a) + f (−2a)  2( f (a) + f (−a)) = 0 implies f (2a) + f (−2a) = 0, and then f (2a) = 2 f (a), f (−2a) = 2 f (−a). Whether or not a is multipliable, we can now apply the third point of Lemma 7.1.5 if Ua, x, f (a)+  Ua, x, f (a) , and the first point of that lemma otherwise. Assume finally that a is multipliable, f (2a) + f (−2a) = 0, but f (a) + f (−a) > 0. If U2a, x, f (2a)+ = U2a, x, f (2a) or U−2a, x, f (−2a)+ = U−2a, x, f (−2a) we can apply the second point of Lemma 7.1.5. If U2a, x, f (2a)+  U2a, x, f (2a) and U−2a, x, f (−2a)+  U−2a, x, f (−2a) , then we may apply the fourth point of Lemma 7.1.5, for we know that at least one of f (2a) < 2 f (a) and f (−2a) < 2 f (−a) must hold and we may assume without loss of generality that f (2a) < 2 f (a). Note that according to that lemma the group generated by Ua, x, f (a) , U2a, x, f (2a) , U−2a, x, f (−2a) , and Z(k)0 , contains U−a, x, f (a)− f (2a) , which contains U−a, x, f (−a) since f (−a) + f (2a)  f (a). 

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Proposition 7.3.12 Let f be a concave function and x ∈ A. Let Φ+ ⊂ Φ be a positive system of roots, Φ+,nd the subset of indivisible positive roots and Φ−,nd = −Φ+,nd . Let U + and U − be the unipotent radicals of the Borel subgroups ± = U ± (k)∩G(k) , N  = N(k)∩G(k) , determined by Φ+ and Φ− . Set Ux, f x, f x, f x, f   and Z x, f = Z(k) ∩ G(k) x, f .

− · U + · N  , where the product is taken in G(k). (1) G(k)x, f = Ux, f x, f x, f  (a) (2) The group Nx, f is generated by Nx, f for all a ∈ Φ. (3) The product morphism  ± Ua, x, f → Ux, f a ∈Φ±, nd

is a bijection, where the factors are taken in an arbitrary fixed order.   (4) If f (0) > 0 then Nx, f = Z x, f .   (5) If Nx, f = Z x, f then the product morphism    Ua, x, f × Z x, × Ua, x, f → G(k)x, f f a ∈Φ−, nd

a ∈Φ+, nd

is bijective, where again the factors in each product are taken in an arbitrary fixed order. Proof Let Xa = Ua, x, f for all a ∈ Φnd , with U2a = {1} if 2a  Φ. Let X ± be the product of Xa for all a ∈ Φ±,nd , taken with respect to an arbitrary fixed order. It is a group by Lemma 7.3.11. Let N  be the subgroup of N(k) generated (a) − +  by Nx, f for all a ∈ Φ. We claim that the set X · X · N is independent of the + choice of Φ . To prove the claim, it is enough to show that if we exchange a simple root b(∈ Φ+ ) by its negative, the product X − · X + · N  is unaffected. Let X ±b be the product of X±a for all a ∈ Φ+,nd not equal to b. Then X b and X −b are subgroups of X + and X − respectively by Lemma 7.3.11(2). Using assertion (1) of that lemma we see that X b and X −b are normalized by both Xb and (b) (b) X−b . The same Lemma 7.3.11 implies that X−b Xb Nx, f = G(k) x, f ; therefore, X−b Xb N  = Xb X−b N . With this we see that X − X + N  equals X −b X−b Xb X b N  = X −b X b X−b Xb N  = X −b X b Xb X−b N  = X −b Xb X−b X b N  . The claim is thus proved. It implies that the set X − · X + · N  is invariant under conjugation by elements of N , and also equal to X + · X − · N , which ultimately implies that it is a subgroup of G(k)x, f . Furthermore, this claim implies that the subgroup X − · X + · N  is invariant under multiplication (say on the left) by

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elements of Ua, x, f (a) for all a ∈ Φ. This implies that this subgroup equals all of G(k)x, f .

Consider now x − · x + · n  ∈ X − · X + · N  = G(k)x, f . If x − · x + · n  ∈ U − (k) then x + · n  ∈ U − (k), which by Bruhat decomposition ([Bor91]) implies n  = 1 − . In the same way one sees X + = and hence x + = 1. This shows X − = Ux, f + − +  Ux, f . Finally if x · x · n lies in N(k), then Bruhat decomposition implies   x − · x + · n  = n , showing Nx, f = N . This proves the first two points. The third point follows from Lemma 7.3.11(2). The fourth point follows from Proposition 2.11.3(2), noting that f (0) > 0 implies f (a) + f (−a) > 0 for all a ∈ Φ. For the final point, surjectivity follows from the preceding discussion, and injectivity from Proposition 2.11.3(2).   Corollary 7.3.13 If f takes only non-negative values, then the action of Nx, f on A fixes the point x.

Proof

(a) According to Proposition 7.3.12 it is enough to show that each Nx, f

(a) aff is trivial unless fixes the point x. By Lemma 7.3.11(4) the image of Nx, f in W either f (a) + f (−a) = 0 or f (2a) + f (−2a) = 0. If f (a) + f (−a) = 0, then also f (2a)+ f (−2a) = 0, and the assumption that f takes non-negative values shows f (a) = f (−a) = f (2a) = f (−2a) = 0. If Ua, x,0 = Ua,x,0+ then replacing f (a) (a) by f (a)+ and f (2a) by f (2a)+ does not change Nx, f and again its image in W aff is trivial. Otherwise, the third point of Lemma 7.1.5 shows that the image (a) of Nx, f is the group of order 2 generated by the reflection along the affine hyperplane {z | a(x − z) = 0}. If f (a) + f (−a) > 0 but f (2a) + f (−2a) = 0, we (a) aff have f (2a) = f (−2a) = 0. If U2a,x,0 = U2a,x,0+ then the image of Nx, f in W is trivial. Otherwise the fourth point of Lemma 7.1.5 shows that this image is generated by the reflection along the affine hyperplane {z | a(x − z) = 0}. In all cases, this image fixes the point x. 

Corollary 7.3.14 The subgroup G(k)x, f of G(k) is bounded. If f does not take the value +∞, then G(k)x, f is also open. Proof Openness follows from Corollary 8.2.6. To prove boundedness, apply Lemma 7.3.9 and Fact 7.3.8 to replace x by x + v and f by f + v without changing G(k)x, f and assume that f takes non-negative values. Each group

 aff fixes x by Corollary Ua, x, f is bounded by definition. The image of Nx, f in W 7.3.13. Therefore it does not contain any non-trivial translation and hence is  finite. Since Z(k)0 is bounded it follows that Nx, f is bounded. Proposition 7.3.12 allows us to write G(k)x, f as the product of Ua, x, f for all

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 a ∈ Φnd , Nx, f , and Z(k) f (0) . Each of these being bounded, the boundedness of  G(k)x, f follows from Lemma 2.2.6.

Remark 7.3.15 As we saw in Example 8.2.7, the concavity of f is essential for the boundedness of G(k)x, f . Lemma 7.3.16 G(k)x, f ∩ Z(k) = Z(k) f (0) . Proof By construction the inclusion Z(k) f (0) ⊂ G(k)x, f ∩ Z(k) holds. We will now prove the opposite inclusion. For this, it is enough to show that G(k)x, f ∩ Z(k) ⊂ Z(k) f (0) .

Assume first that f (0) = 0. Since G(k)x, f lies in G(k) , we may assume without loss of generality that G is semi-simple and simply connected. By Corollary 7.3.14 the intersection G(k)x, f ∩ Z(k) is a bounded subgroup of Z(k), thus contained in Z(k)1 , but Z(k)1 = Z(k)0 by Lemma 2.6.27.   Assume now that f (0) > 0. Recall the notation Z x, f = G(k) x, f ∩ Z(k).

 (a) Proposition 7.3.12 shows that Z x, f is generated by Z x, f , while Proposition 7.1.1 shows that the latter is generated by the Z-components of commutators of elements of Ua, x, f and U−a, x, f . These lie in Z(k) f (0) due to the assumption that the filtration of Z(k) satisfies Definition 7.2.1. 

In the remainder of this section we will discuss the following question: if  → R is a concave function taking non-negative values, x ∈ A, under what f:Φ conditions on f is G(k)x, f normalized by G(k)x,0 ? We begin with a positive result.  is constant, then G(k)x, f is invariant under →R Proposition 7.3.17 If f : Φ conjugation by G(k)x,0 and N(k)x . Proof The case f = +∞ is trivial, because then G(k)x, f = {1}. Using Lemma 7.3.10 we may assume that f is valued in R. It is clear that G(k)x, f is invariant under conjugation by Z(k)0 . Being invariant under conjugation by G(k)0x is thus equivalent to being invariant under conjugation by Ua, x,0 for every a ∈ Φ. This is equivalent to conjugation by Ua, x,0 sending Ub, x, f (b) into G(k)x, f for all a, b ∈ Φ, which is in turn equivalent to the commutator group (Ua, x,0, Ub, x, f (b) ) being contained in G(k)x, f . According to Axiom V 3 of Definition 6.1.2, given u ∈ Ua, x,0 and v ∈ Ub, x, f (b) we  have (u, v) ∈ p,q Upa+qb, x, q f (b) . If f (pa+qb)  q f (b) then [u, v] ∈ G(k)x, f . The condition f (pa + qb)  q f (b) is satisfied if f is constant, for then this constant must be non-negative due to f (0)  0, cf. Lemma 7.3.2. Therefore, if f is constant, G(k)x, f is invariant under conjugation by G(k)x, f . On the other

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hand, it is clear that for any n ∈ N(k) we have n · G(k)x, f · n−1 = G(k)nx, n f , which shows that G(k)x, f is normalized by N(k)x provided f is constant.  One may ask whether the converse of Proposition 7.3.17 holds – if G(k)x, f is normalized by G(k)x,0 and N(k)x , then f is constant. While normality does pose significant restriction on f , it does not force it to be constant in general. To explore this phenomenon we will assume that G is split and A is the equipollence class of Chevalley valuations of Definition 6.1.15. These results will not be used in the remainder of the book and can be skipped on first reading.  → R be a concave Definition 7.3.18 Let x := (xa )a ∈Φ ∈ A and f : Φ   → R defined by function. The optimization of f at x is the function f : Φ  f (a) = min{r ∈ R | r  f (a), Ua, x,r+  Ua, x,r }  If  for a ∈ Φ. f = f , we will say that f is optimal at x. It is obvious from the definition of  f that Ua, x, f (a) = Ua, x, f (a) for every  For a ∈ Φ we have the alternative expression a ∈ Φ.  f (a) = min{xa (u) | u ∈ Ua (k)∗, xa (u)  f (a)}. Lemma 7.3.19

Assume that p  2, 3. Then  f is concave.

Proof Let a, b ∈ Φ and assume that a + b is also a root. Let u ∈ Ua (k)∗ and v ∈ Ub (k)∗ . We will denote the a + b component of the commutator (u, v) by (u, v)a+b . If x ∈ A is a Steinberg valuation, then the Chevalley–Steinberg commutator identities reviewed in §2.9(d) imply that xa+b ((u, v)a+b ) = xa (u) + xb (v). This identity is visibly unchanged if we replace x by x + v for any v ∈ V(S ), and therefore holds for any x ∈ A. For every a ∈ Φ we fix an element ua ∈ Ua (k)∗ such that xa (ua ) =  f (a). Then ua ∈ G(k)x, f . Consider a, b ∈ Φ such that a  ±b and a + b ∈ Φ, then the commutator (ua, ub ), and hence also the a + b component (ua, ub )a+b of (ua, ub ), lie in the subgroup G(k)x, f , as well as in Ua+b (k). According to Proposition 7.3.12, (ua, ub )a+b lies in Ua+b, x, f = Ua+b, x, f . Since according to the preceding paragraph, xa+b ((ua, ub )a+b ) = xa (ua ) + xb (ub ) =  f (a) +  f (b),    we conclude that f (a + b)  f (a) + f (b). Consider next the case of a, −a ∈ Φ. This case reduces to an immediate computation in SL2 which shows that the S-component of the commutator of  ua and u−a lies in S(k)r − S(k)r+ , where r = xa (ua ) + x−a (u−a ). Lemma 7.3.20 Assume that f is optimal at x and takes non-negative values.

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Let Ψx = {ψ ∈ Ψ | ψ(x) = 0} and let Φx = ∇Ψx . If G(k)x, f is normal in G(k)0x , then for all a ∈ Φ and b ∈ Φx such that a + b ∈ Φ, we have f (a + b) = f (a). Proof Choose u ∈ G(k)x, f ∩ Ua (k) and v ∈ G(k)0x ∩ Ub (k) such that xa (u) = f (a) and xb (v) = 0. Then the valuation of (u, v)a+b is f (a) and so it lies in G(k)x, f ∩ Ua+b (k) = Ua+b, x, f , from which we see that f (a + b)  f (a). Applying this with a  = a + b and b = −b we conclude that f (a)  f (a + b), hence the result.  Corollary 7.3.21 Assume that f is optimal at x and takes non-negative values. If G(k)x, f is normalized by G(k)0x , then f is constant on Φx . Proof Take a, b ∈ Φx . By applying the above lemma to the pairs (a, b) and  (b, a) we find the f is constant on Φx . It need not be true that f is constant outside of Φx . Even more, it need not be true that there exists a constant whose optimization at x equals f . To see this, choose a chamber whose closure contains x and a special vertex o in the closure of that chamber. This gives the identification Ψ = Φ × Z and a choice of a Borel subgroup containing S. The set of jumps of the filtration Ua, x,r is x, a + Z. Lemma 7.3.22 For a ∈ Φ and b ∈ Φ − Φx we have f (a + b) = f (a) + rb − δa,b for some δa,b ∈ {0, 1}, where rb is the smallest non-negative member of x, b + Z. Proof Choose u ∈ G(k)x, f ∩ Ua (k) and v ∈ G(k)0x ∩ Ub (k) such that xa (u) = f (a) and xb (v) = rb . As in the previous lemma one obtains f (a+b)  f (a)+rb . Applying the same computation with a  = a + b and b = −b one obtains f (a)  f (a + b) + r−b . Combining this leads to f (a) − r−b  f (a + b)  f (a) + rb . Note that rb + r−b = 1, and that f (a) − r−b and f (a) + rb are consecutive elements of x, a + b + Z.  Example 7.3.23 Consider the split group Sp4 . Let o be the special point corresponding to the standard integral structure and let x be the barycenter of the standard alcove. If {a, b} is the standard basis of Φ, with a short and b long, then a, x = b, x = 1/4. The following gives an example of a concave function f that is optimal for x and such that G(k)x, f is normalized by G(k)0x .

7.4 Parahoric Subgroups 1 4

+9 b

3 4

+8

1 4

+9

1 2

+9

3 4

a1

1 2

+9

255

+9

4

+ 10

3 4

+9

7.4 Parahoric Subgroups In this section we define the parahoric subgroups of G(k). They are special cases of the concave function groups introduced in §7.3 and inherit the properties of these groups. We will prove here further properties that are specific to parahoric subgroups. Let Ω ⊂ A be a non-empty subset. Definition 7.4.1

(1) Ua,Ω,0 = x ∈Ω Ua, x,0 .  (2) G(k)Ω is the subgroup of G(k) generated by Ua,Ω,0 for all a ∈ Φ.   (3) NΩ = N(k) ∩ G(k)Ω , ZΩ = Z(k) ∩ G(k)Ω .  (4) PΩ = G(k)0Ω is the product Z(k)0 · G(k)Ω .

(5) NΩ0 = N(k) ∩ G(k)0Ω = NΩ · Z(k)0 .

If Ω = {x} we write G(k)x ⊂ G(k)0x in place of G(k){x } ⊂ G(k)0{x } . Remark 7.4.2 Even when G is simply connected it need not be true that G(k)x = G(k)0x . If we take G = SL2 and let x be the barycenter of a chamber, then " " # # 1+m o o o G(k)x =  G(k)0x = . m 1+m m o Lemma 7.4.3

For any subset Ω ⊂ A we have Z(k) ∩ G(k)0Ω = Z(k)0 .

Proof

This is a special case of Lemma 7.3.16.



Recall from Definition 4.1.14 the notation cl(Ω) for the union of all facets of B, equivalently of A, that intersect Ω non-trivially. Lemma 7.4.4

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(1) The group G(k)x depends only on the facet containing x. More generally,  depends only on cl(Ω). the group G(k)Ω (2) Let x ∈ A be any point and f be the function fx,Ω of Example 7.3.7(4).  Then G(k)Ω = G(k)x, f .  is the subgroup of G(k) generated by Uψ for all ψ ∈ Ψ (3) The group G(k)Ω with ψ(Ω)  0.

Proof Lemma 6.3.20 states that for each a ∈ Φ the group Ua, x,0 depends only on the facet of A containing x, from which the first claim follows. The second point follows from Example 7.3.7, where we saw that Ua, x, f = Ua,Ω,0 .  Ua (k)ψ . This would imply the third We also saw that this equals ψ=a,ψ(Ω)0  point once we argue that it is enough to take the union over affine roots instead of all affine functionals. It is clear that we may restrict ψ to the set Ψa of Notation 6.3.6 because then the set of subgroups of Ua (k) obtained from all affine functionals is the same as the one obtained from the set Ψa . If a is not multipliable then elements of Ψa and Ψa agree. If a is multipliable this is not  the case, but the subgroup of Ua (k) generated by ψ ∈Ψa ,ψ(Ω)0 Ua (k)ψ and    ψ ∈Ψ2 a ,ψ(Ω)0 U2a (k)ψ is equal to the subgroup ψ ∈Ψa , ψ(Ω)0 Ua (k)ψ . Definition 7.4.5 (1) If Ω ⊂ A is a facet, the group G(k)0Ω is called the parahoric subgroup of G(k) associated to Ω. (2) When Ω is a chamber, G(k)0Ω is called the Iwahori subgroup of G(k) associated to Ω. (3) A parahoric subgroup of G(k) is any G(k)-conjugate of a subgroup of the form G(k)0Ω for a facet Ω. (4) An Iwahori subgroup of G(k) is any G(k)-conjugate of a subgroup of the form G(k)0Ω for a chamber Ω. In Definition 4.1.3 we gave a different characterization of parahoric subgroups and Iwahori subgroups. That the two definitions coincide will be shown in Proposition 7.6.4. There is yet another characterization of Iwahori subgroups in the special case when the field k is local, cf. Proposition 13.5.2. Fact 7.4.6 Let F ⊂ A be a facet and let C ⊂ A be a chamber whose closure contains F. Let ΨF (C)+ be the positive system of roots of Proposition 1.3.35 and let Φ+ be a positive system of roots that contains ∇ΨF (C)+ . Let f = fΦ+ be the concave function of Example 7.3.7(2). For any x ∈ F we have  = G(k)x, f . G(k)C

Lemma 7.4.7 The image of the map NΩ → W aff is the subgroup WΩaff of W aff

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257

that fixes each point of Ω. In particular, if ψ is an affine root, the group NΩ contains an element realizing the reflection along ψ if and only if ψ(x) = 0. Proof Choose an arbitrary x ∈ A and let f = fΩ,x be the function of Example 7.3.7(4). Apply Lemma 7.4.4(2) and Proposition 7.3.12 to conclude that NΩ is generated by NΩ(a) for all a ∈ Φ. Lemma 7.1.5 shows that the image of NΩ(a) in W aff is trivial unless there exists an affine root ψ ∈ Ψ with derivative a or 2a that vanishes on Ω, in which case that image is of order 2 and generated by the reflection along ψ. Thus the image of NΩ in W aff is the subgroup generated by reflections along those affine roots that vanish on Ω. According to [Bou02, Chapter V, §3, no. 3, Proposition 2] this is precisely the pointwise stabilizer of  Ω in W aff . Corollary 7.4.8 Let x ∈ A be a vertex. The fixed point set for the action of G(k)x ∩ NG (T)(k) on A is equal to {x}. Proof Since x is a vertex, there is a set {ψ1, . . . , ψn } of affine roots that vanish at x and whose derivatives are a basis for the vector space underlying A. If ri is the reflection along ψi , the subgroup of the affine Weyl group generated by {r1, . . . , rn } has a unique fixed point in A and this fixed point is x. According  to Lemma 7.4.7, the group G(k)x ∩ N(k) realizes each reflection ri . Corollary 7.4.9 Let C ⊂ A be a chamber, x ∈ C. Let Φ+ ⊂ Φ be a positive system of roots. Then Nx = Z x and with r = 0 or r = 0+ we have    = Ua, x,r · Z x · Ua, x,r . G(k)C a ∈Φ−, nd

a ∈Φ+, nd

If F is a facet contained in the closure of C and if Φ+ contains the positive system of roots ΦF (C)+ determined by C as in Proposition 1.3.35, then    = Ua,F,0+ · Z x · Ua,F,0 . G(k)C a ∈Φ−, nd

a ∈Φ+, nd

Proof Since x does not lie on any affine root hyperplane, for each a ∈ Φ we have Ua, x,0 = Ua, x,0+ . For the same reason, Lemma 1.3.17 implies that Wxaff is trivial. The image of Nx in W, which by Lemma 7.4.7 equals the image of Wxaff in W, is therefore trivial, which implies Nx = Z x . The first equation  defined by →R follows from Proposition 7.3.12 applied to the function f : Φ f (a) = 0+. The second equation follows from the first and 6.3.21.  Lemma 7.4.10  . G(k)F 1

 ⊂ (1) If F1, F2 ⊂ A are facets with F1 ⊂ F2 , then G(k)F 2

  (2) If F1, F2 ⊂ A are facets and F1  F2 , then G(k)F  G(k)F . 1 2

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Proof If ψ ∈ Ψ satisfies ψ(F2 )  0 and F1 ⊂ F2 , then ψ(F1 )  0, hence the first point. For the second point, the assumption F1  F2 implies the existence of an affine root ψ such that ψ(F1 ) = 0 and ψ(F2 )  0. Replacing ψ by −ψ if  . Proposition necessary we may assume ψ(F2 ) < 0. By definition Uψ ⊂ G(k)F 1  , hence the 7.3.12 applied to a point x ∈ F2 and f = 0 shows that Uψ  G(k)F 2 second point. 

 is an NGad (Sad )(k)-equivariant Corollary 7.4.11 The assignment F → G(k)F order-reversing injection from the set of facets in A to the set of subgroups of G(k).

Proof The NGad (Sad )(k)-equivariance is immediate. Lemma 7.4.10 implies that it is order reversing and injective.  As discussed in Lemma 7.4.4, for a given facet F the group PF = G(k)0F coincides with the group G(k)x,0 of Definition 7.3.3 for the concave function  → R that is constant equal to 0 and any x ∈ F. We may write PF,0 for this Φ group. In analogous fashion we may define PF,0+ to be the group G(k)x,0+ associated to the constant concave function equal to 0+ and any x ∈ F. Lemma 7.4.12 Let F1, F2 be facets such that F1 is contained in the closure of F2 . Then the following inclusion relations hold: PF1 ,0+ ⊂ PF2 ,0+ ⊂ PF2 ,0 ⊂ PF1 ,0 . Proof For every a ∈ Φ we have Ua,F1 ,0+ ⊂ Ua,F2 ,0+ ⊂ Ua,F2 ,0 ⊂ Ua,F1 ,0 according to 6.3.21. The claim follows from Definition 7.3.3.  The following Lemma is a complement to Lemma 7.4.7. It describes what happens when reflections fail to lie in a parahoric subgroup. Lemma 7.4.13 Let z ∈ A. Given u ∈ Ua (k) − {1}, let u , u  ∈ U−a (k) such that m(u) = u uu  normalizes S. (1) If u  Pz , then u , u  ∈ Pz . (2) If u ∈ Pz,0+ , then u  and u  do not belong to Pz . Proof (1) We have Ua (k)∩Pz = Ua,z,0 by Proposition 7.3.12. The assumption u  Pz implies that za (u) < 0. Then z−a (u ) = z−a (u ) = −za (u) > 0 by Definition 6.1.2. Therefore u , u  ∈ Pz . (2) Assume that u ∈ Pz,0+ . Then za (u) > 0, implying ϕ−a (u ) < 0 and  ϕ−a (u ) < 0, and hence u , u  cannot belong to Pz .

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7.5 The Iwahori–Tits System Assume for a moment that G is semi-simple and simply connected. Let C ⊂ A be a chamber and I = G(k)0C the corresponding Iwahori subgroup. Let G = G(k), N = N(k), Z = Z(k)0 = Z(k)1 , cf. Lemma 2.6.27. Let R ⊂ W aff be the set of reflections along the affine roots whose hyperplanes contain a face of codimension 1 of C. Note that R is in bijection with the set of walls of C. When G is absolutely simple, R is in bijection with the set of vertices of C, cf. Proposition 1.3.22(6). Theorem 7.5.1 (1) The tuple (G, I, N, R) is a saturated Tits system. (2) The action of N on A identifies the Weyl group of (G, I, N, R) with the Weyl group of the affine root system Ψ. (3) Up to conjugation by G, this Tits system is independent of the choices of S and C. Proof Corollary 7.4.9 shows that I ∩ N = Z, which is normal in N. According to Proposition 6.6.2 the quotient N/Z is isomorphic to W aff . The set R evidently consists of elements of W aff of order 2 and Proposition 1.3.22(1) states that W aff is generated by R. Let a ∈ Φ. By construction the group I contains the subgroup Ua (k)C,0 . We  have Ua (k) = t ∈Z(k) t · Ua,C,0 · t −1 . This shows that the subgroup of G(k) generated by I and Z(k) ⊂ N contains each Ua (k). Therefore it equals all of G(k). Let s ∈ R and let α ∈ Ψ(C)0 be the corresponding affine root, that is, the unique affine root whose reflection is s and such that α(x) > 0 for x ∈ C. We have Uα ⊂ I. By Proposition 7.3.12 we have Ua (k) ∩ sIs = Uα+ , where Uα+  Uα is the next smallest filtration subgroup. Thus Uα ∩ sIs  Uα , and hence sIs  I. Let Ψ  be the set of smallest indivisible affine roots with β(C) > 0. Applying Corollaries 7.4.9 and 6.3.11 (with a system of positive roots Φ+ that contains the  derivative of ψ) we have I = β ∈Ψ  −{α } Uβ · Z · Uα . The action of s preserves the set of positive affine roots not proportional to α (Proposition 1.3.25), as well as the partial order on the set of affine functionals (Definition 1.2.17). Thus s preserves the set Ψ  − {α, −α + k}, negates α, and sends −α + k to α + k, where −α + k is the smallest affine roots greater than −α. Therefore sI ⊂ IsUα . At the same time, Lemma 7.1.5(3) implies that X = (Uα sZUα ) ∪ (Usα+ ZUα )

(7.5.1)

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is a group. We claim that X wI ⊂ IswI ∪ IwI.

(7.5.2)

Since X contains an element realizing the reflection s, we have X wI = X swI. Thus, we may replace w by sw if necessary to achieve that [w −1 α](C) > 0. Assuming this, we see Uα ⊂ wIw −1 , hence Uα sZUα wI ⊂ IswI. At the same time, Usα+ ZUα ⊂ I, and the claim is proved. Now sUα ⊂ X and (7.5.2) imply sUα wI ⊂ IswI ∪ IwI, which together with (7.5.1) implies sIw ⊂ IsUα wI ⊂ IswI ∪ IwI completing the proof that (G, I, N, R) is a Tits system. The argument used above to prove I ∩ sIs  I can be iterated to prove

n∈N I = Z. Together with the equality I ∩ N = Z this implies that the Tits system is saturated. That the Weyl group of the Tits system is isomorphic to that of the affine root system Ψ is stated in Proposition 6.6.2. Since any two maximally split maximal tori of G are conjugate under G, and any two chambers are conjugate under the affine Weyl group by Lemma 1.3.17, the final claim follows.  We now consider a general connected reductive group G. Let G = G(k) , G = G(k)0 , N = N(k) ∩ G , N = N(k) ∩ G. Let Z  be the preimage of Z in Gsc . Let Z be the image of Z (k)0 = Z (k)1 in G(k) and let Z = Z(k)0 . Again we fix a chamber C ⊂ A and let I = G(k)0C be the Iwahori subgroup,  I = G(k)C , and I = I · Z . Let R ⊂ W aff be the set of reflections along the affine roots whose hyperplanes contain the faces of C.

Lemma 7.5.2 The subgroup G ⊂ G is normal. We have the following equalities. (1) (2) (3) (4) (5) (6)

I ∩ N = Z. I ∩ N  = Z . Z ∩ G = Z . Z · G = G. I = I · Z. N = N · Z.

Proof The identity I ∩ N = Z follows from Corollary 7.4.9. The group I is generated by subgroups of root groups and therefore lies in G(k) . The intersection I ∩ Z(k) then lies in the image of Z (k). Since I is bounded, its preimage in Z (k) lies in Z (k)1 = Z (k)0 , thus I ∩ Z(k) lies in Z . From this

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I ∩N = Z follows. Since the preimage of Z in Z (k) lies in Zsc (k)1 = Zsc (k)0 we have Z = Z ∩ G . The equality Z · G = G is just Definition 2.6.23 and the equality I = I · Z follows from Definition 7.4.1. The equality N = N · Z is immediate from Definition 2.6.23.  Theorem 7.5.3 (1) The tuples (G, I, N, R) and (G , I , N , R) are saturated Tits systems. (2) The action of NG (T)(k) on A identifies the Weyl groups of these Tits systems with the Weyl group of the affine root system Ψ. (3) Up to conjugation by G , these Tits systems are independent of the choices of T and C. Proof The map Gsc (k) → G has central kernel contained in Z (k)0 = Z (k)1 so the claim about (G , I , N , R) follows from Theorem 7.5.1 and the first part of Lemma 1.4.12. The claim about (G, I, N, R) follows from the second part of Lemma 1.4.12 and Lemma 7.5.2.  Definition 7.5.4 The tuple (G, I, N, R) is called the Iwahori–Tits system. In Definition 1.5.17 we introduced the concept of an admissible parabolic subgroup of a Tits system. We take a moment to examine this concept in the special case of the Iwahori–Tits system. Proposition 7.5.5 Let F ⊂ A be a facet contained in the closure of a chamber C ⊂ A. Let WFaff be the subgroup of W aff generated by the reflections along those C-simple affine roots that vanish on F. Then WFaff is the stabilizer of F in W aff and G(k)0F = G(k)0C · WFaff · G(k)0C . Proof That WFaff is the stabilizer of F is stated in Lemma 1.3.17. According to Theorem 7.5.3, the right-hand side is a group, namely the standard parabolic subgroup of the Tits system corresponding to the subset of R consisting of those reflections that fix F. Since every affine root that is non-negative on C is also non-negative on F we have the inclusion G(k)0C ⊂ G(k)0F . The inclusion WFaff ⊂ G(k)0F follows from Lemma 7.4.7. This shows that the right-hand side of the asserted equality is contained in the left-hand side. For the opposite inclusion, observe that we may generate the left-hand side by choosing for each a ∈ Φ the unique affine F with that derivative that is smallest on F, see Remark 6.3.17. If such root ψa,0 an affine root ψ is strictly positive on F, then it is also strictly positive on C,  . Otherwise ψ vanishes identically on F. Then so does and then Uψ ⊂ G(k)C  −ψ. Exactly one of ψ, −ψ is strictly positive on C, say ψ. Thus Uψ ⊂ G(k)C .

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The reflection along Hψ lies in WFaff . Since this reflection sends ψ to −ψ we see that U−ψ is also contained in the right-hand side. The opposite inclusion is proved.  Proposition 7.5.6 (1) The admissible parabolic subgroups of the Iwahori–Tits system are precisely the parahoric subgroups of G(k). (2) An admissible parabolic subgroup of the Iwahori–Tits system is standard if and only if it is a parahoric subgroup G(k)0F for a facet F of C. Proof We will first prove (2). Assume first that the affine root system Ψ is irreducible, equivalently that the Iwahori–Tits system is irreducible. Given a non-empty subset X ⊂ R we associate to it the facet of C whose vertices are the set X c = R−X. This establishes a bijection between the facets of C, by definition required to be non-empty , and the proper subsets of R. If X and F correspond in this way, then the stabilizer W(Ψ)F of F in the affine Weyl group is equal to the subgroup W(Ψ)X generated by X, according to Proposition 1.3.35. The standard parabolic subgroup of type X for the Iwahori–Tits system is then by definition I · W(Ψ)X · I, which by Proposition 7.5.5 equals the parahoric subgroup G(k)0F . Conversely, Proposition 7.5.5 implies that any parahoric subgroup associated to a facet of C is of this form. This establishes the claim when Ψ is assumed irreducible. Now drop the assumption that Ψ is irreducible. Lemma 1.4.14 states that the standard parabolic subgroups of the Iwahori–Tits system are in bijection with the standard parabolic subgroups of the Tits system (G , I , N , R), the bijection sending P to P · Z, and its inverse sending P to P ∩ G . We have the  ·Z analogous bijection on the level of parahoric groups, since G(k)0F = G(k)F  and G(k)F ⊂ G . We may thus replace (G, I, N, R) by (G , I , N , R). Applying the same argument to the surjective homomorphism Gsc (k) → G with finite central kernel we reduce to the case that G is simply connected. Write G = G1 × · · · × G n , with Gi a simply connected k-simple group. The Iwahori–Tits system of G decomposes as a product of the Iwahori–Tits systems for each Gi . Example 1.4.11 shows that a standard parabolic subgroup for (G, I, N, R) has the form P1 × · · · × Pn , where Pi is a standard parabolic subgroup for (Gi , Ii , Ni , Ri ). By definition P is admissible if and only if each Pi is. At the same time, the chamber C decomposes as C1 × · · · × Cn , and a facet of C is of the form F1 × · · · × Fn , where each Fi is a facet of Ci , cf. Proposition 1.3.21 and Remark 1.3.16. By construction G(k)0F = G1 (k)0F1 × · · · × G n (k)0Fn . The claim now follows from the case that Ψ is irreducible. The proof of (2). For (1), a parabolic subgroup of the Iwahori–Tits system is a G-conjugate of

7.6 The (Reduced) Building

263

a standard parabolic subgroup, hence a parahoric subgroup by what was just proved. Conversely, a parahoric subgroup P of G(k) is a G(k)-conjugate of G(k)0F for a facet F of A. Since Z(k)0 acts trivially on A, Fact 2.6.22 implies that P is also G-conjugate to G(k)0F . Since all chambers in A are conjugate under the affine Weyl group, P is also G-conjugate to G(k)0F for a facet F of C. This completes the proof of (1).  Proposition 7.5.7 Let P ⊂ G be a parabolic subgroup for the Iwahori–Tits system and let X ⊂ R be its type. The following are equivalent. (1) P is admissible. (2) P is a bounded subgroup of G(k). (3) The subgroup W(Ψ)X of W(Ψ) = W aff is finite. Proof Using the same argument as in the proof of Proposition 7.5.6 we reduce to the case that G is simply connected and k-simple, so that Ψ is irreducible. If P is admissible, it is a parahoric subgroup of G(k) by Proposition 7.5.6, hence bounded by Lemma 7.4.4 and Corollary 7.3.14. Moreover, its type X is a proper subset of R and Proposition 1.3.35 implies that W(Ψ)X is the Weyl group of a finite root system, hence is itself finite. If P is not admissible, then P = G, which is not bounded, and W(Ψ)X = W(Ψ), which is infinite, cf. Proposition 1.3.11. 

7.6 The (Reduced) Building We continue with a connected reductive k-group G with non-empty root system. Let S ⊂ G be a maximal k-split torus and C ⊂ A a chamber. According to Proposition 1.5.6, there is a building associated to the Tits system of Theorem 7.5.3. However, unless G is k-simple, this is not the Bruhat–Tits building of G(k). Rather, the Bruhat–Tits building of G(k) is the restricted building of Definition 1.5.20. We will not have use for the non-restricted building that Proposition 1.5.6 assigns to the Iwahori–Tits system. Definition 7.6.1 The Bruhat–Tits building of G(k), denoted by B(G), or simply B, is the restricted building of the Iwahori–Tits system of G. Remark 7.6.2 The Iwahori–Tits system, and hence the Bruhat–Tits building, were constructed here by taking as an input a system of apartments. When G is quasi-split, a canonical such system was constructed in §6.1. Hence the building is also canonical. When G is not quasi-split, we have taken the existence of such a system as an unspecified input so far. Only in §9.6 will we be able to obtain such a system canonically, based on the building of the quasi-split group G K .

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Recall from Proposition 1.5.18 that the building B(G) is a polysimplicial complex, which is simplicial if and only if G is k-simple. Recall further that the building associated to a Tits system comes equipped with a standard apartment and a standard chamber. Since the Iwahori–Tits system is independent of the choices of S and C up to conjugation by G(k), the building B(G) is also independent of these choices. The standard apartment of course depends on the choice of S, but not on the choice of C. The standard chamber of B(G) depends on the choice of C. Recall that the facets of the polysimplicial complex B(G) are the admissible parabolic subgroups of the Iwahori–Tits system. These are not to be confused with the parabolic subgroups of G. The facets of the standard apartment are those admissible parabolic subgroups that contain a conjugate of I under N(k)∩ G(k)0 . The standard chamber is the group I itself. Therefore, a priori, the abstract polysimplicial complex that is the standard apartment and the affine space A with its polysimplicial structure are not related. In Proposition 7.6.3 below we will show that the affine space A with its polysimplicial structure induced by the affine root system Ψ of Definition 6.3.4 is a geometric realization of the standard apartment of B(G). Under this identification, C is identified with the standard chamber of B(G). By construction, B(G) comes equipped with an action of G(k)0 . This action easily extends to an action of G(k). Indeed, Lemma 6.3.19 shows that the action of G(k) by conjugation on the set of subgroups of G(k)0 preserves the set of (admissible) parabolic subgroups of the Iwahori–Tits system. The various stabilizers for this action will be studied more carefully in §7.7. The natural maps Gsc → Gder → G → Gad induce maps between the associated Tits systems that satisfy the conditions of Lemma 1.4.13. Therefore they induce bijections B(Gsc ) → B(Gder ) → B(G) → B(Gad ).

(7.6.1)

In particular, the group Gad (k) acts on B(G) = B(Gad ). Proposition 7.6.3 Let S ⊂ G be a maximal k-split torus. (1) The affine space A with the polysimplicial structure coming from the affine root hyperplanes as in Remark 1.3.16 is a geometric realization of the polysimplicial complex that is the standard apartment of B(G). In particular, the apartments of B(G) are in G(k)-equivariant bijective correspondence with the maximal k-split tori of G. (2) Let F ⊂ A be a facet. The parabolic subgroup of the Iwahori–Tits system corresponding to F equals the parahoric subgroup G(k)0F of Definition 7.4.5.

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265

Proof Let A ⊂ B(G) be the standard apartment of B(G). Let G = G(k)0 , N = N(k) ∩ G, and let I = G(k)0C as in Definition 7.4.5. Then (G, I, N, R) is the Iwahori–Tits system, where R is the set of reflections along the affine root hyperplanes that are the walls of C. The facets of A are the admissible parabolic subgroups of the Iwahori–Tits system that contain an N-conjugate to I. By Lemma 6.3.19 this is equivalent to containing an N(k)-conjugate of I. According to Propositions 7.5.6 and 6.3.13 these are precisely the parahoric subgroups of the form G(k)0F for facets F ⊂ A. Corollary 7.4.11 now implies that this correspondence is a bijection between the facets of A and the facets of A, and that this bijection preserves the closure relations between facets.  We now have a building B(G) equipped with an action of G(k)0 , and even of G(k). For a subset Ω of A(S) ⊂ B(G) the notation G(k)0Ω is suggestive of the stabilizer of Ω for the action of G(k)0 on B(G). On the other hand, we used this notation for a subgroup of G(k)0 defined in §7.4 “by hand.” The following proposition reconciles this conflict. Proposition 7.6.4 Let Ω ⊂ A be a subset of a facet. Consider the groups  and G(k)0Ω of Definition 7.4.1. G(k)Ω (1) The group G(k)0Ω is the stabilizer, as well as the pointwise stabilizer, of Ω in G(k)0 .  (2) The groups G(k)Ω and G(k)0Ω act transitively on the set of apartments containing Ω.  and G(k)0Ω depend only on the subset Ω of B(G), and (3) The groups G(k)Ω not on the apartment A. For a more general subset Ω ⊂ A see Proposition 7.7.5.  Proof Let F be the facet containing Ω. The groups G(k)Ω and G(k)0Ω do not change if we replace Ω by F. Since the action of G(k) on B preserves the polysimplicial structure, the stabilizer of Ω equals the stabilizer of F. Proposition 1.5.13 implies that G(k)0F is the stabilizer of F in G(k)0 and it stabilizes every face of it. Thus, it is the pointwise stabilizer of F in G(k)0 . The same proposition implies that, given two apartments A1, A2 containing F, there exists g ∈ G(k)0 sending A1 to A2 and fixing every point of F. By the first point g ∈ G(k)0Ω . Thus G(k)0Ω acts transitively on the set of apartments containing Ω. Fixing one such apartment A corresponding to a maximal k-split  · Z(k)0 . Since Z(k)0 stabilizes A, we torus S ⊂ G, we have G(k)0Ω = G(k)Ω

  · A is the set of all apartments containing Ω, so G(k)Ω also acts see that G(k)Ω transitively on all apartments containing Ω. The third point follows immediately from the second. 

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Remark 7.6.5 The above Proposition completes the proof of Axioms 4.1.1, 4.1.4, and 4.1.9. In particular, we may apply Proposition 4.4.4 to see that a geometric realization of B(G), compatible with the action of Gad (k), is obtained by taking the quotient of G(k) × A under the following equivalence relation: (g, x) ∼ (h, y) ⇔ ∃n ∈ N(k) : nx = y, g −1 hn ∈ G(k)x .

(7.6.2)

This quotient remains unchanged if we replace G by Gad , or by Gsc , or by Gder , as well as on replacing G(k)x by any larger subgroup contained in G(k)x . In light of Proposition 7.6.3, we will from now on not distinguish between the polysimplicial complex B(G) and its geometric realization. In line with this, a facet F of B(G) will be considered as a subset, and written as F ⊂ B(G), rather than F ∈ B(G), which was the notation used in the case of abstract buildings. Some of the main properties of B were listed in Propositions 1.5.6 and 1.5.13. We now collect further properties. Proposition 7.6.6 If f is finite, then B is locally finite. Proof It is enough to show that a given facet F ⊂ B is contained in only finitely many chambers. Propositions 1.5.6 and 1.5.13 imply that PF acts transitively on the set of chambers whose closure contains F. Fix one such chamber C. Then PF /PC is in bijection with the set of chambers containing F in their closure. The groups PF and PC are open and bounded by Corollary 7.3.14. Since f is finite, k is locally compact, and therefore PF and PC are compact by Fact 2.2.3.  The quotient PF /PC is thus finite. Fact 7.6.7 Let ψ ∈ Ψ. The affine root subgroup Uψ of Definition 6.3.1 fixes pointwise the closed half-apartment Aψ0 of Definition 1.2.17. Proof Immediate from (7.6.2).



Proposition 7.6.8 The stabilizer of A in G(k) is N(k), and the pointwise stabilizer of A in G(k)1 is Z(k)1 . Proof We know from Proposition 1.5.13 that the stabilizer of A in G(k)0 is N(k) ∩ G(k)0 . On the other hand it is clear that N(k) stabilizes A. The first claim follows from Fact 2.6.22, and the second claim follows from the first and Proposition 6.2.4.  Corollary 7.6.9 The map S → A(S) is a G(k)-equivariant bijection between the set of maximal k-split tori of G and the set of apartments of (the geometric realization of) B(G).

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Proposition 7.6.10 Let S ⊂ G be a maximal k-split torus, Ω ⊂ A a non-empty bounded subset. Then Z(k) ∩ G(k)0Ω = Z(k)0 . Proof The inclusion ⊇ holds by construction of G(k)0Ω . For the opposite inclusion we use that Z(k) acts on A(S) by translations through the negative of the valuation homomorphism Z(k) → V(S ). The left-hand side is thus contained in the kernel of that homomorphism, but also in G(k)1 , hence by  Lemma 2.6.19 in Z(k)1 , hence by Corollary 2.6.28 in Z(k)0 .

7.7 Disconnected Parahoric Subgroups Let A ⊂ B be an apartment. Let Ω ⊂ A be a non-empty subset. We have the  ⊂ G(k)0Ω of Definition 7.4.1. We will define a chain subgroups G(k)Ω b ⊂ G(k)1Ω ⊂ G(k)†Ω ⊂ G(k)Ω G(k)0Ω ⊂ G(k)Ω

(7.7.1)

of open subgroups of G(k). For this we consider the chain Z(k)0 ⊂ Z(k)1 ⊂ N(k)1Ω ⊂ N(k)†Ω ⊂ N(k)Ω

(7.7.2)

of subgroups of the stabilizer N(k)Ω of Ω in N(k), where N(k)†Ω is the stabilizer of Ω in N(k)1 = N(k) ∩ G(k)1 and N(k)1Ω is the pointwise stabilizer of Ω in N(k)1 .  is normalized by N(k)Ω . We define each of the groups in The group G(k)Ω  and the corresponding the chain (7.7.1) as the product, within G(k), of G(k)Ω group in the chain (7.7.2). It is clear that in either chain all containments except for the last one are of finite index. When Ω = {x} consists of a single element x we will write G(k)x instead of G(k){x } , etc. We have G(k)1x = G(k)†x .

b , G(k)1 , and G(k)† are bounded subProposition 7.7.1 The groups G(k)Ω Ω Ω groups of G(k). If Ω is bounded, then these groups are open, and G(k)0Ω is of finite index in G(k)†Ω .

Proof Openness follows from Corollary 7.3.14, since all these groups contain G(k)0Ω . Assume now that Ω is bounded. Then the image of N(k)†Ω in W ext does not contain a non-trivial translation and is therefore finite. According to Proposition 6.2.4, this image is isomorphic to N(k)†Ω /Z(k)1 . Since Z(k)1 is bounded, we see that N(k)†Ω is bounded. The boundedness of G(k)†Ω now follows from Corollary 7.3.14 and Lemma 2.2.6. The index of G(k)0Ω in G(k)†Ω is less than or equal

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to the index of Z(k)0 in N(k)†Ω , which is finite, since Z(k)1 /Z(k)0 is finite by Lemma 2.6.29.  The group G(k)Ω is not bounded unless it equals G(k)†Ω , which is the case if and only if the center of G(k) is bounded. If Ω is bounded, then G(k)Ω is open. Remark 7.7.2 Proposition 7.3.12(1) holds for any of the groups in (7.7.1) in  , by construction of those groups. place of G(k)Ω Lemma 7.7.3 Let a ∈ Φ. Then Ua (k) ∩ G(k)Ω = Ua,Ω,0 , cf. Definition 7.4.1. Proof Choose a system of negative roots Φ− containing a. According to Lemma 7.4.4 and Proposition 7.3.12(1) we have G(k)Ω = U(k)−Ω · U(k)+Ω · N(k)Ω,  . Let u ∈ Ua (k)∩G(k)Ω and write it as u = v − v + n. where U(k)±Ω = U(k)∩G(k)Ω Then u−1 v − v + = n−1 and the Bruhat decomposition ([Bor91]) implies n = 1, hence v + = 1, and u = v − . Thus u ∈ U(k)−Ω . Proposition 7.3.12(3) implies the claim. 

Lemma 7.7.4

Let Ω ⊂ A be a non-empty subset. For ∗ ∈ {0, b, 1} we have G(k)∗x . G(k)∗Ω = x ∈Ω

Proof The inclusion ⊂ is clear by construction of the groups, so it is enough to show the opposite inclusion. Let NΩ∗ = G(k)∗Ω ∩ N(k). We claim that NΩ∗ =

∗ x ∈Ω Nx . The kernel of NΩ0 → W aff equals Z(k)1 ∩ G(k)0 by Proposition 6.2.4, which in turn equals Z(k)0 by Corollary 2.6.28. As Z(k)0 ⊂ G(k)0Ω by construction, Lemma 7.4.7 shows that NΩ0 is the preimage in N(k) ∩ G(k)0 of the subgroup

of W aff that fixes Ω pointwise. This implies NΩ0 = x ∈Ω Nx0 . We have NΩb = [G(k)0Ω · Z(k)1 ] ∩ N(k) = NΩ0 · Z(k)1 and NΩb /Z(k)1  NΩ0 /Z(k)0 , from which

we see NΩb = x ∈Ω Nxb . We also have NΩ1 = [G(k)0Ω · N(k)1Ω ] ∩ N(k) = NΩ0 · N(k)1Ω = N(k)1Ω,

and the equality NΩ1 = x ∈Ω Nx1 is immediate. The claim is thus proved. Next we claim that given a non-empty subset Ω ⊂ A and x ∈ A we have G(k)∗Ω∪ {x } = G(k)∗Ω ∩ G(k)∗x . Choose y ∈ Ω and let Φ+ be a positive system of roots, all of which are non-negative on x − y. Apply Proposition 7.3.12 to write G(k)∗Ω = UΩ+ · UΩ− · NΩ∗ .

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Let g ∈ G(k)∗Ω ∩ G(k)∗x . Write g = uvn according to the above decomposition. For every a ∈ Φ+ we have Ua,Ω,0 ⊂ Ua, y,0 = Ua, x, a(x−y) ⊂ Ua, x,0 since a(x − y)  0. Thus UΩ+ ⊂ Ux+ , hence u ∈ G(k)∗Ω ∩ G(k)∗x , hence vn ∈ G(k)∗Ω ∩ G(k)∗x . We now apply Proposition 7.3.12 and the negative of Φ+ to write G(k)∗x = Ux− · Ux+ · Nx∗ and decompose accordingly vn = v u  n . Then n(n )−1 = v −1 v  · u  ∈ U(k)− · U(k)+ . By the Bruhat decomposition we have n = n . Proposition 2.11.3(2) ∗ and implies v = v  and u  = 1. In other words, n  ∈ Nx∗ ∩ NΩ∗ = NΩ∪ {x } − − − ∗ v ∈ Ux ∩ UΩ = UΩ∪ {x } . It follows that g = uvn ∈ G(k)Ω∪ {x } . The claim we have just proved implies by induction the statement of the lemma for all non-empty finite sets Ω. Since a general set Ω can be written as  an increasing union of finite sets, it remains for us to prove that if Ω = i ∈I Ω i is an increasing union of non-empty sets, then G(k)∗Ω = G(k)∗Ω i .

We may assume that all Ω i contain a common point x. Let g ∈ G(k)∗Ω i . We fix arbitrarily a positive system of roots Φ+ and write g = ui vi ni according to the decomposition G(k)∗Ω i = UΩ+ i · UΩ− i · NΩ∗ i . Let Z ∗ = G(k)∗Ω ∩ Z(k), so that Z 0 = Z(k)0 and Z b = Z 1 = Z(k)1 . The quotient Nx∗ /Z(k)∗ is finite, so passing to a cofinal family of the index set I we may assume that the image of ni in Nx∗ /Z ∗ is constant. Fix a lift n ∈ Nx∗ of that image and write ni = ti n with ti ∈ Z ∗ . Then we have ui vi t = u j v j t for all i, j ∈ I. Then Proposition 2.11.3(2) shows that each of ui , vi , and ti is constant in i. Writing u, v, and t for

these elements we see u ∈ UΩ+ i = UΩ+ , v ∈ UΩ− i = UΩ− , and t ∈ Z ∗ . Thus  g ∈ G(k)∗Ω . Proposition 7.7.5

Let Ω ⊂ A be a non-empty bounded subset.

(1) The group G(k)∗Ω is the pointwise stabilizer of Ω in G(k)∗ for ∗ = 0, b, 1. (2) The group G(k)Ω is the stabilizer of Ω in G(k). (3) The group G(k)†Ω is the stabilizer of Ω in G(k)1 . Thus G(k)†Ω = G(k)Ω ∩ G(k)1 .  acts transitively on the set of apartments containing Ω. (4) The group G(k)Ω (5) Each group in the chain (7.7.1) depends only on Ω, and not on the apartment A.

Proof For (1) Lemma 7.7.4 reduces the case of a general set Ω to the case of Ω = {x} for some x ∈ A. If g ∈ G(k)∗ fixes x then gA and A are two apartments containing x. By Proposition 7.6.4 there exists h ∈ G(k)0x

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such that hgA = A. Proposition 7.6.8 implies hg ∈ N(k)x ∩ G(k)∗ . For ∗ = 1 we have N(k)x ∩ G(k)1 = N(k)1x ⊂ G(k)1x by definition. For ∗ = 0, N(k) ∩ G(k)0 maps onto W aff with kernel Z(k)0 by Theorem 7.5.3. According to Lemma 7.4.7, N(k) ∩ G(k)0 ⊂ G(k)0x . For ∗ = b we have N(k)x ∩ G(k)b = [N(k)x ∩ G(k)0 ] · Z(k)1 ⊂ G(k)0x · Z(k)1 = G(k)bx . This proves (1). For (4), given an apartment A containing Ω, Proposition 4.2.24 implies the existence of g ∈ G(k)0Ω with the property gA = A and gx = x for all x ∈ Ω.  · N(k)1Ω By the previous point we conclude g ∈ G(k)1Ω . Since G(k)1Ω = G(k)Ω

 we see that there is g  ∈ G(k)Ω with g A = A. This proves (4), and (5) follows at once. For (2), let g ∈ G(k) be such that gΩ = Ω. Since A and gA both contain Ω we  such that hgA = A. Proposition 7.6.8 implies hg ∈ N(k)Ω . obtain h ∈ G(k)Ω Therefore g ∈ G(k)Ω . This proves (2), and the same argument also proves (3). 

Remark 7.7.6 The above proposition shows that the groups G(k)1Ω , G(k)†Ω , and G(k)Ω have a natural interpretation in terms of the action of G(k) on B(G). When Ω lies in a facet, the group G(k)0Ω also has such a natural interpretation, b does not have such a natural in terms of Proposition 7.6.4. The group G(k)Ω interpretation. The most natural interpretation of G(k)0Ω for general Ω is that the integral model of G it corresponds to, via the constructions we will discuss in Chapter 8, has connected special fiber. In fact, we will construct smooth affine group schemes GΩ0 , GΩb , GΩ1 , GΩ† over o with generic fiber G and group of o-points given by the corresponding member of (7.7.1). The special fiber of GΩ0 will be connected, while that of GΩ† will not be. All the groups in the chain (7.7.1) account for the various reasons this special fiber is disconnected. There are two main sources for the disconnectedness of the special fiber of GΩ† . The first is that the special fiber of the finite type Néron-Raynaud model of T can be disconnected. This reason is reflected in the difference between T(k)0 and T(k)b , hence in the disconnectedness of the special fiber of GΩb . The second reason is that non-trivial elements of the affine Weyl group may preserve, or even pointwise fix, the set Ω. The existence of Weyl elements b and G(k)1 . that fix Ω pointwise accounts for the difference between G(k)Ω Ω The existence of Weyl elements that preserve Ω without fixing it pointwise accounts for the difference between G(k)1Ω and G(k)†Ω . That latter difference becomes more pronounced when G is replaced by a central quotient, which has  1 farther away from the affine Weyl group and closer the potential of moving W  1 ⊂ W ext to the extended affine Weyl group, in the chain of inclusions W aff ⊂ W discussed in §6.6.

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Remark 7.7.7 Recall from Lemma 7.4.4 that the group G(k)x , and hence also G(k)0x and G(k)bx , depends only on the facet containing x. This does not hold for G(k)1x . Indeed, consider the group PGL2 . The element " # 0 1 , π 0 where π ∈ k is a uniformizing element, preserves the standard apartment in the building of PGL2 , as well as a chamber of it, but acts non-trivially on it. Thus it fixes the mid-point of that chamber, but no other point. Lemma 7.7.8

Let g ∈ G(k).

(1) For a non-empty bounded subset Ω of an apartment of B one has has g · G(k)∗Ω · g −1 = G(k)∗gΩ, where ∗ denotes any of the decorations in the chain (7.7.1).  (2) For any maximal k-split torus S, x ∈ A, and concave function f : Φ → R, −1 −1  one has g · G(k)x, f · g = G(k)gx, g f , where g f : Φ(gSg ) → R is the transport of f . Proof For the first point we may choose an apartment A containing Ω. Then gA contains gΩ, and the claim follows directly from the definitions of the various groups in (7.7.1). The proof of the second claim is also immediate.  Lemma 7.7.9 Assume that G is semi-simple and simply connected, and Ω is b = G(k)1 = G(k)† = contained in a facet. Let x ∈ Ω. Then G(k)0Ω = G(k)Ω Ω Ω G(k)x . Proof We have G(k) = G(k)0 by definition. Since B(G) is by definition the restricted building of a Tits system in G(k)0 , the action of G(k)0 on B(G) is via polysimplicial automorphisms and fixes pointwise every facet that it preserves, cf. Proposition 1.5.13. Since G(k)x preserves the facet containing x, it fixes it pointwise, in particular it fixes Ω pointwise. The claim follows from Proposition 7.7.5.  Lemma 7.7.10

If G is quasi-split and either adjoint or unramified, then b . G(k)0Ω = G(k)Ω

Proof Let T ⊂ G be a maximally split maximal torus. If G is adjoint then T is induced, while if G is unramified then TK is split. In both cases, TK is an  induced torus, and Lemma 2.5.18 implies T(k)1 = T(k)0 . Proposition 7.7.11 If x is a special point, then G(k)bx = G(k)1x = G(k)†x .

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Proof By construction G(k)bx is normal in G(k)†x and we have G(k)†x = G(k)bx · N(k)†x . Let n ∈ N(k)†x . By Lemmas 1.3.42 and 7.4.7 there exists an element n  ∈ G(k)0x ∩ N(k) with the same image in W as n. Then t := n−1 n  ∈ Z(k) ∩ G(k)1 is an element whose action on A fixes x. Since this action is by translation, this translation must be trivial. Proposition 6.2.4 implies that  t ∈ Z(k)1 ⊂ G(k)bx . 0 = Given a maximal k-split torus S ⊂ G recall from §6.6 the group W 0 N(k)/Z(k) . Proposition 7.7.12 Let A be an apartment in B(G), C a chamber in A, I the corresponding Iwahori subgroup, and N the normalizer of I in G(k). The  0 ) → N/I is an isomorphism. In particular, natural homomorphism Stab(C, W if G is quasi-split and adjoint, we obtain the isomorphism N/I → Ξ, where Ξ is the stabilizer of C in the extended affine Weyl group, cf. Remark 1.3.76. Proof Lemma 7.5.2 gives the equality I ∩ N(k) = Z(k)0 (recall the notation N = N(k) and Z = Z(k)0 of that lemma). Therefore, the embedding N(k)C → N induces an embedding N(k)C /Z(k)0 → N/I. This embedding is in fact a bijection due to Proposition 7.6.4. Assume now that G is quasi-split and adjoint. Proposition 6.6.2 implies ad = W ext and the second claim follows. 0 = W  W Remark 7.7.13 Assume that G is adjoint and k-simple. Then Ξ is described in Remark 1.3.76. When G is quasi-split and T is a maximally split maximal torus, one sees that it is isomorphic to (P/Q)Θ , where P = X ∗ (Tsc ) is the weight lattice, Q = X ∗ (Tad ) is the root lattice, and Θ = Gal(k s /k).

7.8 The Iwahori–Weyl Group Let S ⊂ G be a maximal k-split torus. Recall from Definition 6.6.1 the Iwahori–  0 = N(k)/Z(k)0 and the affine Weyl group W aff of the affine root Weyl group W  0 that are useful in applications, system Ψ. We collect here some facts about W following [Ric16a]. Theorem 7.8.1

The inclusion N → G induces

 0 → G(k) / G(k)0 → 1; (1) an exact sequence 1 → W aff → W

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(2) a bijection  0 → I \ G(k) / I, W where I = G(k)0C is the Iwahori subgroup associated to a chamber C ⊂ A; (3) a bijection  0 / W aff → G(k)0 \ G(k) / G(k)0 WFaff1 \ W F2 F1 F2 for any two facets F1 , F2 contained in the closure of a given chamber C.  0 → G(k)/G(k)0 as well as of the map in (2) Proof The surjectivity of W  0 → G(k)/G(k)0 ) = (N(k)∩G(k)0 )/Z(k)0 , follow from Fact 4.1.25. Now ker(W which according to Theorem 7.5.3 is identified with W aff . The injectivity of the map in (2) follows from I ∩ N(k) = Z(k)0 by Lemma 7.5.2. The bijectivity of the map of (3) follows from the bijectivity of the map of (2) together with Proposition 7.5.5.  Recall from 1.3.72 that the embedding of the affine Weyl group into the extended affine Weyl group has a retraction, and one can specify such a retraction by choosing a chamber. The same applies to the Iwahori–Weyl group.  0 . Then  0 be the stabilizer of C in W 7.8.2 Let C ⊂ A be a chamber and let W C  0 = W aff  W  0 . In particular, the exact sequence Lemma 1.3.17 implies that W C  0 → G(k)/G(k)0 is an of Theorem 7.8.1(1) splits, and the homomorphism W C isomorphism. A further interpretation of the quotient G(k)/G(k)0 can be given using the Kottwitz homomorphism when dim(f)  1, cf. Corollaries 11.6.2, 11.7.6.

7.9 Change of Base Field and Automorphisms It is natural here to work with a discrete valuation ω : k × → R that may not satisfy ω(k × ) = Z, cf. §6.5. As before, we will assume that k is Henselian.

(a) Change of Base Field 7.9.1 Restriction of scalars Let /k be a separable field extension. Let H be a connected reductive quasi-split -group. Then G = R/k H is a connected reductive quasi-split k-group. We will construct a natural isomorphism B(H ) = B(G)

(7.9.1)

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that is equivariant with respect to the identification G(k) = H(). For the case of general groups, cf. §9.11. Let S be a maximal -split torus of H and let T be its centralizer. Then T is a maximal torus of H. Then R/k T is a maximal torus of G. The split torus S arises as an -torus, but, being split, has a natural structure of a k-torus. Therefore, there is a natural embedding of k-tori S → R/k S that identifies S with the maximal k-split torus of R/k S. Applying the functor R/k to the inclusion S → H we obtain an identification of S with the maximal k-split torus of G = R/k H contained in R/k S. Since every maximal k-split torus of G arises this way (see [CGP15, Pro.A.5.15(2)]), we obtain a natural bijection between the set of maximal k-split tori of H and the set of maximal k-split tori of G. This bijection is equivariant for the identification G(k) = H(). Fix now a maximal -split torus S of H and let T be its centralizer. As discussed in §2.6(b), the relative root system Φ(S, H) is identified with the set of Gal(k s /)-orbits in the absolute root system Φ(T, H). The absolute root system s /k) Φ(T, H). The Φ(R/k T, R/k H), as a Gal(k s /k)-set, is the induction IndGal(k Gal(k s /) relative root system Φ(S, G) is the set of Gal(k s /k)-orbits in this induced set, and is therefore naturally identified with Φ(S, H). Let a ∈ Φ(S, H) and let UaH be the corresponding relative root subgroup of H. It is a closed connected unipotent -subgroup of H. It is normalized by S and its Lie algebra is the a-weight space for S in Lie(H). Set UaG = R/k UaH ⊂ R/k H = G. Then UaG is a closed connected unipotent k-subgroup of G. It is normalized by S and its Lie algebra is the a-weight space for S in Lie(G). Thus UaG is the relative root subgroup for the root a. We have the identification UaG (k) = UaH (). Let A(S, G) and A(S, H) be the apartments associated to S in the buildings B(G) and B(H ), respectively. The identification of the previous paragraph allows us to view a valuation of the root datum of (G, S) as a valuation of the root datum of (H, S), and vice versa. Moreover, an -pinning for H induces a k-pinning for G and one sees easily that the corresponding Chevalley valuations for G and H match under this identification. We thus obtain an identification A(S, G) = A(S, H) of affine spaces over V(S ); where S  is the maximal subtorus of S contained in the derived subgroup of H. For a point x ∈ A(S, G) = A(S, H), a ∈ Φ(S, G) = Φ(S, H), and r ∈ R, the G H identification UaG (k) = UaH () identifies Ua, x,r with Ua, x,r . The formation of the Iwahori subgroup of a torus is compatible with restriction of scalars, hence T()0 = (R/k T)(k)0 . These statements imply G(k)0x = H()0x . Proposition 7.6.3 now provides the identification (7.9.1).

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7.9.2 Extension of scalars Let G be a connected reductive quasi-split kgroup. Let /k be a finite field extension. Let G = G ×k . The buildings B(G) and B(G ) are both defined. We will show that there is a natural embedding B(G) → B(G ).

(7.9.2)

Let T be a maximally split maximal torus of G. Then T = T ×k  is a maximally split maximal torus of G . We have the natural embedding of real vector spaces V(T) → V(T ) whose image is equal to V(T )Gal(/k) in the case that /k is Galois. We have the apartments A and A(T ), both constructed with respect to the valuation ω, where on  we are using here the unique extension ω :  × → R of ω : k × → R. We will now see that there is a canonical injection A → A(T ). Since any k-pinning of G is also an -pinning of G , given a k-pinning of G we obtain a point o ∈ A, and also a point o ∈ A(T ). The injection V(T) → V(T ) then gives an injection A → A . Another k-pinning would give points o ∈ A and o ∈ A(T ). Applying Proposition 6.1.22 to o, o and o , o we see that o − o = o − o . This implies that the injection A → A(T ) is independent of the choice of k-pinning of G. When /k is Galois there is an obvious action of Gal(/k) on the set of all valuations of the relative root datum of G with respect to T . This action preserves A(T ) and the image of A in A(T ) is equal to A(T )Gal(/k) . The embedding A → A(T ) induces an embedding B(G) → B(G ), for example via the geometric realization of Proposition 7.6.3. Since all choices of T are conjugate under G(k), this embedding is independent of the choice of T. Remark 7.9.3 If /k is Galois, then the image of B(G) in B(G ) is contained in B(G )Gal(/k) , but can be a proper subset. Equality holds when /k is unramified, cf. Chapter 9, or even when /k is tamely ramified, cf. Chapter 12. But when /k is wildly ramified the containment B(G) ⊂ B(G )Gal(/k) is proper already for G = SL2 and k = Q2 . Proposition 7.9.4 Let /k be a finite Galois extension. Let G be a connected reductive quasi-split k-group. Let H = Res/k (G ). Then B(H) contains a chamber invariant under Θ = Gal(/k). Proof Since G is a k-group we have the embedding G → H of k-groups and G = H Θ . Let S be a maximal k-split torus of G. Then T = Z is a maximal torus of G and Z = R/k (T ) is a maximal torus of H and equals the centralizer of S in H according to [CGP15, Proposition A.5.15(1)]. The maximal torus Z of H is maximally split, and using it to index the corresponding apartment of B(H) we have A(Z, H)Θ = A(S, G) as discussed in 7.9.2. The equality Z = Z H (S) implies that no root of Z in H restricts trivially to S. This implies that no affine root

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in Ψ(Z, H) ⊂ A(Z, H)∗ has constant restriction to A(S, G). Therefore, A(S, G) is not contained in any affine root hyperplane. We conclude that A(S, G) must meet a chamber of A(Z, H). Such a chamber is then Θ-stable. 

(b) Automorphisms Proposition 7.9.5 Consider two Henselian valued fields k1 and k2 and an isomorphism f : k 1 → k 2 . Let Gi be a connected reductive ki -group and let fG : G1 → G2 be an isomorphism covering f , that is, fG : G1 ×k1 k 2 → G2 is an isomorphism of k2 -groups. The pair ( f , fG ) induces an isomorphism of buildings fB : B(G1, k1 ) → B(G2, k 2 ). It satisfies fB (g1 x1 ) = fG (g1 ) fB (x1 ) for g1 ∈ G1 (k1 ) and x1 ∈ B(G1, k1 ). Proof The facets in these buildings are in 1-1 correspondence with the parahoric subgroups, and the face-relation between facets is the opposite of the inclusion relation between parahoric subgroups. It is immediate from the definitions that fG induces an isomorphism of abstract groups G1 (k1 )0 → G2 (k2 )0 and that P1 ⊂ G1 (k1 )0 is a parahoric subgroup if and only if P2 = f (P1 ) ⊂ G2 (k2 )0 is a parahoric subgroup. This gives an isomorphism B(G1, k1 ) → B(G2, k 2 ) of chamber complexes. If S1 is a maximal k 1 -split torus of G1 , then S2 = fG (S1 ) is a maximal k2 -split torus of G2 , and the above isomorphism restricts to an isomorphism of chamber  complexes A(S1, k1 ) → A(S2, k 2 ).

7.10 Passage to Completion Let K be a discretely valued Henselian field and G be a connected reductive K-group. We assume that G is quasi-split. Then the same is true for the base  of K. The Bruhat–Tits buildings B(G K ) and change of G to the completion K B(G K ) are thus defined and we have a natural embedding B(G K ) → B(G K ) of (7.9.2). We will presently show that this embedding is in fact an isomorphism of polysimplicial complexes. We warn the reader that this isomorphism of polysimplicial complexes is not  The reason is that a an isomorphism of buildings (unless of course K = K). building, as defined in Definition 1.5.5 is a polysimplicial complex equipped with a collection of apartments. While every apartment of B(G K ) will be shown to be an apartment of B(G K ) as well, the building B(G K ) has many more apartments than the building of B(G K ), because G K has many more maximally split maximal tori than G. The fact that B(G K ) = B(G K ), which

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we will prove below, means that for any chamber in B(G K ) there exists an apartment in B(G K ) that contains it, in addition to any further apartments in B(G K ) that may also contain it. We begin by proving the following well-known result. Proposition 7.10.1

 The K-rank of G is equal to the K-rank of G K .

Proof Let T be a maximal K-split torus of G and Z be its centralizer in G. Let Za be the maximal K-anisotropic connected normal subgroup of Z. Then     K-rankG  = K-rank Z = dim(T)+ K-rank Za = K-rankG+K-rank Za . K

 So to prove the proposition, it suffices to show that Za is anisotropic over K.  if and only if Za (K)  is But according to Theorem 2.2.9, Za is anisotropic over K bounded. The same theorem implies that Za (K) is bounded. As Za (K) is dense  we see that Za (K)  is bounded.  in Za (K), Proposition 7.10.2 (Guy Rousseau) The polysimplicial complex B(G K ) can be identified with the polysimplicial complex B(G K ).  := T  is a Proof Let T ⊂ G be a maximally split maximal torus. Then T K  := G  according to Proposition 7.10.1. maximally split maximal torus of G K  constructed in §6.1. We first compare the apartments A and A(T)  is the splitting extension Let L/K be the splitting extension of T. Then  L/K  Restriction from   → Gal(L/K) and of T. L to L gives an isomorphism Gal( L/K)  We obtain this isomorphism is equivariant for the actions on X∗ (T) and X∗ (T). the natural isomorphism of real vector spaces V(T) → VT . A weak Chevalley–  G)  Steinberg system for (T, G) is also a weak Chevalley–Steinberg system for (T,  Moreover, and this gives a natural identification of affine spaces A → A(T).  → Gal(L/K) identifies the sets Ja (respectively the isomorphism Gal( L/K)  T),  Ja/2a ) in the case of (T, G) with the corresponding sets in the case of (G,

see Fact 6.4.5. Thus the affine root system Ψ for (T, G) is identified with that  G),  so the identification A → A(T)  of affine spaces also identifies the for (T, polysimplicial structures. Using Proposition 7.6.3 we obtain an injective map B(G K ) → B(G K ). To  show that this map is also surjective it is enough to check that for any g ∈ G(K) −1 0  and x ∈ A there exists h ∈ G(K) such that g h ∈ G(K)x , where we are using  (cf. Corollary 7.3.14) while  0x is open in G(K) Remark 7.6.5. But since G(K)  G(K) is dense in G(K) (cf. Proposition 2.3.4), this is immediate. Since the polysimplicial structure on B(G K ) and B(G K ) comes from the polysimplicial structure on A, the fact that the bijection B(G K ) → B(G K ) respects it is immediate. Finally, it is clear that the bijection B(G K ) → B(G K )

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is independent of the choice of T, since all such choices are conjugate under G(K). 

7.11 Absolutely Special Points In this Section we assume that k is a discretely valued Henselian field with perfect residue field and Axiom 4.1.27. This axiom will be proved in Chapter 9. This section will not be used in the proofs of Chapter 9. Let G be a connected reductive k-group and let /k be a separable algebraic extension. Let L be a maximal unramified extension of  and K be the maximal unramified extension of k contained in L. Then K is a maximal unramified extension of k. Let k  be the maximal unramified extension of k in . Then L = K ⊗k   and hence, Gal(L/) = Gal(K/k ) ⊂ Gal(K/k) and the embedding B(G K ) → B(G L ) of (7.9.2) induces an embedding B(G) → B(G ). This allows us to make the following definition. Definition 7.11.1

Let x ∈ B(G). We say that x is

(1) superspecial, if x is special over every finite unramified extension k /k, (2) absolutely special, if x is special over every finite separable extension /k, (3) hyperspecial, if x is superspecial and G splits over an unramified extension. Recall from Proposition 1.3.43 that every special point is a vertex. We will call the parahoric subgroup Px superspecial, or absolutely special, or hyperspecial, if the point x is such. Remark 7.11.2 If G splits over an unramified extension, then the notions of superspecial, absolutely special, and hyperspecial are equivalent. If G does not split over an unramified extension, then, by definition, there are no hyperspecial points. Every absolutely special point is clearly superspecial. The converse is not always true, as we discuss in Example 7.11.6. We will show in Proposition 7.11.7 that when G is quasi-split and its root system Φ is reduced, every special point is automatically absolutely special and hence also superspecial. Thus, these notions are subtle only when G has a unitary component, that is, a component of type 2 A2n . Remark 7.11.3 In some parts of the literature, a superspecial vertex has been called very special, see for example [Zhu11, Definition 6.1]. We have chosen the name “superspecial” instead, in part because “very special” has already been

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279

defined in [BP89, A.4] with a different meaning. For the rest of this remark, we will use the term “very special” in the sense of Borel–Prasad. The definitions of “very special” and “superspecial” are quite different, but turn out to be related. To discuss this relation, let us extend the Borel–Prasad notion from the case of absolutely almost simple groups to the case of general reductive groups in the obvious way – the original definition applies to k-simple groups just as well, and we extend it to arbitrary groups by defining that a vertex is very special if and only if, under the direct product decomposition of the building according to k-simple factors, all components are very special. We now point out the difference between “superspecial” and “very special”: we will see in Proposition 10.2.1 that, assuming dim(f)  1, the existence of a superspecial vertex implies that G is quasi-split; this is not the case for very special vertices, which exist for any connected reductive group. As for their relationship, assuming that G is quasi-split, one can check that a vertex is superspecial if and only if it is very special. To see this, one reduces immediately to the case that G is k-simple, after which one can use the tables in [Tit79] to check that every special vertex is very special unless G is an unramified odd unitary group, in which case only the hyperspecial vertex is very special. That the same is true for superspecial vertices can be seen using Proposition 7.11.7 below for all groups except the unitary groups, for which the claim is immediate from the definition. We will now show that absolutely special points always exist when G is quasi-split. The converse is true if dim(f)  1, see Proposition 10.2.1. Proposition 7.11.4 Assume that G is quasi-split. Let T ⊂ G be a maximally split maximal torus and let A be the corresponding apartment. Let o ∈ A be a Chevalley valuation. The following statements are equivalent for a point x = o + v ∈ A. (1) (2) (3) (4)

The point x is absolutely special. a(v) ∈ va Z for all non-divisible a ∈ Φ, with va as in Fact 6.5.1. x = t · o for some t ∈ Tad (k). x is a Chevalley valuation.

In particular, absolutely special points exist.  be an absolute root Proof (1) ⇒ (2): Let a ∈ Φ be non-divisible and let  a∈Φ a (cf. Notation 2.6.11). The point lifting a. Let k a be the field of definition of  k a(v) ∈ J aa according to Proposition 6.4.11, x remains special in Ak a . Thus  k

a is a where J aa is the set J a relative to the base field k a. Over that base field 

relative root of type R1 and hence J aa = ω(k ×a) = va Z. k

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The Bruhat–Tits Building for a Valuation of the Root Datum

 It (2) ⇒ (3): Choose a Θ-invariant set of simple absolute roots  Δ ⊂ Φ. provides an isomorphism  Rk a /k Gm, t → ( a(t)) a. Tad → a ∈Δ/Θ

Choose t a ∈ k ×a with ω(t a) = a(v) and tσ( a) = σ(t a) for all σ ∈ Θ. The collection (t a) is a k-point of the right-hand side and determines t ∈ Tad (k) with  a(t) = a(v) for all  a ∈ Δ with image a ∈ Φ. Then x = t · o. (3) ⇒ (4): Immediate. (4) ⇒ (1): Immediate.  Corollary 7.11.5 If G is quasi-split and adjoint, any two absolutely special points are conjugate. Example 7.11.6 We return to the example of the group SU3 associated to a separable quadratic extension /k and its standard apartment A discussed in §3.2. This example illustrates that the notions of special, superspecial, and absolutely special vertices, can all be distinct from each other. Assume first that /k is unramified. The set of special vertices is then 12 Z ⊂ R. The vertices in Z are Chevalley valuations, hence they are absolutely special, and therefore hyperspecial. The vertices in Z + 12 are not Chevalley valuations, so they are not absolutely special; but they are special. Assume next that /k is ramified and p  2. Then the set of special vertices is 1 1 Z 4 ⊂ R. The vertices in 2 Z are Chevalley valuations, hence they are absolutely special. The vertices in 12 Z + 14 are not Chevalley valuations, so they are not absolutely special; but they are superspecial. Finally if /k is ramified and p = 2, then we recall that we have introduced the shift by μ to ease bookkeeping. Thus the vertices 12 (Z + μ) are Chevalley valuations and hence they are absolutely special, while the vertices 12 (Z + μ+ 12 ) are not Chevalley valuations, hence they are not absolutely special, but they are superspecial. The following results shows that in many cases, the notions of special, superspecial, and absolutely special, are equivalent. Proposition 7.11.7 If G is quasi-split and Φ is reduced then every special vertex is absolutely special. Proof The claim follows from Propositions 6.4.11, 7.11.4, and Fact 6.5.1, given that every a ∈ Φ is of type R1 by assumption.  Remark 7.11.8

The relationships between the various notions applied to a

7.11 Absolutely Special Points

281

point x ∈ B(G) can now be summarized as follows. vn absolutely special fn

G quasi-split, Φ reduced

+3 superspecial

false; counterexample Sp4

t| +3 special

+3 vertex

G K split; then x hyperspecial

We will prove in Proposition 9.9.2 that a point of B(G) that is special in B(G K ) is automatically special in B(G), and therefore superspecial. We will prove in Proposition 10.2.1 that, under the assumption dim(f)  1, such a point exists if and only if G is quasi-split. Remark 7.11.9 We can also relate the Lie-theoretic notion of “extra special” (cf. Definition 1.3.39) to the arithmetic notions of superspecial and absolutely special, again assuming that G is quasi-split. When Φ is reduced, then extra special is equivalent to special by Proposition 1.3.43. In the case that Φ is not reduced, let us first assume that Φ is irreducible. Thus G = R/k SU2n+1 , where SU2n+1 is the special unitary group associated to a quadratic extension  /. From the discussion of §3.2 one extracts the following two cases. (1)  / is ramified and there exists a uniformizing element  ∈   with  2 ∈ ; then “extra special” coincides with “absolutely special.” (2)  / is ramified but there does not exist a uniformizing element  ∈   with  2 ∈ ; then “extra special” coincides with “superspecial but not absolutely special.” Note that (1) includes the case that  / is unramified or tamely ramified. The case that  / is wildly ramified (in particular p = 2), is spread over cases (1) and (2). The part that is in (1) is rather particular – in that case, the characteristic of k is zero and the unique jump of the lower ramification filtration of Gal( /) is as large as possible, namely equal to the absolute ramification degree of  . For more information, cf. §3.22.8. A pictorial summary is given in Figure 7.11.1. To drop the assumption that Φ is irreducible, one observes that a vertex in A is a tuple of vertices, one for each irreducible component of Φ, and such a vertex has the property of being special, or extra special, or absolutely special, or superspecial, if and only if each of its coordinates does.

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The Bruhat–Tits Building for a Valuation of the Root Datum

Ramified

a=0 2s

absolutely special (and superspecial)

a=1 2a = 1

2a = 3

superspecial (not absolutely special)

absolutely special (and superspecial)

2 s

a=1

a=2

superspecial (not absolutely special)

Unramified

2a = 2 a=1

2a = 0 a=0 2a = 1

hyperspecial 2a = 3

special, but not hyperspecial

Figure 7.11.1 The apartment of SU3 (k) as a subset of the apartment of SL3 ().

8 Integral Models

In this chapter we continue with a non-archimedean discretely valued field k with Henselian ring of integers o, maximal ideal m, and residue field f. We fix a generator π of m. In the first three sections of this chapter, we do not assume that the residue field f is perfect. We will denote by K a maximal unramified extension of k and by O its ring of integers. The maximal ideal of O will be denoted by M. The residue field O/M of K is a separable closure of f, so we will denote it by fs . We fix an algebraic closure f of fs . Also, the completion of o will be denoted by  o. There are natural isomorphisms Gal(K/k) → Aut(O/o) → Gal(fs /f). We will identify these groups and denote each of them by Γ. Throughout this chapter, G will denote a connected reductive k-group.

8.1 Preliminaries Lemma 8.1.1 Let Y be a scheme, and X ⊂ Y an open subscheme. If for a local ring R, f : Spec(R) → Y is a map carrying the closed point into X , then f factors through X . Proof Since X is an open subscheme of Y , the property of f factoring through X is purely topological; that is, it is equivalent to show that the open subset f −1 (X ) ⊂ Spec(R) is the entire space. Our hypothesis says that this latter open subset contains the closed point, so our task reduces to showing that the only open subset of a local scheme that contains the unique closed point is the entire space. Said equivalently in terms of its closed complement, we want to show that the only closed subset Z of Spec(R) not containing the closed point 283

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is the empty set. For an ideal J ⊂ R defining Z, this is the obvious assertion that if J is not contained in the unique maximal ideal of R then J = (1).  Lemma 8.1.2 Let f : B → C be a morphism between o-schemes. Assume that f is of finite type and is smooth between the generic fibers of B and C . If B( o ) → C ( o ) is surjective, then B(o) → C (o) is surjective. Proof Consider a point c ∈ C (o). Using the map c : Spec(o) → C we form the fiber product Bc of B → C ← Spec(o). Then Bc is an o-scheme of finite type. It is enough to show that Bc (o)  ∅. The generic fiber of Bc is smooth by assumption. The assumed surjectivity o-valued point  b. Let X be an of B( o ) → C ( o ) implies that Bc has an  b. By open affine subset of Bc containing the image of the closed point of  Lemma 8.1.1 the point  b: Spec( o ) → Bc factors through X . Thus X is an affine o-scheme of finite type with smooth generic fiber that has an  o-point. By Proposition 2.3.4 there is a sequence of k-points of X converging to this  o-point. Considering coordinates under a fixed affine embedding we see that this sequence of k-points eventually becomes a sequence of o-points. Thus  ∅  X (o) ⊂ Bc (o). Lemma 8.1.3 Let X be a smooth o-scheme. For any natural number n the canonical reduction map X (o) → X (o/m n ) is surjective. In particular, the reduction map X (o) → X (f) is surjective. Proof In the case of n = 1 so that o/m n = f this is is [EGAIV4 , 18.5.17] or [BLR90, §2.3, Proposition 5]. The argument in the general case is the same. We reproduce it here for the convenience of the reader. For a given point x0 in X (o/m n ), let x 0 be its underlying reduced rational point in the special fiber. By [EGAIV4 , 17.11.4] there is an open subscheme U in X around x 0 for which there is an étale o-map f : U → Aor ; we may and do shrink U around x 0 so it is affine. Since x0 : Spec(o/m n ) → X has image x 0 and the space Spec(o/m n ) has only one point, x0 factors through the open subscheme U . We rename U as X so that now there is an affine étale o-map f : X → Aor . The map f carries x0 to an o/m n -point y0 of Aor , and we can easily lift y0 to an o-point y of Aor , by lifting each of its coordinates arbitrarily. Since f is étale, the pull-back f −1 (y) = Spec(o) ×Aor X of f along y is an étale o-scheme, and by the meaning of such a fiber product and the design of y in terms of x0 we see that x0 factors through f −1 (y) → X . It then suffices to solve our problem for the o-étale f −1 (y) in place of the o-smooth X , so that reduces us to the case where X is o-étale and affine. Now we will use that o is Henselian: by [EGAIV4 , 18.5.12], for any o-finite

8.1 Preliminaries

285

Z the natural map Homo (Z , X ) → Hom f (Zf, Xf ) is bijective. In particular, taking Z to be both Spec(o) and Spec(o/m n ), we see that X (o) → X (o/m n ) is bijective (as each has bijective reduction map to X (f)).  Lemma 8.1.4 Let G be a smooth affine o-group scheme with connected fibers. Then the map H1 (Γ, G (O)) → H1 (Γ, G (f)) is injective. In particular, if dim(f)  1, then H 1 (Γ, G (O)) is trivial. Proof We first prove that only the trivial element of H1 (Γ, G (O)) maps to the trivial element of H1 (Γ, G (f)). By unramified Galois descent over discrete valuation rings [BLR90, §6.2, Ex.B], the cohomology set H1 (Γ, G (O)) classifies G -torsors over o which admit an O-point. The cohomology set H1 (Γ, G (f)) classifies G f -torsors over f which admit an f-point. The map between the two sets is given by changing base from o to f. Let X be a G -torsor over o admitting an O-point. Assume that this torsor lies over the trivial element of H1 (Γ, G (f)), which is to say that it admits an f-point. Since X inherits o-smoothness from G , Lemma 8.1.3 implies that X admits an o-point and thus corresponds to the trivial element of H1 (Γ, G (O)). We note here that in fact any G f -torsor admits an f-point, and the same argument implies that any G -torsor admits an O-point. To prove the injectivity claim we apply the usual method of twisting. If X is any G -torsor over o we can consider the form of G twisted by X , which we denote by G . There is a natural bijective map H1 (Γ, G (O)) → H1 (Γ, G (O)) sending X to the trivial element. This reduces the triviality of the fiber through X for G to the triviality of the trivial fiber for G . The vanishing of H1 (Γ, G (O)) under the assumption of dim(f))  1 now  follows from the vanishing of H1 (Γ, G (f)) due to Theorem 2.3.3. In the following lemma, F is an arbitrary field of characteristic p  0. Lemma 8.1.5 Let U be a smooth connected unipotent F-group and Θ a finite group acting by automorphisms on U. We assume that p does not divide the order of Θ. Then U Θ is smooth and connected. Proof To prove the lemma, we may replace F with its algebraic closure. Smoothness of U Θ follows from a general result, see Proposition 2.11.5 or [Edi92, Proposition 3.4]. Connectedness of U Θ will now be proved by induction on the dimension of U as follows. In case p = 0, let V be the center of U, and if p > 0, let V be the maximal smooth connected p-torsion central subgroup of U. Then V is a vector group that is stable under the action of Θ. We consider the short exact sequence 1 → V → U → U/V → 1.

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This short exact sequence gives rise to the exact sequence 1 → V Θ → U Θ → (U/V) Θ → H1 (Θ,V) → · · · .

 Now we note that as p does not divide the order of Θ, the map v → θ ∈Θ θ(v) is a homomorphism of V onto V Θ , and moreover, H1 (Θ,V) is trivial. We conclude that V Θ is connected and we get the following short exact sequence 1 → V Θ → U Θ → (U/V) Θ → 1. The first term V Θ is connected, and by induction on dimension, (U/V) Θ is also  connected. Hence, U Θ is smooth and connected.

8.2 General Properties of Smooth Models of G In this section we will study general properties of smooth models of G and their special fibers. Proposition 8.2.1 Let G be a smooth affine o-group scheme, G := Gf its special fiber, Rus,f (G ) the f-split unipotent radical of G , and G = G /Rus,f (G ) the maximal quasi-reductive (Definition 2.11.16) quotient of G . (1) Let S be a closed o-torus of G . The centralizer M of S in G is a smooth closed (hence affine) o-subgroup scheme of G . Let T be an f-torus of G that contains the special fiber S of S . Then there exists a closed o-torus T of G that contains S and whose special fiber is T . Consequently, S is a maximal closed o-torus of G if and only if S is a maximal f-torus of G , and S is a maximal o-split torus of G if and only if S is a maximal f-split torus of G . (2) Let T1 and T2 be two closed o-tori of G and T 1 and T 2 be their special fibers. If there is an element g ∈ G (f)(= G (f)) that conjugates T 1 into T 2 , then there exists a g ∈ G (o) lying over g that conjugates T1 into T2 . (3) Let T be a closed o-torus of G . Then the normalizer NG (T ) of T in G is a closed smooth o-subgroup scheme of G . Hence the natural homomorphism NG (T )(o) → NG (T )(f) is onto. (4) Let S be a closed o-split torus of G whose special fiber S is a maximal f-split torus of G . We denote by S the isomorphic image of S in G. Then f-Weyl group WG (S) := NG (S)/ZG (S) is constant and the normalizer of S in G (o) maps onto WG (S). (5) Let T1 , T2 , T 1 and T 2 be as in (2). If the images of T 1 and T 2 in G are conjugate to each other under an element g ∈ G(f), then there is an element g ∈ G (o) lying over g that conjugates T1 onto T2 .

8.2 General Properties of Smooth Models of G

287

(6) Given an f-torus T in G, there is a closed o-torus T in G whose special fiber maps isomorphically onto T. In assertion (1), since the special fiber of T is T , the character groups of TO and T fs are isomorphic as Γ-modules. Hence, T is split if and only if T is split. Proof (1) That the centralizer M of S is a smooth closed o-subgroup scheme is a consequence of a general fact; see [SGA3, Exp.XI, Cor.5.3] or Proposition 2.11.5. The special fiber M of M contains T since T contains S and hence it commutes with the latter. Let X := Xf∗s (T fs ) be the character group of T fs considered as a Γ-module under the natural action of Γ = Gal(K/k) = Gal(fs /f). As T splits over a finite Galois extension of f, the Galois group Γ acts on X through a finite quotient. Let fs [X] (respectively O[X]) be the group ring of X with coefficients in fs (respectively O). Then according to [Bor91, Proposition in §8.12], the affine ring of the f-torus T is (fs [X])Γ . Let T be the o-torus whose affine ring is (O[X])Γ . Using a suitable modification of the argument given in [Bor91, AG, §14.2] one can see that (O[X])Γ ⊗o O = O[X]. Therefore, the affine ring of the special fiber Tf of T is Γ

Γ

(O[X])Γ ⊗o f = (O[X])Γ ⊗o O ⊗o f = (fs [X])Γ ⊗ f fs = (fs [X])Γ . Hence the special fiber of T is isomorphic to T and the character group of TO is isomorphic as a Γ-module to X.(Hence, in particular, TO is Osplit.) We fix an f-isomorphism ι : Tf → T (⊂ M ) and view it as a closed immersion of Tf into M . According to a result of Grothendieck [SGA3, Exp.XI, 4.2], the homomorphism functor HomSpec(o)−gr (T , M ) is representable by a smooth o-scheme X . Clearly, ι ∈ X (f). Now since o is Henselian, the natural map X (o) → X (f) is surjective by Lemma 8.1.3, and hence there is an ohomomorphism ι : T → M lying over ι, that is, ι f = ι. As ι is a closed immersion, using [SGA3, Exp.IX, 2.5 and 6.6] we see that ι is also a closed immersion. We identify T with a closed o-torus of M (⊂ G ) in terms of ι. Then the special fiber of T is T and it clearly contains S . We will now prove the second assertion of (1). If S were contained in a larger closed o-torus (respectivelyo-split torus) of G , then its special fiber would contain S properly. On the other hand, if G contains an f-torus (respectivelyf-split torus) that contains S properly, then from the part of (1) that has already been proved, we infer that S is contained in a larger closed o-torus (respectivelyo-split torus) of G . (2) The transporter scheme T := Transp G (T1, T2 ), consisting of points of the scheme G that conjugate T1 into T2 , is a closed smooth o-subscheme of G (see [Con14, Proposition 2.1.2] or [SGA3, Exp.XI, 2.4bis]). Let T be the special

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Integral Models

fiber of T. Then g belongs to T(f). Now as o is Henselian, the natural map T(o) → T(f) is surjective by Lemma 8.1.3. Therefore, there exists a g ∈ T(o) lying over g. This g will conjugate T1 into T2 . (3) In the proof of assertion (2), by taking T1 = T2 = T we conclude (3). (4) Let NG (S ) be the normalizer and ZG (S ) be the centralizer of S in G . It follows from [CGP15, Lemma 3.2.1] that the natural homomorphism NG (S )/ZG (S ) → NG (S)/ZG (S) = WG (S) is an isomorphism. It is also well known that the f-Weyl group WG (S) is constant and the homomorphism NG (S)(f)/ZG (S)(f) → WG (S)(f) is an isomorphism (see [CGP15, Proposition C.2.10] in the general case, and [Bor91, Theorem 21.2] when f is perfect). Now (4) follows from (3). (5) Since Rus,f (G ) is f-split, H1 (f, Rus,f (G )) is trivial. This implies that the natural homomorphism G (f) → G(f) is surjective. Since o is Henselian, G (o) → G (f) is surjective too. Hence, after replacing T1 by a conjugate under an element of G (o) lying over g, we may (and do) assume that T 1 and T 2 have the same image, say T, in G. Now let H be the inverse image of T in G under the natural homomorphism G → G. Then H is a smooth connected solvable f-subgroup that contains both T 1 and T 2 as maximal f-tori. Using [Bor91, Theorem 19.2] we see that there exists an element g ∈ Rus,f (H )(f) = Rus,f (G )(f) that conjugates T 1 onto T 2 . Now (2) implies (5). (6) Let H be the inverse image of T under the natural homomorphism G → G. Then, as in (5), H is a connected solvable f-subgroup of G that maps onto T. Let T be a maximal f-torus of H . This torus maps isomorphically  onto T. Now (6) follows from (1) by taking S to be the trivial torus. Proposition 8.2.2 Let G be a smooth affine o-group scheme with generic fiber G. We assume that G contains a o-split torus S whose generic fiber S is a maximal k-split torus of G. Then the following hold. (1) The o-torus S is the schematic closure of S in G , so it is closed. It is a maximal closed o-split torus of G and the special fiber S of S is a maximal f-split torus of G . (2) Let S  be a closed o-split torus of G , S  (⊂ G) be its generic fiber and S  be its special fiber. Then S  is a maximal closed o-split torus of G and S  is a maximal k-split torus of G if and only if S  is a maximal f-split torus of G . Proof (1) Since S ⊂ G , S(K)b = S (O) ⊂ G (O), and as S(K)b is the maximal bounded subgroup of S(K) and the subgroup G (O) of G(K) is bounded, cf. Remark 2.10.3, S(K) ∩ G (O) = S (O). This implies that S is the schematic closure of S in G .

8.2 General Properties of Smooth Models of G

289

If S were contained in a strictly larger closed o-split torus T of G , then the generic fiber T of T would be a k-split torus of G strictly larger than S. This would contradict the fact that S is a maximal k-split torus of G. So S is a maximal o-split torus of G . Now using the second assertion of Proposition 8.2.1(1), we see that S is a maximal f-split torus of G . (2) By the conjugacy of maximal f-split tori in G ([CGP15, Theorem C.2.3]), ¯  g¯ −1 is contained in S . Then by we see that there is a g¯ ∈ G (f) such that gS Proposition 8.2.1(2), there exists a g ∈ G (o) lying over g¯ such that gS  g −1 is contained in S . Thus after replacing S  by a conjugate under an element of G (o) we can (and do) assume that S  ⊂ S . Now (2) follows easily from the last assertion of (1).  Lemma 8.2.3 Let X and Y be affine flat o-schemes. Let ϕ : X → Y be an o-morphism. Assume that the ring map ϕk∗ is an isomorphism and the ring map ϕf∗ is injective. Then ϕ is an isomorphism. Proof

In view of flatness, the fact that the ring map ϕk∗ : k ⊗o o[Y ] = o[Y ][ π1 ] → o[X ][ π1 ] = k ⊗o o[X ]

is an isomorphism implies that the ring map ϕ∗ : o[Y ] → o[X ] is injective. We identify o[Y ] with an o-subalgebra of o[X ] in terms of ϕ∗ . Then o[X ][ π1 ] = o[Y ][ π1 ]. If ϕ∗ were not surjective, there would exist an f ∈ o[X ] such that f is not in o[Y ] but π f is in this subalgebra. Then the image of π f in o[Y ]/πo[Y ] is non-zero whereas its image in o[X ]/πo[X ] is zero. This contradicts the  assumption that ϕf∗ is injective and proves that ϕ∗ is an isomorphism. Proposition 8.2.4 Assume that G is quasi-split, S is a maximal k-split torus of G, and T is the centralizer of S in G. Let G be a smooth affine o-group scheme with generic fiber G and connected special fiber. We assume that G (O) contains the Iwahori subgroup T(K)0 of T(K). Then G contains a unique closed o-torus S with generic fiber S. The centralizer T of S in G is a closed smooth o-subgroup scheme; T is the relative identity component of the ft-Néron model of T. Proof The first assertion follows from Proposition B.2.4 since the subgroup G (O) of G(K) contains the maximal bounded subgroup S(K)b ⊂ T(K)0 of S(K). The centralizer Z of S in G is smooth by Proposition 8.2.1(1). Its generic fiber equals the centralizer T of S in G (hence, Z is the schematic closure of T in G ), and its special fiber is connected since the centralizer of a torus in any connected linear algebraic group is connected. Moreover, Z (O) = T(K)∩G (O)

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by Lemma A.2.1. By assumption this groups contains T(K)0 . On the other hand, since G (O) is a bounded subgroup of G(K) (Remark 2.10.3), T(K) ∩ G (O) is a bounded subgroup of T(K) so it is contained in T(K)1 . Corollary 2.10.10 and Proposition B.7.2 imply that the identity on T extends to a morphism Z → T ft of group schemes, where T ft is the ft-Néron model of T. Since Z has connected special fiber, this map factors through the relative identity component T 0 of T ft . Corollary B.8.7 implies that Z (O) is contained in T(K)0 . We thus conclude that Z (O) = T(K)0 . Corollary 2.10.11 now implies  that Z = T 0 . Theorem 8.2.5 Let G be a reductive k-group, S a maximal k-split torus of G, and Z the centralizer of S in G. Let S be the o-split torus with generic fiber S and let G be a smooth affine o-group scheme with generic fiber G. Assume that the conjugation action S × G → G of S on G extends to an action φ : S × G → G of S on G by o-group scheme automorphisms. (1) G contains a closed maximal o-split torus that commutes with S . (This maximal o-split torus of G may be trivial!) (2) Let Φ be the set of roots of G with respect to S, Φ+ (⊂ Φ) be a positive system of roots, and Φ− = −Φ+ . For a ∈ Φ, we will denote the a-root group of G with respect to S by Ua and Z will denote the centralizer of S in G . Upon taking the order of the factors in any way in the following product over Φ±,nd , we obtain the same o-smooth closed subgroup scheme of G  under multiplication a ∈Φ±, nd Ua → G , and these subgroup schemes have unipotent fibers. Moreover, the product morphism   Ua × Z × Ua → G (8.2.1) a ∈Φ −, nd

a ∈Φ +, nd

is an open immersion. According to Corollary 2.10.10, the conjugation action S × G → G extends to S × G → G if and only if the conjugation action of S (O) ⊂ S(K) on G(K) preserves the subgroup G (O) ⊂ G(K). The image of (8.2.1) is called the “open cell” with respect to S and Φ+ , or simply an “open cell.” Proof (1) Let G be the special fiber of G and S that of S . We form a semi-direct product H := S  G using the action of S on G . The special fiber H of H is then the semi-direct product S  G . We choose a maximal f-split torus of H containing the f-split torus S . Then using the conjugacy of maximal f-split tori in H ([CGP15, Theorem C.2.3]), we see that this maximal f-split torus of H equals S × T , where T is a maximal f-split torus of G .

8.2 General Properties of Smooth Models of G

291

Now using Proposition 8.2.1(1) we conclude that there is a closed maximal o-split torus T of G that commutes with S . (2) Let λ : Gm → S be a 1-parameter subgroup of S such that b, λ > 0 for all b ∈ Φ+ . For simplicity we will denote the o-subgroup schemes UG (λ) and UG (−λ) defined in §2.11 by U (λ) and U (−λ) respectively. These subgroup schemes are smooth, closed, and have connected fibers, see Propositions 2.11.1, 2.11.2, and 2.11.3. We will now show that the subgroup U (λ) is independent of the choice of λ. Let λ  be another 1-parameter subgroup such that b, λ  > 0 for all b ∈ Φ+ . Observe that the Lie algebra of U (λ), as well as that of UU (λ) (λ ), is the sum of root spaces for all roots b ∈ Φ+ and the fibers of UU (λ) (λ ) are connected so the subgroup scheme UU (λ) (λ ) is equal to U (λ). As UU (λ) (λ ) ⊂ U (λ ), we conclude that U (λ) is contained in U (λ ). By symmetry they are equal. A similar argument shows that U (−λ) is independent of λ. We will denote the smooth closed o-subgroup schemes U (λ) and U (−λ) by U + and U − respectively. It is clear that the generic fiber U + (respectively U − ) of U + (respectively U − ) is the unipotent radical of the minimal parabolic k-subgroup of G containing Z corresponding to the set of positive (respectively negative) roots Φ+ (respectively Φ− ). Applying Proposition 2.11.3 we see that the product map U−×Z ×U+ →G is an open immersion. The ±a root groups U±a of G are contained in U ± (Re mark 2.11.11). We will now show that the product morphism a ∈Φ+, nd Ua → U + is an isomorphism, where the product on the left side can be taken in any order. Using Theorem 2.11.15 over k and over f, we see that this map is an isomorphism after base change to k or to f. The claim then follows from  Lemma 8.2.3. An analogous argument implies that a ∈Φ−, nd Ua → U − is an isomorphism, where the product on the left side again can be taken in any order.  Corollary 8.2.6 Assume that k is a discretely valued field. For all a ∈ Φ(S), let Ua ⊂ Ua (k), and Z ⊂ Z(k) be open subgroups. The subgroup of G(k) generated by the Ua s and Z is an open subgroup of G(k). Example 8.2.7 Even if Z and each Ua are bounded, the subgroup of G(k) they generate may fail to be bounded. As an example, consider G = SL2 , with " " " # # # 1 ∗ 1 0 ∗ 0 T= , U−a = . , Ua = 0 1 ∗ 0 0 ∗ If we take T to be the subgroup of T whose entries are units, Ua the subgroup

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whose upper right entry has valuation  0, and U−a the subgroup whose lower left entry has valuation  −1, then taking x = 1 and x = π a uniformizer in the calculation # " # " # " # " 1 x 0 x 1 x 1 0 · = · 0 1 −x −1 0 −x −1 1 0 1 shows that the element " π 0

# " 0 0 = −π −1 π −1

# " π 0 · 0 −1

1 0

# −1

lies in the subgroup G generated by T, Ua , U−a . Therefore, T(k) ⊂ G, and hence Ua (k), U−a (k) ⊂ G, thus G = G(k). One can see more generally that if one defines Ua and U−a by the condition that their corner entries have valuation bounded by ra ∈ Z and r−a ∈ Z, respectively, then the condition that G is bounded is equivalent to ra + r−a  0. We studied this phenomenon systematically for general quasi-split groups in §7.3. In this and the next section, given an o-group scheme H , we will denote by H its special fiber, by π : H (o) → H (f) the natural morphism, and by H (m) ⊂ H (o) the kernel of π. Proposition 8.2.8 We use the notation and hypothesis of Theorem 8.2.5. We   denote the open cell a ∈Φ−, nd Ua × Z × a ∈Φ+, nd Ua of G by X and its special fiber by X . (1) G (m) = ker π is contained in X (o), and for each element g ∈ G (m), all its components under the given decomposition of the open cell X lie in G (m). Thus G (m) admits the following decomposition:   Ua (m) × Z (m) × Ua (m). G (m) = a ∈Φ −, nd

a ∈Φ +, nd

(2) Assume that the special fiber G of G is connected. Then the group G (o) is generated by Z (o) and the root groups Ua (o) = Ua (k) ∩ G (o), for a ∈ Φ. Proof (1) Let g ∈ G (m). Since π(g) = 1 lies in X (f), we infer from Lemma 8.1.1 that g lies in X (o). Now as the special fiber X admits the following compatible product decomposition.   Ua × Z × Ua → X , a ∈Φ −, nd

a ∈Φ +, nd

where for every a ∈ Φ, U a is the special fiber of Ua and Z is the special fiber

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293

of Z , we see at once that every component of g under the above decomposition lies in G (m). (2) Recall from Proposition 2.11.9 that Ua (o) = Ua (k) ∩ G (o). We observe now that since G and X are smooth and o is Henselian, the homomorphism π : G (o) → G (f) is surjective and π(X (o)) = X (f). Since ker π is contained in X (o), to prove (2), it is enough to note that since G has been assumed to be connected, G (f) is generated by Z (f) and the root groups U a (f), for a ∈ Φ.  We will now establish some notation for stating Theorem 8.2.9. Let G denote a smooth affine o-group scheme with generic fiber a reductive k-group G, and connected special fiber G . We assume that G contains a closed o-split torus S whose generic fiber is a maximal k-split torus S of G. Let Φ = Φ(S, G) be the root system of G with respect to S. For a ∈ Φ, let Sa be the subtorus of S of codimension 1 contained in the kernel Ca of a in S and let Ma be the identity component of the centralizer of Ca in G. The root groups of Ma are U±a , and also U±2a if a is multipliable. The 1-dimensional subtorus of S contained in the derived subgroup of Ma will be denoted by S a . Let Ca be the kernel of a in S and Sa (⊂ Ca ) be the o-split closed torus whose generic fiber is Sa . Let Ma be the relative identity component of the centralizer of Ca in G ; Ma is a smooth affine o-subgroup scheme (Proposition 2.11.5) that contains S , and its generic fiber is Ma . We will denote the special fibers of G , S and Ma by G , S and Ma respectively. The f-split unipotent radicals of G and Ma will be denoted by Rus,f (G ) and Rus,f (Ma ) respectively. We will denote the natural surjective homomorphism G (o) → G (f) by π. We will denote the maximal quasi-reductive quotient G /Rus,f (G ) of G and the maximal quasi-reductive quotient Ma /Rus,f (Ma ) of Ma by G and Ma respectively and let ϕ denote both the projection maps G → G and Ma → Ma . The isomorphic image of S in G, as well as in Ma , will be denoted by S. Let u ∈ Ua (k)∗ = Ua (k) − {1}. Recall from Proposition 2.11.17 the unique  u , u  ∈ U−a (k) such that the element m(u) = u uu  ∈ G(k) normalizes S. The elements u , u  necessarily lie in U−a (k)∗ and the element m(u) acts as the reflection along the root a. Theorem 8.2.9 Let u ∈ Ua (k)∗ ∩ G (o). Then u , u , and hence also m(u), belong to G (o) if and only if π(u)  Rus,f (G )(f). Proof Write u = π(u). We assume first that u  Rus,f (G )(f). From Proposition 2.11.5(4) we infer that Rus,f (Ma ) ⊂ Rus,f (G ). Hence, u  Rus,f (Ma )(f). This implies that the maximal quasi-reductive quotient Ma of Ma is not a torus. We will show now that u , u  belong to Ma (o). Let λ : Gm → S a be the

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isomorphism such that a, λ < 0. This isomorphism extends to an o-group scheme homomorphism Gm → S , we will denote this extension again by λ. We will now use the notation introduced in §2.11. The subgroups P := PMa (λf )Rus,f (Ma ) and P  := PMa (−λf )Rus,f (Ma ) are minimal pseudo-parabolic f-subgroups of Ma containing the special fiber S of S , so by Theorem C.2.5 of [CGP15], there is an element n of Ma (f) that normalizes S and conjugates P onto P  and hence it maps a onto −a. Now using Proposition 8.2.1(3) we conclude that there is an element n ∈ Ma (o) lying over n that normalizes S , and hence S, and maps a onto −a. According to Proposition 2.11.3 (with A replaced by o and G replaced by Ma ), the multiplication map UM a (λ) × ZM a (λ) × UM a (−λ) → Ma is an open immersion of o-schemes. We will denote UM a (λ), UM a (−λ) and ZM a (λ) by U−a , Ua and Z respectively, and the special fibers of these osubgroup schemes by U −a , U a and Z respectively. Note that U±a are the ±a-root groups of Ma with respect to S , and U ±a are the ±a-root groups of Ma with respect to S . We will now show that U−a (o)uU−a (o) contains an element of Z (o)n. Since nU−a n−1 = Ua , we see that X := U−a Z nU−a is an open subscheme of Ma satisfying X (o) = U−a (o) · Z (o) · n · U−a (o). Our claim reduces to showing that u ∈ X (o). Lemma 8.1.1 further reduces this to showing that u ∈ X (f), where X ⊂ Ma denotes the special fiber of X . The special fiber X equals U −a Z n U −a , so its image under the projection map ϕ : G → G decomposes as ϕ(X ) = ϕ(U −a ) ϕ(Z ) ϕ(n) ϕ(U −a ). Note that ϕ(U ±a ) are the ±a-root groups of the quasi-reductive f-group Ma with respect to the maximal f-split torus S, cf. Proposition 2.11.1(5). The assumption u  Rus,f (Ma )(f) implies that ϕ(u) is a non-trivial element of ϕ(U a )(f). Using [CGP15, Proposition C.2.24(i)] we conclude that ϕ(u) lies in ϕ(X )(f). The claim u ∈ X (f) will be established once we show X · Rus,f (Ma ) = X . According to Proposition 2.11.3(2), the open immersion (Rus,f (Ma ) ∩ U a ) × (Rus,f (Ma ) ∩ Z ) × (Rus,f (Ma ) ∩ U −a ) → Rus,f (Ma ), defined by multiplication, is an isomorphism of schemes. Using this, and the normality of Rus,f (Ma ) in Ma , we see that X · Rus,f (Ma ) = U −a Z n U −a · Rus,f (Ma ) = U −a Z n · Rus,f (Ma )U −a = U −a Z n (Rus,f (Ma ) ∩ U a ) (Rus,f (Ma ) ∩ Z )U −a = U −a Z n U −a = X .

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295

We have thus shown that U−a (o)uU−a (o) contains an element of Z (o)n. Write this element as u  uu  with u , u  ∈ U−a (o). From the uniqueness assertion in Proposition 2.11.17, we conclude that these u  and u  are the same as in that proposition. Thus m(u) := u uu  ∈ Z (o)n ⊂ NM a (S )(o) ⊂ Ma (o) ⊂ G (o). To prove the converse, we assume that u ∈ Rus,f (G )(f) and u , u  ∈ G (o). Then m(u) = u uu  lies in G (o) and its image m(u) = u  u u , where u  = π(u ) and u  = π(u ), lies in G (f), normalizes S , and carries a to −a. But ϕ(m(u)) = ϕ(u  u ) is an f-rational element of the −a root group of the quasi-reductive group Ma with respect to the maximal f-split torus S; it normalizes this torus and acts non-trivially on its character group. This is of course impossible. 

8.3 Parahoric Integral Models We will assume in this section that G K is quasi-split (this is automatic by Corollary 2.3.8 if f is perfect). Then the Bruhat–Tits building B(G K ) of G(K) is available and the results of the preceding chapters hold over K. Since we will exclusively work over K in this section, we will denote G K and B(G K ) by G and B(G) respectively. So, G will denote a quasi-split connected reductive K-group. We will construct various smooth affine O-models of G associated to a nonempty bounded subset Ω of an apartment A(⊂ B(G)). As will be discussed in 9.2.5, these models descend readily to o when Ω is a non-empty bounded subset of an apartment of B(G K )Γ in the sense of Axiom 4.1.27. More precisely, we will prove that the open bounded subgroups G(K)0Ω ⊂ b ⊂ G(K)1 ⊂ G(K)† of (7.7.1) are schematic, and moreover G(K)0 is G(K)Ω Ω Ω Ω connected. The corresponding smooth integral models will be denoted GΩ0 ⊂ GΩb ⊂ GΩ1 ⊂ GΩ† . Recall that they are unique, cf. Corollary 2.10.11. In addition, we will define a (usually not affine) smooth group scheme GΩ with generic fiber G and GΩ (O) = G(K)Ω . Proposition 8.3.1 Assume that Ω is a union of facets. There exists a smooth integral model GΩ1 of G(see §2.8) such that GΩ1 (O) = G(K)1Ω . Proof Let x1, . . . , xn be the vertices of all facets contained in Ω. By Theorem 4.2.15 each Ki = G(K)1xi is a maximal bounded subgroup of G(K). The intersection K = K1 ∩ · · · ∩ Kn is a bounded subgroup of G(K) and equals G(K)1Ω by Proposition 7.7.5. By Corollary A.6.6 it is enough to show that each

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Ki is schematic. This reduces the proof to the case Ω = {x} is a singleton set containing a vertex. In this case, K = G(K)1x . Choose a faithful algebraic representation ρ : G → GL(V). Then ρ(K) is a bounded subgroup of GL(V). Using Proposition 2.2.13, choose a lattice L ⊂ V  = GL(L). Since K is a maximal bounded subgroup we such that ρ(K) ⊂ K −1  have K = ρ (K). Let G be the schematic closure of ρ(G) in the smooth group scheme GL(L). Lemma A.2.1 implies that G is a model of G and  = K. By Corollary A.6.5, K is schematic, that is, there exists G (O) = ρ−1 (K)  a smooth affine O-group scheme Gx1 such that Gx1 (O) = K. Theorem 8.3.2 Let Ω be a non-empty bounded subset of A. The groups G(K)1Ω and G(K)†Ω are schematic and share the same relative identity component. Proof Let Ω  be the union of all facets of A that intersect Ω. Since the action of G(K) on B(G) maps facets to facets, we have the chain of inclusions G(K)1Ω ⊂ G(K)1Ω ⊂ G(K)†Ω ⊂ G(K)†Ω . The smallest of these groups is of finite index in the largest, due to Proposition 7.7.1. The first one, that is, G(K)1Ω , is schematic due to Proposition 8.3.1. This allows us to apply Proposition A.7.1 to the integral model for U := G(K)1Ω provided by Proposition 8.3.1, and V being any of the three other groups in the above chain.  b is schematic. Proposition 8.3.3 The group G(K)Ω

Proof This follows from Proposition A.7.1 applied to the finite index inclusion b.  G(K)0Ω ⊂ G(K)Ω In the sequel, we will denote G(K)0{x }, G(K)b{x }, G(K)1{x } , and G(K)†{x } by

G(K)0x , G(K)bx , G(K)1x , and G(K)†x respectively.

Remark 8.3.4 There also exists a smooth group scheme GΩ with generic fiber G such that GΩ (K) = G(K)Ω . It is obtained by applying Theorem 8.3.2 and Proposition A.7.1 to U = G(K)1Ω and V = G(K)Ω . It is generally not affine, and only locally of finite type. Example 8.3.5 Consider G = SL2, K and choose a uniformizing element π ∈ K. Let A ⊂ B be the standard apartment and x ∈ A the vertex corresponding to the canonical integral model. There are two edges in A containing x in their closure. Let y, z ∈ A be the other vertices contained in the closures of these edges, so that " " " # # # O M O O O M−1 = = , G(K) . , G(K) G(K)y = x z M O M−1 O O O

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297

The corresponding integral models are given by the coordinate rings O[Gx ] = O[a, b, c, d]/(ad − bc − 1), O[Gy ] = O[a, π −1 b, πc, d]/(ad − bc − 1), O[Gz ] = O[a, πb, π −1 c, d]/(ad − bc − 1). To obtain the integral model for the standard Iwahori subgroup G(K)x ∩ G(K)z , we follow the proof of Corollary A.6.6 and take the schematic closure in Gx × Gz of the diagonal in G × G. The map K[G] ⊗K K[G] → K[G] dual to the inclusion of the diagonal is the multiplication map, and we see that this schematic closure is given by the coordinate ring O[a, b, π −1 c, d]/(ad − bc − 1) = O[a, b, c , d]/(ad − πbc  − 1). The special fiber of this integral model has coordinate ring f[a, b, c , d]/(ad −1). It is thus isomorphic to Gm ×A2 , in particular it is smooth. Thus no smoothening is required, and the integral model G(x, z) of the standard Iwahori subgroup has coordinate ring O[a, b, π −1 c, d]/(ad − bc − 1). In the special fiber, the group operation is (a1, b1, c1 , d1 )(a2, b2, c2 , d2 ) = (a1 a2, a1 b2 + b1 d2, c1 a2 + d1 c2 , d1 d2 ). In the same way we obtain the coordinate ring for the pointwise stabilizer of the union of the two edges G(x, y, z) , and it is given by O[a, π −1 b, π −1 c, d]/(ad − bc − 1). 8.3.6 Let S be the maximal K-split torus of G corresponding to the apartment A. Then S(K)1 ⊂ G(K)1Ω . Let S be the standard integral model of S, cf. §B.4. The inclusion S → G extends to a morphism S → GΩ1 . Proposition 8.2.2(1) implies that it is a closed embedding. Recall from 2.11.7 that the set of S -roots of GΩ1 coincides with Φ(S, G). For a ∈ Φ(S, G), let Ua,Ω,0 be the corresponding root group of GΩ1 of Definition 2.11.10. Let T be the centralizer of S in G. Then T is a maximal torus of G. Let S and T 0 be the connected Néron model of S and T respectively, in the sense of §B.8. Note that, since S is split, S is also the standard integral model of S in the sense of §B.2, which is an (automatically split) O-torus. The inclusion S → T extends to a closed immersion S → T 0 due to Lemma B.7.11. Since T(K)0 ⊂ G(K)0Ω , the inclusion T → G extends to a morphism T 0 → 0 GΩ . This morphism is a closed immersion since T 0 is clearly the centralizer of the closed torus S in GΩ0 . Lemma 8.3.7 Recall the open bounded subgroup Ua,Ω,0 of Ua (K) of Definition 7.4.1. We have Ua,Ω,0 (O) = Ua,Ω,0 . Thus Ua,Ω,0 is the smooth model of Ua,Ω,0 . Proof Lemma 7.7.3 states Ua,Ω,0 = Ua (K) ∩ G(K)Ω . But Ua,Ω,0 ⊂ G(K)1Ω , so Ua,Ω,0 = Ua (K) ∩ G(K)1Ω = Ua (K) ∩ GΩ1 (O). The claim now follows from Proposition 2.11.9.  Remark 8.3.8 Using the preceding lemma for Ω = {x}, x ∈ A, we see that the subgroup Ua, x,0 is schematic. We denote its smooth model by Ua, x,0 . We

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will now show that for any r ∈ R, Ua, x,r is also schematic. For this purpose, we consider a v ∈ V := R⊗Z X∗ (S) such that a(v) = −r, and let y = v + x ∈ A. Then Ua, x,r = Ua, y,0 . As Ua, y,0 is schematic, we infer that Ua, x,r is also schematic. We will denote the smooth model of Ua, x,r by Ua, x,r in what follows. 8.3.9 For x ∈ A, a ∈ Φ, and r ∈ R, we have the open bounded subgroup Ua, x,r ⊂ Ua (K). For varying r, these groups form a descending exhaustive separated filtration of Ua (k) by bounded open subgroups. The filtration groups are independent of the choice of x; only their indexing depends on x, and the dependence is given by Ua, v+x,r+a(v) = Ua, x,r according to Lemma 6.1.6. We 

have the conventions Ua, x,r+ = s>r Ua, x, s and Ua,Ω,0 = x ∈Ω Ua, x,0 . Remark 8.3.10 Let Ω be a non-empty bounded subset of A and we fix an x ∈ Ω. Then for any y ∈ Ω we have Ua, y,0 = Ua, x, a(x−y) by Lemma 6.1.6,

so Ua,Ω,0 = y ∈Ω Ua, x, a(x−y) . The jumps of the filtration Ua, x,r , that is, the real numbers r for which Ua, x,r+  Ua, x,r , form a discrete subset of R, more precisely a torsor under the discrete subgroup Ja ⊂ R discussed in Fact 6.4.5. Therefore, Ua,Ω,0 is equal to Ua, x,r for some suitable r ∈ R. Hence, Ua,Ω,0 = Ua, x,r for some r ∈ R. 8.3.11 For r, s ∈ R, x ∈ A and a ∈ Φ, we define the bounded open subgroup Ua, x,r , s of Ua (K) to be the subgroup Ua, x,r · U2a, x, s (= Ua, x,r if either 2a  Φ or s  2r). Remark 8.3.8 provides a smooth affine integral model Ua, x,r of Ua, x,r . We will now show that the subgroup Ua, x,r , s admits a smooth affine integral model. For this purpose, we consider the normal subgroup Ua, x,r of Ua, x,r , s and apply Proposition A.7.1 to H = Ua , U = Ua, x,r ⊂ V = Ua, x,r , s to obtain a smooth separated group scheme, which we denote by Ua, x,r , s , such that Ua, x,r , s (O) = Ua, x,r , s . Its generic fiber is Ua , and it comes equipped with an open immersion Ua, x,r → Ua, x,r , s that extends the inclusion Ua, x,r → Ua, x,r , s . We will now show that Ua, x,r , s is of finite type. As the generic fiber Ua of this group scheme is of finite type, to show that the group scheme itself is of finite type, it would suffice to show that its special fiber U a, x,r , s is of finite type. Now to show that U a, x,r , s is of finite type, we consider the O-group scheme homomorphism p : Ua, x,r ×U2a, x, s → Ua, x,r , s defined by the product in Ua . It is obvious that under this homomorphism, Ua, x,r (O) × U2a, x, s (O) maps onto Ua, x,r , s (O). Hence, by Hensel’s lemma, U a, x,r (fs ) × U 2a, x, s (fs ) maps onto U a, x,r , s (fs ). Since the induced map p fs between special fibers is an fs -morphism whose source is an fs -scheme of finite type, as a map it is quasicompact (since all subsets of a noetherian topological space are quasi-compact). Hence, by [SGA3, Exp.VIB , Prop.1.2] applied over the field fs , the image of p fs

8.3 Parahoric Integral Models

299

is closed. By what we have just seen, this closed image contains all fs -points of U a,x,r ,s . Invoking the fact that as U a, x,r , s is smooth, the group of its fs -points is Zariski-dense, we see that the induced map U a, x,r × U 2a, x, s → U a, x,r , s is surjective. This implies that U a, x,r , s is of finite type. As Ua, x,r , s is separated, of finite type, and its generic fiber Ua is affine, Proposition 3.1 of [PY06] implies that Ua, x,r , s is affine. Hence, it is a smooth affine integral model of Ua, x,r , s . We state this result as the first part of the following proposition. Proposition 8.3.12 Let r, s ∈ R, a ∈ Φ and x ∈ A. (1) The bounded open subgroup Ua, x,r , s of Ua (K) admits a smooth affine integral model Ua, x,r , s with generic fiber Ua . (2) The conjugation action S × Ua → Ua of S on Ua extends to an action S × Ua, x,r , s → Ua, x,r , s . (3) The special fiber U a, x,r , s of Ua, x,r , s is connected. Proof The second assertion follows from the fact that under the conjugation action of S(K) on Ua (K), the subgroup Ua, x,r , s is stable under S (O) = S(K)b . Now the third assertion follows from the fact that the weights for the adjoint action of S on the Lie algebra of the smooth affine group U a, x,r , s are contained in {a, 2a}.  Theorem 8.3.13 The relative identity component GΩ0 of the smooth model GΩ1 (equivalently, that of GΩ† ) satisfies GΩ0 (O) = G(K)0Ω . Proof Let S ⊂ G be a maximal K-split torus whose apartment contains Ω and T be its centralizer. The group G(K)0Ω is generated by T(K)0 and Ua,Ω,0 for all roots a ∈ Φ(T). It is open by Proposition 7.7.1. Let T 0 be the connected Néron model of T, cf. §B.8. It satisfies T 0 (O) = T(K)0 according to Corollary B.8.7. The embeddings Ua,Ω,0 → G(K)1Ω and T(K)0 → G(K)1Ω extend to morphisms T 0 → GΩ1 and Ua,Ω,0 → GΩ1 of group schemes according to Corollary 2.10.10 and Remark 2.10.19. The O-points of their images thus generate the open subgroup G(K)0Ω ⊂ G(K)1Ω = GΩ1 (O). Since by Proposition 7.7.1, [G(K)1Ω : G(K)0Ω ] < ∞, Lemma A.4.26 shows that G(K)0Ω = GΩ0 (O).  0 by G 0 . In what follows, we will always denote G {x x }

As in 8.3.6, S and T 0 are the connected Néron model of S and T respectively. For a K-split torus S, we will often denote the maximal bounded subgroup S(K)b of S(K) by S(K)0 . Proposition 8.3.14

(1) S is a maximal closed torus of GΩ0 .

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(2) For each a ∈ Φ, the open bounded subgroup Ua,Ω,0 of Ua (K) is schematic and connected; the corresponding smooth model Ua,Ω,0 is the a-root group of GΩ0 . (3) We choose a system of positive roots Φ+ ⊂ Φ and let Φ− = −Φ+ . Upon taking the order of the factors in any way in the following product over Φ±,nd , we obtain the same o-smooth closed subgroup scheme of G under  multiplication a ∈Φ±, nd Ua,Ω,0 → G , and these subgroup schemes have unipotent fibers. Moreover, the product morphism   Ua,Ω,0 × T 0 × Ua,Ω,0 → GΩ0 a ∈Φ −, nd

a ∈Φ +, nd

is an open immersion. Proof The first assertion follows from Proposition 8.2.2(1) (applied over O in place of o) since the generic fiber S of S is a maximal K-split torus of G. We have S(K)0 ⊂ G(K)0Ω , so the conjugation action of S(K)0 on G(K) preserves the subgroup G(K)0Ω . Therefore the conjugation action of S on G extends to an action of S on GΩ0 . This allows us to apply Theorem 8.2.5 and obtain the open immersion   Ua × Z × Ua → GΩ0, a ∈Φ −, nd

a ∈Φ +, nd

where Ua is the a-root group for the action S on GΩ0 and Z is the centralizer of S in GΩ0 . Since T 0 (O) = T(K)0 ⊂ G(K)0Ω = GΩ0 (O), the embedding T → G extends to a morphism T 0 → GΩ0 . According to Proposition 8.2.4, this morphism is a closed embedding with image Z . According to Proposition 2.11.9, the a-root group Ua satisfies Ua (O) = Ua (K) ∩ GΩ0 (O), which equals  Ua,Ω,0 by Lemma 7.7.3. Hence, Ua = Ua,Ω,0 . Proposition 8.3.15 Let G be a smooth integral model of G with connected special fiber. Then G (O) ⊂ G(K)0 . Proof The subgroup G (O) ⊂ G(K) is bounded according to Remark 2.10.3, and therefore lies in G(K)x1 for some x ∈ B(G) according to Theorem 4.2.15. It follows from Corollary 2.10.11 that the identity on G extends to a homomorphism G → Gx1 of group schemes. The connectedness of the special fiber of G and Theorem 8.3.13 imply that this homomorphism factors through Gx0 , hence  G (O) ⊂ Gx0 (O) ⊂ G(K)0 . We recall from [Con14, Definition 3.1.1] the notion of a reductive o-group scheme: it is a smooth affine o-group scheme whose geometric fibers are connected reductive groups. In fact, the connectedness of the special fiber need not be assumed, cf. Proposition A.8.2.

8.3 Parahoric Integral Models Proposition 8.3.16 generic fiber.

301

Let G be a reductive o-group scheme and let G be its

(1) G splits over K. Hence, by the construction in the preceding chapter, there is a Bruhat–Tits building B(G K ) of G(K). (2) There exists a unique point x of B(G K ) that is fixed by G (O). The identity automorphism of G K extends to a O-isomorphism GO → Gx0 . The point x is a special vertex of B(G K ). Proof The claim that G splits over K follows from Proposition A.8.2. Using that proposition we choose a closed maximal o-torus T in G . Let T be the generic fiber of T and T be its special fiber; TK is a split maximal torus of G K and T is a maximal torus of the special fiber G of G . The subgroup G (O) of G(K) is bounded, cf. Remark 2.10.3. Moreover, T (O) is the maximal bounded subgroup T(K)0 of T(K). According to Corollary 4.2.14, G (O) fixes a point x ∈ B(G K ). Proposition 8.2.1(4) implies that the normalizer NG (T )(O) of T in G (O) maps onto the Weyl group of G K and then Lemma 1.3.42 implies that x is a special vertex. Now if y is another point of B(G K ) fixed by G (O), then the same argument applies to y, and hence y is also a special vertex. Now since every point of the geodesic [x, y] is fixed by G (O), we see that each point of this geodesic is a special vertex; but this is of course impossible unless x = y. Thus we see that x is the unique point of B(G K ) fixed by G (O). As G (O) is stable under the action of Γ = Gal(K/k) on G(K), x is fixed under the action of Γ on B(G K ) (cf.Proposition 7.9.5). Now we consider the group scheme Gx1 . Since G (O) fixes x, G (O) ⊂ Gx1 (O). Corollary 2.10.10 implies that the identity automorphism of G extends to a homomorphism GO → Gx1 of O-group schemes. As G has connected fibers, this homomorphism factors through ρ : GO → Gx0 . On the other hand, T(K)0 = T (O) ⊂ G (O) ⊂ Gx0 (O), so T (O) = T(K)0 = T(K) ∩ Gx0 (O). Proposition B.2.4 implies that the schematic closure of TK in Gx0 is an O-torus and this schematic closure is the image of TO under ρ. We conclude that the restriction of ρ to TO is a closed immersion TO → Gx0 . Consider the homomorphism ρ : G fs → G x0 . Its restriction to T fs is a closed immersion. The reduced subscheme ker( ρ)red of the kernel is a smooth closed normal subgroup of G . This subgroup does not contain any non-trivial semisimple f-rational element, because (after possibly conjugating by an element of G (f)) such an element would belong to T (f). Therefore, the identity component of ker( ρ)red is a closed smooth connected unipotent normal subgroup of G . Since G is reductive, this identity component is necessarily trivial. Hence ker( ρ) is a 0-dimensional group scheme. As both GO and Gx0 are smooth and have the same generic fiber, their special fibers have the same dimension. In

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addition, the special fiber of Gx0 is connected. So ρ : G fs → G x0 is surjective. Using Lemma 8.2.3 we now conclude that ρ is an isomorphism.  8.3.17 Consider two non-empty bounded subsets Ω, Ω  of A such that Ω is contained in the closure of Ω . Then G(K)0Ω ⊂ G(K)0Ω , and Corollary 2.10.10 implies that the identity on G extends to a homomorphism ρΩ,Ω : GΩ0 → GΩ0

(8.3.1)

of O-group schemes. We will write ρΩ,Ω : G Ω0  → G Ω0 for the special fiber of this homomorphism.

8.4 The Structure of the Special Fiber of GΩ0 In the rest of this chapter, we will assume that the residue field f of k is perfect. Then the residue field of K is an algebraic closure of f; it will be denoted by f. Corollary 2.3.8 says that G K is quasi-split, and the building B(G K ) of G(K) has been constructed in the preceding chapter. As in the preceding section, we will denote G K by G and B(G K ) by B(G). Let S be a maximal K-split torus of G and Φ be the root system of G with respect to S. For a ∈ Φ, Ua is the a-root group of G. Let S be the o-split o-torus with generic fiber S. Let A be the apartment of B(G) corresponding to S. For a non-empty and bounded subset Ω of A, the integral model constructed in Theorem 8.3.13 has connected special fiber. In this section we will study the structure of this smooth connected group over the residue field f of K. The structure over f can be obtained from that over f without much difficulty, and this will be done in Chapter 9. We will first discuss the special fiber of the model Ua, x,r and the action of the special fiber of S on it. When r   r, the inclusion Ua, x,r  ⊂ Ua, x,r leads, via Corollary 2.10.10, to a homomorphism of models Ua, x,r  → Ua, x,r , and hence a homomorphism U a, x,r  → U a, x,r between their special fibers. We will consider the case when the image of this homomorphism is normal (this is obvious unless 2a ∈ Φ) and the cokernel is 1-dimensional. This will be used in the computation of the root system of the reductive quotient of the special fiber of G Ω0 , where said cokernel will be the corresponding root group. Slightly more generally, when 2a ∈ Φ and r, r , s, s  ∈ R, with r   r and s   s, then Ua, x,r  , s ⊂ Ua, x,r , s and we obtain analogously a O-group scheme homomorphism ϕ : Ua, x,r  , s → Ua, x,r , s .

8.4 The Structure of the Special Fiber of GΩ0 Lemma 8.4.1 (1) (2) (3) (4) (5)

303

The following assertions are equivalent to each other.

ϕ is injective. ϕ is an isomorphism. ϕ is surjective. ϕ is an isomorphism. Ua, x,r  , s = Ua, x,r , s .

Proof We observe that the groups U a, x,r , s and U a, x,r  , s are equidimensional and have the same generic fiber Ua . Hence, assertions (1) and (2) are equivalent. (2) trivially implies (3). Lemma 8.2.3 implies the equivalence of (3) and (4), and of course (4) implies (2). Thus (2), (3) and (4) are equivalent to each other. Corollary 2.10.11 implies the equivalece of (4) and (5).  8.4.2 Let Ω be a non-empty bounded subset of the apartment A corresponding to the maximal K-split torus S of G. We will denote the Bruhat–Tits group scheme associated to Ω by GΩ0 , and its special fiber by G Ω0 . The unipotent radical of G Ω0 is denoted by Ru (G Ω0 ) and the maximal reductive quotient G Ω0 /Ru (G Ω0 ) of G Ω0 by G Ω . Let S be the special fiber of S considered as a maximal torus of G Ω0 and S be its isomorphic image in G Ω . The root system of G Ω with respect to S is reduced and its root groups are non-canonically isomorphic to Ga since f is algebraically closed. There are natural identifications X∗ (S) = X∗ (S ) = X∗ (S ) = X∗ (S) and X∗ (S) = X∗ (S ) = X∗ (S ) = X∗ (S). Since U a,Ω,0 is the a-root group of G Ω0 , the a-root group of Ru (G Ω0 ) is U a,Ω,0 ∩ Ru (G Ω0 ).  8.4.3 For a ∈ Φ define Ua,Ω,0+ = x ∈Ω Ua, x,0+ ∩Ua,Ω,0 . As in Remark 8.3.10 one concludes that Ua,Ω,0+ is schematic. The inclusion Ua,Ω,0+ ⊂ Ua,Ω,0 induces by Corollary 2.10.10 a morphism Ua,Ω,0+ → Ua,Ω,0 of integral models. + We denote by U a,Ω,0 ⊂ U a,Ω,0 the image of the special fiber of the morphism Ua,Ω,0+ → Ua,Ω,0 . Analogously, we denote by T special fiber of the morphism T0+ → T 0 .

+

the image of the

Lemma 8.4.4 Let a ∈ Φ and u ∈ Ua,Ω,0 (O) = Ua,Ω,0 . The image u ∈ U a,Ω,0 (f) of u lies in Ru (G Ω0 )(f) if and only if u ∈ Ua,Ω,0+ (O) = Ua,Ω,0+ . Proof Let u ∈ Ua,Ω,0 (O) = Ua,Ω,0 and let u ∈ U a,Ω,0 (f) be its image. According to Theorem 8.2.9, u lies in Ru (G Ω0 )(f) if and only if at least one of the elements u , u  ∈ U−a (k) of Proposition 2.11.17 does not lie in GΩ0 (O)∩U−a (k) = U−a,Ω,0 (this equality is provided by Lemma 7.7.3). Recall from Definition

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6.1.2 that for any x ∈ A, x−a (u ) = x−a (u ) = −xa (u). Thus the assertions u   U−a,Ω,0 and u   U−a,Ω,0 are equivalent to each other, and to the existence  of x ∈ Ω with xa (u) > 0. This in turn is equivalent to u ∈ Ua,Ω,0+ . Corollary 8.4.5

For a ∈ Φ the following two statements hold.

+ . (1) The a-root group U a,Ω,0 ∩ Ru (G Ω0 ) of Ru (G 0Ω ) equals U a,Ω,0 + (2) The preimage of U a,Ω,0 (f) in Ua,Ω,0 (O) = Ua,Ω,0 equals Ua,Ω,0+ .

Proof (1) is an immediate consequence of Lemmas 8.4.4 and 8.1.3, and (2) follows at once from Lemma 8.4.4.  ΨΩ 8.4.6 Consider the annihilator A∗Ω = {ψ ∈ A∗ | ψ(Ω) = {0}}. In this vector space we have the finite root systems ΨΩ = Ψ ∩ A∗Ω and ΨΩ = Ψ  ∩ A∗Ω . The derivative map restricts to an injection A∗Ω → V(S )∗ and we denote by  (⊂ Φ) the isomorphic images of Ψ ⊂ Ψ  under this injection. ΦΩ ⊂ Φ Ω Ω Ω  and let ψ ∈ Ψ  be such that a = ψ.  Lemma 6.1.6 Assume that a ∈ ΦΩ Ω implies that for any z ∈ Hψ we have Ua,Ω,0 = Ua, z,0 and Ua,Ω,0+ = Ua, z,0+ . Lemma 8.4.7 ΦΩ is a reduced root system and for a non-divisible a ∈ Φ  one has the following dichotomy: belonging to ΦΩ  , (1) a ∈ ΦΩ ⇔ 2a  ΦΩ  . (2) 2a ∈ ΦΩ ⇔ 2a ∈ ΦΩ

Proof Recall from Proposition 6.4.8 that Ψ is reduced. This implies that ΦΩ is reduced. Let ψ ∈ ΨΩ be such that ψ = a. Then 2ψ is the unique affine  if and only if functional with derivative 2a that vanishes on Ω. So 2a ∈ ΦΩ  2ψ ∈ Ψ . Proposition 6.3.8 shows that exactly one of ψ, 2ψ lies in Ψ, and that  2ψ lies in Ψ if and only if it lies in Ψ . Proposition 8.4.8 Let a ∈ Φ. +  . (1) U a,Ω,0  U a,Ω,0 if and only if a ∈ ΦΩ  , then U + (2) If a is non-divisible in Φ and belongs to ΦΩ a,Ω,0 /U a,Ω,0 is

non-canonically isomorphic to Ga and the weight of the S -action of the Lie algebra of this quotient is the unique element of {a, 2a} ∩ ΦΩ . (3) The root system of G Ω , with respect to S, is ΦΩ .  if and only if U Proof (1) By definition, a ∈ ΦΩ a,Ω,0+  Ua,Ω,0 , which by +  U a,Ω,0 . Corollary 8.4.5 is equivalent to the assertion U a,Ω,0 Assertions (2) and (3) will now be proved together.

8.4 The Structure of the Special Fiber of GΩ0

305

 . According to (1) the quotient U + Assume a ∈ ΦΩ a,Ω,0 /U a,Ω,0 is non-trivial. By Corollary 8.4.5, this quotient is the root group corresponding to the unique positive integral multiple of a which is a root of G Ω . So it is non-canonically isomorphic to Ga . Unless 2a ∈ Φ, a is a root of G Ω and it belongs to ΦΩ . We assume now that 2a ∈ Φ. Then 2a is a root of G Ω if and only if +  . U 2a,Ω,0  U 2a,Ω,0 . This in turn is equivalent by (1) to the assertion 2a ∈ ΦΩ But then by Lemma 8.4.7(2), 2a ∈ ΦΩ . On the other hand, a is a root of G Ω  , but if and only if 2a is not its root. Hence, if a is a root of G Ω , then 2a  ΦΩ  then by Lemma 8.4.7(1), a ∈ ΦΩ . + is trivial if and only if Ua is 1-dimensional. Remark 8.4.9 The group U a,Ω,0 + → U a,Ω,0 → Ga → 1. The Indeed, we have the exact sequence 1 → U a,Ω,0

smoothness of Ua,Ω,0 implies that the dimensions of Ua and U a,Ω,0 are equal. The claim now follows from Proposition 8.4.8(2). Let Φ∨Ω = {a∨ ∈ Φ∨ | a ∈ ΦΩ }. Theorem 8.4.10 roots Φ+ ⊂ Φ.

Let G Ω and S be as in 8.4.2. Choose a system of positive

(1) The open immersion of Proposition 8.3.14 restricts to an isomorphism   + + U a,Ω,0 ×T +× U a,Ω,0 → Ru (G Ω0 ). a ∈Φ −, nd

a ∈Φ +, nd

(2) The root datum of G Ω with respect to S is (X∗ (S), ΦΩ, X∗ (S), Φ∨Ω ). For a ∈ ΦΩ , the composite of the maps U a,Ω,0 → G Ω0 → G Ω0 /Ru (G Ω0 ) + and the a-root group provides an isomorphism between U a,Ω,0 /U a,Ω,0 of G Ω . Proof (1) According to Proposition 8.3.14, S is a maximal torus of G Ω0 . Let a ∈ Φnd . According to Proposition 2.11.9, U a,Ω,0 is the a-root group of G Ω0 with respect to this maximal torus. The subgroup Ru (G Ω0 ) is normalized by S . The a-root group of Ru (G Ω0 ) is Ru (G Ω0 ) ∩ U a,Ω,0 , which by Corollary 8.4.5 + . The zero weight space equals Ru (G Ω0 )S , which according to equals U a,Ω,0

[Bor91, §13.7, Corollary 1] equals Ru ((G Ω0 )S ), which in turn equals Ru (T 0 ) by the last assertion in 8.3.6. The latter equals T + by Proposition B.10.12. The assertion now follows from Theorem 2.11.15. (2) According to Proposition 8.4.8(3) the root system of G Ω , with respect to S is ΦΩ . From this assertion (2) is obvious.  Remark 8.4.11

Consider a point x ∈ B(G K ). Theorem 8.4.10 implies that

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G x is semi-simple if and only if G does not contain a non-trivial K-split central

torus and x is a vertex, cf. Proposition 1.3.35. Assume now that G is semisimple and simply connected and that x is a vertex. Then G x need not be simply connected. For example, when G is the group SU3 , necessarily ramified since we are working over the maximal unramified extension K of k, and x is an extra special vertex, then G x is SO(3) = PGL2 . Corollary 8.4.12 The preimage of the unipotent radical Ru (G Ω0 )(f) under the surjective map G(K)0Ω = GΩ0 (O) → G Ω0 (f) equals G(K)Ω,0+ . Proof This follows at once from Theorem 8.4.10, Corollary 8.4.5(2), Lemma B.10.11, and Proposition 7.3.12.  Corollary 8.4.13 The action of G(K)F,0+ on B(G K ) fixes an open neighborhood of F. Proof Let F  be a facet whose closure contains F. The observation in 6.3.21 implies that G(K)F,0+ ⊂ G(K)F  ,0+ ⊂ G(K)F  ,0 ⊂ G(K)F,0 . Therefore G(K)F,0+ fixes F . But the union of all F  whose closure contains F is an open neighborhood of F.  Proposition 8.4.14 Assume that G splits over K, then the O-group scheme GF0 is reductive if and only if F is a hyperspecial vertex. In case F is a hyperspecial vertex, GF0 = GF1 . Proof Since the group scheme GF0 is smooth, affine, has connected geometric fibers, and reductive generic fiber, it is reductive if and only if its special fiber is reductive. We will now prove that its special fiber is reductive if and only if F is a hyperspecial vertex. Let ΦF be as in 8.4.6 for Ω = F. According to Theorem 8.4.10, ΦF is the root system of the maximal reductive quotient G F of G F0 . So the dimension of this reductive quotient is dim(S) + #ΦF . By smoothness of GF0 , the dimension of G F0 equals the dimension of G, which equals dim(S) + #Φ. We conclude that G F0 is reductive, that is, it equals G F , if and only if #ΦF = #Φ, which is equivalent to ΦF = Φ. This in turn is equivalent to F being a hyperspecial vertex in the sense of Definition 1.3.39 since G has been assumed to be K-split. The equality GF0 = GF1 is equivalent, via Corollary 2.10.11, to the equality G(K)0F = G(K)1F . The latter follows from Lemma 7.7.10 and Proposition 7.7.11.  Proposition 8.4.15 Assume that Ω ≺ Ω ⊂ A. The kernel of ρΩ,Ω is a smooth connected unipotent group and the multiplication map induces an isomorphism

8.4 The Structure of the Special Fiber of GΩ0 of f-schemes



307

ker(U a,Ω ,0 → U a,Ω,0 ) → ker(ρΩ,Ω ).

a ∈Φnd

Proof We begin by showing that ker(ρΩ,Ω ) is unipotent. We recall that restricted to S , the homomorphism ρΩ,Ω is injective, and hence it is an injective homomorphism restricted to any torus of GΩ0 . This implies that ρΩ,Ω restricted to any torus of G Ω0  has trivial schematic kernel. This implies at once that ker(ρΩ,Ω ) is unipotent. Fix a system of positive roots Φ+ in Φ = Φ(S, G). We claim that the kernel of ρΩ,Ω is contained in the special fiber of the open cell described in Proposition 8.3.14(3) for Ω  in place of Ω. Let g ∈ ker(ρΩ,Ω )(f)(⊂ GΩ0 (f)) and choose a lift g ∈ GΩ0 (O) = G(K)0Ω . Corollary 8.4.12 implies that g ∈ G(K)0Ω ∩ G(K)Ω,0+ . By Proposition 7.3.12(5) we have   Ua,Ω,0+ × T(K)0 × Ua,Ω,0+ . g∈ a ∈Φ−, nd

In particular, g ∈ G(K)0Ω ∩ =



a ∈Φ+, nd



Ua (K) × T(K) ×

a ∈Φ−, nd

Ua,Ω ,0 × T(K)0 ×

a ∈Φ−, nd





Ua (K)



a ∈Φ+, nd

Ua,Ω ,0 .

a ∈Φ+, nd

We conclude that g lies in the special fiber of the open immersion, as claimed. Therefore, the kernel of ρΩ,Ω is the product of the kernels of the morphisms U a,Ω ,0 → U a,Ω,0 over all a ∈ Φnd , as well as the identity morphism T → T . We see thus that the kernel of ρΩ,Ω is isomorphic as as a scheme to the following product:  ker(U a,Ω ,0 → U a,Ω,0 ). a ∈Φnd

Lemma C.5.3 implies that each factor of the above product is smooth and connected.  Proposition 8.4.16 In addition to Ω being contained in the closure of Ω , assume that Ω and Ω  span the same subspace of A. Then the kernel of the composition π of ρΩ,Ω with the projection G Ω0 → G Ω is Ru (G Ω0  ), and π induces an isomorphism G Ω → G Ω . Moreover, ρΩ,Ω (Ru (G Ω0  )) = Ru (G Ω0 ). Proof The kernel K of π is a unipotent f-group scheme since the kernel of the projection G Ω0 → G Ω is the unipotent radical Ru (G Ω0 ) of G Ω0 , and according to Proposition 8.4.15, ker(ρΩ,Ω ) is a (smooth connected) unipotent group. Let U

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be the identity component of the underlying reduced subscheme of K. Then U is a smooth connected unipotent normal subgroup of G Ω0  , so it is contained in the unipotent radical Ru (G Ω0  ) of G Ω0  . Therefore, dim(G Ω ) = dim(G Ω0  /Ru (G Ω0  ))  dim(G Ω0  /U) = dim(G Ω0  /K)  dim(G Ω ). But Theorem 8.4.10(2) implies that G Ω and G Ω have the same root datum (which depends only on the affine subspace of A spanned by Ω and Ω ). So, in particular, these two reductive groups have equal dimension. Hence, U = Ru (G Ω0  ), the inclusion G Ω0  /K → G Ω is equality, and G Ω → G Ω is an isogeny with kernel K/Ru (G Ω0  ). Now as G Ω and G Ω have the same root datum, this isogeny is an isomorphism. Therefore, K = Ru (G Ω0  ). The last assertion of the proposition is seen at once by dimension count.  8.4.17 Let Ω 1 , Ω 2 and Ω 3 be three non-empty bounded subsets of A such that the affine subspace of A spanned by any of the Ω i , for i  3, is the same as the affine subspace of A spanned by Ω 1 ∪ Ω 2 ∪ Ω 3 . Then there is an isomorphism f j,i j : G Ω i ∪Ω j → G Ω j , for i, j  3 obtained by setting Ω  = Ω i ∪Ω j and Ω = Ω j in the preceding proposition. Let ϕΩ j ,Ω i : G Ω i → G Ω j be the isomorphism −1 . f j,i j ◦ fi,i j Corollary 8.4.18

We have ϕΩ3 ,Ω1 = ϕΩ3 ,Ω2 ◦ ϕΩ2 ,Ω1 .

Proof For i, j  3, there is an isomorphism fi j : G Ω1 ∪Ω2 ∪Ω3 → G Ω i ∪Ω j obtained from the preceding proposition by setting Ω  = Ω 1 ∪ Ω 2 ∪ Ω 3 and Ω = Ω i ∪ Ω j . On the other hand, taking Ω = Ω j , we obtain the isomorphism f j : G Ω1 ∪Ω2 ∪Ω3 → G Ω j . It is obvious that f j,i j = f j ◦ fi−1 j . Hence, −1 −1 −1 −1 ϕΩ j ,Ω i = f j,i j ◦ fi,i j = f j ◦ fi j ◦ fi j ◦ fi = f j ◦ fi .

Therefore, ϕΩ3 ,Ω2 ◦ ϕΩ2 ,Ω1 = f3 ◦ f2−1 ◦ f2 ◦ f1−1 = f3 ◦ f1−1 = ϕΩ3 ,Ω1 .



Now we consider the case of two facets F ≺ F  ⊂ A. Recall the notation F ≺ F , which signifies that F is contained in the closure of F . As above we obtain the homomorphism ρF,F  : GF0  → GF0 of O-group schemes, and its special fiber ρF,F  : G F0  → G F0 . Recall from Proposition 1.3.35 the parabolic subset ΨF (F ) of the finite root system ΨF = Ψ ∩ A∗F . Via the embedding ∇ : A∗F → V(S )∗ we identify ΨF with ΦF and consider ΨF (F ) as a parabolic subset of the latter. As in 8.3.6, S and T 0 denote the connected Néron model of S and T respectively, and their special fibers will be denoted by S and T 0 respectively.

8.4 The Structure of the Special Fiber of GΩ0

309

Theorem 8.4.19 (1) The kernel of ρF,F  is a smooth connected unipotent subgroup isomorphic as an f-scheme to the product:  ker(U a,F  ,0 → U a,F,0 ). a ∈ΦF  −ΨF (F  )

(2) The image pF (F ) of ρF,F  is the parabolic subgroup of G F0 that is associated to the parabolic subset ΨF (F ) of the finite root system ΦF of GF := G F0 /Ru (G F0 ). (3) The preimage of pF (F ) under the surjective map GF0 (O) → G F0 (f) equals GF0  (O)(= G(K)0F  ). (4) The map F  → pF (F ) between the set of facets {F  | F ≺ F  } of B(G K ) and the set of parabolic subgroups of G F0 is bijective. Proof (1) We apply Proposition 8.4.15 and observe that, according to 6.3.21,  − Ψ (F  ). the kernel of U a,F  ,0 → U a,F,0 is trivial unless a ∈ ΦF F  (2) The homomorphism ρF,F  commutes with the closed immersions of T 0 into G F0 and G F0  . Therefore, its image pF (F ) is a smooth closed connected subgroup of G F0 containing (the image of) T 0 , and as such it is determined by the root groups for the action of S . These are the images of the root groups of G F0  , that is, im(U a,F  ,0 → U a,F,0 ) for non-divisible a ∈ Φ. According to + . This implies, via Theorem 8.4.10(1), 6.3.21 this image always contains U a,F,0

that pF (F ) contains Ru (G F0 ). It is therefore enough to analyze the image of pF (F ) in the reductive quotient G F = G F0 /Ru (G F0 ). The root groups for the + for action of S on this image are given by im(U a,F  ,0 → U a,F,0 )/U a,F,0 a ∈ Φ non-divisible. According to 6.3.21 this quotient is non-trivial precisely when there exists ψ  ∈ Ψa such that ψ (F) = 0 and ψ (F )  0. This is equivalent to the existence of ψ ∈ Ψa ∪ Ψ2a with ψ(F) = 0 and ψ(F )  0. + Moreover, the weight for the action of S on im(U a,F  ,0 → U a,F,0 )/U a,F,0 is precisely the derivative of ψ by Proposition 8.4.8(2). We conclude that the set of roots of S in the group pF (F ) is the subset ΨF (F ) of the finite root system ΦF . This subset is parabolic by Proposition 1.3.35. On the other hand, ΦF is the root system of the reductive group G F and the maximal torus S by Theorem 8.4.10. This proves (2). (3) The image of the composite of the maps GF0  (O) → GF0 (O) → G F0 (f) equals pF (F )(f) by construction. The claim follows from Corollary 8.4.12 and the fact that G(K)F,0+ ⊂ G(K)0F  according to 6.3.21. (4) The set of parabolic subgroups of G F0 is in bijection with the set of parabolic subgroups of G F . Those parabolic subgroups of G F that contain the maximal torus S are in bijection with the parabolic subsets of the root

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system of this connected reductive group relative to the maximal torus S . This root system is ΦF by Theorem 8.4.10. The parabolic subsets of this root system are precisely the subsets ΨF (F ) for facets F  of A whose closure contains F, according to Proposition 1.3.35. Therefore the map of (4) induces a bijection between the facets F  of A whose closure contains F, and the parabolic subgroups of G F0 that contain S . Since every facet F  whose closure contains F lies in an apartment that contains F, and since the homomorphism ρF,F  is independent of this apartment, (4) follows. 

8.5 Integral Models Associated to Concave Functions As in the previous section, we will work over the maximal unramified extension K of k. We assume that the residue field f of k is perfect. Hence, the residue field of K is an algebraic closure of f, denoted f. We keep the notation of the previous section: G is a connected reductive K-group, automatically quasi-split due to Corollary 2.3.8, S is a maximal K-split torus of G, T is its centralizer N  = Φ ∪ {0}, A its normalizer; Φ = Φ(S, G) be the relative root system of G, Φ ∗ the apartment of B(G) corresponding to S, Ψ ⊂ A the affine root system, T the ft-Néron model of T. For a ∈ Φ we denote by Ua the corresponding root  we set U0 = T. For an O-group scheme H with special group, and for 0 ∈ Φ fiber H , as before, we will denote the kernel of the reduction homomorphism H (O) → H (f) by H (M).  − {∞} be a concave function. →R 8.5.1 Let x ∈ A be a point and let f : Φ Recall from Definition 7.3.3 the bounded open subgroup G(k)x, f ⊂ G(k) generated by the subgroups T(k) f (0) ⊂ T(k) and Ua, x, f = Ua, x, f (a) · U2a, x, f (2a) ⊂ Ua (k). Its construction involves a choice of a functorial filtration of T(k) as in §7.2. We now make the further assumption that this functorial filtration is admissible, schematic, connected, and congruent, cf. Definitions B.10.2, B.5.2, B.5.3. We recall that when T is weakly induced (cf. Definition B.6.2) then there is a unique such filtration, namely the standard filtration (cf. Propositions B.6.4 and B.10.5). For general tori, we will use the minimal congruent filtration of Definition B.10.8. For r ∈ R0 , the bounded open subgroup T(k)r ⊂ T(k) is schematic and connected according to Proposition B.10.10; we denote by Tr the corresponding integral model. According to Proposition 8.3.12, for r, s ∈ R the open bounded subgroup Ua, x,r , s = Ua, x,r · U2a, x, s of Ua (K) is schematic and connected. As above, we will denote its smooth affine integral model by  − {∞} we have the notation  → R Ua, x,r , s . For a concave function f : Φ

8.5 Integral Models Associated to Concave Functions

311

Ua, x, f = Ua, x, f (a) · U2a, x, f (2a) and denote the corresponding smooth model by Ua, x, f . We fix a system of positive roots Φ+ in Φ. Let Φ− = −Φ+ . For a ∈ Φ, we will write a > 0 if a ∈ Φ+ and a < 0 if −a > 0. The following theorem generalizes Theorems 8.3.13 and 8.2.5 to groups associated to concave functions.  − {∞} be a concave function. The subgroup →R Theorem 8.5.2 Let f : Φ G(k)x, f is schematic and connected. Write Gx, f for the corresponding smooth  the morphism Ua → G extends to model with connected fibers. For each a ∈ Φ, a closed immersion Ua, x, f → Gx, f . Upon taking the order of the factors in any way in the following product over Φ±,nd , we obtain the same o-smooth closed  subgroup scheme of G under the multiplication a ∈Φ±, nd Ua, x, f → G , and these subgroup schemes have unipotent fibers. Moreover, the product morphism   Ua, x, f × T f (0) × Ua, x, f → Gx, f a ∈Φ −, nd

a ∈Φ +, nd

is an open immersion. Before we give the proof of this theorem we collect a few lemmas. Lemma 8.5.3 Let Gi , i = 1, . . . , n and G be smooth group schemes over O. Assume given homomorphisms Gi → G such that the product morphism   i Gi → G is an open immersion. Then it induces a bijection i Gi (M) → G (M).  Proof Let U ⊂ G be the open subscheme that is the image of Gi . Let g ∈ G (M). Thus g : Spec(O) → G , when composed with Spec(f) → Spec(O), becomes the identity section of the special fiber of G . Since each Gi → G is a homomorphism of group schemes, g maps the closed point of Spec(O) into U . Lemma 8.1.1 implies that g factors through the inclusion U → G  hence pulls back to a morphism Spec(O) → i Gi . Its composition with Spec(f) → Spec(O) is still the identity section of the special fiber of the group     scheme i Gi . Therefore, it lies in ( i Gi )(M) = i Gi (M). In the following for an O-scheme X , its special fiber Xf will be denoted by X.  assume given connected schematic subgroups Lemma 8.5.4 For each a ∈ Φ Wa ⊂ Va ⊂ Ua (K), say with models Wa and Va . Assume the following. (1) Va (M) ⊂ Wa .  is schematic, say with (2) The subgroup V of G(K) generated by Va , a ∈ Φ, model V .

312

Integral Models   (3) The product morphism a0 Va → V is an open immersion, say with image V .

 is also schematic, say with Then the subgroup W generated by Wa , a ∈ Φ, model W . Assume in addition the following.   (4) The image of a0 Wa → V  equals W ∩ V .   Then the product morphism a0 Wa → W is an open immersion. Proof The morphism Ua → G extends to a morphism Wa → V by Corollary 2.10.10 and hence descends to a morphism between the special fibers Wa → V . The image of this morphism is Zariski closed and connected by [Bor91, Corollary 1.4(a)]. The group W (⊂ V ) generated by these images is also Zariski closed and connected and W (f) is the image of W in V (f) by [Bor91, Proposition 2.2]. By assumption and Lemma 8.5.3, V (M) ⊂ W. Therefore, Proposition A.5.23 implies that W is schematic. More precisely, Proposition A.5.23 states that W is the dilatation of W within V . Let W  be the dilatation of W ∩ V  within V . By Lemma A.5.11, W  is an open subscheme of W . Since dilatation commutes with products (Lemma A.5.6), our assumption implies that W  is isomorphic to the dilatation     of a0 Wa within a0 Va . By Proposi  tion A.5.23(2), this dilatation is precisely a0 Wa and the isomorphism between this product and W  is just the product isomorphism.  Lemma 8.5.5 Let X , X1, . . . , Xn be smooth affine O-group schemes of finite type equipped with morphisms Xi → X . Assume the following. (1) Each morphism Xi ×O k → X ×O k is a closed immersion. (2) For each i we have X (O) ∩ Xi (k) = Xi (O).  (3) The product morphism i Xi → X is an open immersion. Then each morphism Xi → X is a closed immersion. Proof Let Xi  be the schematic closure of Xi ×O k in X . Then Xi  is a closed subgroup scheme of X by Lemma A.2.3. By assumption the morphism Xi → Xi  is an open immersion. The smoothness of Xi implies the smoothness of Xi . By Lemma A.2.1 we have Xi (O) = Xi (k) ∩ X (O), which equals Xi (O) by assumption. Lemma 2.10.9 now implies that Xi → Xi  is an isomorphism.  Proof of Theorem 8.5.2 We apply Lemma 7.3.10 to assume f takes values in R. Next we apply Lemma 7.3.9 to f and let g(a) = f (a) + a, v . Then g is

8.5 Integral Models Associated to Concave Functions

313

concave and it takes only non-negative values. We have G(k)x, f = G(k)v+x, g by Lemma 6.1.6. We may therefore assume without loss of generality that f takes only non-negative real values.  there is a finite sequence We will induct on the values of f . For each a ∈ Φ 0 = ra,0 < ra,1 < · · · < ra, na = f (a) such that Ua, x,ra , i+1 (O)  Ua, x,ra , i (O) and for any 0 r  f (a) there exists i  na such that Ua, x,r (O) = Ua, x,ra , i (O). This implies that there is a finite sequence 0 = t0 < t1 < · · · < tn = 1 with the following property: for each a and each i there exists j such that Ua, x,ra , i (O) = Ua, x,t j f (O). In particular, we have for each a and i the inclusions Ua, x,ti f (M) ⊂ Ua, x,ti+1 f (O) ⊂ Ua, x,ti f (O). This allows us to proceed by induction. The base case is that of 0 f ≡ 0, which is handled by Theorems 8.3.13 and 8.2.5 applied to the facet F containing x. The induction step will be furnished by Lemma 8.5.4, applied with Wa = Ua, x,ti+1 f (= Ua, x,ti+1 f (O)) and Va = Ua, x,ti f (= Va, x,ti f (O)). The first three assumptions of this lemma are already verified, with V = Gx, ti f , and imply that W = G(k)x,ti+1 f is schematic with model W = Gx, ti+1 f . In order for the induction to proceed we need to verify the fourth assumption as well. Let W be the Zariski closure of the image of W in V (f) of W and let W ± and W 0  be the images of a ∈Φ±, nd W a and W 0 in W ∩ V . Proposition 7.3.12 implies that W + and W − are groups, normalized by W 0 . Let g be a f-rational point of W ∩ V . Consider the morphism W

−

×W 0×W

+

→ W ∩ V ,

(w −, w 0, w + ) → w − · g · w 0 · w + .

It induces an isomorphism between the tangent space at (1, 1, 1) on its source and the tangent space at g on its target. Therefore, its image contains a nonempty open subset. The group W is connected by [Bor91, Proposition 2.2], and so is its open subvariety W ∩ V . Therefore, for any two g1, g2 ∈ W ∩ V , the images of the two versions of above morphism have a non-trivial intersection. Taking g1 = g and g2 = 1 we see that g is a f-rational point of W − · W 0 · W + , as claimed. We have thus verified the fourth assumption of Lemma 8.5.4 and hence proved by induction that G(k)x, f is schematic and that the product morphism   Ua, x, f × T f (0) × Ua, x, f → Gx, f a ∈Φ −, nd

a ∈Φ +, nd

is an open immersion. That the morphism Ua, x, f → Gx, f is a closed immersion follows from Lemma 8.5.5, in which the assumption Gx, f (O) ∩ Ua (K) =  Ua, x, f (a) (O) follows from Proposition 7.3.12.

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Remark 8.5.6 The above theorem implies that Ua, x, f is the schematic closure of Ua in Gx, f , hence equal to the a-root group of Gx, f according to Proposition 2.11.9. We will now prove an analog of Theorem 8.4.10 for the group Gx, f . Write G x, f for the special fiber of Gx, f and G x, f for the reductive quotient of G x, f . We will denote the special fibers of S by S and the special fiber of the a-root group Ua, x, f by U a, x, f . Then U a, x, f is the a-root group of G x, f . Definition 8.5.7 (1) (2) (3) (4) (5)

 − {∞} be a concave function. Define →R Let f : Φ

 f = {a ∈ Φ  | f (a) + f (−a) = 0} . Φ  f ∩ Φ. Φf = Φ Ψ f = {ψ ∈ Ψ | ψ ∈ Φ f }.  Ψx, f = {ψ ∈ Ψ f | ψ(x) = f (ψ)}. Φx, f = ∇Ψx, f .

Definition 8.5.8

 − {∞} by →R Define the function f + : Φ ) f f (a)+, a ∈ Φ f + (a) = f . f (a), aΦ

Lemma 8.5.9 The function f + is concave. Proof Given the construction of f + and the concavity of f the only possible  a + b ∈ Φ,  failure of the concavity of f + would be the existence of a, b ∈ Φ, + + + such that f (a + b) = f (a) + f (b), f (a + b) > f (a + b), f (a) = f (a), f (b) = f (b). The last three of these four relations imply f (a + b) + f (−a − b) = 0, f (a) + f (−a) > 0, f (b) + f (−b) > 0. The concavity of f implies f (−a)  f (−a − b) + f (b). Combining these inequalities, we obtain the contradiction 0 < f (a) + f (−a)  f (a) + f (−a − b) + f (b) = f (a + b) + f (−a − b) = 0.  Since f +  f , we see that Ua, x, f + ⊂ Ua, x, f , and hence G(k)x, f + ⊂ G(k)x, f . These inclusions induces homomorphisms Ua, x, f + → Ua, x, f and Gx, f + → Gx, f of O-group schemes that are the identity on the common generic fiber Ua respectively G. The image of U a, x, f + → U a, x, f will be denoted by U a,+ x, f . Analogously, the image of T f + (0) → T f (0) will be denoted by T f+ . We will also set U0, x, f = T f (0) and U 0,+ x, f = T f+ for uniformity. In the following arguments, extra care has to be taken for roots a ∈ Φ for which 2a ∈ Φ. We have tried to formulate the arguments in a way that allows the reader who is only interested in the case that Φ is reduced to skip these parts.

8.5 Integral Models Associated to Concave Functions Lemma 8.5.10

315

Assume that a, 2a ∈ Φ.

(1) If a ∈ Φ f , then 2a ∈ Φ f , and f (2a) = 2 f (a), f (−2a) = 2 f (−a) = −2 f (a). (2) If 2a ∈ Φ f and a  Φ f , then 2 f (a) > f (2a) and 2 f (−a) > f (−2a). Proof (1) We have f (2a)  2 f (a), f (−2a)  2 f (−a), and 0  f (2a) + f (−2a) by concavity, and f (a) + f (−a) = 0 by assumption. In the resulting chain 0  f (2a) + f (−2a)  2 f (a) + 2 f (−a) = 0 all inequalities must be equalities, and the claim follows. (2) By assumption f (2a) + f (−2a) = 0 and f (a) + f (−a) > 0. By contradiction, assume (without loss of generality) that 2 f (a) = f (2a). Combining f (a) = 1 1 2 f (2a) = − 2 f (−2a) with the inequality f (−a)  f (−2a)+ f (a) implied by the concavity of f we conclude f (−a)  f (−2a) + f (a) = −2 f (a) + f (a) = − f (a), which contradicts f (−a) + f (a) > 0.  Lemma 8.5.11 Let a ∈ Φ. Let u ∈ Ua, x, f and let u ∈ U a, x, f be its image. Then u ∈ Ru (G x, f ) if and only if u ∈ Ua, x, f + . Proof According to Theorem 8.2.9, u does not lie in Ru (G )(f) if and only if the elements u , u  ∈ U−a (k) of Proposition 2.11.17 belong to G x, f (O) ∩ U−a (k). This intersection equals U−a, x, f by Proposition 7.3.12. Recall from Definition 6.1.2 that x−a (u ) = x−a (u ) = −xa (u). We consider two cases. The first case is when either 2a  Φ, or when 2a ∈ Φ and f (2a) = 2 f (a) and f (−2a) = 2 f (−a). Under these conditions we have Ua, x, f = Ua, x, f (a) and U−a, x, f = U−a, x, f (−a) . Therefore u  ∈ U−a, x, f is equivalent to x−a (u )  f (−a). We then have the chain − f (−a)  −x−a (u ) = xa (u)  f (a). On the other hand, the concavity of f implies f (a) + f (−a)  0, which in turn makes all inequalities in the above displayed chain into equalities, that is, − f (−a) = −x−a (u ) = xa (u) = f (a). Thus u does not lie in Ru (G )(f) if and only if a ∈ Φ f and xa (u) = f (a). The latter is equivalent to u ∈ Ua, x, f (a) −Ua, x, f (a)+ . The second case is when 2a ∈ Φ, and in addition f (2a)  2 f (a) or f (−2a)  2 f (−a). By concavity we thus have f (2a) < 2 f (a) or f (−2a) < 2 f (−a). By concavity we also have f (2a) + f (−2a)  0. These inequalities imply f (a) + f (−a) > 0. In particular, f + (a) = f (a) and f + (−a) = f (−a). We now claim that U a, x, f (a) , which is a possibly proper subgroup of U a, x, f , lies in Ru (G x, f ). Admitting this claim (which, by symmetry, can be used for both a and −a), we complete the proof of the lemma as follows. If u ∈ U a, x, f (f) lies in

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Ru (G )(f), then writing u = u1 · u2 according to Ua, x, f = Ua, x, f (a) · U2a, x, f (2a) we know from the claim u1 ∈ Ru (G )(f) and conclude that u2 ∈ Ru (G )(f). From the previously proved case applied to 2a ∈ Φ we conclude u2 ∈ U2a, x, f + (2a) , and hence u ∈ Ua, x, f + . Conversely, if u ∈ U a, x, f (f) does not lie in Ru (G )(f), then writing u  = u1 · u2 according to U−a, x, f = U−a, x, f (−a) · U−2a, x, f (−2a) and applying Theorem 8.2.9 to u , we see u   Ru (G )(f). By the claim, u1 ∈ Ru (G )(f), so we conclude u2  Ru (G )(f). Apply the previous case to the root −2a to conclude that u2 ∈ U−2a, x, f (−2a) − U−2a, x, f + (−2a) , equivalently f (−2a)  f + (−2a) and x−2a (u2 ) = f (−2a). By Lemma 8.5.10(2) we have x−a (u2 ) = 12 x−2a (u2 ) = 12 f (−2a) < f (−a), while x−a (u1 )  f (−a). Remark 6.1.5 implies x−a (u ) = x−a (u2 ) and hence xa (u) = −x−a (u ) = − 12 f (−2a) = 1 1 + + 2 f (2a) < 2 f (2a), hence u ∈ Ua, x, f − Ua, x, f , as claimed. It remains to prove the claim. Take u ∈ Ua, x, f (a) and assume by way of contradiction that u  Ru (G )(f). Then u  ∈ U−a, x, f ⊂ U−a,x, 1 f (−2a) , hence xa (u ) 

1 2

2

f (−2a). We obtain 1 1 − f (−2a)  −xa (u ) = xa (u)  f (a)  f (2a). 2 2

Again, combining this with f (2a) + f (−2a)  0 implies that all inequalities in the chain are equalities. This contradicts Lemma 8.5.10(2).  Corollary 8.5.12 (1) The open immersion of Theorem 8.5.2 restricts to an isomorphism of affine schemes   U a,+ x, f × T f+ × U a,+ x, f → Ru (G x, f ), a ∈Φ −, nd

a ∈Φ +, nd

upon taking the products over Φ −,nd and Φ +,nd in any order. (2) If f (0) > 0, then G x, f is unipotent, and the open immersion of Theorem 8.5.2 is an isomorphism of affine schemes   U a, x, f × T f × U a, x, f → G x, f , a ∈Φ −, nd

a ∈Φ +, nd

upon taking the products over Φ −,nd and Φ +,nd in any order. Proof Lemma 8.5.11 implies U a, x, f ∩ Ru (G x, f ) = U a,+ x, f for any a ∈ Φ. On the other hand, Ru (T f ) equals T f+ according to Lemma B.10.11. The first claim follows from Proposition 2.11.3. If f (0) > 0, then for any a ∈ Φ the concavity of f implies f (a) + f (−a)   We conclude f (0) > 0. Therefore f = f + , so U a,+ x, f = U a, x, f for all a ∈ Φ. that the image of the open immersion of Theorem 8.5.2, which is a Zariski-open

8.5 Integral Models Associated to Concave Functions

317

subset of G x, f , is contained in the Zariski-closed subgroup Ru (G x, f ). Thus  Ru (G x, f ) = G x, f . Corollary 8.5.13 Under the surjective map G(K)x, f = Gx, f (O) → G x, f (f) the preimage of Ru (G x, f )(f) equals G(K)x, f + . Proof Corollary 8.5.12 implies that this preimage is equal to the product over  of the preimage of U + (f) in Ua, x, f (O) = Ua, x, f . For a ∈ Φ all a ∈ Φ a, x, f Lemma 8.5.11 implies that this preimage is Ua, x, f + . The same follows for a = 0 from Proposition B.10.12. The claim now follows from Proposition  7.3.12, using that f + (0) > 0. Theorem 8.5.14 Assume f (0) = 0 and let Gx, f = G x, f /Ru (G x, f ) be the maximal reductive quotient of G x, f . (1) The closed immersion S → Gx, f identifies S with a maximal torus of Gx, f and the special fiber S of S with a maximal torus of G x, f . (2) The centralizer of S in Gx, f is T . The centralizer of S in G x, f is T . (3) The set Φx, f is a reduced root system. Let S be the image of S in G x, f . The root datum of G x, f with respect to the maximal torus S is (X∗ (S), Φx, f , X∗ (S), Φ∨x, f ). (4) For each b ∈ Φx, f let a ∈ Φ be the unique non-divisible root of which b is a positive multiple. Then the b-root group of G x, f is canonically isomorphic to U a, x, f /U a,+ x, f . Proof The connected Néron model T 0 of T is embedded into Gx, f as a closed subgroup scheme by Theorem 8.5.2. The natural inclusion S → T extends to a closed immersion S → T 0 by Lemma B.7.11. The resulting closed immersion S → Gx, f is an integral extension of the natural inclusion S → G. We use it to identify S with a closed O-torus of Gx, f . This proves (1), and (2) follows from Proposition 8.2.4. We come to (3) and (4). Theorem 8.5.2 implies that the set of roots of G x, f (with respect to S) is contained in Φ and for a ∈ Φ the a-root group (that is, the connected unipotent subgroup normalized by S and whose Lie algebra is the sum of weight spaces for weights  a) is the image of U a, x, f . Lemma 8.5.11 implies further that this a-root group equals U a, x, f /U a,+ x, f . It is therefore enough to check that this quotient is non-trivial if and only if there is a positive integral multiple b ∈ Φx, f of a, and that the weight of the S -action on the Lie algebra of U a, x, f /U a,+ x, f equals b. By Lemma 8.4.1 this quotient is non-trivial if and only if Ua, x, f +  Ua, x, f . Assume first that 2a  Φ. Then Ua, x, f + (a) = Ua, x, f +  Ua, x, f = Ua, x, f (a) is equivalent to a ∈ Φ f and the existence of ψ ∈ Ψa with ψ(x) = f (a). Since

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Integral Models

2a  Φ, 2ψ  Ψ and Proposition 6.3.8 implies ψ ∈ Ψ. Hence Ua, x, f +  Ua, x, f is equivalent to a ∈ Φx, f . Assume now that 2a ∈ Φ. Note first that {a, 2a} ∩ Φx, f has at most one element: if both a, 2a ∈ Φx, f , then f (2a) = 2 f (a) by Lemma 8.5.10(1), and letting ψ1, ψ2 ∈ Ψ be such that ψ i = i · a and ψi (x) = f (ψ i ) we see ψ2 = 2ψ1 , contradicting the reducedness of Ψ of Proposition 6.4.8. Next, arguing as for the case 2a  Φ we see that U2a, x, f + (2a)  U2a, x, f (2a) (which implies Ua, x, f +  Ua, x, f ) is equivalent to 2a ∈ Φx, f . On the other hand, if U2a, x, f + (2a) = U2a, x, f (2a) and Ua, x, f + (a)  Ua, x, f (a) , then a ∈ Φ f and there exists ψ ∈ Ψa such that ψ(x) = f (a). Lemma 8.5.10(1) implies that f (2a) = 2 f (a), hence Ua, x, f + = Ua, x, f + (a)  Ua, x, f (a) = Ua, x, f . The assumption U2a, x, f + (2a) = U2a, x, f (2a) shows that 2ψ  Ψ2a and Proposition 6.3.8 implies that ψ ∈ Ψa . Therefore a ∈ Φx, f and 2a  Φx, f . This argument is reversible. We have thus proved that Ua, x, f +  Ua, x, f if and only if {a, 2a} ∩ Φx, f  ∅, and have moreover computed that intersection. At the same time, the quotient U a, x, f /U a,+ x, f , when non-trivial, is non-canonically isomorphic to Ga , being the root group of the connected reductive group Gx, f over an algebraically closed field. The weight of the action of S on the Lie algebra of U a, x, f /U a,+ x, f is precisely the unique element of {a, 2a} ∩ Φx, f .  Remark 8.5.15 We can describe the Weyl group and the spherical building of G x, f in terms of the affine building B in a way similar to the description in the special case f = 0 given in Theorem 8.5.14. For this it is most convenient to use Lemmas 7.3.10, 7.3.9, and Fact 7.3.8 to reduce to the case that f takes  | f (a) = 0 = f (−a)} and  f = {a ∈ Φ only non-negative real values. Then Φ Φx, f = Φx ∩ Φ f . Let us assume Φ f  ∅, for otherwise G x, f is solvable. It is easily seen that Φ f is a closed subsystem of Φ. Then Ψ f = {ψ ∈ Ψ | ψ ∈ Φ f } is a reduced affine root system in (A/Vf )∗ , where Vf ⊂ VT is the space annihilated by Φ f . This affine root system decomposes A/Vf into a disjoint union of facets. Proposition 1.3.35 states that the Weyl group of Φx ∩ Φ f is the stabilizer of x in the Weyl group of the affine root system Ψ f . Let F f be the facet in A/Vf containing the image of x. The link of that facet (that is, the set of facets of A/Vf whose closure contains F) is in bijection with the set of parabolic subgroups of G x, f that contain S . The proof is similar to that given for Theorem 8.4.19 and is left to the reader. Before stating the next proposition, we recall the convention ω(k × ) = Z and the notation M for the maximal ideal of O.

8.5 Integral Models Associated to Concave Functions

319

Proposition 8.5.16 For any integer n  0, the nth congruence group scheme (Gx, f )n equals Gx, f +n . In particular, Gx, f (Mn ) = Gx, f +n (O). Proof The second statement follows from the first and Lemma A.5.13. To prove the first statement we may assume n = 1, since the case n > 1 follows by induction and the case n = 0 is trivial. By Corollary 2.10.11 it is enough to prove Gx, f (M) = Gx, f +1 (O). We have Gx, f +1 (O) = G(k)x, f +1 =   a ∈Φ+, nd Ua, x, f +1 × T f (0)+1 × a ∈Φ−, nd Ua, x, f +1 by Proposition 7.3.12(4)(5). On the other hand, Lemma 8.5.3 and Theorem 8.5.2 show that Gx, f (M) is equal   to a ∈Φ+, nd Ua, x, f (M) × T f (0) (M) × a ∈Φ−, nd Ua, x, f (M). Since the filtration on T(k) is congruent, we have T f (0) (M) = T f (0)+1 (O), while Ua, x, f (M) =  Ua, x, f +1 follows from Proposition C.5.1.  − {∞} be concave functions such that f (a)  g(a) →R 8.5.17 Let f , g : Φ  Then G(K)x, g ⊂ G(K)x, f and Corollary 2.10.10 shows that for all a ∈ Φ. the identity of G extends to a homomorphism ρ f , g : Gx,g → Gx, f of group schemes. Denote by ρ f , g : G x,g → G x, f the homomorphism between special fibers. Proposition 8.5.18 The kernel of ρ f , g is a smooth connected unipotent subgroup scheme and the multiplication map induces an isomorphism of f-schemes  ker(U a, x,g → U a, x, f ) → ker(ρ f , g ).  a ∈Φ

Proof The proof is the same as for Proposition 8.4.15, where now in addition to Lemma C.5.3 we use Proposition 8.4.15.  Let M ⊂ G be a Levi subgroup, S ⊂ M a maximal split torus, x a valuation of the root datum for S and G. Recall from §6.7 the valuation π(x) of the root datum for S and M obtained from x by “restriction.” Lemma 8.5.19

 − {∞} be a concave function. Then  G) → R Let f : Φ(S, M(K)π(x), f = M(K) ∩ G(K)x, f .

Proof Let T be the centralizer of S in G. Then T is a maximal torus of G contained in M. Let AM ⊂ S be the maximal split torus in the center of M. Let A M and S be the standard models of AM and S, respectively, cf. §B.2. The inclusion AM → S extends to a closed immersion A M → S according to Lemma B.4.7. The morphism S × G → G given by the conjugation action of S on G restricts to a homomorphism S(K)0 × G(K)x, f → G(K)x, f and hence induces a morphism of schemes S × Gx, f → Gx, f by Corollary 2.10.10. Let M be the fixed point scheme for the action of A M on Gx, f . Then M is a

320

Integral Models

smooth closed subscheme of G with generic fiber M by Proposition 2.11.5, in particular an integral model of M. Since M is flat, it is the schematic closure in Gx, f of its generic fiber, and we have M (O) = M(K) ∩ G(K)x, f , cf. Lemma A.2.1. Let T be the fixed point scheme for the action of S on Gx, f . Then T (O) = T(K) ∩ G(K)x, f = T(K) f (0) by Lemma 7.3.16. For a ∈ Φ(M, S) let Ua be the a-root group for the action of S on M . Then Ua is the schematic closure of Ua in M , cf. Proposition 2.11.9, equivalently in Gx, f , since M is closed in Gx, f . Thus Ua (O) = Ua (K) ∩ G(K)x, f = Ua, x, f by Proposition 7.3.12. By Proposition 8.2.8 the group M (O) is generated by T (O) = T(K) f (0) and Ua (O) = Ua, x, f for all a ∈ Φ(M, S), hence equal to M(K)π(x), f .  8.5.20 Description of the Lie algebra Consider given a concave function  − {∞}. The Lie algebra gx, f of the group scheme Gx, f is  G) → R f : Φ(S, an O-lattice in the Lie algebra g of the K-group G. Theorem 8.5.2 gives the isomorphism of O-modules * * u a, x, f ⊕ t f (0) ⊕ u a, x, f → gx, f , a ∈Φ −, nd

a ∈Φ +, nd

where u a, x, f is the Lie algebra of Ua, x, f and t f (0) is the Lie algebra of T f (0) . According to Proposition 8.5.16 and Lemma A.5.13, gx, f +n = Mn · gx, f and the above displayed direct sum decomposition is compatible with this identity. When T is weakly induced in the sense of Definition B.6.2, we have the explicit description e f (0) Gal(L/K)

t f (0) = {X ∈ t(K) | ∀ χ ∈ X∗ (T), ω( χ(X))  f (0)} = (X∗ (T) ⊗Z M L

)

of Lemma B.6.7 and Remark B.6.8, where L/K is any finite tamely ramified Galois extension such that TL is induced, and e is the (ramification) degree of L/K. When G is simply connected or adjoint, then T is induced, and the description of t f (0) simplifies further. For example, when G is simply connected, let Δ be a system of simple roots for the absolute root system Φ(T, G). Choosing a set an } for the orbits of Gal(L/K) in Δ we obtain the of representatives { a1, . . . ,  isomorphism n * e f (0) Mi i → t f (0), i=1

where Mi is the maximal ideal of the ring of integers of the field extension ai in Gal(L/K), ei Ki /K that is the fixed subfield of L of the stabilizer of  is the (ramification) degree of Ki /K, and the map sends z ∈ Mi to the sum i ). σ ∈Gal(L/K)/Gal(L/Ki ) σ(z · Ha To describe the lattices u a, x, f , the identity u a x, f = u a, x+v, f +v of Lemma

8.6 Passage to Completion

321

6.1.6 allows us to assume that x is a Chevalley valuation. Let {Xa} be a corresponding Chevalley basis. The discussion of C.5.4 then shows that if a is of type R1,  a ∈ Φ(T, G) is a lift of a, Ka/K the associated extension, Ma the maximal ideal in the ring of integers, ea the (ramification) degree, and r  = inf{h ∈ e−1 a Z| h  f (a)}, then   Meaa r → u a, x, f , z → σ(z · Xa) σ ∈Gal(L/K)/Gal(L/K a )

is an isomorphism of O-modules. On the other hand, if a is of type R2, we fix lifts  a,  a  ∈ Φ(T, G) such that  b =  a+ a  ∈ Φ(T, G). Let Ka and Kb be the corresponding fields, so that Ka/Kb is a quadratic Galois extension, whose (ramification) degree we denote by ea/2a , and whose non-trivial automorphism we denote by σa . Let Ma and Mb be the maximal ideals in the rings of integers of these two fields. Let e be the (ramification) degree of Ka/K. Let r  = inf{h ∈ Ja/2a | h  f (a) − 21 μa } and t  = inf{h ∈ J2a | h  min(2 f (a), f (2a)) − μa }, then an isomorphism of O-modules 

Mer ) a  ⊕ (λ)(Mb

t  −e−1 a/2a

→ u a, x, f

is given by (z1, z2 ) →

⎧ ⎪ ⎨ ⎪





⎫ ⎪ ⎬ ⎪

σ(z1 · Xa), σ(z2 · Xb) , ⎪ ⎪ ⎪ σ ∈Gal(Ks /K)/Gal(Ks /K a ) ⎪ σ ∈Gal(K /K)/Gal(K /K ) s s  b ⎩ ⎭

where λ ∈ Ka is an element of valuation μa satisfying λ + σa (λ) = 1 and  ∈ Ka is a uniformizing element satisfying λ + σa (λ) = 0.

8.6 Passage to Completion  be the completion of O. Recall  of K. Let O Consider again the completion K from Proposition 7.10.2 that there is a natural isomorphism B(G K ) → B(G K ) of polysimplicial complexes. Proposition 8.6.1 Let Ω be a bounded subset of an apartment of B(G K ) and  be its image in B(G  ). The Bruhat–Tits group scheme G ∗ with generic let Ω K  Ω fiber G K associated to Ω, where ∗ ∈ {0, 1, †}, is obtained by base change along  from the Bruhat–Tits group scheme G ∗ . O → O Ω Proof The proof for the different choices of ∗ is the same, so take ∗ = 1 as an  Proposition 2.3.4(2)  1 be the base change of G 1 along O → O. example. Let G Ω Ω

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Integral Models

 is open and closed in G(K)  Since G  1 (O)  1 (O). 1 implies that GΩ1 (O) is dense in G Ω Ω 1 1 and contains the pointwise stabilizer in G(K) of Ω, which is GΩ (O) according  ⊂ G(K)  1 (O)  1 is to Proposition 7.7.5, as a dense subgroup, we conclude that G Ω  1 . By Proposition 7.7.5 again, as well as the pointwise stabilizer of Ω in G(K) Corollary 2.10.11, the proof is complete. 

9 Unramified Descent

We recall some assumptions and notation. We assume that k is a field equipped with a discrete valuation ω, normalized by ω(k × ) = Z. The valuation ring o of ω is assumed to be Henselian. In this, and the subsequent chapters, we assume that the residue field f of k is perfect. Let K be the maximal unramified extension of k contained in a fixed separable closure k s of k. We will denote the unique valuation of a fixed algebraic closure k of k containing k s , extending the given valuation of k, also by ω. The valuation ring of K will be denoted by O. The residue field f of K is an algebraic closure of f. We identify Gal(K/k), Aut(O/o) and Gal(f/f) with each other, and denote each of them by Γ. For a subset Y of a set Z given with an action of Γ, we will denote by Y Γ the set of elements of Y fixed under Γ. Throughout this chapter, G will denote a connected reductive k-group. This chapter is devoted to deriving Bruhat–Tits theory for a general connected reductive k-group G from this theory for the group G K . More precisely, assuming that Bruhat–Tits theory is available for G K in the sense of Definition 4.1.26, we will prove that the same is true for the group G, and we will prove that Axiom 4.1.27 holds. Since we are assuming that the residue field f of k is perfect, Corollary 2.3.8 implies that the group G K is quasi-split. The results of Chapters 6, 7, and 8 imply that Bruhat–Tits theory is available for G K . The process of deriving Bruhat–Tits theory for G from that for G K is called the unramified descent (“descente étale” in French) and is originally due to Bruhat and Tits ([BT72, §9] and [BT84a, §5]). For this, they used descent of a valuation of the root datum from K to k: given a valuation of the root datum of G K , they proved the existence of a valuation of the root datum of G. The material laid out in Chapters 6 and 7 then produces the desired structures – apartments, the building, etc. Our approach here is different. It was developed in a recent paper [Pra20b] 323

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Unramified Descent

and appears to be conceptually simpler and more geometric, even for reductive groups over locally compact non-archimedean fields. It does not use descent of valuation of the root datum from K to k. Instead, it relies on the geometry of the building for G K and the structure of the associated group schemes to produce directly the building for G and its apartments. Once this is done and the main results of Bruhat–Tits theory for G have been derived from those for G K , we will use these results to obtain a valuation of the root datum of G over k in §9.6. The material in Chapters 6 and 7 can then be used to provide a different interpretation of the building for G/k and its apartments. We mention here that in [Pra20b] unramified descent is given without the assumption that the residue field f is perfect. However, if the residue field is not perfect, G K may not be quasi-split, so in [Pra20b] it is assumed that Bruhat–Tits theory is available for G K .

9.1 Preliminaries We will denote the derived subgroup (G, G) of G by Gder . It is the maximal connected normal semi-simple subgroup of G. The group of k-rational characters on G will be denoted by X∗k (G). Let S be a maximal k-split torus of G, Z its centralizer in G and Zder = (Z, Z) the derived subgroup of Z. Then Zder is a connected semi-simple group which is anisotropic over k since S is a maximal k-split torus of G. Hence, by Theorem 2.2.9, Zder (k) is bounded, and so according to Proposition 2.2.12, Z(k) contains a unique maximal bounded subgroup Z(k)b . This subgroup admits the following description: Z(k)b = {z ∈ Z(k) | χ(z) ∈ o× for all χ ∈ X∗k (Z)}. The restriction map X∗k (Z) → X∗k (S) is injective and its image is of finite index in X∗k (S). Let X∗ (S) = Homk (Gm, S) and V(S) = R ⊗Z X∗ (S). Let ν : Z(k) → V(S) be the negative of the valuation homomorphism (2.6.4). Thus χ(ν(z)) = −ω( χ(z)) for z ∈ Z(k) and χ ∈ X∗k (Z)(→ X∗k (S)), and Z(k)b is the kernel of ν. As the image of ν is isomorphic to Zr , r = dim S, we conclude that Z(k)/Z(k)b is isomorphic to Zr . For a smooth connected linear algebraic group H defined over a perfect field F, its unipotent radical (that is, the maximal smooth connected normal unipotent F-subgroup) will be denoted by Ru (H). The quotient H red := H/Ru (H) is reductive; it is the maximal reductive quotient of H.

9.2 Statement of the Main Result

325

For a k-variety X and a field extension  of k, X will denote the -variety obtained from X by base change k → . 0 be the maximal K-split torus contained in the center of G. This torus Let Z G 0 for the maximal torus in the center of is defined over k. Indeed, if we write ZG 0 is defined over k since G is. We have X ( Z 0 0 Gal(k s /K) G, then ZG ∗ G ) = X∗ (ZG ) and the claim follows from the normality of Gal(k s /K) in Gal(k s /k). 0 ×k K. There is a natural action of the Galois group 0 = Z Let us write Z G,K G 0 ) and HomK (Gm, Z 0 )Γ = Homk (Gm, Z 0 ). Let Γ of K/k on HomK (Gm, Z G,K

G,K

G

0 ) = R ⊗Z HomK (Gm, Z 0 ). The action of Γ on HomK (Gm, Z 0 ) exV( Z G,K G,K G,K 0 ), and V( Z 0 )Γ = R ⊗Z Homk (Gm, Z 0 ). tends to an R-linear action on V( Z G,K

G,K

G

There is a natural bijective correspondence between the set of maximal K-split 0 T. tori of (Gder )K and the set of maximal K-split tori of G K given by T → Z G,K Recall the normal subgroup G(K)1 of G(K) consisting of elements that act 0 ). This subgroup contains Gder (K) and also every bounded trivially on V( Z G,K subgroup of G(K), and has the following description: G(K)1 = {g ∈ G(K) | χ(g) ∈ O× for all χ ∈ X∗K (G K )}.

9.2 Statement of the Main Result In this section we describe the building of G and all relevant structures, such as apartments, polysimplices, chambers, etc. In the following §9.3 we will prove that these structures do indeed satisfy all the axioms stated in §4.1 and therefore Bruhat–Tits theory is available for G in the sense of Definition 4.1.26. 9.2.1 Action of the Galois group of K/k on the building Chapter 7 provides a building B(G K ) for G K . Proposition 7.9.5 endows B(G K ) with an action of Γ by building automorphisms such that for all g ∈ G(K), x ∈ B(G K ), γ ∈ Γ, we have γ(g · x) = γ(g) · γ(x). Thus there is an action of Γ  G(K) on B(G K ),  K ) = V( Z 0 ) × B(G K ) considered in as well as on the enlarged building B(G G,K §4.3. For any apartment A of B(G K ), and γ ∈ Γ, γ(A) is an apartment and the action map A → γ(A) is affine. Let T be a maximal K-split torus of G K with corresponding apartment AT and let k /k be a finite unramified extension over which T is defined and split. We claim that AT is fixed pointwise by Gal(K/k ). Indeed, using Proposition 4.4.3 it is enough to show that Gal(K/k ) acts trivially 0 ) by the Weyl R . This group is an extension of V = R⊗Z X∗ (T/ Z on the group W G group W = NG (T)(K)/ZG (T)(K). We know that Gal(K/k ) acts trivially on R and σ ∈ Gal(K/k ) we have vσ := x −1 σ(x) ∈ V. both V and W. For any x ∈ W

326

Unramified Descent

Since H1 (Gal(K/k ),V) = 0 there exists v ∈ V such that vσ = v −σ(v), therefore R is fixed by Gal(K/k ). But since Gal(K/k ) fixes all of V, it must x+v ∈W also fix x, and the claim is proved. If TK is defined over k, then AT is stable under the action of Γ, and Γ acts on AT by affine transformations through the finite quotient Gal(k /k) for any finite unramified extension k /k splitting T. We claim that the orbit of each point x ∈ B(G K ) under Γ is finite. To see this, choose a maximal K-split torus T in G with apartment A and g ∈ G(K) such that gx ∈ A. There exists a finite unramified extension k /k such that T is defined and split over k  and g ∈ G(k ). Since the action of Gal(K/k ) on B(G K ) fixes A pointwise, the orbit map of Γ through gx thus factors through Gal(k /k). The finiteness of the orbits of Γ on B(G K ) implies, via Corollary 4.2.12, that  K )Γ = V( Z 0 )Γ × B(G K ) contains a point fixed under Γ. It is obvious that B(G G,K  K ) sending x to (0, x) is equivariant B(G K )Γ . The embedding ι : B(G K ) → B(G

under ΓG(K) and provides us a G(k)-equivariant embedding ιΓ : B(G K )Γ →  K )Γ . B(G

9.2.2 Special k-tori and special k-apartments A special k-torus of G is a k-torus T (⊂ G) that contains a maximal k-split torus of G and TK is a maximal K-split torus of G K . We will prove below that G contains special k-tori (Proposition 9.3.4). 0 · T  is a special k-torus of G, and Given a special k-torus T  of Gder , Z G 0 · T  is a bijective correspondence between the sets of special k-tori T  → Z G of Gder and G.  K ), corresponding to TK , for a special The apartment of B(G K ), or of B(G k-torus T, will henceforth be called a special k-apartment corresponding to the (special) k-torus T. Since the actions of Γ on B(G K ) and G(K) are compatible, the bijection between maximal K-split tori of G K and apartments in B(G K ) implies that every special k-apartment is stable under the action of Γ. If x  y are two points of a Γ-stable apartment A which are fixed under Γ, then the whole straight line in A passing through x and y is pointwise fixed under Γ. 9.2.3 The building of G and its apartments We will denote B(G K )Γ by B in what follows. Then B is closed and convex and is stable under the action of G(k) on B(G K ). We equip it with the metric restricted from B(G K ). One of our main goals is to show that B is a building. To that end, we define the apartments of B to be A = AΓ for special k-apartments A of B(G K ).

9.2 Statement of the Main Result

327

If T is a special k-torus containing the maximal k-split torus S of G, and A is the apartment of B(G K ) corresponding to T, we will see in Proposition 9.3.16 that the actions of N(k) and V(S) on A preserve A and fulfill Axiom 4.1.4. Proposition 9.3.17 will show that the apartments of B are in bijective correspondence with maximal k-split tori of G. 9.2.4 Polysimplicial structure on B = B(G K )Γ We will call a facet F of the building B(G K ) a K-facet, to distinguish it from the facets of B that we are about to define. A K-facet that meets B will be called a Γ-facet. If a K-facet is stable under the action of Γ, then its barycenter is fixed under Γ and hence belongs to B. Conversely, if a K-facet F contains a point x of B, then being the unique facet containing x, F is stable under the action of Γ. Thus Γ-facets are precisely the Γ-invariant K-facets. A Γ-facet that is maximal among all Γ-facets will be called a Γ-chamber, and a Γ-facet that is minimal among all Γ-facets will be called a Γ-vertex. A Γ-chamber may not be a K-chamber. In other words, there may not exist a Γinvariant K-chamber. This is related to the notion of “residual quasi-splitness”, cf. Definition 9.10.2. We will prove that a Γ-chamber is always a K-chamber provided dim(f)  1, cf. Proposition 9.10.4. A Γ-vertex need not be a K-vertex. In other words, there may not exist a Γ-invariant K-vertex. There are many examples of this, even when dim(f)  1, cf. Example 9.2.9. However, when dim(f)  1 and a Γ-fixed special K-vertex does exist, then G must be quasi-split, cf. Proposition 10.2.1. We define the facets of B, which we may also call the k-facets, to be the subsets F := F ∩ B for Γ-facets F of B(G K ). For a facet F = F ∩ B of B, the closure F of F equals F ∩ B. Indeed, the inclusion F ⊂ F ∩ B is obvious. To see the opposite inclusion we observe that F is just the union of F and all its faces. Now fix a point x in F, and for any point y ∈ B lying in a proper face of F, consider the geodesic [x, y] joining x to y. The half-open geodesic [x, y) is entirely contained in F and y is a limit point of [x, y), and the claim is proved. We will use the ordering defined in Notation 1.3.34 for the facets of B. The maximal facets will be called chambers, and minimal facets will be called vertices. We will show in Proposition 9.3.10 that the facets defined here make B into a polysimplicial complex. A priori it is clear that B is the disjoint union of its facets, since this is true for B(G K ). 9.2.5 Descent of Bruhat–Tits group schemes Given a non-empty bounded subset Ω of an apartment of B(G K ), Axiom 4.1.20 provides affine models GΩ1 and GΩ0 of G, uniquely determined by the conditions GΩ1 (O) = G(K)1Ω and GΩ0 (O) = G(K)0Ω , cf. Corollary 2.10.11. The relative identity component of GΩ1

328

Unramified Descent

is GΩ0 . Assuming that Ω is stable under the action of Γ, the O-group schemes G Ω1 and GΩ0 admit unique descents to smooth affine o-group schemes with generic fiber G according to Fact 2.10.16; the coordinate rings of these descents are (O[G Ω1 ])Γ = O[GΩ1 ] ∩ k[G] and (O[GΩ0 ])Γ = O[GΩ0 ] ∩ k[G]. As it is unlikely to cause confusion, henceforth, whenever Ω is stable under Γ, we will write G Ω1 and GΩ0 for these smooth affine o-group schemes, and denote the special fibers of these group schemes by G Ω1 and G Ω0 respectively. Then G 1Ω is a smooth affine f-group and G Ω0 is its identity component. The maximal reductive quotient of G Ω0 will be denoted by G Ω . 1 , G0 , G0 and For a point x ∈ B(G K ) fixed under Γ, we will denote G {x } {x } {x }

G {x } by G x1 , G x0 , G x0 and G x respectively.

Let T be a k-torus of G such that TK is a maximal K-split torus of G K . Let Ω be a non-empty bounded subset of the apartment of B(G K ) corresponding to TK . We assume that Ω is stable under the action of Γ. Then the O-torus T of Axiom 4.1.20(3) admits a unique descent to a closed o-torus of GΩ0 ; from now on we will denote this o-torus also by T . The generic fiber of T is T, its special fiber T is a maximal f-torus of G Ω0 , and GΩ0 (o) ∩T(k) is equal to T (O)Γ and is thus the maximal bounded subgroup of T(k), equivalently the Iwahori subgroup of T(k), cf. Lemma 2.5.18. If the k-torus T contains a maximal k-split torus S of G, then the generic fiber of the maximal o-split subtorus S of T is S and the special fiber S (⊂ T ) of S is a maximal f-split torus of G Ω0 . In addition to GΩ0 and GΩ1 , we constructed in §8.3 the integral models GΩb , GΩ† , and (the usually non-affine) GΩ . In the same way, these O-group schemes descend to o when Ω is Γ-stable. This is in particular the case when Ω lies in an apartment A of B and is thus Γ-fixed. In that case it is immediate from Proposition 7.7.5 and the relation G(k)1 = G(K)1 ∩ G(k) that GΩ1 (o) is the pointwise stabilizer of Ω in G(k)1 , GΩ† (o) is the stabilizer of Ω in G(k)1 , GΩ (o) is the stabilizer of Ω in G(k). The relation G(k)0 = G(K)0 ∩G(k) will be proved in Proposition 9.3.24 and will likewise imply, via Proposition 7.7.5, that GΩ0 (o) is the pointwise stabilizer of Ω in G(k)0 . On the other hand, it is not true that G(k)b = G(K)b ∩ G(k), cf. Example 2.6.31. Given a Γ-facet F of B(G K ) and F = F Γ , we have G(K)0F = G(K)0F and hence GF0 = G F0 . On the other hand, the identity G(K)1F = G(K)1F is generally false, cf. Example 9.2.9 and Remark 7.7.7. As the subset of points of B(G K ) fixed under GF0 (O) = GF0 (O) is F (Axiom 4.1.20 and 4.1.12), the subset of points of B fixed under GF0 (O) is F ∩ B = F. Using the description of parabolic f-subgroups of G F up to conjugacy, we

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will see (9.3.2) that the f-rank of the derived subgroup of G F is equal to the codimension of F = F ∩ B in B. 9.2.6 Parahoric and Iwahori subgroups of G(k) For x ∈ B, we will refer to the o-group scheme Gx0 with connected fibers, described in 9.2.5, as the Bruhat–Tits parahoric o-group scheme, and to Px := Gx0 (o) as the parahoric subgroup of G(k), both associated to the point x. If F = F ∩ B is the facet of B containing x, then Gx0 = GF0 = GF0 . The generic fiber of Gx0 is G, and the subgroup Gx0 (o) = GF0 (O)Γ = GF0 (O) ∩ G(k) of G(k) fixes F pointwise. Since F is the unique facet of B(G K ) containing F, the stabilizer of F also stabilizes F. But GF0 (O) is of finite index in the stabilizer of F in G(K)1 . Therefore, Gx0 (o) = GF0 (o) is of finite index in the stabilizer of F in G(k)1 . For a Γ-chamber C of B(G K ), let C = C Γ denote the corresponding chamber of B. The subgroup GC0 (o) is then a minimal parahoric subgroup of G(k), and all minimal parahoric subgroups of G(k) arise this way. The minimal parahoric subgroups of G(k) are called the Iwahori subgroups. We will see in Proposition 9.3.23 that Gder (k) acts transitively on the set of Iwahori subgroups of G(k), and hence also on the set of chambers of B. Let P be a parahoric subgroup of G(K) which is stable under the action of Γ on G(K), then the facet F of B(G K ) corresponding to P is Γ-stable, that is it is a Γ-facet. Let F = F ∩ B be the corresponding facet of B, and GF0 be the associated o-group scheme with generic fiber G and with connected special fiber. Then GF0 (o) = GF0 (O)Γ = P Γ is a parahoric subgroup of G(k). Thus the parahoric subgroups of G(k) are the subgroups of the form P Γ , for Γ-stable parahoric subgroups P of G(K). The following is the main theorem on unramified descent and it will be proved in this chapter. Theorem 9.2.7 The set B = B(G K )Γ equipped with the apartments and polysimplicial structure described above is a building. Its apartments are in bijective correspondence with the maximal k-split tori of G. Its chambers are C Γ = C ∩ B for Γ-chambers C of B(G K ), and its facets are F Γ = F ∩ B for Γ-facets F of B(G K ). The group G(k) acts on B by polysimplicial isometries. This building, and the group schemes of 9.2.5, fulfill Axioms 4.1.1, 4.1.4, 4.1.6, 4.1.9, 4.1.17, 4.1.20, and 4.1.22; thus, Bruhat–Tits theory is available for G. Moreover, Axiom 4.1.27 holds. The formal proof of this theorem will be given in §9.5. Definition 9.2.8 B = B(G K )Γ is called the Bruhat–Tits building of G(k). We shall denote it by B(G) in what follows.

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Example 9.2.9 Let k = Q p and let G be the anisotropic k-group whose quasi-split inner form is PGL2 . Then G K = PGL2 , and G(k) = D× /k × , where D is the unique quaternion algebra over k. We claim that the action of Γ on B(G K ) cannot fix a vertex. If we assume to the contrary that x ∈ B(G K ) is a Γ-fixed vertex, then the special fiber of Gx0 is a connected f-group which over f is isomorphic to PGL2 . By Lang’s theorem, it must be isomorphic to PGL2 already over f. Applying Proposition 8.2.1(1) to S = {1} and T a split torus in PGL2 we obtain a copy of Gm in Gx , hence a copy of Gm in G, contradicting the fact that G is anisotropic. From this we conclude at once that B(G K ) contains a unique Γ-facet, which is both a Γ-chamber and a Γ-vertex, and it is a K-chamber. In particular, B = B(G K )Γ consists of a single point. Indeed, we already know that B(G K )Γ is non-empty, and that no K-vertices are fixed, so at least one K-chamber is Γ-invariant and its barycenter belongs to B(G K )Γ . Given x  y ∈ B(G K )Γ , the unique geodesic [x, y] is pointwise Γ-fixed, but it must contain a vertex of the chamber that contains x, contradiction.

9.3 The Building and its Apartments We now begin to establish the fact that B = B(G K )Γ is an affine building on which G(k) acts by building automorphisms. Recall from Definition 1.5.1 that there is a notion of dimension of an abstract polysimplex, defined as the longest chain of proper face relations. In a geometric realization, one also has the dimension of the affine subspace spanned by a polysimplex, as well as the affine space spanned by any open neighborhood of a polysimplex (with respect to the relative topology of the polysimplex). All of these notions of dimension coincide. 9.3.1 Let X be a non-empty convex subset of B(G K ) and F be the set of facets of B(G K ), or facets lying in a given apartment A of this building, that meet X. Since any two facets lie in an apartment and every apartment is an affine space under a finite-dimensional real vector space, one sees the following (1) All maximal facets in F are of equal dimension and a facet F ∈ F is maximal if and only if dim(F ∩ X) is maximal. (2) Let F be a facet lying in an apartment A. Assume that F is maximal among the facets of A that meet X, and let AF be the affine subspace of A spanned by F. Then every facet of A that meets X is contained in AF and A ∩ X is contained in the affine subspace of A spanned by F ∩ X.

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In particular, for any facet F  in A, dim(F ∩ X)  dim(F  ∩ X). As B is a non-empty convex subset of B(G K ), the above assertions hold for X = B. In particular, the Γ-chambers are of equal dimension, and moreover, for any Γ-chamber C, C := C Γ = C ∩ B is a chamber of B. Conversely, given a chamber C of B, the unique facet C of B(G K ) that contains C is a Γ-chamber and C = C ∩ B. 9.3.2 Given non-empty Γ-stable bounded subsets Ω, Ω  of an apartment of B(G K ) such that Ω is contained in the closure Ω  of Ω , that is Ω ≺ Ω , the homomorphism ρΩ,Ω described in Axiom 4.1.22 descends to an o-group scheme homomorphism GΩ0 → GΩ0 which is the identity homomorphism on the common generic fiber G. We shall denote this o-homomorphism also by 0 → G 0 between the special ρΩ,Ω ; it induces an f-homomorphism ρΩ,Ω : G Ω  Ω  fibers. In particular, if F ≺ F are two Γ-facets of B(G K ), then there is an o-group scheme homomorphism GF0  → GF0 that is the identity homomorphism on the common generic fiber G. The image of the induced homomorphism GF0 → GF0 is a parabolic f-subgroup pF (F ) of GF0 , and F  → pF (F ) is an order preserving bijective map of the partially ordered set {F  | F ≺ F  } onto the set of parabolic f-subgroups of GF0 partially ordered by opposite of inclusion (Axiom 4.1.22). Thus, F  is a maximal Γ-facet (that is, it is a Γ-chamber) if and only if pF (F ) is a minimal parabolic f-subgroup of GF0 . Let G F be the maximal reductive quotient of GF0 . Note that the projection map GF0 → G F induces an inclusion-preserving bijective correspondence between the parabolic f-subgroups of GF0 and the parabolic f-subgroups of its maximal reductive quotient G F , cf. Proposition 2.4.5. Hence, a Γ-facet C of B(G K ) is a Γ-chamber if and only if the reductive f-group GC does not contain a proper parabolic f-subgroup, or, equivalently, this reductive group contains a unique maximal f-split torus (this torus is central so it is contained in every maximal torus of GC ), cf. Proposition 2.4.7. Proposition 9.3.3 Let F ≺ F  be two Γ-facets. The inverse image of the subgroup pF (F )(f) of GF0 (f) in GF0 (o) is the subgroup GF0  (o). Proof Since o is Henselian, the natural homomorphisms GF0 (o) → GF0 (f) and GF0  (o) → GF0 (f) are surjective. Recall that according to Axiom 4.1.22 (Theorem 8.4.19), the kernel of the natural homomorphism π : GF0 (O) → GF0 (f) is contained in the subgroup GF0  (O), the kernel U of the homomorphism ρF , F  : GF0 → GF0 is a smooth connected unipotent f-group, and its image is the parabolic subgroup pF (F ). So the kernel of πo := π |G 0 (o) is contained F

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in GF0  (o). The Galois cohomology set H1 (Γ, U (f)) is trivial by Lemma 2.4.2. 

These observations imply that ρF , F  GF0 (f) = pF (F )(f) and the proposition follows.  Proposition 9.3.4 Every maximal k-split torus S of G is contained in a special k-torus. Proof The maximal bounded subgroup S(K)1 of S(K) is stable under the action of Γ. By Proposition 4.2.13 the action of S(K)1 on B(G K ) is bounded. In 9.2.1 we showed that Γ acts on the apartments of B(G K ) through finite quotients. Therefore the orbits in B(G K ) under the action of Γ  S(K)1 are bounded. Theorem 1.1.15 applied to any orbit of this action provides a fixed point. So there exists a point, say x, of B(G K ) that is fixed under Γ and S(K)1 . In particular, x ∈ B = B(G K )Γ and S(K)1 ⊂ G(K)1x . Let Gx1 be the smooth affine o-group scheme with generic fiber G associated to x in 9.2.5 and let S be the schematic closure of S in Gx1 . Then S is a torus according to Proposition B.2.4. Applying Proposition 8.2.1(1), we conclude that Gx1 contains a closed o-torus T that contains S and whose special fiber T is a maximal f-torus of the special fiber G x1 of Gx1 . Therefore, the generic fiber T of T is a k-torus of G containing the maximal k-split torus S. Furthermore, as T is a maximal f-torus of G x1 and it splits over f, TK is a maximal K-split  torus of G K . Thus T is a special k-torus of G. The following proposition provides useful criteria for a non-empty bounded subset of an apartment of B(G K ) to be contained in another apartment. It will be used in the proof of Proposition 9.3.6 below. Proposition 9.3.5 Let T and T  be maximal K-split tori of G K with apartments A and A, respectively. A non-empty bounded subset Ω of A is contained in A if and only if one of the following four equivalent conditions holds. (1) There is an element g ∈ GΩ0 (O) such that T  = gT g −1 . This element carries A to A and fixes Ω pointwise. (2) GΩ0 contains a closed O-torus with generic fiber T . (3) GΩ0 (O) ∩ T (K) is the maximal bounded subgroup of T (K). (4) The identity component T  of the Zariski closure of the image of T (K) ∩ GΩ0 (O) in G Ω0 (f) is a maximal torus of G Ω0 . Proof We will denote GΩ0 by G , and its special fiber by G , and use Proposition 8.2.1, with O in place of o, in this proof. Let T be the closed O-torus of G with generic fiber T and T be the special fiber of T . If Ω is contained in A, then G contains a closed O-torus with generic fiber T . Let us assume now that

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G contains a closed O-torus T  with generic fiber T . The special fibers T and T  of T and T  are maximal tori of G , and hence there is an element g of G (f) that conjugates T onto T  [Bor91, Theorem 18.2]. Now Proposition 8.2.1(2) implies that there exists a g ∈ G (O) lying over g that conjugates T onto T . This element fixes Ω pointwise and conjugates T onto T  and hence carries A to A. Hence Ω is contained in A. Conversely, if there is an element g ∈ G (O) such that T  = gT g −1 , then T  := gT g −1 is a closed O-torus of G with generic fiber T , and g carries A to A fixing Ω pointwise. By Lemma B.2.3, G (O) ∩ T (K) is the maximal bounded subgroup of T (K) if and only if the schematic closure of T  in G is an O-torus. Assertion (2) clearly implies (4). To prove that (4) implies (2), let T  be the schematic closure of T  in G . The O-group scheme T  is faithfully flat and of finite type. We will show that it is an O-torus. Let T  be the special fiber of T   )0 be the identity component of the reduced subgroup scheme of T  . and (T red According to Lemma 2.10.6, all irreducible components of the special fiber  ) 0 contains of T  have dimension equal to the dimension of T . Since (T red  the maximal torus T , it must be equal to it by dimension considerations. Let +  be the smoothening of T  (see §A.6). The generic fiber of T +  is T ; T  , . Then there is a canonical O-scheme we will denote its special fiber by T   + homomorphism T → T that is the identity on the generic fiber T . It ,  → T  of special fibers. Since the induced induces an f-homomorphism T  + (O) + homomorphism T (O) → T (O) is an isomorphism, the image of T    , (f) → T (f) is same as the + (O) → T under the composite of the maps T   image of T (O) → T (f). Now using the hypothesis in (4) we see that the  )0 = T  . Since these two , ) 0 onto (T red ,  → T  maps (T homomorphism T  )0 = T  is , ) 0 → (T red groups are of equal dimension, the homomorphism (T  0 , ) is a torus. an isogeny. Hence, (T +  are tori, + ) 0 of T As both the fibers of the relative identity component (T according to [SGA3, Exp.X, Corollary4.9] or [Con14, TheoremB.4.1] the O+ ) 0 is an O-torus. Since the residue field of O is algebraically group scheme (T + ) 0 (O) is a maximal bounded subgroup closed, every O-torus splits. Then (T  + (O) contains (T + ) 0 (O) and so of T (K). The bounded subgroup T (O) = T  it equals the maximal bounded subgroup of T (K). Now Lemma B.2.3 implies that T  is a closed O-torus in G = GΩ0 with generic fiber T . Thus we have proved that (4) implies (2).  Proposition 9.3.6 ber. Proof

Every special k-apartment of B(G K ) contains a Γ-cham-

Let A be a special k-apartment and T be the corresponding special

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k-torus. Then T contains a maximal k-split torus S of G and TK is a maximal K-split torus of G K . As A is stable under the action of Γ, it contains a point x which is fixed under Γ. Let F be the facet lying on A which contains x. Then, by definition, F is a Γ-facet. Let GF0 be the smooth affine o-group scheme, with connected fibers, associated to Ω = F in 9.2.5 and G F0 be the special fiber of GF0 . Let T be the closed o-torus of GF0 with generic fiber T, and let S be the maximal o-split subtorus of T (cf.9.2.5). Then the generic fiber of S is S. Let S and T be the special fibers of S and T respectively. We fix a minimal parabolic f-subgroup P of G F0 containing S , then P contains the centralizer of S by Proposition 2.4.9, and so it contains T . Let P be the inverse image of P(f) in GF0 (O)(⊂ G(K)) under the natural homomorphism GF0 (O) → G F0 (f). Then P is a parahoric subgroup of G(K) contained in the parahoric subgroup GF0 (O) (Axiom 4.1.22(5)); P contains T (O) and is clearly stable under the action of Γ on G(K). Let C be the facet of the Bruhat–Tits building B(G K ) fixed by P. Then C contains F in its closure and is stable under Γ, that is it is a Γ-facet; it is a Γ-chamber since P is a minimal parabolic f-subgroup of G F0 (9.3.2). Moreover, as P contains the maximal bounded subgroup T (O) of T(K), C lies in the apartment A (Proposition 9.3.5(3)).  Remark 9.3.7 Let A be a special k-apartment of B(G K ). Proposition 9.3.6 and the discussion of (9.3.1) imply that, among the facets of A that meet B, the maximal ones are Γ-chambers. Proposition 9.3.8 Given points x, y of B, there is a special k-apartment in B(G K ) that contains both x and y. Therefore, given any two Γ-facets of B(G K ) (which may not be different), there is a special k-apartment containing them. Proof We will first reduce to the following claim. If C is a Γ-chamber that lies in a special k-apartment and z ∈ B, then there exists a special k-apartment containing C and z. To see that this implies the proposition, let C be a Γ-facet that contains y in its closure and is maximal among such facets. Then C is a Γ-chamber. Let z ∈ C Γ . We apply the claim to this point z and a Γ-chamber lying in a special k-apartment (cf.Proposition 9.3.6), and see that there is a special k-apartment which contains z, and hence also C. We then apply the claim to the Γ-chamber C and the point x(∈ B) to obtain a special k-apartment which contains C and x. This apartment then contains C, and hence also y. We now prove the claim. Let T be the special k-torus corresponding to a special k-apartment A that contains C. Then T contains a maximal k-split torus S of G. We fix a point y of C Γ , then Gy0 = GC0 . Let S be the closed o-split torus in GC0 with generic fiber S. Let S be the special fiber of S and

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S be the image of S in the maximal reductive quotient GC of G C0 (= G y0 ). As C is a Γ-chamber, S is central and so every maximal torus of GC contains

it (9.3.2). By the uniqueness of the geodesic [y, z], every point on it is fixed under Γ, that is [y, z] ⊂ B. Let π be the composite of the f-homomorphisms 0 G [y,z] → G y0 → GC , where the first homomorphism is the f-homomorphism ρΩ,Ω of 9.3.2 for Ω = {y} and Ω  = [y, z], and the second homomorphism is the natural projection. The kernel of this homomorphism is a smooth connected unipotent group (Axiom 4.1.22(1) that is Proposition 8.4.15), so the restriction 0 is an isomorphism onto a maximal f-torus of π to any maximal f-torus of G [y, z] 0 . Then the maximal f-split of GC . Let T [y, z] be a maximal f-torus of G [y,z]

subtorus of T [y, z] is isomorphic to S since this is true for the isomorphic image of T [y, z] under π. 0 According to Proposition 8.2.1(1), G[y, contains a closed o-torus T[y,z] z] 0 whose special fiber (as an f-subgroup of G [y, ) is T [y,z] . The generic fiber z] T[y, z] of T[y, z] is then a k-torus of G that splits over K and contains a maximal k-split torus of G, so it is a special k-torus. The special k-apartment of B(G K ) determined by T[y,z] contains [y, z] by Proposition 9.3.5(2), and hence contains C and z. 

Proposition 9.3.9 If Gder is anisotropic over k, then B consists of a single point. Therefore, B(G K ) contains a unique Γ-chamber and its barycenter is the unique point of B. Proof The statement is unchanged if we replace G by Gder . We therefore assume that G is semi-simple. Let x, y ∈ B. According to Proposition 9.3.8 there is a special k-apartment A of B(G K ) which contains both x and y. Let T be the special k-torus of G corresponding to A. Then v = x − y ∈ V(TK )Γ . Since G is anisotropic, V(TK )Γ = {0}, hence x = y. This proves that B contains a single point, and the remainder of the proposition is immediate.  Proposition 9.3.10 The set B equipped with the facets described in 9.2.4 is a polysimplicial complex and the action of G(k) on B is via polysimplicial automorphisms. If G is k-simple, then B is a simplicial complex. Proof We begin by studying the relationship between facets of B and Γfacets of B(G K ). Let F be a minimal Γ-facet (minimal in the ordering defined in Axiom 4.1.22) of B(G K ). Then F ∩ B is a minimal facet of B according to 9.2.4, so it is a vertex. We assert that F ∩ B is a single point; equivalently, F contains a unique point fixed under Γ. To show this, let A be a special kapartment containing F (Proposition 9.3.8). This apartment is stable under the action of the Galois group Γ which acts on it by affine automorphisms. Now if x

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and y are two distinct points in F Γ , then the whole straight line in the apartment A passing through x and y is pointwise fixed under Γ. This line must meet the boundary of F, and hence meet a facet of dimension strictly smaller than that of F. That facet is then also a Γ-facet, contradicting the minimality of F. Let F be a (not necessarily minimal) Γ-facet in B(G K ) and let VF be the set of vertices of B contained in F. For v ∈ VF , let Fv be the face of F that contains v. Since v is a vertex of B, Fv is a minimal Γ-facet. We claim that if x and y are two distinct vertices in VF , then Fx ∩ Fy is empty. Indeed, this intersection is convex and stable under Γ and hence if it is non-empty, it would contain a Γ-fixed point (that is, a point of B). This would contradict the minimality of the Γ-facets Fx and Fy . The claim just proved implies that the sets of K-vertices of the facets Fx and Fy are disjoint, and each one of these sets is Γ-stable. Next we claim that the union of the sets of K-vertices of Fv , for v ∈ VF , is the set of K-vertices of F. To see this, we observe that any K-vertex of F is a K-vertex of a face of F which is a minimal Γ-facet and so it contains a (unique) point of VF . We now begin the proof of the statements in the proposition. Since B(G K ) is unchanged by passing to the adjoint group, we may assume that G is adjoint. The product structure on B(G K ) coming from decomposing G as a product of k-simple factors induces a product structure on B. Therefore it is enough to prove the proposition in the case that G is k-simple. Assuming that G is k-simple, G K is a product of copies, permuted transitively by Γ, of a K-simple group. Then the Bruhat–Tits building B(G K ) is a product of copies, permuted transitively by Γ, of a simplicial complex. A Γ-facet F of B(G K ) is then the product of copies of a single simplex permuted by Γ. A facet of B is of the form F := F ∩ B = F Γ . We claim that F is a simplex. By replacing Γ by the stabilizer of a single copy of the simplicial complex, and F by the corresponding single copy of a simplex, we do not change F. This allows us to reduce to the case that G is K-simple. Then F is a simplex. Given a non-empty subset V  of VF , the Γ-facet F  whose set of K-vertices is the union of the set of K-vertices of Fv , for v ∈ V , is a face of F, so F  := F  ∩ B is a face of F and its set of vertices is V . This shows that B is indeed an abstract simplicial complex. That this structure is compatible with the geometric realization is seen easily by arguing by induction on the dimension of a given Γ-facet F to see that F ∩ B is the convex hull of the set V of vertices of B contained in F. Let F be a Γ-facet of B(G K ), and F = F ∩ B be the corresponding facet of B. Then, for g ∈ G(k), g · F is also a Γ-facet and g ·F = g ·(F ∩B) = (g · F)∩B is the facet of B corresponding to g · F. Thus the action of G(k) on B is by polysimplicial automorphisms. 

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Let S be a k-split torus of G and S(K)b be the maximal bounded subgroup of S(K). The set B(G K )S of points of B(G K ) fixed under S(K)b is a non-empty closed convex subset. As S(K)b is stable under the action of the Galois group Γ on G(K), B(G K )S is Γ-stable and hence B(G K )SΓ is a non-empty closed convex subset. Proposition 9.3.11 (1) Let x, y be two points of B(G K )SΓ . Then there is a special k-torus T of G containing S such that both x and y lie in the apartment of B(G K ) corresponding to TK . (2) B(G K )S is the union of the apartments of B(G K ) corresponding to maximal K-split tori of G K that contain SK . It will be shown in §9.7 that B(G K )S can be identified with the enlarged building of the Levi subgroup ZG (S)K . Proof Since the maximal bounded subgroup S(K)b of S(K) fixes both x and y, it fixes the unique geodesic [x, y], hence S(K)b ⊂ G(K)1[x, y] . Proposition 1 B.2.4 implies that the schematic closure S of S in G[x, is a o-split torus. The y] 1 1 special fiber S of S is then an f-split torus in the special fiber G [x, of G[x, . y] y] Proposition 9.3.8 implies that [x, y] lies in a special k-apartment. Therefore, 1 , where T is the there is a special k-torus T and a closed immersion T → G[x, y] 1 standard model of T. The special fiber T of T is a maximal f-torus of G [x, , y] 1 and the maximal f-split subtorus of T is a maximal f-split torus of G [x, . y] 1 Hence a conjugate of T under an element of G [x, (f) contains the f-split torus y]

S . Proposition 8.2.1(1) allows us to assume, after replacing T by a conjugate under an element of G(k)1[x, y] , that S ⊂ T. According to Proposition 9.3.5(2), the geodesic [x, y] lies in the apartment corresponding to T. This proves (1), and (2) follows at once by working over K in place of k and taking x = y.  Proposition 9.3.12 Let S1 and S2 be two maximal k-split tori of G and S be a k-torus contained in S1 ∩ S2 . Let Ω be a non-empty Γ-stable bounded subset of B(G K )S1 ∩ B(G K )S2 . There exists an element of G(k)0Ω that commutes with S and maps B(G K )SΓ1 onto B(G K )SΓ2 . Proof Let G := GΩ0 be the Bruhat–Tits o-group scheme associated with Ω, and for i = 1, 2, let Si be the schematic closure of Si in G . According to Proposition B.2.4, S1 and S2 are o-tori. Let S be the closed o-torus contained in S1 ∩ S2 whose generic fiber is S. The centralizerM of S in G is a closed smooth o-subgroup scheme (Proposition 8.2.1(1)) and its fibers are connected since the centralizer of a torus in a connected smooth affine algebraic group is connected [Bor91, Corollary11.12]. Using Proposition 8.2.1(1) for M in

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place of G , and the remark following that proposition, we see that the special fibers S 1 and S 2 of S1 and S2 respectively are maximal f-split tori in the special fiber M of M . Hence there exists an element g ∈ M (f) that conjugates S 1 onto S 2 [Bor91, Theorem15.14]. By Proposition 8.2.1(2), there exists an element g ∈ M (o)(⊂ G(k)0Ω ) lying over g that conjugates S1 onto S2 . As  gS1 g −1 = S2 , we infer that gS1 g −1 = S2 , so g · B(G K )SΓ1 = B(G K )SΓ2 . For a k-torus S of G, S  will denote the maximal subtorus of S contained in Gder . Proposition 9.3.13 Let S be a maximal k-split torus of G. If T is a special k-torus containing S and A is the corresponding special k-apartment, we have B(G K )SΓ = AΓ . Moreover, AΓ is an affine space under V(S ). So B(G K )SΓ carries a natural structure of an affine space under V(S ). Proof The apartment A corresponding to T is contained in B(G K )S and it is stable under Γ. So it contains a point, say x, which is fixed under Γ. Let V(S ) = R ⊗Z X∗ (S ) and V(T ) = R ⊗Z X∗ (TK). As A = V(T ) + x, and V(T )Γ = V(S ) since S  is the maximal k-split torus in T , we infer that AΓ = V(S ) + x, so AΓ is an affine space under V(S ). Let y be an arbitrary point of B(G K )SΓ . According to Proposition 9.3.11(1), there exists a special k-apartment A ⊂ B(G K )S that contains both x and y. Working with A in place of A, we see that A Γ = V(S )+x, and hence A Γ = AΓ . In particular, AΓ contains y. Therefore, B(G K )SΓ = AΓ . Using Proposition 9.3.12 we see that the action of V(S ) on B(G K )SΓ induced by its action on AΓ is independent of the special k-apartment A ⊂ B(G K )S . So there is a well-defined action of V(S ) on B(G K )SΓ making the latter an affine  space under V(S ). 9.3.14 Let S be a maximal k-split torus of G. According to Proposition 9.3.13, B(G K )SΓ carries a natural structure of an affine space space under V(S ). We will denote this affine space by A(S) in what follows. Let N (respectively, Z) be the normalizer (respectively, the centralizer) of S in G and Z  be the centralizer of S in Gder . Then B(G K )SΓ is stable under the natural action of N(k) ⊂ N(K) on B(G K )S . For n ∈ N(K), the action of n carries an apartment A ⊂ B(G K )S to the apartment n · A by an affine transformation. Corollary 9.3.15 (1) The dimension of A(S) is equal to the k-rank of Gder . (2) Any facet of B that meets A(S) is contained in A(S). (3) Any chamber contained in A(S) is open in A(S), and hence of dimension equal to the k-rank of Gder .

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Proof (1) The dimension of the affine space A(S) is equal to that of its translation space V(S ). (2) Let F be a facet of B that meets A(S) and let F be the facet of B(G K ) containing F. According to Proposition 9.3.11 there is a special k-apartment in B(G K )S containing a point of F, hence a point of F. Then A contains all points of F, so A(S) = AΓ (Proposition 9.3.13) contains all points of F = F Γ . (3) According to (2) the affine space A(S) is the union of all facets of B that meet A(S). These are polysimplices by Proposition 9.3.10, and the largest among them are chambers by Proposition 9.3.6. The claim follows from 9.3.1(2).  Proposition 9.3.16 The affine space A(S) = B(G K )SΓ satisfies Axiom 4.1.4. Moreover, Z(k)b consists of all elements of N(k) ∩ G(k)1 that act trivially on A(S). Proof Axiom A 0 has already been shown. Axiom A 2 follows from the same axiom applied to G K . It remains to prove Axiom A 1. Let T be a special k-torus of G containing S and, as above, T  denotes the maximal subtorus of T contained in the derived subgroup of G. The action of n ∈ N(k) on B(G K ) carries the special k-apartment A = AT via an affine isomorphism ϕ(n) : A → AnT n−1 to the special k-apartment AnT n−1 corresponding to the special k-torus nT n−1 containing S. As (AnT n−1 )Γ = B(G K )SΓ = AΓ by Proposition 9.3.13, we see that ϕ(n) keeps AΓ stable and so f (n) = ϕ(n)| AΓ is an affine automorphism of AΓ . The apartments A and AnT n−1 are affine spaces under the vector spaces V(T  K ) and V(nT  K n−1 ) := R ⊗Z X∗ (nT  K n−1 ) respectively. The derivative dϕ(n) : V(T  K ) → V(nT  K n−1 ) of the affine isomorphism ϕ(n) is induced from the map X∗ (T  K ) → X∗ (nT  K n−1 ) defined by λ → Intn · λ, where Intn is the inner automorphism of G determined by n. So, the restriction df (n) : V(S ) → V(S ) is induced from the homomorphism X∗ (S ) → X∗ (S ), λ → Intn · λ. This implies that df is trivial on Z(k). The action of the bounded subgroup Z(k)b on AΓ admits a fixed point by Theorem 1.1.15. Let x ∈ AΓ be a point fixed under Z(k)b . Then AΓ = V(S ) + x. For all z ∈ Z(k) and v ∈ V(S ), as df (z) is trivial, z · (v + x) = v + z · x. Let z · x = φ(z) + x, where φ is a V(S )-valued function on Z(k). It is easily seen that φ is a homomorphism trivial on Z(k)b . Thus z ∈ Z(k) acts on AΓ by translation by φ(z) and Z(k)b acts trivially. We will now prove the formula in Axiom A 1 by showing that φ = ν. Since the image of S(k) in Z(k)/Z(k)b  Zr is a subgroup of finite index,

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to prove that φ = ν, it would suffice to prove this equality on S(k). But for s ∈ S(k), sT s−1 = T, and f (s) is a translation of the apartment A by the element ν(s) ∈ V(T ) which satisfies χ(ν(s)) = −ω( χ(s)) for all χ ∈ X∗K (TK ). This implies that φ = ν on S(k), since the restriction map X∗K (TK ) → X∗K (SK ) = X∗k (S ) is surjective. The proof of Axiom A 1 is complete. To prove the last assertion, let z be an element of N(k) ∩ G(k)1 that acts trivially on A(S). Then the derivative of f (z) is trivial, and hence z is contained in Z(k) ∩ G(k)1 . But then z acts by translation by the image ν(z) ∈ V(S ) of z under the negative of the valuation homomorphism Z(k) → V(S) composed  with the projection V(S) → V(S ). Lemma 2.6.19 implies z ∈ Z(k)b . Proposition 9.3.17 The map S → B(G K )SΓ is a G(k)-equivariant bijection between the set of maximal k-split tori of G and the set of apartments of B. In particular, the stabilizer of A in G(k) is N(k). Proof It is clear that the given map is G(k)-equivariant. Proposition 9.3.4 implies the existence of a special k-torus T containing S. If A is the special k-apartment corresponding to T, then AΓ = B(G K )SΓ . This shows that B(G K )SΓ is an apartment of B according to the definition of apartments given in 9.2.3. Conversely, given a special k-torus T, the maximal k-split subtorus S is a maximal k-split torus of G. We conclude that the map S → B(G K )SΓ is surjective. To prove injectivity, fix S and let A = B(G K )SΓ . Observe that as N(k) acts on A and the maximal bounded subgroup Z(k)b of Z(k) acts trivially (Proposition 9.3.16), the subgroup H of G(k) consisting of elements that fix A pointwise is a bounded subgroup of G(k) containing Z(k)b and it is normalized by N(k). Therefore, for any h ∈ H, the orbit {shs−1 | s ∈ S(k)} of S(k) through h, under the conjugation action, is contained in H and hence it is bounded. Now, using Lemma 2.2.8 we see that S(k) commutes with h. This implies that H commutes with S(k) and hence H ⊂ Z(k). As H contains Z(k)b , the Zariski closure of H is Z (= ZG (S)). Since S is the unique maximal k-split torus of G contained in Z, injectivity follows.  Proposition 9.3.18 Let A be an apartment of B, and C, C  two chambers in A. Then there is a gallery C = C0, C1, . . . , Cm = C 

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joining C and C  and consisting of chambers in A. (By definition of a gallery, for i with 1  i  m, Ci−1 and Ci share a face of codimension 1.) Proof Let A2 be the codimension 2-skeleton of A, that is the union of all facets in A of codimension at least 2. Then A2 is a closed subset of the affine space A of codimension 2, so A − A2 is arcwise connected. This implies that given points x ∈ C and x  ∈ C , there is a piecewise linear curve in A − A2 joining x and x . Now the chambers in A that meet this curve make a gallery  joining C to C . In the following we will use the word panel to refer to a facet of B of codimension 1. Proposition 9.3.19 Let A be an apartment of B and S be the maximal k-split torus of G corresponding to this apartment (then A = B(G K )SΓ ). The group N(k) acts transitively on the set of chambers of A. Proof According to Proposition 9.3.18, given any two chambers in A, there exists a minimal gallery in A joining these two chambers. So to prove the proposition by induction on the length of a minimal gallery joining two chambers, it suffices to prove that given two different chambers C and C  in A which share a panel F, there is an element n ∈ N(k) such that n · C = C . Let G := GF0 be the smooth o-group scheme associated with the panel F and S ⊂ G be the closed o-split torus with generic fiber S. Let G be the special fiber of G , S the special fiber of S . Then S is a maximal f-split torus of G (9.2.5). The chambers C and C  correspond to minimal parabolic f-subgroups P and P  of G , see 9.3.2. Both of these minimal parabolic f-subgroups contain S since the chambers C and C  lie on A. But then by Theorem 2.4.6, there is an element n ∈ G (f) which normalizes S and conjugates P onto P . Now from Proposition 8.2.1(3) we conclude that there is an element n ∈ NG (S )(o) lying over n. It is clear that n normalizes S and hence it lies in N(k); it fixes F  pointwise and n · C = C . Proposition 9.3.20 B is thick, that is any panel is a face of at least three chambers, and every apartment of B is thin, that is any panel lying in an apartment is a face of exactly two chambers of the apartment. Proof Let F be a Γ-facet of B(G K ) that is not a Γ-chamber, and C be a Γchamber of which F is a face. Then there is an o-group scheme homomorphism GC0 → GF0 . The image of G C0 in G F0 , under the induced homomorphism of special fibers, is a minimal parabolic f-subgroup of G F0 , and, conversely, any minimal parabolic f-subgroup of the latter determines a Γ-chamber with F as a

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face. The minimal parabolic f-subgroups of G F0 are the f-points of a projective variety that contains a copy of the 1-dimensional projective space P1 . Therefore, there are at least |P1 (f)|  3 such subgroups, and we see that F is a face of at least three distinct Γ-chambers. To prove the second assertion, we assume that F = F Γ is a panel and let A be an apartment of B containing F. Let S be the maximal k-split torus of G corresponding to A by Proposition 9.3.17. Let S be the closed o-split torus of GF0 with generic fiber S. Then the chambers of B lying in A, with F as a face, are in bijective correspondence with minimal parabolic f-subgroups of the reductive quotient G F of G F0 that contain the image S of the special fiber of S , cf. Axiom 4.1.22 and Proposition 9.3.5(4)). The f-rank of the derived subgroup of G F is 1 since F is of codimension 1 in B (9.2.4). This implies that G F has exactly two minimal parabolic f-subgroups containing S. The second assertion also follows at once from the following well-known result in algebraic topology: in any polysimplicial complex whose geometric realization is a topological manifold without boundary (such as an apartment A in B), any facet of codimension 1 is a face of exactly two chambers (that is, maximal dimensional facets).  Theorem 9.3.21 The set B, equipped with the apartments and facets as in 9.2.3 and 9.2.4, is a building in the sense of Definition 1.5.5. The group G(k) acts on B via polysimplicial automorphisms. The apartments are in G(k)equivariant bijective correspondence with the maximal k-split tori of G. Proof Propositions 9.3.10 and 9.3.18 show that B is a chamber complex on which G(k) acts by polysimplicial automorphisms. Proposition 9.3.20 shows that axioms BL1 and BL2 hold. Propositions 9.3.8, 9.3.13, and Corollary 9.3.15 establish BL3. Proposition 9.3.12 establishes BL4. Proposition 9.3.17 establishes a G(k)-equivariant bijection between the apartments of B and the maximal k-split tori of G.  Proposition 9.3.22 The group G(k) acts transitively on the set of ordered pairs (A, C) consisting of an apartment A of B and a chamber C lying in the apartment A. Proof

The claim follows from Propositions 9.3.17 and 9.3.19.

Proposition 9.3.23 other under G(k).



The Iwahori subgroups of G(k) are conjugate to each

Proof The Iwahori subgroups of G(k) are the subgroups GC0 (o) for chambers C in the building B. Proposition 9.3.22 implies that G(k) acts transitively on the set of chambers of B. 

9.3 The Building and its Apartments Proposition 9.3.24 Proof

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The identity G(k)0 = G(K)0 ∩ G(k) holds.

We recall from Definition 2.6.23 that G(k)0 = Z(k)0 · G(k) = [T(K)0 · Z(K) ]Γ · G(k) ,

where Z is the centralizer of a maximal k-split torus S of G, and T ⊂ Z is a maximal K-split torus. We may of course take T to be a special k-torus of G containing S. On the other hand, G(K)0 ∩ G(k) = [T(K)0 · G(K) ]Γ . The inclusion G(k)0 ⊂ G(K)0 ∩ G(k) is immediate. To obtain the reverse inclusion, let g ∈ G(K)0 ∩ G(k) and consider a chamber C lying in the apartment A of B corresponding to S. By Proposition 9.3.22 applied to Gsc we can multiply g on the left by an element of G(k) to ensure that g preserves A and C. Since G(k) ⊂ G(K)0 ∩ G(k) we see that g ∈ G(K)0 ∩ G(k). Now g preserves A, hence lies in N(k) by Proposition 9.3.17. Moreover g preserves C, hence also the unique Γ-chamber C of B(G K ) containing C. Since g lies in G(K)0 , it fixes every point of C, and hence also of C. Since C spans the affine space A by Corollary 9.3.15, it follows that g acts trivially on A. By Proposition 9.3.16, g lies in Z(k)1 . Thus g ∈ G(K)0 ∩ Z(k)1 = Z(k)0 ⊂ G(k)0 by Corollary 2.6.28.  Proposition 9.3.25 Fix an apartment A of B and a chamber C lying in A. Let S be the maximal k-split torus of G corresponding to A and N be the normalizer of S in G. Let G = G(k)0 , I = GC0 (o), and N = N(k) ∩ G. Then (I, N) is a saturated Tits system in G and B is the (restricted) building of that Tits system. In particular, F → G(k)0F is an order-reversing bijection from the set of facets of B to the set of parabolic subgroups of G, and the subset of points of B fixed under G(k)0F is precisely the closure F of F. Proof Proposition 9.3.17 implies that the stabilizer of A in G is N. We now claim that if an element of G stabilizes a facet of B, then it fixes it pointwise. Indeed, let g ∈ G stabilize a facet F. Let F be the unique facet of B(G K ) containing F. Then g also stabilizes F. Since B(G K ) is the building of a Tits system of G(K) 0 and G(k)0 ⊂ G(K)0 , we see that g fixes every point of F, in particular of F = F Γ . This proves the claim. In particular, the stabilizer of C in G equals I. Since G contains the image of the natural map Gsc (k) → G(k) and B(G K ), hence B, is unchanged if we replace G by Gsc , Proposition 9.3.22 implies that G acts transitively on the set of ordered pairs consisting of an apartment of B and a chamber lying in the apartment. It now follows from Proposition 1.5.28 that (I, N) is a saturated Tits system

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in G, and B is the restricted Tits building corresponding to this Tits system. The remaining claims follow from Proposition 1.5.13.  9.3.26 Assume that G is semi-simple and simply connected. Let F be a facet of B and F the Γ-facet of B(G K ) containing F. Since G(k) 0 = G(k), the stabilizer of F in G(k) is GF0 (o) = GF0 (o), hence the stabilizer of F in G(k) fixes F and F pointwise. The normalizer of a parahoric subgroup P of G(k) is P itself, for if P is the stabilizer of the facet F of B, then the normalizer of P also stabilizes F, and hence it coincides with P. Proposition 9.3.27 Let A be an apartment of B and C1, C2 two chambers of A, and x ∈ C1 ∩ C2 . There exists an element of G(k)0x that stabilizes A and maps C1 to C2 . Proof Let S ⊂ G be the maximal k-split torus whose apartment is A. The schematic closure of S in Gx0 is a maximal closed o-split torus of Gx0 . The chambers C1 and C2 correspond to minimal parabolic subgroups P 1 and P 2 of the special fiber G x0 that contain the maximal torus S . According to Theorem 2.4.6 there exists an element n ∈ G x0 (f) normalizing S and mapping P 1 to P 2 . By Proposition 8.2.1(3) there exists a lift n ∈ N(k) of n. This element has the desired property.  We now record a result that will not be used in this book, but may be of independent interest. Lemma 9.3.28 Let G be a reductive o-group scheme with generic fiber G. Then G contains a split closed o-torus S of dimension equal to k-rank(G). Proof According to Proposition 8.3.16 there is an isomorphism ρ : G → Gx0 for a special point x ∈ B(G) = B(G K )Γ . Every point of B(G K ) which is fixed under the action of Γ lies in a special k-apartment, that is, an apartment corresponding to a k-torus T of G that contains a maximal k-split torus of G and TK is a maximal split torus of G K . Thus the point x ∈ B(G K ) fixed by G (O) lies on a special apartment corresponding to a special torus T. This torus splits over K. Let T be its o-smooth model (the “standard model” of Appendix B). Since x is fixed by Γ, the parahoric group scheme Gx0 descends to o (see 9.2.5) and gets identified with the o-group scheme G using ρ. Now with this identification, there is a closed immersion of T in G . The generic fiber T of T contains a maximal k-split torus S of G. Let S be the schematic closure of S in G , then S is a closed o-split torus of G (Proposition B.2.4); its dimension clearly equals k-rank of G. 

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Remark 9.3.29 The above lemma is valid without assuming that f is perfect. Indeed, Proposition 8.3.16 does not require this assumption, and the existence of special k-tori is proved without this assumption in [Pra20b], provided G K is quasi-split, which is the case here according to Proposition 8.3.16.

9.4 The Affine Root System We continue with a connected reductive k-group G. We fix a maximal k-split torus S of G, and denote by Φ = Φ(S, G) the root system of G with respect to S. For a ∈ Φ, Ua denotes the a-root group of G. As before, we will denote B(G K )Γ by B, and A will denote the apartment of B corresponding to S. The set of real valued affine functions on A will be denoted by A∗ . For a ∈ Φ, Sa will denote the maximal subtorus of S contained in kera, and S  a will denote the 1-dimensional subtorus of S contained in the connected semi-simple group G a generated by U±a . Definition 9.4.1

Let ψ ∈ A∗ with derivative a ∈ Φ.

(1) Hψ = {x ∈ A | ψ(x) = 0}. (2) Aψ = {x ∈ A | ψ(x)  0}. (3) Uψ is the subgroup of Ua (k) consisting of elements fixing Hψ pointwise. Lemma 9.4.2 (1) The stabilizers in Ua (K) of the points of Hψ are open bounded subgroups of Ua (K). They are all equal to each other. (2) The isotropy subgroup in Ua (k) of any point of Hψ is Uψ , so Uψ is an open bounded subgroup of Ua (k). Proof It is enough to prove (1). For s ∈ S(k), Proposition 9.3.16 implies that the action of s on A is by translation by ν(s) ∈ V(S ) given by the formula in Axiom 4.1.4. For z ∈ A we thus have (s · ψ)(z) = ψ(s−1 z) = −a(ν(s)) + ψ(z) = ω(a(s)) + ψ(z). If s ∈ Sa (k) and z ∈ Hψ , then (s · ψ)(z) = 0, that is s−1 · z lies in Hψ . Since Sa is a subtorus of S of codimension 1, we see that the convex hull of Sa (k) · z is the hyperplane Hψ . Now as Sa (k) commutes with Ua (K), we see that the subgroup of Ua (K) consisting of elements that fix z fixes Hψ pointwise. That this subgroup is open and bounded follows from the fact that it equals the intersection of Ua (K) with G(K)1z and the latter is an open and bounded subgroup of G(K) by Axiom 4.1.20.  Proposition 9.4.3 (1) Given a neighborhood U of the identity in Ua (k), there is a real valued affine function ψ on A, with derivative a, such that Uψ ⊂ U.

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(2) Uψ is the subgroup of Ua (k) consisting of elements that fix Aψ pointwise. Hence, if x is a point of A such that ψ(x)  0 (that is, x ∈ Aψ ), then it is fixed by Uψ . (3) Given u ∈ Ua (k)∗ , there is a real valued affine function ψau on A, with derivative a, such that the set of points of A fixed by u is the half-apartment Aψau . −1 (4) For u ∈ Ua (k)∗ and s ∈ S(k), ψasus = ω(a(s)) + ψau . (5) Let z ∈ Hψ . Then Uψ − {1} = {u ∈ Ua (k)∗ | ψ  ψau } = {u ∈ Ua (k)∗ | ψau (z)  0}. u = 2ψ u . This implies at (6) If a, 2a ∈ Φ and u ∈ U2a (k) ⊂ Ua (k), then ψ2a a once that Uψ ∩ U2a (k) ⊂ U2ψ .

(Note that Uψ may fix every point of an half-apartment that is strictly larger than Aψ . For example, if for a positive real number , Uψ = Uψ+ , then Uψ fixes Aψ+ pointwise.) Proof Let λ : Gm → Sa (⊂ S) be the isomorphism such that a, λ > 0. We fix a uniformizer π ∈ o. (1) Let ψ  be a real valued affine function on A with derivative a. Then as Uψ  is a bounded subgroup of Ua (k), for all sufficiently large positive integers n, Uλ(π n )·ψ  = λ(π n )Uψ  λ(π −n ) ⊂ U. We fix one such integer, say n0 , and set ψ = λ(π n0 ) · ψ  = ψ  + n0 a, λ . Then Uψ ⊂ U. (2) For any integer n, and z ∈ Hψ , λ(π −n ) · z = nλ + z. By Lemma 9.4.2, the subgroup λ(π −n )Uψ λ(π n ) of Ua (k) is the isotropy subgroup of the point nλ + z. But as a, λ > 0, the subgroups λ(π n )Uψ λ(π −n ) shrink as n increases and form a basis of neighborhoods of {1} in Ua (k). Since Uψ is open by Lemma 9.4.2, it contains λ(π n )Uψ λ(π −n ) for all sufficiently large positive integers n. This implies that Uψ fixes nλ + Hψ pointwise for all sufficiently large positive integers n. But as the convex hull of the union of Hψ and nλ + Hψ , for all large positive integers n, is the closed half-apartment Aψ , we see that Uψ fixes every point of Aψ . Since, by definition, Uψ is the subgroup of Ua (k) consisting of elements that fix Hψ pointwise, Uψ can be described also as the subgroup of Ua (k) consisting of elements that fix Aψ pointwise. (3) The set of points of A fixed by u is a closed convex subset. We claim that it is non-empty. Let x be any point of A. Since G(k)1x is an open bounded subgroup, Ua (k) ∩ G(k)1x is an open bounded subgroup of Ua (k). So, for all large positive integers n, λ(π n )uλ(π −n ) belongs to Ua (k)∩G(k)1x , hence it fixes x. Then u fixes λ(π −n ) · x = nλ + x ∈ A. According to Proposition 9.3.17, the stabilizer of A in G(k) is N(k), hence

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u (which is not in N(k)) cannot fix A pointwise. Now let ψ be any real valued affine function on A, with derivative a, such that the vanishing hyperplane Hψ contains a point fixed by u. Then by (2), we see that u fixes the half-apartment Aψ . Any proper closed convex subset of A that contains Aψ is of the form Aψ  , for a real valued affine function ψ  on A with derivative a, such that ψ   ψ. Hence, there is a real valued affine function ψau , with derivative a, such that the set of points of A fixed by u is precisely the half-apartment Aψau . (4) For u ∈ Ua (k)∗ and s ∈ S(k), it is obvious that the half-apartment pointwise fixed by sus−1 is s · Aψau = {x ∈ A | s−1 x ∈ Aψau } = {x ∈ A | ψau (s−1 x)  0}. −1

But ψau (s−1 x) = ω(a(s)) + ψau (x). Therefore, ψasus = ω(a(s)) + ψau . (5) We know from (2) that an element u of Ua (k) belongs to Uψ if and only if it fixes the closed half-apartment Aψ pointwise. On the other hand, according to (3), for u ∈ Ua (k)∗ , Aψau is precisely the set of points of A fixed by u. Therefore, u belongs to Uψ if and only if Aψ ⊂ Aψau , or, equivalently, ψ  ψau . We note that as ψ and ψau have equal derivative, ψau − ψ is constant. Now since ψ(z) = 0, ψ  ψau if and only if ψau (z)  0. u and Aψ u are equal to the set of points of A fixed by u. On the (6) Both Aψ2a a u = 2a = 2∇ψ u . The claim follows.  other hand, ∇ψ2a a If ψ  is another real valued affine function on A with derivative a, and   ψ, then Aψ ⊆ Aψ  and hence Uψ  ⊆ Uψ . Let Uψ+ =  >0 Uψ+ Thus Uψ+ ⊂ Uψ . ψ

Corollary 9.4.4 For u ∈ Ua (k)∗ , ψau ∈ A∗ is the unique affine function such that u ∈ Uψau − Uψau + . 9.4.5 Let a ∈ Φ and ψ be a real valued affine function on A with derivative a. Let Hψ be the vanishing hyperplane of ψ and z be a point on it. Let Gz1 be the Bruhat–Tits smooth affine o-group scheme with generic fiber G such that Gz1 (O) consists of elements of G(K)1 that fix z. Let Gz0 be the relative identity component of Gz1 , and Pz := Gz0 (o) be the parahoric subgroup of G(k) associated with z. Let S be the o-split torus of Gz0 with generic fiber S. Let Uψ be the a-root group of Gz1 with respect to S (see Definition 2.11.10). Since both the fibers of Uψ are connected, it is actually contained in Gz0 . Lemma 9.4.6 (1) The group scheme Uψ depends only on ψ, and not on z. (2) Uψ (o) = Uψ . Proof

According to Remark 2.11.11, Uψ is the schematic closure of Ua in

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Gz1 . Therefore Uψ (O) = Ua (K)∩Gz1 (O) = Ua (K)∩G(K)1z . Lemma 9.4.2 shows that this subgroup of Ua (K) is independent of the choice of z ∈ Hψ , and (1) follows from Corollary 2.10.11. Moreover, Uψ (o) = Ua (k) ∩ G(k)1z = Uψ , the last equality again by Lemma 9.4.2.  Lemma 9.4.7 Let u1, u2 ∈ Ua (k)∗ and u = u1 u2 . If ψau1  ψau2 , then ψau  ψau2 . On the other hand, if ψau1 > ψau2 , then ψau = ψau2 . Proof If ψau1  ψau2 , then Uψau2 contains Uψau1 , and hence, in particular, it contains both u1 and u2 , and therefore, it contains u. This implies that ψau  ψau2 . We now assume that ψau1 > ψau2 . Let ψ be the smaller of ψau and ψau1 . Then the subgroup Uψ contains both u and u1 , and hence also u2 . This implies that ψau2  ψ. Therefore, ψ = ψau , and as ψau  ψau2 , we conclude that ψau = ψau2 .  Proposition 9.4.8 Let ψ and η be real valued affine functions on A, with derivative a and b in Φ. We assume that b is not a multiple of a. Then the commutator subgroup (Uψ , Uη ) is contained in the group generated by Umψ+nη , where m, n are positive integers such that ma + nb ∈ Φ. Proof According to Proposition 9.4.3(2), the subgroups Uψ and Uη fix the half-apartments Aψ and Aη respectively pointwise. We will denote the intersection Aψ ∩ Aη by Aψη ; this subset of A consists precisely of the points where both ψ and η take non-negative values. For a root c = ma + nb, where m and n are positive integers, we denote by ψc the smallest real valued affine function on A with derivative c that takes only non-negative values on Aψη . Then as Aψη contains a point where both ψ and η vanish, we see that ψc = mψ + nη. We note that Aψc ⊃ Aψη , so Uψc fixes every point of Aψη (Proposition 9.4.3(2)). Also, if v ∈ Uc (k)∗ fixes Aψη pointwise, then the real valued affine function ψcv takes only non-negative values on Aψη . So ψcv  ψc . Therefore, v lies in Uψc . Hence, Uψc is the subgroup of Uc (k) consisting of all the elements which fix Aψη pointwise. Since the root b is not a multiple of a, there exists a 1-parameter group λ : Gm → S such that both a, λ and b, λ are positive. Let U = UG (λ) be as in §2.11. This subgroup is a smooth connected unipotent subgroup normalized by S. Let Φ+λ be the set of roots of U with respect to S; this subset consists of roots c ∈ Φ = Φ(S, G) such that c, λ > 0 so it contains a and b. For c ∈ Φ+λ we denote the c-root group of U by Uc . Let z be any point of Aψη and G := Gz1 be the smooth affine o-group scheme with generic fiber G and G (O) = G(K)1z . Let S be the closed o-split torus of G whose generic fiber is S. Since λ carries the maximal bounded subgroup Gm (O) of Gm (K) into the maximal bounded subgroup of S(K), there is an extension

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of λ to an o-group scheme homomorphism Gm → S ; we will denote this extension again by λ. Now let U = UG (λ), where the latter is as in §2.11. Propositions 2.11.1 and 2.11.3 imply that U is smooth, it is normalized by S , its generic fiber is U, and its special fiber is a connected unipotent f-group. The set of roots of U with respect to S is Φ+λ as the generic fiber of U is U. For c ∈ Φ+λ , let Uc be the c-root group of U (Definition 2.11.10). We fix  an order on the subset of non-divisible roots in Φ+λ . Let ϕ : c Uc → U be the product map, where the product is over the non-divisible roots in Φ+λ in the order we have fixed on this subset. Applying Lemma 8.2.3 to this map of  o-schemes, we see that ϕ is an isomorphism since ϕk : U → c Uc and ϕf are isomorphisms by Theorem 2.11.15. Since z ∈ Aψ ∩ Aη we have Ua (o) ⊃ Uψ and Ub (o) ⊃ Uη , so the commutator (u, u ) = uu u−1 u −1 of u ∈ Uψ and u  ∈ Uη is contained in U (o)(⊂ G (o) = G(k)1z ). Hence, in terms of the isomorphism   ϕ : c Uc → U , we can express the commutator (u, u ) as the product c uc , with uc ∈ Uc (o)(⊂ U (o) ⊂ G(k)1z ). For all c, uc fixes z.  The uniqueness of decomposition of (u, u ) as an element in c Uc (k), the product over non-divisible elements of Φ+λ in the chosen order, implies that the  above decomposition (u, u ) = c uc is independent of the choice of z ∈ Aψη , so uc fixes all of Aψη pointwise. Hence, uc ∈ Uψc = Umψ+nη . We also conclude that the c-component of (u, u ) is trivial unless c is a non-divisible root of the form ma + nb for some positive integers m, n. Thus we have shown that every commutator (u, u ), and hence the commutator subgroup (Uψ , Uη ), is contained in the group generated by Umψ+nη , for ma + nb ∈ Φ, where m and n are positive integers.  Let u ∈ Ua (k)∗ = Ua (k) − {1}. Recall from Proposition 2.11.17 the unique ∈ U−a (k) such that the element m(u) = u  uu  ∈ G(k) normalizes S. The elements u , u  lie in U−a (k)∗ and the element m(u) acts as the reflection along the root a.

u , u 

Proposition 9.4.9 Let a ∈ Φ and u ∈ Ua (k)∗ . Let z be a point of Hψau , that is ψau (z) = 0, and let Gz0 be the parahoric o-group scheme associated to this point. Let Gz0 be its special fiber and Ru (Gz0 ) be the unipotent radical of the special fiber. Then the following two equivalent assertions hold. (1) The image u of u in Gz0 (f) does not lie in Ru (Gz0 )(f). (2) Let m(u) = u  uu  be as above. Then u , u  and m(u) belong to Gz0 (o). Moreover, for every u ∈ Ua (k)∗ , m(u) acts on A by reflection in the hyperplane Hψau and the derivative of this action is the reflection ra in a. Proof

As ψau (z) = 0, u fixes z (Proposition 9.4.3(3)), so u ∈ Gz0 (o) by Lemma

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9.4.6. From Axiom 4.1.22(4) (Corollaries 8.4.12 and 8.4.13) we see that every element of Gz0 (o) whose image in Gz0 (f) lies in Ru (Gz0 )(f) fixes every point of every facet in B(G K ) that contains z in its closure; the union of these facets is a neighborhood of z in B(G K ). Therefore, if the image u of u in Gz0 (f) lies in Ru (Gz0 )(f), u fixes a neighborhood of z in A. But as z lies on the boundary Hψau of the half-apartment Aψau , we see that the set of points of A fixed by u is strictly larger than Aψau . This contradicts Proposition 9.4.3(3). So u  Ru (Gz0 )(f). Theorem 8.2.9 implies the equivalence of assertions (1) and (2). To prove the last assertion, we choose a point z ∈ Hψau and note that according to (2), m(u) lies in Gz0 (o), and hence it fixes z. Since m(u) commutes with the codimension 1 torus Sa contained in the kernel of a in S, m(u) fixes every point of Sa (k) · z. As the hyperplane Hψau is the convex hull of Sa (k) · z, m(u) fixes this hyperplane pointwise and then, clearly, it acts on A by reflection in this hyperplane, and the derivative of this action is ra since m(u) acts on X∗ (S) by reflection in a.  Corollary 9.4.10 Let u ∈ Ua (k)∗ and m(u) = u  uu  be as above. Then u = −ψ u = ψ u and ψ u = ψ u −1 . ψ−a a a −a a Proof From Remark 2.11.18 we know m(u) = m(u ) = m(u ). The last u = −ψ u = ψ u . The assertion assertion of Proposition 9.4.9 implies that ψ−a −a a −1 that ψau = ψau follows at once from the fact that a point of A is fixed by u if  and only if it is fixed by u−1 .

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The following is a converse of Proposition 9.4.9. Proposition 9.4.11 z ∈ A.

Let u ∈ Ua (k)∗ and m(u) = u  uu  be as above, and

(1) If m(u) lies in Gz0 (o), then u, u , u  also lie in Gz0 (o) and ψau (z) = 0. (2) If ψau (z) > 0, then the image u of u in Gz0 (f) lies in Ru (Gz0 )(f). Proof Since m(u) lies in Gz0 (o), it fixes z. We know from Proposition 9.4.9 that m(u) acts on A by reflection in the hyperplane Hψau . Therefore, every point of A that is fixed under the action of m(u) lies in Hψau . In particular, z ∈ Hψau u (z) = 0 = ψ u (z). and hence ψau (z) = 0. Now Corollary 9.4.10 implies that ψ−a −a   u = ψ u = −ψ. By Proposition 9.4.3 we have u ∈ U Letting ψ = ψau we see ψ−a ψ −a and u , u  ∈ U−ψ . By Lemma 9.4.6 all the three elements u, u , u  lie in Gz0 (o). We will prove the second assertion by contradiction. So assume that u  Ru (Gz0 )(f). Then by Theorem 8.2.9, m(u) lies in Gz0 (o) and then assertion (1)  implies that ψau (z) = 0. If a ∈ Φ, but 2a  Φ, then U2a will denote the trivial group. 9.4.12 Affine roots In Definition 6.3.4 we defined two subsets Ψ  and Ψ of A∗ in the setting where A was an affine space whose points are valuations of the root datum of (G, S). In our current setting we do not yet have this interpretation of the points of A; we will obtain it in §9.6. Nonetheless, we can use the same definition as in Definition 6.3.4, because this definition relies on the existence of open bounded subgroups Uψ ⊂ Ua (k) associated to ψ ∈ A∗ with ψ = a with the property Uψ1 ⊂ Uψ2 if ψ2  ψ1 . Definition 9.4.1 provides these open bounded subgroups in our current context. Therefore, following Definition 6.3.4, let Ψ  = {ψ ∈ A∗ | ψ ∈ Φ, Uψ+  Uψ } and Ψ = {ψ ∈ A∗ | ψ ∈ Φ, Uψ  Uψ+ · U2a (k)}. As was already remarked in Definition 6.3.4, the assertion Uψ  Uψ+ · U2a (k) is equivalent to U ψ+  U ψ , where the overline denotes image in Ua (k)/U2a (k). We will prove in Proposition 9.4.19 below that both Ψ  and Ψ are affine root systems in A∗ with derivative root system Φ. The set Ψ will play the more important role. It will be called the (relative) affine root system of G, and its elements will be called the (relative) affine roots. The relationship between Ψ and Ψ  will be discussed in Proposition 9.4.18 below.

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Remark 9.4.13 It is clear from the definition that Ψ ⊂ Ψ . If ψ ∈ Ψ  and u ∈ Uψ − Uψ+ , then ψ = ψau . On the other hand, if u ∈ Ua (k)∗ and ψ = ψau , then u ∈ Uψ − Uψ+ . Therefore we have the alternative description Ψ  = {ψau | a ∈ Φ, u ∈ Ua (k)∗ }. Notation 9.4.14 Given a ∈ Φ we write Ψa (respectively, Ψa ) for the elements of Ψ  (respectively, Ψ) whose derivative is a. Lemma 9.4.15 The natural action of N(k)/Z(k)b on A∗ preserves the subsets Ψ  and Ψ. More precisely, given a ∈ Φ, u ∈ Ua (k)∗ , and n ∈ N(k), we have nun−1 = n · ψ u . ψna a Proof It is enough to show that for every ψ ∈ A∗ with derivative a ∈ Φ and n ∈ N(k), we have nUψ n−1 = Un·ψ . Let b = n · a. Then the derivative of n · ψ is b. Moreover, n · Aψ = An·ψ . As Uψ is the subgroup of Ua (k) consisting of elements that fix Aψ pointwise and Un·ψ is the subgroup of Ub (k) consisting of elements that fix An·ψ = n · Aψ pointwise, we infer that Un·ψ = nUψ n−1 .  Proposition 9.4.16 Given a ∈ Φ, there exists a positive rational number ca such that ψ  − ψ ∈ ca Z for any ψ, ψ  ∈ Ψ  with derivative a. Proof As both ψ and ψ  have derivative a, there exists a c ∈ R, such that  ψ  = ψ + c. Choose u, u  ∈ Ua (k)∗ such that ψ = ψau and ψ  = ψau . We know from Proposition 9.4.9 that m(u) and m(u ) act on A by reflections rψ and rψ  in the hyperplanes Hψ and Hψ  respectively. Then rψ · rψ  maps onto the trivial element of the Weyl group, and hence s := m(u) · m(u ) lies in Z(k). As the homomorphism Q ⊗Z X∗k (Z) → Q ⊗Z X∗k (S), obtained by restricting characters of Z to S, is an isomorphism, we can (and will) view X∗k (S) as a subgroup of Q ⊗Z X∗k (Z). Now (rψ · rψ  ) · ψ  = −rψ · ψ  = −rψ · (ψ + c) = ψ − c. On the other hand, (rψ · rψ  ) · ψ  = s · ψ  = s · (ψ + c) = ψ + ω(a(s)) + c. So ψ − c = ψ + ω(a(s)) + c, therefore, c = − 12 ω(a(s)). It is obvious that { 12 ω(a(s)) | s ∈ Z(k)} is a cyclic subgroup of Q. Let ca be the positive generator  of this subgroup. Then c ∈ ca Z. Consider a ∈ Φ such that 2a ∈ Φ. The following lemma shows that any u ∈ Ua (k) − U2a (k) can be broken up into a product u = u0 · v0 with u0 ∈ Ua (k) and v0 ∈ U2a (k) such that u0 is as small as possible.

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Lemma 9.4.17 Assume that u ∈ Ua (k)−U2a (k). Then there exist u0 ∈ Ua (k)∗ , and v0 ∈ U2a (k), such that u = u0 · v0 , and u0  Uψau0 + · U2a (k). Thus ψau0 ∈ Ψ v0 = 2ψau . and ψau0  ψau . If ψau  Ψ, then ψ2a Proof If u  Uψau + ·U2a (k), then we take u0 = u and v0 = 1. We will now prove the existence of the decomposition u = u0 · v0 by contradiction. So assume that u ∈ Uψau + · U2a (k). We write u = u1 · v1 with u1 ∈ Uψau + , and v1 ∈ U2a (k). We have ψau < ψau1 , so u1  u and hence v1  1, while at the same time u1  U2a (k) since u  U2a (k). As we are assuming that a decomposition u = u0 · v0 does not exist, we have u1 ∈ Uψau1 + · U2a (k). We can then apply the same reasoning to u1 in place of u. Inductively we obtain sequences of elements ui ∈ Ua (k) − U2a (k) and vi ∈ U2a (k)∗ such that u = ui · vi and ψaui is a strictly increasing sequence of affine functions with derivative a. Using Proposition 9.4.16 we see that there exists a strictly increasing infinite sequence {ni } of positive integers such that ψaui = ψau + ni ca . From Proposition 9.4.3(1) we infer that the sequence {ui } converges to 1. Then the sequence {vi } converges to u. Now since U2a (k) is a closed subgroup of Ua (k), u must lie in U2a (k), a contradiction. We have thus established the existence of decomposition u = u0 · v0 with u0  Uψau0 + · U2a (k). The latter condition implies ψau0 ∈ Ψ and ψau0  ψau . If ψau0 > ψau , then Lemma 9.4.7 shows that ψav0 = ψau , and Proposition 9.4.3(6) v0 = 2ψau .  shows ψ2a Proposition 9.4.18

Let ψ ∈ Ψ  with derivative a ∈ Φ.

(1) If 2a  Φ, then ψ ∈ Ψ. (2) If ψ  Ψ, then 2ψ ∈ Ψ and there exists ψ  > ψ such that ψ  ∈ Ψ. (3) If ψ  ∈ Ψ is the smallest such that ψ   ψ (cf. Proposition 9.4.16), then Uψ = Uψ  · U2ψ . Proof (1) When 2a  Φ then the condition on ψ ∈ A∗ to lie in Ψ is the same as the condition to lie in Ψ . (2) The subgroup Uψ of Ua (k) is open, and so is Uψ+ . Thus Uψ − Uψ+ is a non-empty open subset of Ua (k). It cannot be contained in U2a (k), since the latter is the group of k-points of the Zariski-closed subgroup U2a of Ua . Therefore there exists u ∈ Uψ − Uψ+ that does not lie in U2a (k). We apply Lemma 9.4.17 to this u and obtain u = u0 · v0 . Then ψ  = ψau0 ∈ Ψ and since v0 . According to (1), 2ψ ∈ Ψ. ψ  Ψ we have 2ψau = ψ2a (3) The inclusion Uψ  · U2ψ ⊂ Uψ is immediate from U2ψ ⊂ Uψ . The converse inclusion Uψ ⊂ Uψ  · U2ψ is obvious if ψ  = ψ. So we assume that ψ  > ψ, that is ψ  Ψ. Since Uψ ∩ U2a (k) = U2ψ , it is sufficient to show that Uψ − U2a (k) is contained in Uψ  · U2ψ . Let u ∈ Uψ − U2a (k). Then ψau  ψ. According to Lemma 9.4.7, u can be written as u = u0 · v0 with ψau0 ∈ Ψ,

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v0 = 2ψau  2ψ. Hence, v0 ∈ U2ψ . As ψ  is the smallest ψau0 > ψ and ψ2a affine root larger than ψ, we conclude that ψau0  ψ , so u0 ∈ Uψ  . Therefore,  u = u0 · v0 ∈ Uψ  · U2ψ

Proposition 9.4.19 The sets Ψ  and Ψ constitute affine root systems in A∗ with derivative root system equal to Φ. Proof To prove the proposition, it is enough to verify the three conditions of Proposition 1.3.12. For each a ∈ Φ, Propositions 9.4.3(4) and 9.4.16 show that Ψa is an infinite set without accumulation points, and Proposition 9.4.18(2) extends this to Ψa . This verifies the third condition, and also shows that ∇Ψ = ∇Ψ  = Φ, thereby verifying the first condition. The second condition is verified by Remark 9.4.13 and Lemma 9.4.15.  Lemma 9.4.20 Let S be a maximal k-split torus of G and Z := ZG (S) be its centralizer in G. Let A be the apartment of B associated to S. Then there exists a unique o-smooth affine group scheme Z with generic fiber Z, and connected special fiber, such that for every non-empty bounded subset Ω of A, G(K)0Ω ∩ Z(K) = Z (O). Moreover, Z (o) = Z(k)0 . Proof We claim that the subgroup G(K)0x ∩ Z(K) is independent of the choice of x ∈ A. To prove this we observe that since S(k) commutes with Z(K), for every x ∈ A and s ∈ S(k), G(K)0s ·x ∩ Z(K) = G(K)0x ∩ Z(K). Now fix a point x0 ∈ A. Then the affine subspace of A spanned by the orbit S(k) · x0 is all of A. This implies the claim. 

Now G(K)0Ω ∩ Z(K) = x ∈Ω G(K)0x ∩ Z(K) equals G(K)0x0 ∩ Z(K) for any x0 ∈ A. Let G := Gx00 be the parahoric o-group scheme associated with x0 and S be the closed o-split torus of G with generic fiber S provided by Proposition B.2.4. Let Z be the centralizer of S in G . Then Z is a closed o-smooth subgroup scheme of G with generic fiber Z (see Propositions 2.11.2 and 2.11.3). The special fiber of Z is connected since the centralizer of any torus in a linear algebraic f-group is connected. Moreover, Z (O) = G (O) ∩ Z(K) = G(K)0x0 ∩ Z(K) ⊂ G(K)0 ∩ Z(K) = Z(K)0 . In view of the identity Z (O) = G(K)0x0 ∩ Z(K), the uniqueness of Z is an immediate consequence of Corollary 2.10.11. Since Z (O) ⊂ Z(K)0 , Z (o) ⊂ Z(k)∩Z(K)0 = Z(k)0 , the latter by Definition 2.6.23. On the other hand, as Z(k)0 is contained in G(k)0 , and it acts trivially on A(S) by Proposition 9.3.16, Z(k)0 ⊂ G(k)0x0 ∩ Z((k) = Z (o). This proves the last assertion. 

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Lemma 9.4.21 Let Ω be a non-empty bounded subset of A(S) and GΩ0 be the associated smooth affine o-group scheme with generic fiber G. Let S be the closed o-split torus of GΩ0 with generic fiber S. For a ∈ Φ, let Ua,Ω be the a-root group of GΩ0 with respect to S . The group G(k)0Ω (= GΩ0 (o)) is generated by Z(k)0 and the groups Uψ for all ψ ∈ Ψ with ψ(Ω) ⊂ R0 . Moreover, for a ∈ Φ, there exists an affine root ψaΩ , with derivative a, that takes only non-negative values on Ω, such that Ua,Ω (o) = UψaΩ · Uψ Ω (= UψaΩ if either 2a is not a root Ω  2ψ Ω ). or ψ2a a

2a

Proof Let Z be as in Lemma 9.4.20; Z has a natural identification with the centralizer of S in GΩ0 and Z (o) = Z(k)0 . According to Proposition 8.2.8(2), G(k)0Ω = GΩ0 (o) is generated by Z (o) and Ua,Ω (o), a ∈ Φ. By construction Ua,Ω (o) = Ua (k) ∩ G(k)0Ω , thus Ua,Ω (o) is the subgroup of elements of Ua (k) that fix Ω pointwise. In view of Proposition 9.4.16, there exists a smallest element ψ ∈ Ψ , with derivative a, such that ψ(Ω) ⊂ R0 . By Proposition 9.4.3(2) and 9.4.12, G(k)0Ω ∩ Ua (k) = Uψ . Using Proposition 9.4.18(3), we see that there is an affine root ψaΩ  ψ, with derivative  a, such that either ψaΩ = ψ, or 2a ∈ Φ and Uψ = UψaΩ · Uψ Ω . 2a

Theorem 9.4.22 Let G be a smooth affine o-group scheme with generic fiber G and connected special fiber G . Let Ru (G ) be the unipotent radical of G , and G be the maximal reductive quotient G /Ru (G ) of G . We assume that G contains a closed o-split torus S with generic fiber S which is a maximal k-split torus of G. We will denote the isomorphic image of the special fiber S of S in G by S. For a ∈ Φ, denote by Ua the a-root group of G with respect to S . We assume further that there exists an affine root ψaG , with derivative a, such that G  2ψ G , then U = U . On the other hand, if if either 2a is not a root or ψ2a a a ψaG G G 2a is a root and ψ2a < 2ψa , then Ua (o) = UψaG (o) · U2a (o). Then a ∈ Φ is a G = −ψ G . root of G with respect to S if and only if ψ−a a (Note that UψaG (o) = UψaG and U2a (o) = Uψ G (o) = Uψ G .) 2a

2a

Proof Since Γ  G (O) is a bounded group of automorphisms of B(G K ), it fixes a point, say z ∈ B(G K ). As z is fixed under Γ, and also under S (O), this point actually lies in the apartment A of B corresponding to the maximal ksplit torus S. Moreover, since G has connected fibers, there is a o-group scheme homomorphism G → Gz0 that is the identity on the common generic fiber G. We now observe that since for every a ∈ Φ, the subgroups Uψ±a G and Uψ G ±2a

G (z)  0 and ψ G (z)  0. Hence, ψ G = −ψ G are contained in Gz0 (o), so ψ±a −a a ±2a G G (z). On the other hand, if ψ G (z) = 0 = ψ G (z), if and only if ψa (z) = 0 = ψ−a a −a

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G and −ψ G have the same derivative and they both vanish at z, we then as ψ−a a G = −ψ G . Thus ψ G = −ψ G if and only if ψ G (z) = 0. Moreover, infer that ψ−a a −a a ±a G G (z) = 0, then as ψ G (z)  0, we conclude that ψ G in case ψ±a ±2a ±2a  2ψ±a . So 0 U±a = Uψ±a G , and Uψ G are the ±a-root groups of Gz in this case. ±a

The special fiber U a of Ua is the a-root group of G . Since o is Henselian, the homomorphisms Ua (o) → U a (f) and U2a (o) → U 2a (f) are surjective. Under the natural projection G → G /Ru (G ) = G, U a and U 2a are mapped respectively onto the a and 2a root groups Ua and U2a of G. The intersections Ru (G ) ∩ U a and Ru (G ) ∩ U 2a are a and 2a-root groups of Ru (G ), hence they are smooth connected f-split unipotent subgroups (see Remark 2.11.11), so the maps U a (f) → Ua (f) and U 2a (f) → U2a (f) are surjective. Therefore, the composite maps Ua (o) → Ua (f) and U2a (o) → U2a (f) are surjective. It is obvious that a is a root of G if and only if Ua (f) − U2a (f) is non-empty

 or, equivalently, U a (f) − U 2a (f) · Ru (G )(f)  ∅. Let us assume first that a(∈ Φ) is a root of G. Then there is an element u ∈ Ua (o) whose image u in U a (f) does not lie in U 2a (f) · Ru (G )(f). As Ua (o) = UψaG (o) · U2a (o), after replacing u with a suitable element in the coset u · U2a (o) we may (and do) assume that u lies in UψaG (o). Then ψau  ψaG . Since u  Ru (G )(f), Theorem 8.2.9 implies that u , u , and so also m(u)(= u  uu ) of Proposition 2.11.17, lie in G (o)(⊂ Gz0 (o)). Hence, according to Proposition 9.4.11, ψau (z) = 0. Thus 0 = ψau (z)  ψaG (z)  0, and we conclude that ψaG (z) = 0. Since −a is also a G (z) = 0. root of the reductive group G, by an analogous argument we see that ψ−a G G Therefore, ψ−a = −ψa . G . For simplicity of notation, To prove the converse, we assume that ψaG = −ψ−a G we will denote ψa by ψ. Then, as we observed above in this proof, U ±a = U ±ψ and these are ±a-root groups of Gz0 . But as ψ is an affine root, Uψ = Uψ (o) cannot be contained in Uψ+ · U2a (k), and hence there is an x ∈ Uψ such that xU2a (k) ∩ Uψ+ = ∅. Therefore, in particular, for any v ∈ U2a (o)(⊂ Uψ ), if we set u = xv then ψau = ψ and hence, ψau (z) = ψ(z) = 0. Now Proposition 9.4.9 implies that u , u  and m(u) = u  uu  belong to Gz0 (o). Hence, u , u  ∈ U−a (o) ⊂ G (o), and according to Theorem 8.2.9, the image of u(= xv) in G (f) cannot lie in Ru (G )(f). Thus U a (f)  U 2a (f) · Ru (G )(f) which implies that a  is a root of G. Let S be a maximal k-split torus of G and A be the corresponding apartment of B. Let Ω be a non-empty bounded subset of A and GΩ0 be the associated Bruhat–Tits o-group scheme. Let G Ω0 be the special fiber of GΩ0 and G Ω be the maximal reductive quotient of the special fiber. Let S be the closed o-split torus of GΩ0 with generic fiber S and S be the isomorphic image of the special

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fiber S of S in G Ω . There are natural identifications of X∗ (S) with X∗ (S ), X∗ (S ) and X∗ (S). Therefore there are also natural identifications of X∗ (S) with X∗ (S ), X∗ (S ) and X∗ (S). Let Φ Ω be the set of derivatives of affine roots that vanish identically on Ω. The set ΦΩ was studied, in the special case of Ω = {x}, in Proposition 1.3.35. For a ∈ Φ, let the coroot a∨ ∈ X∗ (S) = X∗ (S) be as in 2.6.4 and let Φ∨Ω = {a∨ | a ∈ Φ Ω }. Proposition 9.4.23 The relative root datum of G Ω with respect to S is (X∗ (S), Φ Ω, X∗ (S), Φ∨Ω ). Proof According to Lemma 9.4.21 there exists an affine root ψaΩ such that Ω  2ψ Ω ). Now Ua (o) = UψaΩ · Uψ Ω (= UψaΩ if either 2a is not a root or ψ2a a 2a Theorem 9.4.22 is applicable, and we deduce from it that a ∈ Φ is a root of G Ω Ω = −ψ Ω . if and only if ψ−a a Ω take only non-negative values on Ω, if ψ Ω = −ψ Ω , Since both ψaΩ and ψ−a −a a Ω then the affine roots ψ±a must vanish identically on Ω. On the other hand, if Ω vanish identically on Ω, then since the affine roots −ψ Ω and both ψaΩ and ψ−a a Ω ψ−a have the same derivative and coincide on Ω, they are equal. The second assertion of the proposition is now obvious.  Corollary 9.4.24 Let F be a facet of B. Let AG be the maximal k-split torus in the center of G, and let A F be the maximal f-split torus in the center of G F . Then dim(A F ) − dim(AG ) is equal to the dimension of F. Proof Proposition 9.4.23 shows that dim(A F )−dim(AG ) equals the difference of the ranks of Φ and Φ F , which equals dim(A∗ ) − dim(A∗F ) by Proposition 1.3.35, which in turn equals dim(F).  Proposition 9.4.25 Let Gx0 and Gy0 be the Bruhat–Tits group schemes associated to the points x, y ∈ B and Px := Gx0 (o) and Py := Gy0 (o) be the corresponding parahoric subgroups of G(k). We denote the special fiber of Gx0 (respectively, Gy0 ) by G x0 (respectively, G y0 ) and the maximal reductive quotient of this special fiber by G x (respectively, G y ). Let P+y denote the kernel of the natural surjective homomorphism Py → G y (f). Then the image of Px ∩ Py in G x (f) under the natural surjective homomorphism G x0 (o) → G x (f) is the group of f-rational elements of a parabolic f-subgroup Pxy of G x . Moreover, the image of Px ∩ P+y in G x (f) is the group of f-rational elements of the unipotent radical of Pxy . The subgroup denoted above by P+y is also denoted P y,0+ , see 13.2.6.

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Proof Let A be an apartment in B that contains both x and y and let S be the corresponding maximal k-split torus of the derived subgroup of G. Let Z be the centralizer of S in G. We will denote the closed o-split tori of Gx0 and Gy0 with generic fiber S by the same symbol S . Let S be the special fiber of S . Let Z be as in Lemma 9.4.20. The centralizers of S in Gx0 and Gy0 have natural identification with Z . The image of S in both G x and G y will be denoted by S; it is a maximal f-split torus of G x and Gy . For a ∈ Φ = Φ(S, G), let Ua, x (respectively, Ua, y ) be the a-root group of 0 Gx (respectively, Gy0 ) with respect to S . Let Ψx (respectively, Ψy ) be the set of affine roots which vanish at x (respectively, y) and let Φx (respectively, Φy ) be the set of derivatives of affine roots in Ψx (respectively, Ψy ). According to Proposition 9.4.23 (for Ω = {x}, {y}), the root systems of G x and Gy with respect to S are Φx and Φy respectively. For a ∈ Φx , let ψa ∈ Ψx be the affine root with derivative a. Then, in the notation of 9.4.5, Ua, x = Uψa . + = {ψ ∈ Ψ | ψ(y) > 0} and Φ (⊂ Φ ) Let Ψxy = {ψ ∈ Ψx | ψ(y)  0}, Ψxy x xy x be the set of derivatives of the affine roots in Ψxy . We define Φ+xy (⊂ Φxy ) similarly. It is obvious that Φxy is a parabolic subset of Φx . For a ∈ Φxy , as ψa (y)  0, Ua, x = Uψa ⊂ Ua, y . This implies that Px ∩ Py contains Z (o) = Z(k)0 and Ua, x (o) for all a ∈ Φxy . Let Z be the special fiber of Z , and for a ∈ Φ, U a, x be the special fiber of Ua, x . Let Z and Ua, x be respectively the images of Z and U a, x in G x ; Z is the centralizer of S in G x . Since o is Henselian, Z (o) maps onto Z (f) and Ua, x (o) maps onto U a, x (f). Now let R be the unipotent radical of G x0 . Then the centralizer of S in R is a smooth connected f-split unipotent subgroup, and R ∩ U a, x is the a-root group of R, and hence it is also a smooth connected f-split unipotent group (cf.Remark 2.11.11). These observations imply that the images of Z (f) and U a, x (f) in G x (f) are Z(f) and Ua, x (f) respectively. Let Pxy be the parabolic f-subgroup of G x generated by the centralizer of the maximal f-split torus S and the a-root groups for a in the parabolic subset Φxy of Φx . Then Z(f) and Ua, x (f), for a ∈ Φxy , generate Pxy (f). Hence, the image of Px ∩ Py in G x (f) is Pxy (f). To prove the second assertion of the proposition, we observe that a ∈ Φxy lies in Φ+xy if and only if −a  Φxy . Hence, the set of roots of the unipotent radical U xy of Pxy is Φ+xy . For a ∈ Φ+xy , since ψa (y) > 0, Ua, y (o) is clearly contained  in Px ∩ P+y . This implies that the image of Px ∩ P+y in G x (f) is Uxy (f). Remark 9.4.26 Before we proceed, we recall the concepts of absolute and relative root systems. Given a maximal k-split torus S ⊂ G, the root system

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Φ = Φ(S, G) is referred to as the relative root system. On the other hand, if  ⊂ G is a maximal k-torus of G containing S, then Φ(T,  G) is referred to as the T  absolute root system. While Φ(T, G) is always reduced, Φ(S, G) need not be.  G). The elements of Φ are the non-zero restrictions to S of the elements of Φ(T,  G) may well have zero restriction to S (in fact, Note that some elements of Φ(T, when G is anisotropic, then Φ = ∅).  G) and Φ(S, G) there is yet another root system: if we choose In between Φ(T,  be its centralizer, then T  is a maximal a special k-torus T containing S and let T k-torus of G, because TK is a maximal K-split torus in G K and G K is quasi-split.  G) and Φ(S, G) we now also have Φ(T, G). The restriction In addition to Φ(T,   G) → Φ(T, G) → Φ(S, G) ∪ {0}. map Φ(T, G) → Φ(S, G) ∪ {0} factors as Φ(T,  G) has zero restriction to Φ(T, G). Note that no element of Φ(T, From the point of view of the k-group G, the root system Φ(T, G) is neither absolute nor relative. It plays an intermediary role. It is usually reduced, but can also be non-reduced (this happens precisely when G contains an almost direct factor isogenous to the ramified special unitary group in an odd number of variables). 9.4.27 Absolute affine root system We now introduce the affine analog of the absolute root system. Given a special k-torus T (containing the maximal k-split torus S), let A be the corresponding special k-apartment, so that A = AΓ is the apartment corresponding to S. We have the affine root system Ψ(T, G) ⊂ A∗ ( Proposition 6.4.8), as well as the affine root system Ψ = Ψ(S, G) ⊂ A∗ . We shall refer to ΨK = Ψ(T, G) as the absolute affine root system, and to Ψ = Ψk = Ψ(S, G) as the relative affine root system. The absolute affine root system is stable under the action of the Galois group Γ of K/k, since both G and T are defined over k. Moreover, it is always reduced (even if Φ(T, G) is non-reduced), cf. Proposition 6.4.8. We will see many examples in §10.7 where the relative affine root system Ψ(S, G) is non-reduced. The following proposition states that the relative affine root system Ψ(S, G) consists of the non-constant restrictions to A of the elements of the absolute affine root system Ψ(T, G). It is thus the affine analog of the relationship between the absolute and relative root systems recalled in Remark 9.4.26. Proposition 9.4.28 Let A be a special k-apartment of B(G K ) that contains A and let T (⊃ S) be the corresponding special k-torus. Let ψ be an affine root of G with respect to S with derivative a ∈ Φ(S, G), and b ∈ Φ(T, G) be any root which restricts to a. Then there is an absolute affine root ψb ∈ Ψ(T, G) which has derivative b and which restricts to ψ on A.

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Conversely, if the restriction to A of an absolute affine root is not constant, then this restriction is a relative affine root. Proof The affine root ψ vanishes at some point of A, say z. If we consider the point z as a point of A, then the corresponding parahoric O-group scheme is by construction the base change of Gz0 from o to O, hence the maximal reductive quotient of its special fiber is the base change of G z to f. Let S and T (⊃ S) be the f-tori of G z corresponding to S and T respectively. Proposition 9.4.23 (for Ω = {z}) implies that since the affine root ψ, with derivative a, vanishes at z, a is a root of G z with respect to S. Therefore b is a root of (G z ) f with respect to T f . According to the same proposition applied to (G K ,TK ), there is an affine root ψb of G K with respect to TK whose derivative is b and which vanishes at z. The restriction of ψb to A is a real valued affine function on A with derivative a and which vanishes at z, so it coincides with the affine root ψ. To prove the converse, we begin by observing that if a real valued affine function on A does not vanish at any point, then it is constant. Now let ψb be an affine root of G K with respect to TK with derivative b that has non-constant restriction ψ to A. Let a be the restriction of b to S. Then the derivative of ψ is a. Let z be a point of A where the non-constant function ψ vanishes. Then ψb (z) = 0, so according to Proposition 9.4.23 for (G K ,TK ), b is a root of (G z ) f with respect to T f . Then a, being the restriction of b to S, is a root of G z with respect to S. Proposition 9.4.23 (for Ω = {z}) implies that there is an affine root ψa with respect to S that has derivative a and which vanishes at z. Since ψ and ψa are affine functions on A with same derivative and they both vanish  at z, they must be equal. Thus ψ = ψb |A = ψa is an affine root. Recall from Definition 1.3.7 that for an affine root ψ we have the affine root hyperplane Hψ ⊂ A. 9.4.29 The set of all affine root hyperplanes is stable under the action of N(k) on A since the set Ψ of affine roots is stable under the action of N(k) (Lemma 9.4.15). From Proposition 9.4.16, we see that the set of affine root hyperplanes is locally finite. Given an a ∈ Φ and u ∈ Ua (k)∗ , according to Proposition 9.4.18, at least one of ψau or 2ψau is an affine root. Note that Hψau = H2ψau . On the other hand, as we have noted in Remark 9.4.13, given an affine root ψ with derivative a, there is a u ∈ Ua (k)∗ such that ψ = ψau . Thus the sets {Hψ | ψ ∈ Ψ} and {Hψau | a ∈ Φ, u ∈ Ua (k)∗ } are equal. Proposition 9.4.30 (1) Let ψ be an affine root and z be any point of Hψ . Then the facet F of A that contains z is contained in Hψ . Thus Hψ is the

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union of the facets contained in it. This implies that if Hψ intersects a facet then it contains it. (2) Let F be a facet of A of codimension 1. Then there exists an affine root ψ such that Hψ contains F.  (3) The connected components of A − ψ ∈Ψ Hψ are precisely the chambers contained in A. Proof We begin by recalling from Remark 9.4.13 that every affine root equals ψau for an a ∈ Φ and u ∈ Ua (k)∗ . Let ψ = ψau and Gz0 be the parahoric o-group scheme associated to z. Then we have seen that u and the element m(u) lie in Pz := Gz0 (o) (Lemma 9.4.6 and Proposition 9.4.9). Moreover, m(u) acts on A by reflection rψ in Hψ . So m(u) · Aψ = A−ψ . Now since the set of points of A that are fixed by u is Aψ , the set of points of A fixed by m(u)um(u)−1 (∈ Pz ) is m(u) · Aψ = A−ψ . This implies that the set of points of A fixed under the parahoric subgroup Pz is contained in Aψ ∩ A−ψ = Hψ . Since the facet F(⊂ A) containing z is fixed pointwise by Pz , we see that this facet must be contained in Hψ . To prove (2), let z ∈ F, and Gz0 = GF0 be the parahoric o-group scheme associated to the facet F. Since F is of codimension 1, the f-rank of the derived group of G z is 1, cf. 9.3.2. Let a(∈ Φ) be a root of G z . Then according to Proposition 9.4.23, there is an affine root ψ with derivative a that vanishes at z. So Hψ contains z. Now (1) implies that Hψ contains F. To prove (3), it is enough to remark that every chamber is bounded by its faces of codimension 1, and then apply (2).  Corollary 9.4.31 Let A be an apartment of B. The facets of A defined in 9.2.4 coincide with the facets determined by the affine root hyperplanes as in Remark 1.3.16. Proof We may replace G by its adjoint group. Using the product structure of B(G) we may then pass to the case that G is k-simple. Then B is a simplicial complex according to Proposition 9.3.10. We will use Proposition 9.4.30. Part (3) of that proposition gives the desired statement for chambers. Given a facet F of codimension 1, part (2) shows that it is contained in some hyperplane Hψ , and part (1) shows that for ψ  ∈ Ψ such that Hψ   Hψ , then no point of F is contained in Hψ ∩ Hψ  . Let C be a chamber of A of which F is a face and let ψ = ψ0, . . . , ψn be the corresponding simple affine roots. Then each face of F is also a face of C and hence contained not only in Hψ0 but also in a hyperplane Hψi for some i  0. It follows that F  is a full connected component of Hψ − ψ  :Hψ  Hψ (Hψ ∩ Hψ  ). The case of facets of higher codimension now proceeds analogously. 

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9.4.32 Let Ω be a non-empty bounded subset of the apartment A of B corresponding to a maximal k-split torus S of G. Let GΩ0 be the Bruhat–Tits ogroup scheme, corresponding to Ω, with generic fiber G and connected special fiber G Ω0 . Let S be the closed o-split torus of GΩ0 with generic fiber S, and S be the special fiber of S . We denote by G Ω the maximal reductive quotient of G Ω0 , and S will denote the isomorphic image of S in G Ω . We denote by ΨΩ the set of affine roots which vanish identically on Ω. Then,

Ω is contained in the intersection H Ω := ψ ∈ΨΩ Hψ of affine root hyperplanes. Let Φ Ω be the set of derivatives of elements of ΨΩ . According to Proposition 9.4.23, Φ Ω is the set of roots of G Ω with respect to the maximal f-split torus S. Since every affine root hyperplane is a polysimplicial subcomplex of A, H Ω is also a polysimplicial subcomplex of A. Let SΩ be the maximal subtorus of S contained in the intersection of kernels of a ∈ Φ Ω . Let V(SΩ ) be the subspace R ⊗Z X∗ (SΩ ) of V(S) = R ⊗Z X∗ (S). Lemma 9.4.33 We will use the notation introduced above. Let S Ω be the closed o-subtorus of S whose generic fiber is SΩ , and S Ω be the special fiber of S Ω . Then the isomorphic image S Ω of S Ω in G Ω is the central torus of the latter. Moreover, V(SΩ ) acts on H Ω transitively. Proof To see that S Ω (⊂ S) is the maximal central torus of G Ω , it is enough to note that S Ω is the maximal subtorus of S contained in the intersection of kernels of all the roots of G Ω with respect to S. For a ∈ Φ Ω , we denote by ψa the unique affine root in ΨΩ with derivative a. We fix an element x of Ω. Now for v ∈ V(S), the translate v + x lies in H Ω if and only if ψa (v + x) = a(v) + ψa (x) = a(v) = 0 for all a ∈ Φ Ω . This condition is equivalent to the assertion that v lies in V(SΩ ). On the other hand, V(S) acts transitively by translations on the apartment A, so we see that V(SΩ )  acts transitively on H Ω . 9.4.34 Affine Weyl group The subgroup of affine automorphisms of the apartment A generated by reflections in the root hyperplanes is called the affine Weyl group and it is denoted by Waff . We recall that if ψ is an affine root with derivative a ∈ Φ, and u ∈ Ua (k)∗ is such that ψ = ψau , then m(u) acts on A by reflection in the hyperplane Hψ (Proposition 9.4.9). Thus Waff ⊂ N(k)/Z(k)b . According to the following proposition, for every simply connected semi-simple group, this containment is an equality. Proposition 9.4.35 Assume that G is semi-simple and simply connected. Then the inclusion of Waff in N(k)/Z(k)b is an equality. Proof

We fix a chamber C in A and let Δ be the basis of the affine root system

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Ψ determined by C. For ψ ∈ Δ, the root hyperplane Hψ is a wall of C, that is it contains a face of C of codimension 1. According to Proposition 9.4.9, the reflection rψ in the root hyperplane Hψ is the action on A by an element m(u) ∈ N(k). Lemma 2 of [Bou02, Chapter V, §3] implies that Waff is generated by rψ , ψ ∈ Δ. Moreover, according to Theorem 1(iii) of [Bou02, Chapter V, §3], Waff acts simply transitively on the set of chambers of A. We know, from Proposition 9.3.19 that N(k) acts transitively on the set of chambers of A. Note that we are using here Corollary 9.4.31. We assume now that G is simply connected. Then if an element of G(k) stabilizes a facet, it fixes the facet pointwise (9.3.26). Hence, if an element n ∈ N(k) stabilizes C, it fixes it pointwise. As n acts on A by an affine automorphism, and C is a non-empty open subset of A that is fixed pointwise under this action, we conclude that the action of n on A is trivial. So by Proposition 9.3.16, n belongs to Z(k)b . Thus we see that N(k)/Z(k)b also acts simply transitively on the set of chambers of A. This proves the equality of  Waff and N(k)/Z(k)b when G is simply connected.

9.5 Completion of the Proof of the Main Result We now come to the formal proof of Theorem 9.2.7. All claims have already been proved, what remains is just to provide the appropriate references. Axiom 4.1.1 was established in Theorem 9.3.21. Axiom 4.1.2 follows immediately from the corresponding axiom for G K . In Proposition 9.3.16 we proved Axiom 4.1.4. For Axiom 4.1.6, AS 1 was proved in Proposition 9.4.9, AS 2 and AS 3 were proved in Proposition 9.4.19, AS 4 was proved in Proposition 9.4.18, AS 5 was proved in Proposition 9.4.35 (see also the paragraph preceding that proposition), AS 6 was proved in Corollary 9.4.31. Axiom 4.1.8 was proved in Proposition 9.4.3, Lemma 9.4.6, and Lemma 9.4.21. Axiom 4.1.9 was proved in Propositions 9.3.24 and 9.3.25. As mentioned in 9.2.1, Proposition 7.9.5 provides an action of the Galois group Γ of K/k on B(G K ). This verifies Axiom 4.1.17. The group schemes for Axiom 4.1.20 were defined in 9.2.5. Indeed, let S be a maximal k-split torus of G and let T be a special k-torus containing S. Let Ω ⊂ A(S) be a non-empty bounded subset. It is then a Γ-invariant subset of A(TK ). We have GΩ1 (o) = GΩ1 (O) ∩ G(k) = G(K)1Ω ∩ G(k) = G(k)1Ω,

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using G(k)1 = G(K)1 ∩ G(k). In the same way, GΩ0 (o) = GΩ0 (O) ∩ G(k) = G(K)0Ω ∩ G(k) = G(k)0Ω, using the identity G(k)0 = G(K)0 ∩ G(k) proved in Proposition 9.3.24. The remaining properties over k follow from those over K. In Axiom 4.1.22, conditions (1) and (2) are geometric, so follow from the corresponding condition over K. The same is true for condition (7). If F is a facet of B and F is the unique facet of B(G K ) containing F, then G(k)0F = = G(K)0+ G(K)0F ∩ G(k) and G(k)0+ F ∩ G(k), the latter due to the fact that f F 0

is perfect, so the unipotent radical of G F ×f f is defined over f. Conditions (3) and (5) also follow from those over K because of the bijective correspondence between facets of B and Γ-invariant facets of B(G K ) (we are using again Corollary 9.4.31). Condition (4) was established in Proposition 9.3.3. Condition (6) was proved in Proposition 9.4.23. Axiom 4.1.27 has also been established. The properties of the building and integral models hold by definition, the existence of special k-tori was proved in Proposition 9.3.4 (this can be applied to Z for a maximal k-split torus S to ensure that any such S is contained in a special k-torus), and the statements about compatibility of affine roots are proved in Proposition 9.4.28. Henceforth, we will denote B = B(G K )Γ by B(G) and call it the Bruhat–Tits building of G(k).

9.6 Valuation of Root Datum Recall from Definition 6.1.2 the notion of a valuation of a root datum. In Chapter 6 this notion played a fundamental role, and we defined the apartments of the building of a reductive k-group as equipollence classes of valuations of root data. For a quasi-split groups we could assign to a maximal k-split torus a canonical such equipollence class. For non-quasi-split groups we did not have a construction of valuations of root data; the material of Chapters 6 and 7 was then conditional on the existence of valuations of root data. Earlier in this chapter we defined the apartment of B(G)(= B(G K )Γ ) corresponding to a maximal k-split torus S of G as the Γ-fixed points in the apartment of B(G K ) corresponding to any special k-torus T containing S. Thus, by definition, the apartment for S consists of the Γ-fixed valuations of the root datum of (G K ,TK ). We will now show how the points of the apartment of S give rise to valuations of the root datum of (G, S) in a natural way. Thus, we now obtain,

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for any connected reductive group G and a maximal k-split torus S, a canonical equipollence class of valuations of the root datum. With this equipollence class, the material of Chapters 6 and 7 will give a new interpretation of the building B(G) constructed here. Strictly speaking, this material will produce a new building for G, but we will see in 9.6.6 that this new building is canonically identified with B(G). This identification will provide an interpretation of the points of B(G) as valuations of root data, of the affine root system in terms of valuations of root data, and a group-theoretic description of the parahoric groups and concave function groups in terms of the relative root system of G. Let S be a maximal k-split torus of G and A the corresponding apartment of B(G), and N and Z be respectively the normalizer and centralizer of S in G. Construction 9.6.1 Valuation of root groups Given a point x ∈ A, and a ∈ Φ, we construct a valuation ϕax of the root group Ua (k) as follows. For u ∈ Ua (k)∗ , set ϕax (u) = ψau (x), where ψau is the real valued affine function on A as in Proposition 9.4.3(3). We define ϕax (1) = ∞. For x ∈ A, a ∈ Φ, and r ∈ R, we now define Ua, x,r = (ϕax )−1 ([r, ∞]) as in Definition 6.1.2. Let ψ denote the real valued affine function on A with derivative a ∈ Φ such that ψ(x) = r. Then for u ∈ Ua (k)∗ , ϕax (u) = ψau (x)  r if and only if ψau  ψ, which by Proposition 9.4.3(5) is equivalent to the condition u ∈ Uψ . Thus Ua, x,r = (ϕax )−1 ([r, ∞]) = Uψ is a subgroup of Ua (k). Since Uψ is bounded, we infer that ϕax (Ua (k)∗ ) is an unbounded subset of R. Thus we have verified properties V 0 and V 1 of Definition 6.1.2 for (ϕax )a ∈Φ . Remark 9.6.2 Let T be a special k-torus containing S. Let A ⊂ B(G K ) be the corresponding apartment. Since A = AΓ , the point x can be viewed as a point of A, hence by construction (cf. §6.1) as a valuation of the root datum of (G K ,TK ). We will denote this valuation by (ϕbx )b ∈Φ(TK ,G K ) . For b ∈ Φ(TK , G K ) and u ∈ Ub (K)∗ := Ub (K) − {1}, let ψbu denote the real valued affine function on A, with derivative b, associated to u by Proposition 9.4.3(3) (for K in place of k). By Proposition 9.4.9, m(u) acts on A by reflection in the vanishing hyperplane Hψbu of ψbu . On the other hand, according to Proposition 6.2.1, m(u) acts on A by reflection in the hyperplane Hu := {z ∈ A | zb (u) = 0}. Hence, Hψbu = Hu . Now we fix v ∈ V(TK ) := R ⊗Z X∗ (TK ) such that xv := x + v ∈ Hu . Then xv, b (u) = xb (u) + b(v) = 0. But as x + v ∈ Hψbu , ψbu (x + v) = ψbu (x) + b(v) = 0. We conclude that ψbu (x) = xb (u) for u ∈ Ub (K)∗ . In other words, Construction 9.6.1, applied to the case k = K, recovers the intrinsic interpretation of the points of A as valuations of the root datum of (G K ,TK ).

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The valuation (ϕbx )b ∈Φ(TK ,G K ) is Γ-invariant, that is, for b ∈ Φ(TK , G K ), u ∈ Ub (K)∗ and γ ∈ Γ, ϕγx·b (γ(u)) = ϕbx (u). As in Definition 6.1.2, for b ∈ Φ(TK , G K ) and r ∈ R, we have the subgroup Ub, x,r of Ub (K). Recall that Ub, x,r = (ϕbx )−1 ([r, ∞]). The subgroups Ua, x,r of Ua (k) and Ub, x,r of Ub (K) are related as follows. Lemma 9.6.3

Let a ∈ Φ(S, G). For x ∈ A ⊂ A and r ∈ R we have   Ub, x,r , Ua, x,r = Ua (k) ∩ b

where the product runs over those non-divisible b ∈ Φ(TK , G K ) that restrict to either a or 2a. Proof Replacing x by x − (r/2)a∨ ∈ V(S ) we may assume r = 0 by Lemma 6.1.6. According to Lemma 9.4.21, the group G(k)0x equals the group Px of Definition 7.4.1. Therefore we can apply Proposition 7.3.12(3) and conclude that Ua, x,0 = G(k)0x ∩ Ua (k). By Proposition 9.3.24, G(k)0 = G(k) ∩ G(K)0 .  On the other hand, Ua (k) = G(k) ∩ b Ub (K), where the product runs over those non-divisible b ∈ Φ(TK , G K ) that restrict to either a or 2a. Using these facts and Proposition 7.3.12, now for the group G(K)0x , we have     Ub (K) = Ua (k)∩ Ub, x,0 , Ua, x,0 = G(k)0x ∩Ua (k) = G(k)∩G(K)0x ∩ b

b

where the product runs over those non-divisible b ∈ Φ(TK , G K ) that restrict to either a or 2a.  Remark 9.6.4 Let ψ and η be real valued affine functions on A with derivatives a, b ∈ Φ. If b is a real positive multiple of a and ma + nb is also a root for some positive integers m and n, then a = b, m = 1 = n and 2a is a root. It is easily seen from the commutator relations given in §2.9(d), and Lemma 9.6.3, that in this case the commutator subgroup (Uψ , Uη ) is contained in Uψ+η . The following theorem proves that Construction 9.6.1 does indeed give a ω-compatible valuation of the root datum of (G, S). Theorem 9.6.5 For x ∈ A, the functions ϕ x := (ϕax )a ∈Φ of Construction 9.6.1 constitute a valuation of the root datum (Ua (k))a ∈Φ , compatible with ω, in the sense of Definition 6.1.2. The map x → ϕ x is equivariant under V(S)  N(k). In particular, its image constitutes an equipollence class. Proof We have already verified properties V0 and V1 of 6.1.2 above in Construction 9.6.1.

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For a ∈ Φ and v ∈ V(S), the identity ϕax+v = ϕax + a(v) is immediate. For nun−1 = n · ψ u , which implies n ∈ N(k), Lemma 9.4.15 gives the identity ψna a ϕ nx = n · ϕ x . This proves the claimed equivariance. Moreover, we see that −1

ϕax (u) − ϕbx (nun−1 ) = ψau (x) − ψbnun (x) = ψau (x) − ψau (n−1 · x) = a(x − n−1 · x). This implies that the function u → ϕax (u) − ϕbx (nun−1 ) is constant on Ua (k)∗ , verifying property V2. Taking n ∈ Z(k), property V6 (that is, ω-compatibility of the valuation) follows from Proposition 9.3.16. Property V3 is verified by Proposition 9.4.8 and Remark 9.6.4. From the results in Corollary 9.4.10 we see that for u ∈ Ua (k)∗ , if m(u) = x (u  ) = −ϕ x (u) = ϕ x (u  ), and is as in Proposition 2.11.17, then ϕ−a a −a x x −1 x = 2ϕ x on U (k)∗ . ϕa (u) = ϕa (u ). Moreover, if 2a is also a root, then ϕ2a 2a a Thus we have verified properties V4 and V5. 

u  uu 

9.6.6 Comparison of two building constructions We temporarily write B1 for the building (B =)B(G) of G constructed in this chapter, thus B1 = B(G K )Γ , and write B2 for the building of G constructed in Chapter 7, where we supply to each maximal k-split torus S ⊂ G the equipollence class of valuations of the root datum of Theorem 9.6.5. Given S, we write A1 and A2 for the corresponding apartments in the two buildings B1 and B2 . The map z → ϕz of Theorem 9.6.5 provides an isomorphism of affine spaces A1 → A2 . Given a ∈ Φ, and ψ2 ∈ A∗2 with derivative a, the group Uψ2 of Definition 6.3.1 was defined as Ua,ϕ,ψ2 (ϕ) for an arbitrary point ϕ ∈ A2 . This definition unwinds to {u ∈ Ua (k) | ϕa (u)  ψ2 (ϕ)}. If z ∈ A1 is the point corresponding to ϕ, then the condition defining Uψ2 becomes ψau (z)  ψ1 (z), where ψ1 ∈ A∗1 is the element corresponding to ψ2 under the isomorphism A1 → A2 . But the latter equals Uψ1 , cf. Proposition 9.4.3(5). From this we conclude that the affine root systems Ψ and Ψ  of Definition 6.3.4 are identified under the isomorphism A1 → A2 with the affine root systems of 9.4.12. According to Lemma 9.4.21, the stabilizer G(k)0z of z ∈ A1 for the action of G(k)0 is generated by Z(k)0 and the groups Uψ for all affine roots with ψ(z)  0. On the other hand, Proposition 7.6.4 shows that G(k)0ϕ is generated by Z(k)0 and Ua,ϕ,0 for all a ∈ Φ. If ψ is the smallest element of Ψa such that ψ(ϕ)  0, then Ua,ϕ,0 = Uψ . If ψ1, ψ2 ∈ Ψ are the smallest elements with derivatives a respectively 2a that are larger than or equal to ψ respectively 2ψ, then Uψ = Uψ1 · Uψ2 , cf. Proposition 6.3.8. We conclude G(k)0z = G(k)0ϕ . Corollary 4.4.5 shows that the isomorphism A1 → A2 extends to a G(k)equivariant bijection B1 → B2 . Assume now that G is quasi-split. Then, in addition to the buildings B1 and B2 discussed so far, there is a third building B3 , namely the one constructed

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in Chapter 7 for the canonical equipollence classes of valuations of root data that contain the Chevalley valuations, as discussed in Chapter 6. Let S ⊂ G be a maximal k-split torus and write A1 , A2 , and A3 for the associated apartments in the three buildings. In addition to the isomorphism A1 → A2 just discussed, we have the isomorphism A3 → A2 of Construction 6.1.21. The composed isomorphism A3 → A1 is an isomorphism of affine spaces, and the points of both its source and target are by construction valuations of the root datum of (G, S). In terms of that interpretation, this isomorphism is simply the identity: this follows from (6.1.2) and Lemma 9.6.3, which compare the valuations in both affine spaces to the valuations in the space A2 . The preceding argument now produces a natural identification B1  B3 .

9.7 Levi Subgroups Let M ⊂ G be a Levi k-factor of a parabolic K-subgroup of G and Z(M) be its center. Let B(G) M ⊂ B(G) be the union of those apartments of B(G) that correspond to maximal k-split tori of M. In this section, we will relate the set B(G) M to the Bruhat–Tits building of M. Proposition 9.7.1 B(G) M = B(G K )ΓM . Proof Let A be the maximal k-split torus in Z(M) ∩ Gder . According to Proposition 9.3.11(2), the set B(G K ) M equals the set B(G K ) A, and Proposition  9.3.11(1) shows that B(G K )ΓA equals B(G) M . Let S ⊂ M be a maximal k-split torus and let T be a special k-torus of M containing S, which exists according to Proposition 9.3.4. We have the associated apartments A(S, G) in B(G) and A(T, G) in B(G K ), and A(S, G) =  for the A(T, G)Γ . Write A for the maximal k-split torus in Z(M) ∩ Gder , and A  is defined over k and A is the maximal K-split torus in Z(M) ∩ Gder . Then A  Consequently, V(A) = V( A)  Γ. maximal k-split torus in A.  → A(T, M). It restricts Recall from §6.7 the isomorphism πT : A(T, G)/V( A) Γ to the isomorphism πT : A(S, G)/V(A) → A(S, M). In §9.6 the apartments A(S, G) and A(S, M) were interpreted in terms of valuations of the root datum of (G, S) and (M, S). Lemma 9.6.3 shows that the valuation of the root datum of (G, S) associated to a given x ∈ A(S, M), when restricted to M, coincides with the valuation of the root datum of (M, S) associated to πTΓ (x). Therefore πTΓ coincides with the map πS of §6.7. Consider now the composition of the V(A)-invariant map πS : A(S, G) → A(S, M) with the inclusion A(S, M) → B(M). Putting these maps together for

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all maximal k-split tori S of M we obtain a V(A)-invariant and M(k)-equivariant $ map π : S A(S, G) → B(M). Proposition 9.7.2 (1) There is an action of V(A) on B(G) M whose restriction to each A(S, G) is the natural action of V(A) in A(S, G) by translation. $ (2) π factors through the natural map S A(S, G) → B(G) M induced by the inclusions A(S, G) → B(G) M . The resulting map π : B(G) M → B(M) is V(A)-invariant and it induces a bijective M(k)-equivariant map π : B(G) M /V(A) → B(M).

(9.7.1)

Proof In the proof we will use the following claim. Given a maximal k-split torus S ⊂ M and x ∈ A(S, G), G(k)0x ∩ M(k) ⊂ M(k)1π(x) . To see this, choose a system of positive roots Φ+ ⊂ Φ = Φ(S, G). It determines a chamber in A(S, G) whose closure contains x, and a chamber in A(S, M) whose closure contains M π(x). Let IG x and Iπ(x) be the corresponding Iwahori subgroups. It is clear that M G M ·N M Iπ(x) ⊂ Ix . We have M(k) = Iπ(x) M · Iπ(x) , where we have abbreviated N M = NM (S)(k) . Using the N M -equivariance of the map π : A(S, G) → A(S, M) we see M M · N M · Iπ(x) ) G(k)1x ∩ M(k) = G(k)1x ∩ (Iπ(x) M M = Iπ(x) · (G(k)1x ∩ N M ) · Iπ(x)

⊂ M(k)1π(x), proving the claim. Let S1, S2 be two maximal k-split tori of M and let x ∈ B(G) M be a point that lies in both A(Si , G). We apply Proposition 9.3.12 to A ⊂ S1 ∩ S2 and Ω = {x}, to obtain an element m ∈ G(k)0x ∩ M(k) that maps S1 to S2 . Thus m is an affine isomorphism A(S1, G) → A(S2, G) that commutes with the action of A(k). Since this action generates a lattice in V(A), we see that this isomorphism commutes with the action of V(A). This shows that for any v ∈ V(A), the points x + v ∈ A(S1, G) and x + v ∈ A(S2, G) are identified by the inclusions of these apartments into B(G). This proves (1). The claim proved above implies that the images of x under the two maps A(S1, G) → B(M) and A(S2, G) → B(M) coincide, proving that π descends to a V(A)-invariant and M(k)-equivariant map B(G) M → B(M). This map is surjective, since its restriction to each apartment A(S, G) is surjective onto A(S, M) and B(M) is the union of all A(S, M). To prove the injectivity statement, consider x, y ∈ B(G) M with π(x) = π(y). Proposition 9.3.11 applied to the split torus A provides an apartment A ⊂ B(G) M containing both x and y, and we conclude from Lemma 6.7.1 that x − y ∈ V(A). 

370 Corollary 9.7.3 B(G) M .

Unramified Descent Let Ω be a bounded subset of an apartment contained in

(1) G(k)1Ω ∩ M(k) = M(k)1π(Ω) .

(2) G(k)0Ω ∩ M(k) = M(k)0π(Ω) .

(3) Let A be the standard model of A. The centralizer of A in GΩ0 equals 0 . Mπ(Ω) Proof The identities G(k)0 = G(K)0 ∩ G(k) and G(k)1 = G(K)1 ∩ G(k), the former due to Proposition 9.3.24, allow us to assume k = K. For (1) and (2) we reduce to the case Ω = {x} using Lemma 7.7.4. Proposition 9.7.2 and the boundedness of G(K)1 imply G(K)1x ∩ M(K) ⊂ M(K)1π(x) . On the other hand, M(K)1π(x) is a bounded subgroup of M(K) whose action on B(G K ) M preserves the preimage of π(x) under π, that is the set x + V(A). This is a convex subset of B(G K ) M , so by Theorem 1.1.15 the action of M(K)1π(x) must have a fixed point. But since the actions of M(K)1π(x) ⊂ G(K) and V(A) on B(G K ) M commute, we conclude that M(K)1π(x) fixes all points of x + V(A), in particular also x. Thus M(K)1π(x) ⊂ G(K)1x . (2) follows from Lemma 8.5.19 applied to f = 0. (3) follows from Corollary 2.10.11 and (2).  9.7.4 Direct product decomposition Isomorphisms (6.7.1) and (9.7.1) can be upgraded to direct product decompositions as follows. Let S ⊂ M be a maximal  and S  for the maximal tori in k-split torus. Recall from §6.7 the notations SG M  ) = V(S  ) ⊕ V(A). S ∩ Gder and S ∩ Mder and the canonical decomposition V(SG M This direct sum decomposition induces by 1.2.16 a direct product decomposi ). Combining this decomposition tion A(S, G) = A(S, G)/V(A)×A(S, G)/V(SM with (6.7.1) we obtain the canonical direct product decomposition  A(S, G) = A(S, M) × A(S, G)/V(SM ).

(9.7.2)

If T is another maximal k-split torus of M there exists m ∈ Mder (k) such that T = mSm−1 . The map Ad(m) : A(S, G) → A(T, G) respects the direct product decompositions (9.7.2) on both sides and in particular induces an isomorphism  ) → A(T, G)/V(T  ) of affine spaces over the vector space V(A). A(S, G)/V(SM M We claim that this isomorphism is independent of the choice of m. To see this,  ). Choose it is enough to check that NMder (S)(k) acts trivially on A(S, G)/V(SM a special point o ∈ A(S, G). Any element of NMder (S)(k) can be written as zn with z ∈ Z Mder (S)(k) and n ∈ NMder (S)(k) such that no = o. Any point of  ) is of the form o + v with v ∈ V(A) and zn(o + v) = o + nv + ν(z). A(S, G)/V(SM  ), and the claim is proved. This claim implies But nv = v and ν(z) ∈ V(SM

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 ), one for each maximal k-split torus S of that the affine spaces A(S, G)/V(SM M, form a direct (equivalently inverse) system and taking the limit we obtain a canonical affine space over the vector space V(A), which we can denote by A(M, G). The decompositions (9.7.2) for the various maximal k-split tori S of M then induce the decomposition

B(G) M = B(M) × A(M, G).

(9.7.3)

 ), and hence on its limit The group M(k) acts on the system A(S, G)/V(SM A(M, G). The subgroup Mder (k) acts trivially on A(M, G). Since M(k) = Mder (k) · Z M (S)(k) and Z M (S)(k) acts on A(S, G) through the negative of the valuation homomorphism Z M (S)(k) → V(S) composed with the projection V(S) → V(S/AG ), we see that M(k) acts on A(M, G) through the negative of the valuation homomorphism M(k) → V(AM ) composed with the projection V(AM ) → V(AM /AG ) = V(A); here AG and AM are the maximal k-split central tori in G and M, respectively. The identification (9.7.3) is M(k)-equivariant provided we equip the right-hand side with the diagonal action of M(k).  There is an analogous decomposition for subset B(G) M of the enlarged  building B(G) = B(G) × V(AG ) that is the union of the enlarged apartments  for G corresponding to maximal k-split tori of M. Note B(G) M = B(G) M ×  G) = A(S, G) × V(AG ). For each maximal k-split torus S ⊂ M we have A(S,   G)/V(S  ) is an affine space over G) of the system A(S, V(AG ). The limit A(M, M V(AM ). It is naturally isomorphic to A(M, G) × V(AG ) as affine spaces over V(AM ) = V(AM /AG ) ⊕ V(AG ). From (9.7.3) we obtain the M(k)-equivariant decomposition   B(G) (9.7.4) M = B(M) × A(M, G).

9.7.5 Embedding B(M) into B(G) The decomposition (9.7.3) implies that each point y ∈ A(M, G) induces an embedding B(M) → B(G), namely the embedding B(M) → B(G) M sending x ∈ B(M) to (x, y) ∈ B(G) M composed with the tautological inclusion of B(G) M into B(G). Note that such an embedding is equivariant for Mder (k), but not for M(k). The different embeddings, corresponding to different y ∈ A(M, G), have disjoint images.  The point y also gives an embedding B(M/A G ) → B(G), namely via  the bijection B(M/AG ) → B(G) M sending (x, z) to (x, zy). The bijection   B(M/A G ) → B(G) M , and hence also the embedding B(M/AG ) → B(G), are M(k)-equivariant. The different embeddings, corresponding to different y ∈ A(M, G), all have the same image, namely B(G) M .   Finally, the point y gives an embedding B(M) → B(G) as follows. Let AM and AG be the maximal k-split tori in the centers of M and G, respectively. The vector space V(AM ) has the canonical direct product decomposition

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V(AM /AG ) × V(AG ). Using it we have  B(M) = B(M) × V(AM ) = B(M) × V(AM /AG ) × V(AG )  = B(M/A G ) × V(AG ),   and the embedding B(M/A G ) → B(G) now leads to the embedding B(M) →  B(G). This embedding is again M(k)-equivariant. The various such embeddings, corresponding to the various y ∈ A(M, G), all have the same image,  namely the subset B(G) M that is the union of the enlarged apartments corresponding to maximal k-split tori of G contained in M. Note that none of these embeddings is canonical, because the affine space A(M, G) does not have a distinguished point. For the remainder of this section we will not regard M as fixed. 9.7.6

Levi subgroups associated to parahoric subgroups.

(1) Consider a Levi subgroup M ⊂ G and a point x ∈ B(G) M . Let A be the standard model of the maximal k-split torus in the center of M. According 0 . Composing the to Corollary 9.7.3, the centralizer of A in Gx0 is Mπ(x) natural inclusion between special fibers M 0π(x) → G 0x with the projection G 0x → Gx we obtain, using [Bor91, Corollary 2 to Proposition 11.14] a surjective homomorphism from M 0π(x) to the Levi subgroup of Gx that is the centralizer of the isomorphic image of A ⊂ G 0x in Gx . We claim that this surjective homomorphism induces an isomorphism between the reductive quotient M0π(x) and this Levi subgroup of Gx . Indeed, the kernel of this homomorphism is the centralizer of the torus A in the unipotent radical of G 0x , and is therefore a smooth connected unipotent normal subgroup of M 0π(x) . On the other hand, the unipotent radical of M 0π(x) lies in the kernel of that homomorphism, and hence equals that kernel. (2) Conversely, consider a point x ∈ B(G) and a Levi subgroup M of Gx . Let A be the maximal f-split torus in the center of M. There exists a split f-torus A ⊂ G 0x mapping isomorphically to A, and Proposition 8.2.1(6) provides an o-torus A of Gx0 with special fiber A . Proposition 8.2.1(5) shows that A is determined by A up to conjugation by Ru (G 0x )(f), which equals G(k)0+ x by Corollary 8.4.12. Let M be the centralizer of the generic fiber A of A in G. Then M is a Levi subgroup of G whose G(k)0+ x -conjugacy class is determined by M. It is clear that M is the generic fiber of the centralizer M of A in Gx0 . A maximal k-split o-torus S of M is also a maximal k-split o-torus of Gx0 . The generic fiber S of S is a maximal

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k-split torus of M and the apartment A(S, G) contains x by Proposition 9.3.5(2). Therefore x ∈ B(G) M . Construction (1) establishes a G(k)-equivariant surjection between the set of pairs (M, x) consisting of a Levi subgroup M of G and a point x ∈ B(G) M and the set of pairs (x, M) consisting of a point x ∈ B(G) and a Levi subgroup M of G x . Upon fixing x, the surjection between G(k)0+ x -conjugacy classes of Levi subgroups M of G such that x ∈ B(G) M and Levi subgroups of G x has a section given by construction (2). A parabolic subgroup P of G with Levi factor M corresponds to a connected component of the complement of X∗ (A) ⊗Z R by the union of the hyperplanes for the elements of the set Φ(A, G) of the weights for the action of A on Lie(G). Under the identification X∗ (A) = X∗ (A) we have the inclusion Φ(A, G x ) ⊂ Φ(A, G). In this way P specifies a parabolic subgroup P of G x with Levi factor M. The assignment P → P need not be injective. For example, it may happen that M = G x despite M  G. Recall that, given a root system R in a real vector space V and a subset S ⊂ R, the rational closure of S in R is the subset of R consisting of those elements that are Q-linear combinations of elements of S. A subroot system R  ⊂ R is called rationally closed if it equals its own rational closure. If R   R  ⊂ R are subroot systems and both R  and R  are rationally closed in R, then the rank of R  is strictly smaller than the rank of R . Lemma 9.7.7 Let x ∈ B(G) and let M be a Levi subgroup of G x . Let (M, x) be the pair associated to (x, M) by construction (2) of 9.7.6. Let S ⊂ M be a maximal k-split torus whose apartment contains x, let S ⊂ Gx0 be the associated closed o-torus, and S the isomorphic image in G x of the special fiber S of S . Then S is a maximal k-split torus of M and Φ(S, M) is the rational closure of Φ(S, M) in Φ(S, G). Proof By construction Φ(S, M) ⊂ Φ(S, M). It is well known that Φ(S, M) is rationally closed in Φ(S, G). Therefore it suffices to show that the rank of Φ(S, M) does not exceed the rank of Φ(S, M). Let A be the maximal k-split f-torus in the center of M and let AM be the maximal k-split torus in the center of M. Let A ⊂ S be as in 9.7.6(2). Then A ⊂ AM . The rank of Φ(S, M) equals dim(S) − dim(AM ) and is thus less than or equal to dim(S) − dim(A) =  dim(S) − dim(A). The latter is equal to the rank of Φ(S, M). 9.7.8 The Levi subgroup M(x) associated to x ∈ B(G) As a special case of 9.7.6(2) we obtain a natural G(k)0+ x -conjugacy class of Levi subgroups of G associated to a point x ∈ B(G). This construction was described in [MP96, 6.3].

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Namely, it is the conjugacy class of Levi subgroups of G associated by 9.7.6(2) to the pair (x, G x ), where the group G x is considered as a Levi subgroup of itself. Note that M(x) need not equal G. For example, when x lies in a chamber, the conjugacy class is represented by M(x) = ZG (S) for a maximal k-split torus S ⊂ G whose apartment A contains x. We can describe M(x) more explicitly as follows. Let Ψx ⊂ Ψ ⊂ A∗ be the set of affine roots that vanish at x and let Φx ⊂ Φ = Φ(S, G) be the set of derivatives ∇Ψx . Let C be the maximal torus in S on which all elements of Φx vanish. Define M(x) to be the centralizer of C in G. Note that it is not always true that Φ(S, M(x) ) = Φ(S, G x ). From Lemma 9.7.7 we know that Φ(S, M(x) ) is the rational closure of Φ(S, G x ) in Φ(S, G). There are many examples where Φ(S, G x ) is not rationally closed in Φ(S, G). For example, this happens whenever x is a vertex of A that is not extra special. The main property of the Levi subgroup M(x) is the following. Proposition 9.7.9 vertex.

([MP96, Proposition 6.4]). The point π(x) ∈ B(M(x) ) is a

Proof Abbreviate M = M(x) . Choose a maximal torus S ⊂ M such that x ∈ A(S, G). Then π(x) ∈ A(S, M). According to Proposition 1.3.35, π(x) is a vertex in B(M) if and only if the ranks of the root systems Φ(S, M) and Φ(S, M)π(x) are equal. But Φ(S, M)π(x) = Φ(S, M) ∩ Φ(S, G)x . By construction Φ(S, M) contains Φ(S, G)x and, according to Lemma 9.7.7, equals the rational closure of Φ(S, G)x in Φ(S, G). Thus Φ(S, M)π(x) = Φ(S, G)x has rank equal to that of Φ(S, M). 

9.8 Concave Function Groups  Let S ⊂ G be a maximal k-split torus, Φ(S) := Φ(S, G), Φ(S) = Φ(S) ∪ {0}, and Z the centralizer of S in G. In §7.3 we introduced the concept of a concave  and constructed an open bounded subgroup G(k)x, f  function f : Φ(S) →R for any valuation x of the root datum of (G, S). For this, the existence of an appropriate filtration Z(k)r was assumed in §7.2. As a first step in this section, we will construct this filtration. For this, we will assume that a functorial filtration satisfying (1) and (2) of §7.2 has been chosen on all tori. According to Proposition 9.3.9 the building B(Z) consists of a single point  x. For any r ∈ R, the constant function f : Φ(S) → R taking the value r is concave. We apply the constructions of §7.3 to the quasi-split group ZK , the point x ∈ B(Z) ⊂ B(ZK ), and the constant function f ≡ r to obtain the open

9.8 Concave Function Groups

375

bounded Γ-invariant subgroup Z(K)x,r of Z(K). Set Z(k)r = Z(K)x,r ∩ Z(k). We remark that this is a special case of a Moy–Prasad filtration; we will discuss the general case in Chapter 13. According to Propositions 7.3.17 and 7.6.4, the group Z(k)r does not depend on the choice of S. Lemma 9.8.1 The filtration Z(k)r just constructed is commutator friendly in the sense of Definition 7.2.1. Proof By construction, Z(k)0 = Z(K)x,0 ∩ Z(k) ⊂ Z(K)0 ∩ Z(k) = Z(k)0 . Conversely, the action of Z(k) on B(ZK ) preserves B(Z) = {x}, hence Z(k)0 ⊂ Z(k) ∩ Z(K)0x = Z(k)0 . Consider a point x in the apartment A(S, G) of B(G) corresponding to S. Under the projection map B(G) → B(Z) of (9.7.1), this point maps to the unique point of B(Z). According to Corollary 9.7.3 the stabilizer in Z(k)0 of x ∈ B(G) equals Z(k)0 . Let i, j ∈ R with i + j > 0, u ∈ Ua, x,i , u  ∈ U−a, x, j . We want to show that the Z-component of the commutator (u, u ) lies in Z(k)i+j = Z(k) ∩ Z(K)x,i+j . We choose a special k-torus T of Z, hence also of G, that contains S, using  Proposition 9.3.4. According to Lemma 9.6.3 we can express u = b ub with   with u  ∈ U ub ∈ Ub, x,i and u  = −b u−b −b, x, j . We apply Lemma 7.1.2 and −b V 3 of Definition 6.1.2 to the quasi-split group ZK and the valuation x ∈ B(ZK ) to see that the commutator (u, u ) lies in G(K)x,i+j . Note that we are using here the fact that the centralizer of a maximal K-split torus of ZK is a maximal torus of ZK , and on this maximal torus we have a commutator-friendly filtration, cf. Lemma 7.2.3. The Z-component of the commutator (u, u ) therefore lies in  G(K)x,i+j ∩ Z(K), which equals Z(K)x,i+j by Lemma 8.5.19. 9.8.2 The preceding lemma allows the constructions and the results of §7.3 to  leading to the bounded  be applied to G and any concave function f : Φ(S) → R, subgroup G(k)x, f for x ∈ A = A(S). To describe this subgroup explicitly, we recall that Construction 9.6.1 allows us to associate to x ∈ A, the valuation (ϕax )a ∈Φ(S) of root datum (Ua (k))a ∈Φ(S) , where for u ∈ Ua (k)∗ , ϕax (u) = ψau (x). As in part V 1 of Definition 6.1.2, we set Ua, x, f (a) = {u ∈ Ua (k)∗ | ψau (x)  f (a)} ∪ {1}, and Ua, x, f = Ua, x, f (a) · U2a, x, f (2a) . The subgroup G(k)x, f is the subgroup of G(k) generated by Ua, x, f , a ∈ Φ(S), and Z(k) f (0) .  − {∞}, so that G(k)x, f From now on, we will assume that f takes values in R is open. Let T be a special k-torus of G containing S. Let Φ(TK ) = Φ(TK , G K )) and  Φ(TK ) = Φ(TK )∪ {0}. The restriction map X∗ (TK ) → X∗ (SK ) = X∗ (S) induces   K ) → Φ(S). The composition of f with the restriction map a surjective map Φ(T

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 − {∞}. Consider a point x ∈ A(S). Via  K) → R is a concave function F : Φ(T the natural inclusion A(S) ⊂ A(TK ) we can interpret x as a valuation of the root datum of (G K ,TK ) and construct the open bounded subgroup G(K)x, F as above for F in place of f and K in place of k. Since x and F are Γ-invariant, so is G(K)x, F . Therefore, the O-model Gx, F of G K constructed in §8.5, with the characterizing property Gx, F (O) = G(K)x, F , descends to o by Fact 2.10.16. We will denote this descent by Gx, f . Let S and T be the standard smooth models of S and T respectively, cf. §B.2. We will denote the special fibers of Gx, f , S , and T by G x, f , S and T respectively. Proposition 9.8.3 There is a natural action S × Gx, f → Gx, f of S on Gx, f by o-group scheme automorphisms induced by the conjugation action of the k-split torus S on G. For a ∈ Φ(S), let Ua, x, f be the a-root group of Gx, f with respect to S . (1) Ua, x, f (o) = Ua, x, f . (2) G(k)x, f = G(K)x, F ∩ G(k) = Gx, f (o). Proof The morphism S × G → G for the action of S on G by conjugation induces a morphism S(K)0 × G(K)x, F → G(K)x, F and hence extends, by Corollary 2.10.10, to a morphism of schemes S × Gx, f → Gx, f . A priori this is an O-morphism, but it is immediately seen that it descends to o. The identity G(K)x, F ∩ G(k) = Gx, f (o) follows at once from Gx, F (O) = G(K)x, F . It remains to prove G(k)x, f = G(K)x, F ∩ G(k). Let Z be the fixed point subscheme of Gx, f for the action of S . If Z is the centralizer of S in G, then Z (O) = Z(K) ∩ Gx, F (O) = Z(K) ∩ G(K)x, F = Z(K)x, F = Z(K)x, F(0) by Lemma 8.5.19, so Z (o) = Z(k) f (0) . Proposition 8.2.8(2) applied to Gx, f implies that Gx, f (o) is generated by Z (o) and Ua, x, f (o). Moreover,   Ua, x, f (O) = Ua (K) ∩ G(K)x, F = Ub (K) ∩ G(K)x, F = Ub, x, F , b

b

by Proposition 7.3.12, where the product runs over those non-divisible b ∈ Φ(TK ) that restrict to either a or 2a. Now Ua, x, f (o) = Ua, x, f (O) ∩ Ua (k) and Lemma 9.6.3 implies that this equals Ua, x, f . We conclude that Gx, f (o) is generated by Z(k) f (0) and Ua, x, f (a) for all a ∈ Φ(S), hence it equals G(k)x, f (cf.9.8.2).  We will now describe the structure of the special fiber G x, f of Gx, f and the maximal reductive quotient G x, f of G x, f . The natural action of S on Gx, f described in the preceding proposition induces an action of S on G x, f , and so also on G x, f , by f-group automorphisms.

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377

Recall the bounded open subgroup Uψ ⊂ Ua (k) given in Definition 9.4.1 for a real valued affine function on A with derivative a ∈ Φ(S). x, f

Lemma 9.8.4 Let ψa be the smallest affine root with derivative a such that x, f x, f ψa (x)  f (a), and if 2a is also a root define ψ2a analogously. Let Ua, x, f be the a-root group of Gx, f . Then Ua, x, f (o) = Ua, x, f = Uψ x , f · Uψ x , f (= Uψ x , f if 2a  Φ(S)). a

a

2a

Proof The identity Ua, x, f (o) = Ua, x, f is due to Proposition 9.8.3(1). Recall  ∈ Ψ  , i = 1, 2, be the smallest elements Ua, x, f = Ua, x, f (a) · U2a, x, f (2a) . Let ψia  with derivative ia such that ψia (x)  f (ia). Then, by definition of Ψ , Ua, x, f =  . According to Proposition 9.4.18(1), the proof is complete if 2a  Uψa · Uψ2a  ∈ Ψ. If also ψ  ∈ Ψ then we set Φ(S). If 2a ∈ Φ(S), then ψ2a := ψ2a a x, f ψa := ψa and the proof is complete. Otherwise Proposition 9.4.18(2) implies  = ψ x, f . 2ψa ∈ Ψ. From 2ψa (x)  2 f (a)  f (2a) we see that 2ψa  ψ2a 2a x, f Letting ψa ∈ Ψ be the smallest element greater than ψa , Proposition 9.4.18(3)  implies Ua, x, f = Uψ x , f · U2ψa · Uψ x , f = Uψ x , f · Uψ x , f . x, f

a

Definition 9.8.5 (1) (2) (3) (4)

a

2a

2a

 − {∞} be a concave function. →R Let f : Φ

Φ(S) f = {a ∈ Φ(S) | f (a) + f (−a) = 0}.  Ψx, f = {ψ ∈ Ψ | ψ ∈ Φ(S) f and ψ(x) = f (ψ)}. Φ(S)x, f = ∇Ψx, f .  x, f = Φ(S)x, f ∪ {0}. Φ(S)

Note that this is the same as Definition 8.5.7, but now in the setting of a base field k that may not be strictly Henselian. As f (a) + f (−a)  f (0)  0, Φ(S) f , and hence Φ(S)x, f , is empty unless f (0) = 0. If f takes only non-negative values, then a ∈ Φ(S) belongs to Φ(S) f if and only if both f (a) = 0 and f (−a) = 0. Lemma 9.8.6

x, f

Φ(S)x, f = {a ∈ Φ(S) | ψa

x, f

+ ψ−a = 0}.

Proof Note that by definition ψ ∈ Ψx, f if and only if −ψ ∈ Ψx, f . For a ∈ Φ(S) we have x, f

ψa

x, f

x, f

+ ψ−a = (ψa

x, f

x, f

x, f

− f (a)) + (ψ−a − f (−a)) + ( f (a) + f (−a))  0,

since ψa  f (a), ψ−a  f (−a) and f (a) + f (−a)  f (0)  0. Hence, for x, f x, f x, f x, f a ∈ Φ(S), ψa + ψ−a = 0 is equivalent to ψa = f (a), ψ−a = f (−a), and f (a) + f (−a) = 0. 

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Let Z be the fixed point o-subscheme of Gx, f for the action of S described in Proposition 9.8.3. We will denote by Z the special fiber of Z . Define Φ(S)∨x, f = {a∨ | a ∈ Φ(S)x, f } ⊂ X∗ (S), where a∨ is as in 2.6.4. We will now prove the following analog of Proposition 9.4.23 for the group G x, f . Theorem 9.8.7 The root datum of G x, f with respect to the isomorphic image S of S in G x, f is

(X∗ (S), Φ(S)x, f , X∗ (S), Φ(S)∨x, f ). If f (0) > 0, then G x, f is a unipotent group, and for any choice Φ(S)+ of positive system of roots, the product morphism   U a, x, f × Z × U a, x, f → G x, f a ∈Φ(S)−, nd

a ∈Φ(S)+, nd

is an isomorphism upon taking the products over Φ(S)−,nd and Φ(S)+,nd in any order. Proof Since by Proposition 9.8.3(1) and Lemma 9.8.4 Ua, x, f (o) = Ua, x, f = Uψ x , f · Uψ x , f , a

2a

the latter being equal to Uψ x , f if 2a  Φ(S), Theorem 9.4.22 implies that a a

x, f

x, f

is a root of G x, f with respect to S if and only if ψ−a + ψa = 0; that is, if a ∈ Φ(S)x, f . From this we conclude at once that the root datum of G x, f with respect to S is (X∗ (S), Φ(S)x, f , X∗ (S), Φ(S)∨x, f ). Now we assume that f (0) > 0. Then F(0) > 0. As (G x, f )f = G x,F , we have G x, f is a unipotent group if and only if G x,F is a unipotent group. Corollary 8.5.12 implies that G x,F is a unipotent group. Now to establish the final assertion of the proposition, we only need to quote Theorem 8.5.2 and Proposition 2.11.3.  9.8.8 To describe G(k)x, f , we can assume that f takes only non-negative values. In fact, if we replace the pair (x, f ) by the pair (v + x, f + v), where f + v(a) = f (a) + a(v), then it is obvious from the definitions that none  x, f and Ψx, f change and the group G(k)x, f is also of the sets Φ(S)x, f , Φ(S) unaffected. This allows us to apply Lemma 7.3.9 to f and assume that it takes only non-negative values. This motivates the following definition. Definition 9.8.9 Assume that f takes only non-negative values. We define the    f , and f + (a) = f (a) → R−∞ by: f + (a) = f (a)+ if a ∈ Φ(S) function f + : Φ(S) otherwise. The function F + is defined analogously. The functions f + and F + are concave (cf.Lemma 8.5.9).

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379

Since f + > f , F + > F hence G(K)x, F + ⊂ G(K)x, F . This inclusion induces a homomorphism Gx, f + → Gx, f of o-group schemes which is the identity on the common generic fiber G. Proposition 9.8.10 Assume that f (0) = 0. The image of the induced fhomomorphism G x, f + → G x, f is the unipotent radical Ru (G x, f ) of G x, f . Proof Let π : G x, f → Gx, f be the natural projection. The kernel of π is the unipotent radical Ru (G x, f ) of G x, f . So to prove the proposition, it would suffice to show that the image of G x, F + (f) in G x, F (f) is the kernel of π(f). But this assertion is obvious from Corollary 8.5.13. 

9.9 Special, Superspecial, and Hyperspecial Points Recall from Definition 1.3.39 that a point z ∈ C is called special if for every non-divisible root a ∈ Φ there is an affine root with derivative either a or 2a that vanishes at z. Recall further that special points exist, cf. Proposition 1.3.43. For a special point z, the parahoric subgroup Pz is said to be a special parahoric subgroup. Proposition 9.9.1 For any point z ∈ C, the following assertions are equivalent to each other. (1) z is special. (2) The f-Weyl group of G z is canonically isomorphic to the k-Weyl group W = N(k)/Z(k) of G. (3) N(k) ∩ Pz maps onto W. Proof Proposition 9.4.23 (for Ω = {z}) implies that if (1) holds, then given a non-divisible root a ∈ Φ, either a or 2a is a root of G z . So the f-Weyl group of G z contains reflection in every root of Φ. As such reflections generate the k-Weyl group of G and f-Weyl group of G z , we conclude that (1) implies (2). To prove that (2) implies (3), we note first that under the natural surjective homomorphism G z0 → G z = G z0 /Ru (G z0 ), the normalizer N (S )(f) of S in G z0 (f) maps onto the normalizer of S in G z (f). According to Lemma 2.4.8, the f-Weyl group of the former is mapped isomorphically onto the f-Weyl group of the latter, as Ru (G z0 ) is unipotent. Finally, Proposition 8.2.1(3) implies that the natural homomorphism N(k) ∩ Pz → N (S )(f) is surjective. We now prove that (3) implies (1). As noted in the preceding paragraph, N(k) ∩ Pz → N (S )(f) is surjective, and N (S )(f) maps onto the normalizer of S in G z (f). Assuming (3), these two results imply that the f-Weyl group of

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G z is isomorphic to the k-Weyl group W of G. Therefore, for any non-divisible root a ∈ Φ, either a or 2a is a root of G z . This, in view of Proposition 9.4.23

(for Ω = {z}) , implies (1).



Proposition 9.9.2 Let G be a connected reductive k-group. If z ∈ B(G) is special in B(G K ), then it is also special in B(G). Thus, z is superspecial in the sense of Definition 7.11.1. Proof Let S be a maximal k-split torus in G whose apartment A(S) contains z. Using Proposition 9.3.4 let T be a special k-torus of G that contains S. Recall that the image of the restriction map Φ(TK , G K ) → Φ(S, G) ∪ {0} contains Φ(S, G). Since z is special in B(G K ), we see that given a non-divisible root a ∈ Φ(S, G), there is a root b ∈ Φ(TK , G K ) such that the restriction of b to S is a positive integral multiple na of a and there is an affine root ψb of G with respect to T which vanishes at z and has derivative b. The restriction ψ of ψb to A is an affine root of G with respect to S which vanishes at z and has derivative na. Thus we have shown that for every root a ∈ Φ(S, G), there is an affine root ψ of G with respect to S that vanishes at z and has derivative a positive integral multiple of a. This proves that z is a special point of B(G).  Recall from [Con14, Definition 3.1.1] the notion of a reductive o-group scheme: it is a smooth affine o-group scheme whose geometric fibers are connected reductive groups. In fact, the connectedness of the special fiber need not be assumed, cf. Proposition A.8.2. Recall also the concept of a hyperspecial vertex of B(G) from Definition 7.11.1: it is a vertex that remains special in B(G K ) and moreover G K is assumed split. Recall from Corollary 7.11.5 that, if G is quasi-split, then any two hyperspecial points are conjugate under Gad (k). We will prove in Proposition 10.2.1 that, if dim(f)  1 and B(G) contains a superspecial (in particular hyperspecial) point, then G is quasi-split over k. Theorem 9.9.3 (1) Let x ∈ B(G). Then Gx0 is a reductive o-group scheme if and only if x is hyperspecial. (2) If G is a reductive o-group scheme with generic fiber G then there exists a hyperspecial point x ∈ B(G) such that the identity automorphism of G extends to an isomorphism G → Gx0 . Proof (1) The reductivity of Gx0 is a property of the group scheme (Gx0 )O . Proposition 8.4.14 shows that G K is split and x is special in B(G K ). According to Proposition 9.9.2, x is hyperspecial. (2) Proposition 8.3.16 implies that the group G K is split and there exists a special vertex x ∈ B(G K ) such that the identity automorphism of G K extends to an isomorphism GO → Gx0 . In particular, G (O) = Gx0 (O) = G(K)0x . The

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381

o-structure of G implies that G(K)0x is Γ-invariant. Since x is the unique point of B(G K ) fixed by G(K)0x (cf. Proposition 1.5.13(4)), we conclude that x ∈ B(G K )Γ = B(G). In particular, Gx0 is endowed with an o-structure; we will denote the corresponding o-group scheme also by Gx0 . Since the isomorphism GO → Gx0 is uniquely determined by G (O) = Gx0 (O) according to Corollary 2.10.10, it too descends to an isomorphism G → Gx0 of o-group schemes.  Corollary 9.9.4 A quasi-split reductive k-group that splits over K admits a reductive model; that is, there exists a reductive o-group scheme G with generic fiber G. Proof This follows from Theorem 9.9.3 and the existence of absolutely special points asserted in Proposition 7.11.4. 

9.10 Residually Split and Residually Quasi-split Groups Let G be a connected reductive k-group. Proposition 9.10.1 The following statements are equivalent. (1) (2) (3) (4)

There exists a Γ-invariant K-chamber. Every Γ-chamber is a K-chamber. For any Γ-chamber C, the f-group G C0 is solvable. For every k-facet F, the f-group G F0 is quasi-split.

Proof (1) ⇒ (2) All K-chambers have the same dimension. According to 9.3.1, all Γ-chambers have the same dimension. (1) implies that the common dimension of all K-chambers equals the common dimension of all Γ-chambers, and (2) follows. (2) ⇒ (3) According to Axiom 4.1.22, the absolute root system of the 0 is empty. Thus G 0 is solvable. reductive quotient of G C C (3) ⇒ (4) Choose a Γ-chamber C whose closure contains F. The image of G C0 → G F0 is a parabolic f-subgroup. It is solvable by (3), hence a Borel subgroup. (4) ⇒ (1) Let F be a k-facet and consider the f-group G F0 . According to Axiom 4.1.22, the set of f-Borel subgroups is in bijection with the set of Kchambers containing F in its closure. This bijection is Γ-equivariant, and (4) implies the existence of a Γ-invariant K-chamber whose closure contains F.  Definition 9.10.2 We say that G is residually quasi-split if it satisfies the equivalent conditions of Proposition 9.10.1.

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Unramified Descent

Fact 9.10.3 If G is residually quasi-split, the Iwahori subgroups of G(k) are of the form I Γ for Γ-stable Iwahori subgroups I of G(K). Proposition 9.10.4

If dim(f)  1, then G is residually quasi-split.

Proof Choose a facet F of B(G) and apply Theorem 2.3.3(2) to conclude that G F0 is quasi-split.  Proposition 9.10.5

If G is quasi-split, then it is residually quasi-split.

Proof Let S be a maximal k-split torus, let A be the corresponding apartment, and let x ∈ A. Then S is a maximal f-split torus of G x0 and projects isomorphically onto a maximal f-split torus S of the reductive quotient G x of G x0 . Let T be a special k-torus containing S, cf. Proposition 9.3.4, and let T be the image of T in G x , a maximal torus. Since G is quasi-split, no element of Φ(T, G) restricts trivially to S. In particular, no element of Φ(T, G)x restricts trivially to S. According to Theorem 8.4.10, the group Φ(T, G)x is the absolute root system of G x with respect to the maximal torus T. We conclude that the  centralizer of S is T and hence that G x is quasi-split. 9.10.6 Assume that G is residually quasi-split. Let T be a special k-torus of G and S be the maximal k-split torus of G contained in T. Let A be the apartment in B(G K ) corresponding to T. Then A := AΓ is the apartment of B(G) corresponding to S. Let C be a Γ-chamber in A. According to Proposition 9.10.1, C is a K-chamber, hence C = C Γ is a chamber in A. Let T be the maximal o-torus of GC0 whose generic fiber is T and let S be the o-subtorus of T whose generic fiber is S. We will denote the special fibers of S and T by S and T respectively. Given a Γ-facet F contained in the closure of C, we consider the o-group scheme homomorphism GC0 → GF0 and denote by the same symbols the image tori in GF0 and G F0 . Let Ψ be the affine root system of G K with respect to TK and Δ be the basis of Ψ determined by C, cf. Definition 1.3.19. Since T and C are Γ-stable, Γ acts on Ψ and preserves Δ. Let ΔF ⊂ Δ be the type of F, cf. Definition 1.4.6. Since F is Γ-stable, so is ΔF . Remark 4.1.23 gives the Dynkin diagram of the maximal reductive quotient G F of G 0F with respect to the isomorphic image T of the maximal torus T . The action of Γ on ΔF endows the Dynkin diagram with the structure of a Tits index. Note that, since GF is quasi-split, all Γ-orbits in the Tits index are distinguished, that is every root of G F with respect to T has non-trivial restriction to the maximal f-split subtorus of T. 9.10.7 Assume that G is residually quasi-split. We can describe the vertices of the chamber C = C Γ as follows. Each Γ-orbit of vertices of C is the set of

9.10 Residually Split and Residually Quasi-split Groups

383

vertices of a Γ-vertex of C. The barycenter of this Γ-vertex is the unique Γ-fixed point in it and is a vertex of C. This establishes a bijection between the set of Γ-orbits of vertices of C and the vertices of C. If G is K-simple, then C is a simplex, and its vertices are in bijection with the elements of Δ by Proposition 1.3.22(6). This bijection is Γ-equivariant. Proposition 9.10.8 Assume that G is residually quasi-split. Let T be a special k-torus, S ⊂ T the maximal k-split subtorus, A the apartment corresponding to T, Ψ ⊂ A∗ the affine root system for (TK , G K ), A = AΓ and Ψk ⊂ A∗ the apartment and affine root system corresponding to (S, G). Let C ⊂ A be a Γinvariant chamber, Δ the corresponding basis of Ψ. The set Δk of restrictions to A of the elements in Δ form the basis of Ψk associated to the chamber C = C Γ . In particular, no element of Δ has constant restriction to A, and two elements have equal restriction if and only if they are Γ-conjugate. Proof It suffices to prove the proposition for almost simple groups, so we will assume that G is almost simple in the proof. As dim A = dim C = r := k-rankG, and C is a simplex (since G is absolutely almost simple), it has r + 1 vertices. Therefore, Δ is the union of (r + 1) Γorbits (9.10.7). Since the restrictions to A of all the affine roots belonging to a Γ-orbit in Δ are equal, the restriction to A of affine roots in Δ provides a set Δk consisting of r + 1 (not a priori distinct) real valued affine functions on A. Since the non-constant restrictions to A of the affine roots of G K with respect to TK are all the affine roots of G with respect to S (Proposition 9.4.28), we see that the integral span of the affine functions in Δk contains the set Ψk of affine roots of G with respect to S. So the (r + 1)-dimensional R-vector space A∗ of real valued affine functions on A is generated by the r + 1 affine functions belonging to Δk , and hence these affine functions form a basis of A∗ , in particular, they are all distinct and non-zero functions. Now we note that each affine function belonging to Δk vanishes at some vertex of C, so if any of them is a constant function, it would be identically zero on A, which as we have observed above, is not possible. Thus every function in Δk is non-constant, so according to Proposition 9.4.28, they are affine roots of G with respect to S. As every affine root in Ψ is a non-negative or a non-positive integral linear combination of roots in Δ, we conclude that every affine root in Ψk is a nonnegative or non-positive integral linear combination of roots in Δk , and hence Δk is a basis of the affine root system Ψk . Since every affine root in Δk takes  positive values on C, Δk is the basis of Ψk determined by C. Remark 9.10.9 Assume that G is residually quasi-split. We saw above that the restriction to A of every affine root in Δ is non-constant and hence it is an

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affine root of G with respect to S. It is not true in general that the restriction of every affine root in Ψ to A is non-constant. In fact, for a root a of G K with respect to TK , the restriction of a ψ ∈ Ψ, with derivative a, to A is non-constant if and only if the restriction of a to S is non-trivial (this restriction then is a root of G with respect to S). This implies, in particular, that the restriction to A of every affine root in Ψ is non-constant (and then these restrictions are affine roots of G with respect to S) if and only if G is quasi-split, that is the centralizer of S in G is a torus (= the centralizer of T). Proposition 9.10.10 The following statements are equivalent. (1) The action of Γ on any special k-apartment is trivial and the maximal K-split torus in the center of G is k-split. (2) The k-rank of G equals the K-rank of G. 0 is f-split solvable. (3) For every Γ-chamber C, the f-group G C (4) For every k-facet F, the f-group G 0F is f-split. Proof (1) ⇔ (2) Let T be a special k-torus of G, T  the maximal K-split subtorus of T ∩ Gder , S and S  the maximal k-split tori in T and T  respectively, and A and A the apartments corresponding to T and S, respectively, so that A = AΓ . Since the Γ-action on A is trivial, we see that A = A and hence V(S ) = V(T ) and hence S  = T . This and the assumption on the center imply S = T. The argument is reversible. 0 (1) ⇒ (3) Since the Γ-action on A is trivial, every chamber is fixed. As G C is solvable, it is enough to find a maximal f-torus in it that is f-split. The special 0

fiber T is a maximal torus of G C . From the previous paragraph we conclude that T = S , so this maximal torus is f-split. (3) ⇒ (4) Choose a chamber C whose closure contains F. Then the image 0

0

0

of ρ : G C → G F is a Borel subgroup. It is also split by (3), so G F is split. 0 (4) ⇒ (1) Since T is a maximal torus of G F , Hence T is k-split. We conclude that T = S and T 

it is f-split according to (4). = S , which implies (1). 

Definition 9.10.11 A group G is called residually split if it satisfies the equivalent statements of Proposition 9.10.10. Proposition 9.10.12 quasi-split.

A residually split group is quasi-split and residually

Proof The first claim follows from Propositions 9.10.1 and 9.10.10, since the triviality of the Γ-action on a special k-apartment A implies that all chambers are fixed by Γ. The second claim follows from Proposition 9.10.5. 

9.11 Restriction of Scalars

385

9.11 Restriction of Scalars Let /k be a finite Galois extension, o be the ring of integers and f the residue field of . We set A = o ⊗o f; A is a local ring, its maximal ideal is nilpotent and its residue field is f . Let k  be the maximal unramified extension of k contained in , and K be a maximal unramified extension of k . We will denote Gal(K/k) by Γ and Gal(K/k ) by Γ . Let H be a connected reductive -group and let G = R/k H. We have an identification G(k) = H(). Proposition 9.11.1 (1) There is a natural identification of B(G) with B(H) that is equivariant for the identification G(k) = H(). (2) If Ω is a non-empty bounded subset of an apartment of B(G) = B(H ), then GΩ∗ = Ro /o HΩ∗ , where ∗ can be any of the decorations 0, b, 1, †. (3) Writing G Ω and H Ω for the maximal reductive quotients of the special fibers of GΩ0 and HΩ0 , we have G Ω = Rf /f H Ω . Proof Let L = K ⊗k  . Then L is the maximal unramified extension of k containing K. By restricting automorphisms of L to K, we obtain an identification of Gal(L/) with Γ . Let L  = L ⊗k k  = K ⊗k . We have B(G) = B(G K ) Γ and G K = R L /K (HL ). Applying §7.9.1 we obtain a G(K)-equivariant identification of Gal(K/k)-sets B(G K ) = Ind ΓΓ  B(HL ). 

Taking Γ-fixed points on the right-hand side, we see that B(G) = B(HL )Γ = B(H). Given a maximal -split torus S of H, let T be a special -torus of H containing S. As in §7.9.1 we view S as a split k-torus and obtain the identification of S with the maximal k-split subtorus of R/k S, which is a torus in G. In this way, S is identified with a maximal k-split torus of G. The k-torus R/k T of G is in general not K-split, but its maximal K-split subtorus T  is defined over k and is a special k-torus of G containing S. The above displayed identification restricts to an identification A(T , G) = Ind ΓΓ  A(T, H) of Γ-sets. Taking Γ-fixed points we obtain the identifications 

A(S, G) = A(T , G) Γ = A(T, H)Γ = A(S, H). These are equivariant under the action of N(k) = NH (S)(). We have proved (1). A bounded Γ-invariant subset Ω  of A(T , G) is of the form Ind ΓΓ  Ω for a

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bounded Γ -invariant subset Ω of A(T, H). Following §7.9.1 we obtain the identification G(K)0Ω = Ind ΓΓ  H(L)0Ω , which upon taking fixed points under Γ becomes the identification G(k)0Ω = H()0Ω . This can be applied in particular to a bounded subset Ω  ⊂ A(S, G) = A(T , G)Γ . The above identification A(S, G) = A(S, H) then identifies Ω  with Ω. From this, we obtain the identifications G(k)∗Ω = H()∗Ω for all other decorations ∗ by multiplying both sides of G(k)0Ω = H()0Ω with the corresponding subgroup of (7.7.1). Now consider the smooth o -model HΩ∗ . Then Ro /o HΩ∗ is a smooth model for G according to Lemma A.3.12. Furthermore, its group of O-points equals   HΩ∗ (O L ) = H(L)∗Ω = G(K)∗Ω . HΩ∗ (O ⊗o o ) = σ ∈Gal(k  /k)

σ ∈Gal(k  /k)

Corollary 2.10.11 gives the identification GΩ∗ = Ro /o HΩ∗ . We have proved (2). Let G Ω0 and H Ω0 denote the special fibers of GΩ0 and HΩ0 respectively. To prove (3), we apply base change to f and use that Weil restriction of scalars commutes with base change. We obtain the following: G Ω0 = (GΩ0 ) f = (Ro /o HΩ0 ) f = R A/f ((HΩ0 ) A). This reduces the proof of (3) to showing that the maximal reductive quotient of R A/f ((HΩ0 ) A) is Rf /f (H Ω ). Applying [CGP15, Proposition A.5.12] to successive powers of the maximal ideal of A, we see that reduction modulo the maximal ideal of A gives a surjective homomorphism R A/f ((HΩ0 ) A) → Rf /f H Ω0 of f-groups with connected unipotent kernel. The projection H Ω0 → H Ω is a smooth surjective homomorphism of f groups with connected unipotent kernel U . Applying the functor Rf /f to it gives a surjective homomorphism Rf /f H Ω0 → Rf /f H Ω of f-groups with kernel the smooth affine f-group Rf /f U , see [CGP15, Proposition A.5.2(4) and Proposition A.5.14(3)]. The kernel is moreover connected and unipotent, for [f :f]

. We have thus obit is enough to check this over f , where it becomes U tained a surjective homomorphism R A/f ((HΩ0 ) A) → Rf /f H Ω whose kernel is a smooth connected unipotent f-group and the image Rf /f H Ω is a reductive f-group. This proves (3). 

P AR T T H R EE ADDITIONAL DEVELOPMENTS

10 Residue Field f of Dimension  1

Let k be a field given with a valuation ω : k → Z ∪ {∞} normalized so that ω(k × ) = Z. Let o = {x ∈ k | ω(x)  0} be the ring of integers of k, m = {x ∈ k | ω(x) > 0} the maximal ideal of o, and f = o/m the residue field. We will assume throughout this chapter that o is Henselian and f is perfect. Let K be a maximal unramified extension of k. We denote the unique extension of the valuation of k to K again by ω. Let O be the ring of integers of K and M the maximal ideal of O. Then O/M =: f is an algebraic closure of f. Using the natural isomorphism of Gal(K/k) with Aut(O/o) we identify them and denote both by Γ. We will assume throughout this chapter that the residue field f of k is of dimension  1, cf.§2.3. The purpose of this chapter is to derive additional results in Bruhat–Tits theory that hold under this stronger assumption on f. Let G be a connected reductive k-group. For any k-torus S of G, ZG (S) will denote its centralizer and NG (S) its normalizer in G.

10.1 Conjugacy of Special Tori Recall the concept of a special k-torus from 9.2.2. Proposition 10.1.1 Let A1 and A2 be special k-apartments of B(G K ), and T1 , T2 be the corresponding special k-tori. Let S  be a k-split torus contained in T1 ∩ T2 . Let Ω be a non-empty Γ-stable bounded subset of A1 ∩ A2 and GΩ0 be the smooth affine o-group scheme associated to Ω in 9.2.5. Then there is an element g ∈ GΩ0 (o)(⊂ G(k)) that commutes with S  and carries T1 to T2 . Proof According to Proposition 9.3.12 we can find g ∈ GΩ0 (o) that conjugates the maximal k-split torus S1 of T1 to the maximal k-split torus S2 of T2 , and commutes with S . We may thus assume that S1 = S2 and denote it 389

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Residue Field f of Dimension  1

by S. Note that S  ⊂ S. The standard model S  of S  is an o-split torus in GΩ0 . The centralizer M of S  in G is a smooth affine o-subgroup scheme (Proposition 8.2.1(1)) and its fibers are connected since the centralizer of a torus in a connected smooth affine algebraic group is connected [Bor91, Cor.11.12]. Using Proposition 8.2.1(1) for M in place of G , and the remark following that proposition, we see that the special fiber S of the standard model S of S is a maximal f-split torus in the special fiber M of M . On the other hand, the standard models T1 and T2 of T1 and T2 are closed o-tori of M with generic fibers T1 and T2 respectively (see 9.2.5). Their special fibers T 1 and T 2 are maximal f-tori of M . The reductive f-group M := M /Ru (M ) is quasi-split by Theorem 2.3.3(2), and hence any maximal f-split torus of M is contained in a unique maximal torus. Therefore, the images of T 1 and T 2 in M agree. This implies that T 1 is a maximal f-torus of the solvable f-subgroup H := T 2 · Ru (M ). Since any two maximal f-tori of the solvable f-group H are conjugate to each other under an element of H (f) [Bor91, Theorem 19.2], we conclude that T 2 is conjugate to T 1 under an element of M (f). Now Proposition 8.2.1(2) implies that there is an element g ∈ M (o)(⊂ G(k)) that  conjugates T1 onto T2 , so gT1 g −1 = T2 . Proposition 10.1.2 Any two special k-tori of G are conjugate to each other under an element of Gder (k). Proof In view of the natural bijective correspondence between the sets of special k-tori of Gder and G (9.2.2), we may (and do) replace G by Gder . For i = 1, 2, let Ti be a special k-torus of G and Ai the corresponding special k-apartment of B(G K ). If A1 ∩ A2 is non-empty, the first assertion follows immediately from Proposition 10.1.1. So let us assume that A1 ∩ A2 is empty. We fix a Γ-chamber Ci in Ai , for i = 1, 2 (Proposition 9.3.6). According to Proposition 9.3.8, there is a special k-apartment A containing C1 and C2 . Let T be the special k-torus of G corresponding to this apartment. Then using Proposition 10.1.1 twice, first for the pair { A, A1 }, and then for the pair {A, A2 }, we see that T is conjugate to both T1 and T2 under Gder (k). So T1 and T2 are  conjugate to each other under an element of Gder (k). Corollary 10.1.3 Let T be a special k-torus of G and S be the maximal k-split torus of G contained in T. Then NG (T)(k) ⊂ NG (S)(k) = ZG (S)der (k) · NG (T)(k). Hence the natural homomorphism NG (T)(k) → NG (S)(k)/ZG (S)(k)b , induced by the inclusion of NG (T)(k) in NG (S)(k), is surjective. Proof

Any k-automorphism of T carries the unique maximal k-split subtorus

10.2 Superspecial Points

391

S to itself. So NG (T)(k) ⊂ NG (S)(k). Now let n ∈ NG (S)(k), then nT n−1 is a special k-torus that contains S. So T and nT n−1 are special k-tori contained in ZG (S). Now Proposition 10.1.2 applied to ZG (S) in place of G implies that there is a g ∈ ZG (S)der (k)(⊂ ZG (S)(k)b ) such that gT g −1 = nT n−1 . Hence,  g −1 n belongs to NG (T)(k), and n = g · g −1 n.

10.2 Superspecial Points Proposition 10.2.1 If B(G) contains a point z that is special in B(G K ), then G is quasi-split. In particular, if B(G) contains a hyperspecial point, then G is quasi-split. Proof We fix a special k-torus T such that the apartment A of B(G K ) corresponding to TK contains z. Let S be the maximal k-split torus of G contained in T. We consider the parahoric o-group scheme Gz0 corresponding to the point z. Let G z0 be its special fiber and G z be the maximal reductive quotient of G z0 . Let S ⊂ T be the f-tori of G z corresponding to S ⊂ T. The torus S is a maximal f-split torus and T is a maximal f-torus of G z . As f is of dimension  1, G z is quasi-split. Hence, every root of Φ(Tf, (G z )f ) has non-trivial restriction to S. So, in particular, the restriction map Φ(Tf, (G z )f ) → Φ(S, G z ) is surjective. Now since z is special in B(G K ), given a nondivisible root b ∈ Φ(TK , G K ), some positive integral multiple nb of b belongs to Φ(Tf, (G z )f ). But then, the restriction of nb to S, and hence also the restriction of b to S, is non-trivial. This implies that every root of Φ(TK , G K ) has a nontrivial restriction to S. Therefore, the centralizer of S equals the centralizer of T in G, so the centralizer of S is a torus. This implies that G is quasi-split.  As existence of a hyperspecial point in B(G) implies that G is quasi-split and it splits over the maximal unramified extension of k (Propositions 10.2.1 and 8.4.14), Corollary 7.11.5 implies the following at once. Proposition 10.2.2 Hyperspecial points of B(G), and hence hyperspecial parahoric subgroups of G(k), are conjugate to each other under Gad (k).

10.3 Anisotropic Groups The following result is proved in [BT87a, 4.4-4.5] for complete k. Theorem 10.3.1

Assume that G is an absolutely almost simple anisotropic

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Residue Field f of Dimension  1

k-group. Then it splits over the maximal unramified extension K of k and is of type An for some n. Proof We know from Proposition 9.3.9 that B(G) = B(G K )Γ consists of a single point, say x. Let A be a special k-apartment of B(G K ), and C be a Γ-chamber in A (Proposition 9.3.6). It is a K-chamber by Proposition 9.10.4. Then C Γ = C ∩ B(G) is non-empty, and hence it equals {x}. Let I be the Iwahori subgroup of G(K) determined by the chamber C and T be the k-torus of G corresponding to the apartment A. Then I is stable under Γ, and TK is a maximal K-split torus of G K . We consider the (absolute) affine root system of G K with respect to TK and let Δ denote its basis determined by the Iwahori subgroup I. Then Δ is stable under the natural action of Γ on the affine root system and there is a natural Γ-equivariant bijective correspondence between the set of vertices of C and Δ. As B(G) does not contain any facets of positive dimension, we see from the discussion in 9.2.4 that Γ acts transitively on the set of vertices of C, and hence it acts transitively on Δ. Now from the description of irreducible affine root systems given in Proposition 6.4.8, we see that the affine root system of G K with respect to TK is of type An for some n, for otherwise, the action of the automorphism group of the (affine) Dynkin diagram of Δ cannot be transitive  on Δ. This implies that G K is K-split. Remark 10.3.2 The special orthogonal group of a non-degenerate quadratic form in 2n +1 variables is an absolutely simple group of type Bn , and according to the preceding theorem any group of type Bn is isotropic for n > 1. Therefore, every non-degenerate quadratic form in at least 5 variables over k is isotropic, that is, it represents 0 non-trivially. Note that the reduced norm form of a quaternion division algebra is a non-degenerate anisotropic quadratic form in 4 variables. If k is a non-archimedean local field (that is, a non-discrete locally compact non-archimedean field; such a field is complete and its residue field is finite), then any absolutely almost simple k-anisotropic group G is of inner type An for some n. This assertion was proved by Martin Kneser for fields of characteristic zero, and by Bruhat and Tits in general. In view of Theorem 10.3.1, to prove it, we just need to show that any simply connected absolutely almost simple k-group G of outer type An for n  2 is k-isotropic. This can be seen in two ways. Since there does not exist a non-commutative finite-dimensional division algebra with center a quadratic Galois extension of k which admits an involution of the second kind with fixed field k (see [Sch85, Chapter 10, Theorem 2.2(ii)]), if G is of outer type, then there is a quadratic Galois extension  of k and a non-degenerate hermitian form h on  n+1 such that G = SU(h).

10.4 Fixed Points of Large Subgroups of Tori

393

But q(x) := h(x, x) is then a non-degenerate quadratic form on  n+1 considered as a (2n + 2)-dimensional k-vector space. So for n  2, q, and therefore h, represents zero non-trivially, and hence SU(h) is isotropic. A second way to see this involves Corollary 10.6.2 below, which allows us a combinatorial classification of all adjoint k-groups. This classification is given in §10.7. From the description of absolutely almost simple k-anisotropic groups of inner type Ad−1 given in [Wei60], we know that given such a k-group G, there is a division algebra D of degree d (that is, of dimension d 2 ) and center k such that G is isogenous to SL1, D , and if G is simply connected, then it is isomorphic to SL1, D . Since over a non-archimedean local field k, any absolutely almost simple k-anisotropic group is of inner type A, we conclude that any simply connected absolutely almost simple k-anisotropic group is SL1, D for a finitedimensional central division algebra D/k. Example 10.3.3 The following example of an absolutely almost simple kanisotropic group of outer type Ar−1 (over a discretely valued complete field k with residue field of characteristic zero and of dimension  1) was communicated by Philippe Gille. As usual, C will denote the field of complex numbers; for a positive integer r, let μr denote the group of rth roots of unity, F = C(x) √ and F  = C(x ) with x  = x. We take k = F((t)) and  = F ((t));  is an unramified quadratic extension of k. Since F, F  are C1 -fields, their Brauer groups are trivial, and the residue maps induce isomorphisms:     N F  /F N/k  1  1 → ker H (F , μr ) −−−−−→ H (F, μr ) ker r Br() −−−−→ r Br(k) −    r N F  /F r − → ker F × /F × −−−−−→ F × /F × , 

× see [Ser97, §2 of the Appendix after Chapter II]. The element u := 1+x 1−x  ∈ F  has trivial norm over F, and has a pole of order 1 at x = 1, so it cannot be an rth power. It defines a central simple -algebra D which is division and cyclic of degree r. By Albert’s theorem, D carries a /k-involution τ of the second kind. The k-group SU(D, τ) is of outer type Ar−1 and is anisotropic over k. For a more combinatorial point of view on this example, see §10.7.

10.4 Fixed Points of Large Subgroups of Tori In the proof of the next proposition we will make use of the following lemma which is a variant of Lemma 5.1.35 of [BT84a].

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Residue Field f of Dimension  1

Lemma 10.4.1 Let F be a field and G be a quasi-split reductive F-group. Let S be a maximal F-split torus of G. Let Φ := Φ(S, G) be the root system of G with respect to S. Let S be a subset of S(F) such that every a ∈ Φ takes a non-trivial value on S. Let P be a parabolic F-subgroup of G. If P(F) contains S, then S ⊂ P. Proof Let B be a Borel F-subgroup of G containing S. The intersection B ∩ P is a smooth connected solvable F-subgroup that contains a maximal F-torus T of G such that S ⊂ T(F) ([Bor91, Corollary 14.13, Proposition 14.22(i), Theorem 10.6(5)(i)]). The centralizer Z of S in B is a smooth connected subgroup of B ([Bor91, Theorem 10.6(5)(ii)]) that contains the centralizer of S, and it also contains T. The Lie algebra of Z cannot contain a root space (with respect to S) of the Lie algebra of B since according to the hypothesis, every root takes a non-trivial value on S. Thus Z equals the centralizer of S, and hence it is a torus since G is quasi-split. Now as Z contains the maximal torus T, it equals T. So S ⊂ Z = T ⊂ P.  Proposition 10.4.2 Let S be a maximal k-split torus of the derived subgroup Gder of G and A be the corresponding apartment in the Bruhat–Tits building B(G) of G(k). If S is a subset of S(k)b such that for every a ∈ Φ(S, G), there is an element s ∈ S such that (a(s) − 1) ∈ o× , then A is the full fixed point set of S in B(G). The above implies at once that if the residue field f has at least four elements, then the full fixed point set of S(k)b in B(G) is the apartment A. Proof The fixed point set of S in B(G) is a convex subset that contains A. If it is strictly larger than A, then there exist chambers C and C  such that C  lies in A and C does not but it shares a panel (that is, a facet of codimension 1) F with C . Let GF0 be the Bruhat–Tits group scheme associated to F, and G F be the maximal reductive quotient of the special fiber of GF0 . Since f has been assumed to be perfect and of dimension  1, G F is quasi-split by Theorem 2.3.3(2). Let S be the maximal f-torus of G F corresponding to S. In view of our hypothesis on S, its image S in S(f) has the property that the roots of G F with respect to S take a non-trivial value on S. The chambers C and C  correspond to Borel subgroups P and P respectively of G F . Moreover, since S, a subset of S(k), is contained in G(k) and it fixes C pointwise, it is contained in GC0 (o), and hence S is contained in P(f). Now Lemma 10.4.1 implies that S is contained in P and hence S(K)b is contained in GC0 (O), so it fixes C pointwise. This implies that C lies in the apartment A. A contradiction. 

10.5 Existence of Anisotropic Tori

395

10.5 Existence of Anisotropic Tori The following theorem is quite important for the representation theory of, and harmonic analysis on, reductive groups over non-archimedean locally compact fields. For example, it implies, together with Harish-Chandra’s very deep work on representation theory of these groups, that each of them admits “discrete series” representations. According to a theorem of Harish-Chandra, G(R), for a connected semi-simple R-group G, has a discrete series representation if and only if G has an anisotropic maximal torus. Thus, for example, the group SLn (R) for n > 2 does not admit a discrete series representation. We will use the theorem in the proof of Theorem 10.6.8 below. Theorem 10.5.1 We assume that every finite extension of f admits cyclic Galois extensions of every finite degree (for example, f finite). Let G be a semisimple k-group. Then G contains an anisotropic maximal k-torus T which contains a k-subtorus S such that SK is a maximal K-split torus of G K . Proof Let z be a special point of the Bruhat–Tits building B(G) of G(k) and let G := Gz0 be the associated Bruhat–Tits parahoric o-group scheme with generic fiber G and connected special fiber G . Let G denote the maximal reductive quotient of G . As f has been assumed to be perfect and of dimension  1, G and its derived subgroup D are quasi-split by Theorem 2.3.3(2). Since z is a special point, the f-Weyl group of G is isomorphic to the k-Weyl group Wk of G, so the f-ranks of G, and D, are equal to the k-rank of G. Hence, in particular, the torus G/D is anisotropic over f (this torus may be trivial!). Lemma 10.5.2 proved below asserts that D contains an anisotropic maximal f-torus. Assuming this assertion for now, let S be a maximal f-torus of G that contains an anisotropic maximal f-torus of D. Then since the f-torus G/D is anisotropic, we infer that S is f-anisotropic. Consider the natural surjective homomorphism π : G → G. The kernel of π is the smooth connected unipotent radical of G and hence π −1 (S) is a smooth connected solvable f-subgroup of G . We fix a maximal f-torus S of this solvable subgroup; S maps isomorphically onto S under π. Now using Proposition 8.2.1(1), we find a closed o-torus S in G whose special fiber is S . Let S be the generic fiber of S . The character groups of S, S and S are all isomorphic to each other as Γ-modules, where we recall that Γ = Gal(K/k) = Aut(O/o) = Gal(f/f). So since S splits over f, we see that S splits over O. Moreover, S is k-anisotropic since the special fiber S of S is f-anisotropic, and SK splits. As the special fiber S of S is a maximal torus of G , according to Theorem 8.2.5(2), SK is a maximal K-split torus of G K . As G is quasi-split over K, we conclude that the centralizer T of S in G is a maximal k-torus of

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Residue Field f of Dimension  1

G. Since SK is a maximal K-split torus of G K , TK /SK is anisotropic over K, and hence T/S is k-anisotropic. Now using the fact that S is k-anisotropic, we conclude that T is also k-anisotropic.  Lemma 10.5.2 Assume that every finite extension of f admits cyclic Galois extensions of every finite degree. Then every quasi-split semi-simple f-group contains an f-anisotropic maximal torus. Proof It is clearly enough to prove the lemma for simply connected quasisplit semi-simple f-groups and we can moreover restrict to considering only absolutely simple f-groups. So let G be such a quasi-split group. We fix a maximal f-split torus S of G. We will treat first the case where G is split. In this case S is a maximal torus. We choose an element n of finite order in the normalizer of S in G(f) whose image in the Weyl group of G is a Coxeter element. (Note that a Coxeter element does not fix any non-trivial character of S.) Let K be a finite cyclic extension of f of degree equal to the order of the element n. Let γ be a generator of the Galois group Γ of K/f. Then c : γ i → ni is a 1-cocycle on the Galois group Γ. Since H1 (f, G) is trivial (Theorem 2.3.3), there exists a g ∈ G(K) such that n = g −1 γ(g). Then gSg −1 is a maximal f-torus of G and it is anisotropic over f since, as can be easily verified, Γ does not fix any non-trivial character of gSg −1 . Now we assume that G is not split (but it is quasi-split). Then the centralizer Z of the maximal f-split torus S in G is a maximal torus. Let T be the maximal f-anisotropic subtorus of Z. Then S · T = Z. Let H be the derived subgroup of the centralizer of T in G. We claim that H contains S as a maximal torus and hence it is a split semi-simple subgroup, so by the already proven result for split semi-simple groups, we know that H contains an f-anisotropic maximal torus, say T . By conjugacy of maximal tori of H over f, we see that the dimension of T  equals that of S and hence the f-anisotropic torus T · T  is a maximal torus of G. We will now prove the claim that H contains S and hence it is a split semisimple group. For this purpose we observe that for every long root a in the root system Φ(S, G), the corresponding root group Ua is 1-dimensional (since H is quasi-split!) and hence the anisotropic torus T commutes with it. So Ua ⊂ H for all long roots a ∈ Φ(S, G). As the f-Weyl group of G acts irreducibly on the Q-vector space V spanned by Φ(S, G) and the set of long roots is stable under the action of this Weyl group, we see that the long roots span V. As all the long roots belong to Φ(S, S · H), we conclude that f−rank(H) = dim(S), and hence S is contained in H. 

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10.6 Cohomological Results Recall from Proposition 9.10.4 that if G is a connected reductive k-group then any Γ-chamber is a K-chamber. Theorem 10.6.1 Let G be a reductive k-group and I = GC0 (O) be a Γ-stable Iwahori subgroup of G(K), where C is a Γ-chamber of B(G K ). Let N be the normalizer of I in G(K) (N is the stabilizer of C). Then the natural maps H1 (Γ, N/I) ← H1 (Γ, N) → H1 (Γ, G(K)) → H1 (k, G) are bijective. Proof The set H1 (K, G) is trivial by Proposition 2.3.5. The inflation-restriction sequence implies that the inflation map H1 (Γ, G(K)) → H1 (k, G) is bijective. Surjectivity of H1 (Γ, N) → H1 (Γ, G(K)): Let c : γ → c(γ) be a 1-cocycle on the Galois group Γ of K/k with values in G(K) and c G be the Galois-twist of G with the cocycle c. The K-groups G K and c G K are canonically isomorphic. We will identify c G(K) with G(K), and also the Bruhat–Tits buildings of G(K) and c G(K), using this isomorphism and denote both the buildings by B. We recall that with the identification of c G(K) with G(K) as abstract groups, the “twisted” action of Γ on c G(K) is described as follows: For x ∈ c G(K), and γ ∈ Γ, γ ◦ x = c(γ)γ(x)c(γ)−1 , where γ(x) denotes the γ-transform of x considered as an element of G(K). Similarly B admits two actions of Γ, an action induced by its action on G(K), and the “twisted action” induced by the twisted action on c G(K). Applying Proposition 9.10.4 to c G we see that B admits a chamber C  that is stable under the “twisted action” of Γ. Since any two chambers of B are conjugate to each other under an element of G(K), there exists a g ∈ G(K) such that C  = g · C. Then gIg −1 is the Iwahori subgroup corresponding to the chamber C  = g · C. Since C , and so also gIg −1 , is stable under the twisted action of Γ, we see that c(γ)γ(g)Iγ(g)−1 c(γ)−1 = gIg −1 for all γ ∈ Γ. Hence, for γ ∈ Γ, c (γ) := g −1 c(γ)γ(g) ∈ G(K) normalizes the Iwahori subgroup I, and it is a 1-cocycle cohomologous to c. This proves the surjectivity of H1 (Γ, N) → H1 (Γ, G(K)). Injectivity of H1 (Γ, N) → H1 (Γ, G(K)): Consider c, c  ∈ Z 1 (Γ, N) and g ∈ G(K) such that c (γ) = g −1 c(γ)γ(g) for γ ∈ Γ. Since c(γ), c (γ) ∈ N for all γ ∈ Γ, we see that C is stable under c(γ)γ and c (γ)γ. The relation c (γ) = g −1 c(γ)γ(g) implies that both C and gC are stable under c(γ)γ. Therefore, C and gC are Γ-chambers in B(c G K ). Proposition 9.3.22 implies the existence of h ∈ c G(k) such that hC = gC. Thus c(γ)γ(h)c(γ)−1 = h and n := h−1 g ∈ N.

398

Residue Field f of Dimension  1

But then n−1 c(γ)γ(n) = c(γ) and we conclude that c and c  are cohomologous in Z 1 (Γ, N). Injectivity of H1 (Γ, N) → H1 (Γ, N/I): Using the usual twisting argument (see [Ser97, Chapter I, §5.3]), it suffices to note that Lemma 8.1.4 implies that H1 (Γ, c GC0 (O)) is trivial for any N-valued 1-cocycle c of Γ. Surjectivity of H1 (Γ, N) → H1 (Γ, N/I): We assume in this paragraph that k is complete. Let T be the centralizer of a special k-torus whose apartment contains C and let N = NG (T). Note that T is a maximal k-torus of G. Let N(K)C be the stabilizer of C in N(K). We know that I ∩ N(K) = T(K)0 (Lemma 7.5.2). Since I acts transitively on the set of apartments containing C, the injection N(K)C → N induces an isomorphism N(K)C /T(K)0 → N/I. It is therefore enough to show that H1 (Γ, N(K)C ) → H1 (Γ, N(K)C /T(K)0 ) is surjective. According to [Ser97, Proposition 41, Chapter I, §5.6], the latter follows from the vanishing of H2 (Γ, c T(K)0 ) for any N(K)C /T(K)0 -valued 1cocycle c on Γ. Now note that since k has been assumed to be complete in this paragraph, Corollary B.10.14 implies the vanishing of H2 (Γ, c T(K)0 ) (recall that c T(K)0 = c T(K)0 ). We now drop the assumption that k is complete. We have already established the injectivity of the left map and the bijectivity of the right map. Let K  =   of K is also a completion of K . The composite of k ⊗k K. The completion K the maps  H1 (Γ, G(K)) → H1 (Γ, G(K )) → H1 (Γ, G(K)), as well as the second of the two maps, are bijective due to [GGMB14, Proposition 3.5.3(2)]. Therefore, H1 (Γ, G(K)) → H1 (Γ, G(K )) is also bijective. Let I  and N  denote the Iwahori subgroup of G(K ) containing I and its normalizer respectively. Then as the natural maps H1 (Γ, N) → H1 (Γ, G(K)) and H1 (Γ, N ) → H1 (Γ, G(K )) are bijective, we infer that H1 (Γ, N) → H 1 (Γ, N ) is also bijective. Also, since  k is complete, from the surjectivity proved in the preceding paragraph we obtain the surjectivity of H1 (Γ, N ) → H1 (Γ, N /I ). On the other hand, Propositions 7.10.2 and 7.7.5 imply N /I   N/I and we  conclude that H1 (Γ, N) → H1 (Γ, N/I) is surjective. Recall from Proposition 7.7.12 the subgroup Ξ of the symmetry group of the affine Dynkin diagram of the group G K . The k-structure of G endows Ξ with an action of Γ.

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Corollary 10.6.2 Let G be a quasi-split adjoint k-group. Let I be a Γ-stable Iwahori subgroup of G(K) and N its normalizer. The natural maps H1 (Γ, Ξ) ← H1 (Γ, N) → H1 (Γ, G(K)) → H1 (k, G) are bijective. In particular, if G is K-simple and the affine Dynkin diagram of G K has no non-trivial symmetries, then H1 (k, G) is trivial. Proof According to Proposition 7.7.12 the map N → Ξ factors through an isomorphism N/I → Ξ, so the corollary is an immediate consequence of Theorem 10.6.1.  Remark 10.6.3 From Tables 1.3.4 and 6.4.1 we infer that the K-simple adjoint quasi-split k-groups with vanishing cohomology are those of K-type 2 A2n , 3 D4 , 6D , 2E , E , F , G . 4 6 8 4 2 Theorem 10.6.4 Let G be a simply connected semi-simple k-group. Then the Galois cohomology set H1 (k, G) is trivial. Proof The simply connectedness of G implies that G(k)0 = G(k). Since, in the abstract setting of a Tits system, every parabolic subgroup is equal to its own normalizer (Proposition 1.4.5(3)), Axiom 4.1.9 implies that every Iwahori subgroup I of G(K) is equal to its own normalizer. The claim now follows from Theorem 10.6.1.  Remark 10.6.5 Theorem 10.6.4 was first proved by a case-by-case analysis by Martin Kneser for k a non-archimedean local field of characteristic zero. It was proved for all discretely valued complete fields k with perfect residue field of dimension  1 by Bruhat and Tits [BT87a, Theorem in §4.7]. For k, G) is injective the completion  k of k, the natural map H1 (k, G) → H1 ( (Proposition 2.3.5). So the vanishing theorem of Bruhat and Tits over the completion  k also implies the above theorem. 10.6.6 Let k s denote a fixed separable closure of k. Let H be an affine flat algebraic k-group scheme. If H is smooth, then H1 (k, H) denotes as usual the first Galois cohomology set with coefficient in H(k s ). If H is not smooth, then we let H1 (k, H) denote the flat cohomology H1 (Spec(k)fl, H). If H is commutative, then H1 (Spec(k)fl, H) is a group and similar flat cohomology groups are defined in all positive degrees. The usual exact sequence of Galois cohomology sets associated with a short exact sequence of smooth algebraic k-groups is also available for flat cohomology and a short exact sequence of affine flat algebraic k-group schemes. If H is smooth, then the two cohomology sets are canonically isomorphic. For all this, see [Sha72, Chapter VI, §3], or [Mil80, Chapter III].

Residue Field f of Dimension  1

400

The following lemma will be used in the proof of the next theorem. Lemma 10.6.7 Assume that k is a local field. Let T be a k-anisotropic torus. Then H2 (k,T) = 0. Proof By Tate–Nakayama duality for local fields [Sha72, Chapter VI, §5], we know that for any finite Galois extension  of k with Galois group Θ, H2 (Θ,T)  0 (Θ, X∗ (T )) is the 0th Tate cohomology which is  0 (Θ, X∗ (T )), where H = H Θ equal to X∗ (T ) /N(X∗ (T )) with N(X∗ (T )) = { θ ∈Θ θ · λ | λ ∈ X∗ (T )}.  0 (Θ, X∗ (T )) = 0, and hence, However, as T is k-anisotropic, X∗ (T )Θ = 0, so H

H2 (Θ,T) = 0. Now by the taking limit over all finite Galois extensions /k, we  conclude that H2 (k,T) = 0. As a corollary we obtain the following description of the cohomology of semi-simple groups. Its extension to connected reductive groups is stated in Theorem 11.7.7. Theorem 10.6.8 Assume that k is a local field. Let G be a simply connected semi-simple k-group, and C a central k-subgroup scheme. (Then C is a finite group scheme of multiplicative type and it is contained in every maximal torus of G.) Let G  = G/C, and π be the natural projection of G onto G . Then the coboundary map δ : H1 (k, G ) → H2 (k, C), associated with the short exact sequence /G

/C

1

π

/ G

/1

is bijective. In particular, H1 (k, G ) has a canonical abelian group structure. Proof We fix a k-anisotropic maximal torus T  in G . Such a torus is provided by Theorem 10.5.1. Let T = π −1 (T ). Then T is a k-anisotropic maximal torus of G. We have the following commutative diagram of exact sequences 1

/C

/T

π

/ T

/1

1

/C

 /G

π

 / G

/ 1.

From this we get the following commutative diagram of Galois cohomology: H1 (k,T )  H1 (k, G )

δ

/ H2 (k, C)

δ

/ H2 (k, C).

/ H2 (k,T)

Now since T is a k-anisotropic k-torus, by Lemma 10.6.7, H2 (k,T) = 0, and

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hence δ : H1 (k,T ) → H2 (k, C) is surjective. So δ : H1 (k, G ) → H2 (k, C) is also surjective. To show that δ : H1 (k, G ) → H2 (k, C) is injective, we argue as follows. The action of G on itself by conjugation induces an action of G  on G (since C is central). Thus given a 1-cocycle c on Gal(k s /k) with values in G (k s ), we can twist G by c to obtain a simply connected semi-simple k-group c G. As  H1 (k, c G) is trivial by Theorem 10.6.4, we see that δ is injective.

10.7 Classification of Connected Reductive k-Groups In this section we will classify the isomorphism classes of connected reductive k-groups. It is well known that if G is a connected reductive k-group, there exists a unique connected reductive quasi-split k-group G∗ admitting an inner twist ξ : G∗ → G, that is an isomorphism ξ : G∗ks → G ks such that for every σ ∈ Gal(k s /k) the automorphism ξ −1 σ(ξ) of G∗ks is inner. The isomorphism class of G∗ is uniquely determined by its based root datum endowed with a Gal(k s /k)-action; the isomorphism class of G is determined by G∗ and the image in H1 (Gal(k s /k), Autks (G∗ )) of the class of σ → ξ −1 σ(ξ). Slightly more rigidly, one can consider the isomorphism class of the pair (G, ξ), which is determined by G∗ and the element of H1 (k, G∗ad ) that is the class of σ → ξ −1 σ(ξ). This discussion works over an arbitrary field k. The classification of the possible groups G∗ in terms of based root data with Gal(k s /k)-action is of a combinatorial nature. For example, when G∗ is k-simple, a lot of information about G∗ can be encoded in the Tits index, which is either one of the split (that is inner) types 1 An , Bn , Cn , 1 Dn , 1 E6 , E7 , E8 , F4 , G2 , or one of the quasi-split non-split (that is outer) types 2 An , 2 Dn , 3 D4 , 6 D4 , 2 E6 , cf. [Tit66]. Here the left superscript denotes the order of the image of the Galois group in the group of symmetries of the absolute finite Dynkin diagram. Note that any two split adjoint k-groups of a given type are necessarily isomorphic. However, two non-split quasi-split adjoint k-groups of a given type need not be isomorphic, because the order of the image of the Galois group does not determine the splitting field of G∗ . On the other hand, the exact determination of H1 (k, G∗ad ) is generally difficult. When k is discretely valued and Henselian, and dim(f)  1, we can apply Theorem 10.6.1 and Corollary 10.6.2 to obtain a linear algebraic description of H1 (k, G∗ad ) as well. More precisely, we assume that G∗ad has been determined, K the absolute affine let DK be the K-relative Dynkin diagram of G∗ , and D

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Residue Field f of Dimension  1

K from Table 6.4.1 (which we are using Dynkin diagram. One can read off D K by removing any in the case of k = K). The diagram DK is obtained from D extra special vertex (an empty node in the Table 1.3.4). As discussed in Remark 1.3.76, the derivative map Aut(ΨK ) → Aut(ΦK ) induces the extension (1.3.2) K ) → Aut(DK ) → 1. 1 → Ξ → Aut(D The group Ξ, which can be described as the quotient of the extended affine Weyl group by the affine Weyl group, is explicitly described in Remark 1.3.76. It is a commutative group. The k-structures of G∗ and G give homomorphisms f∗, f : Γ → Aut(DK ). These homomorphisms are equal, so will be denoted by f below, and give the so-called ∗-action, cf. [Tit66, §2.3]. The k-structures of G∗ and G also K ) which, upon composition with f : Γ → Aut(D give homomorphisms  f∗,  K ) → Aut(DK ), recover the homomorphism f . Unlike the case of f , we Aut(D f . In fact, sending a 1-cocycle c : Γ → Ξ to c ·  f∗ establishes may have  f∗   1 a 1-1 correspondence between H (Γ, Ξ) and the set of Ξ-conjugacy classes K ) that lift f . The element of H1 (Γ, Ξ) of homomorphisms  f : Γ → Aut(D corresponding to  f is identified under H1 (Γ, Ξ) = H1 (k, G∗ad ) (Corollary 10.6.2) with the element classifying G in terms of G∗ . In the cases listed in Remark 10.6.3 we have Ξ = {1} and hence G = G∗ . We summarize this discussion as follows. A connected reductive k-group G is classified by the following two pieces of data: (1) a based root datum with Gal(k s /k)-action; this induces a K-relative based root datum with Γ-action, and we write f : Γ → Aut(DK ) for the latter action; K ) lifting the (2) a (Ξ-conjugacy class of a) homomorphism  f : Γ → Aut(D homomorphism f : Γ → Aut(DK ). In order to obtain more explicit information from this classification, we would like to compute the k-relative affine and finite root systems Ψk and Φk from the homomorphism  f . Of course, Φk = ∇Ψk , so it is enough to compute Ψk . For this computation, it is enough to assume that G∗ is k-simple and adjoint. So, henceforth in this subsection we will assume that G, so also G∗ , is k-simple and adjoint. The nature of the homomorphism f depends on whether G∗ splits over k, or over K, or not, as follows. K ) is trivial. (1) When G∗ is k-split, the homomorphism  f∗ : Γ → Aut(D 1 Hence H (Γ, Ξ) = Hom(Γ, Ξ) and G is determined by a group homomorphism  f : Γ → Ξ.

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(2) When G∗ is not K-split, the same situation occurs. Indeed, the action of the absolute Galois group on the absolute Dynkin diagram of G∗ is non-trivial, so the K-relative Dynkin diagram is already a folding and has no additional symmetries. Therefore the action of Γ on DK is trivial, and  f∗ is also trivial. Again H1 (Γ, Ξ) = Hom(Γ, Ξ) and G is given by a group homomorphism  f : Γ → Ξ. f∗ and f are (3) When G∗ is not k-split, but K-split, then the homomorphisms  non-trivial. An element of H1 (Γ, Ξ) is the same as a Ξ-conjugacy class of K ) lifting f . The conjugacy class group homomorphisms  f : Γ → Aut(D  of f∗ corresponds to the quasi-split form. f . If G is We will now give an algorithm for computing Ψk in terms of  not K-simple, then ΨK is a product of copies of the same affine root system permuted transitively by Γ. Upon selecting one of these copies and replacing Γ by the stabilizer of that copy, we may assume that G, and hence also G∗ , is K-simple. Given a Γ-invariant chamber C in B(G∗K ) there exists a unique homomorphism Γ → Aut(B(G∗K )) such that the resulting action of Γ on B(G∗K ) preserves C and induces on the set of vertices of C, which are in bijection with the set K , the action given by  f . We shall denote this homomorphism of nodes of D f (γ)F = γ(g)−1  also by  f . For a facet F ∈ B(G K ) we have  f (γ)gF for any g ∈ Gsc (K) such that gF is a facet of C. The identification B(G K ) = B(G∗K ) f. translates the natural Γ-action on B(G K ) to the Γ-action on B(G∗K ) given by  We can thus identify B(G) ⊂ B(G∗K ) as the set of fixed points for the action of Γ given by  f , and the set C of fixed points in C is a chamber in B(G), cf. 9.10.7 and Proposition 9.10.8. Construction 10.7.1 Folding of affine Dynkin diagrams The relative affine k of the affine root system of G relative to k is obtained Dynkin diagram D K and the Γ-action given by  f by from the absolute affine Dynkin diagram D the following process of folding. K . k are the Γ-orbits of vertices in D (1) The vertices of D  (2) Two vertices of Dk are linked by an edge if and only if some member of the Γ-orbit for the first vertex is linked to some member of the Γ-orbit for the second vertex. k let  nvk be the sum of the integers nv (1.3.1) for (3) For each vertex vk of D  all vertices v of DK in the Γ-orbit corresponding to vk . Let nvk be the n vk . quotient of  nvk by the greatest common divisor of all   (4) A vertex of Dk is multipliable if and only if there exist two vertices of K in the corresponding Γ-orbit that are linked. D

404

Residue Field f of Dimension  1

Proposition 10.7.2 The above construction specifies a unique affine Dynkin k associated to the relative affine diagram. It is the affine Dynkin diagram D root system of G. k of G satisfies points Proof Let us first argue that the affine Dynkin diagram D (1)–(4) above. Choose a maximal split torus S of G and a special k-torus T of G containing S by Proposition 9.3.4. We have A(S, G) = A(T, G)Γ ⊂ A(T, G) and can consider the restriction map A(T, G)∗ → A(S, G)∗ . Proposition 9.4.28 shows that the relative affine roots are precisely the non-constant restrictions of the absolute affine roots. According to Propositions 9.10.1 and 9.10.4 there exists a Γ-invariant chamber C of A(T, G). Then C = C Γ is a chamber of A(S, G). Proposition 9.10.8 says that the relative simple roots are the restrictions of the absolute simple roots, that no absolute simple root has a constant restriction, and that two absolute simple roots have the same restriction if and only if they are in the same Γ-orbit. This shows point (1), and point (2) follows from this and the fact that the scalar product of two affine simple roots (absolute or relative) is non-positive, and strictly negative if and only if they are linked. Point (3) is the immediate observation that the restriction of a constant function is constant, and that a function is constant if and only if any scalar multiple of it is. For point (4), we leave the special case that DK is of type A1 as an exercise to the reader. In all other cases, two vertices of DK that are in the same orbit under a symmetry are either not linked, or linked by a simple bond, because a multiple bond will signify that the derivatives have different lengths. If ψ1, ψ2 are the corresponding absolute affine simple roots, then it follows from the arguments of [Bou02, Chapter VI, §1, no. 3] that ψ1 + ψ2 is an absolute affine simple root if and only if ψ1 and ψ2 are linked. But the restriction of ψ1 is multipliable if and only if there is ψ2 with the same restriction such that ψ1 + ψ2 is an affine root. k of G satisfies points (1)– We have argued that the affine Dynkin diagram D (4) above. That there is at most one reduced affine Dynkin diagram that satisfies points (1)–(3) follows from Remark 1.3.48 and Table 1.3.5. In particular, the strength and direction of the bonds are determined by this information. That there is at most one affine Dynkin diagram with a fixed reduced affine Dynkin diagram and prescribed multipliable vertices follows by inspection of Tables 1.3.4 and 1.3.3.  Remark 10.7.3 The Dynkin diagram of the relative finite root system of G k by dropping any one of the extra special vertices. can be read off from D  Since f determines G, it also determines the Tits index of G. In fact, the latter depends only on the image of  f.

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Remark 10.7.4 There is one more bit of information that is determined by K and  D f and it is encoded in a positive integer d(v) associated to a vertex v of  we Dk . If ψ ∈ Ψk is the relative simple affine root corresponding to v and a = ψ, have the integral model Uψ of the root group Ua , cf. 9.4.5. There is a natural homomorphism Uψ+ → Uψ and it induces a homomorphism between the special fibers U ψ+ → U ψ whose image is normal. Let V ψ be the cokernel. This is a smooth connected unipotent f-group and we set d(v) = dim(V ψ ). When ψ is not multipliable (that is, 2ψ  Ψk ) then V ψ has a natural structure of an f-vector group. If ψ is multipliable then V 2ψ is a closed central subgroup of V ψ , and each of the two groups V ψ /V 2ψ and V 2ψ is naturally an f-vector group. K and the image The integer d(v) can be computed combinatorially from D of  f as follows. Assume that G is almost K-simple. If the set of vertices of K that map to v consists of non-linked vertices, then d(v) is the cardinality D of that set; in the other case that set consists of exactly two linked vertices and d(v) = 3. We will see from the classification given below that the relative affine Dynkin k together with the integers d(v) determines uniquely the absolute diagram D K and the image of  f. affine Dynkin diagram D Remark 10.7.5 It is known ([Wei60]) that k-groups of classical types are constructed using central simple algebras. For example, groups of inner type An are described in terms of central simple k-algebras of dimension (n + 1)2 . Every group of outer type An is described in terms of a central simple algebra of dimension (n + 1)2 over a quadratic Galois extension  of k, which is endowed with an involution σ of the second kind (that is σ | is the non-trivial automorphism of /k). Groups of type Cn and groups of inner and outer type Dn are constructed using central simple k-algebras given with an involution of the first kind. When comparing this with the description in terms of a K ), it is helpful to keep in mind the following homomorphism  f : Γ → Aut(D fact about the Brauer group of k: Br(k)  H2 (Γ, K × )  H2 (Γ, Z)  Hom(Γ, Q/Z). The first isomorphism is given by the inflation map, the second by the valuation, and the third by the connecting homomorphism for the exact sequence 0 → Z → Q → Q/Z → 0, cf. [Ser79, Chapter XII, §3, Theorem 2] for the case where k is complete, and apply [GGMB14, Proposition 3.5.3(3)] in general, k realizes the isomorphism Br(k) → Br( k). We have used noting that D → D⊗k  that Br(f) = {0} due to dim(f)  1. Moreover, the degree of the division algebra representing an element of Br(k) is equal to the order of the corresponding

406

Residue Field f of Dimension  1

element in any of these groups, cf. [Ser79, Chapter XII, §3, Exercise 1] for the case of k complete, and use the above isomorphism in general. A central division algebra over k can be equipped with an involution of the first kind if and only if it is isomorphic to its opposite or, equivalently, if and only if its class in the Brauer group of k is of order 2. Therefore, over the fields k under consideration, a central division algebra admits an involution of the first kind if and only if it is of degree 2 (that is, it is a quaternion division algebra). Given a quaternion division k-algebra D, the standard involution of D of the first kind is the following x → Trd(x) − x, where Trd(x) is the reduced trace of x ∈ D. On the other hand, as we saw in Example 10.3.3, over certain discretely valued complete fields (with residue field of dimension  1), there exist division algebras of all degrees, admitting an involution of the second kind. But if k is a local (that is, non-discrete locally compact) field, as in Remark 10.3.2, there does not exist any non-commutative division algebra over a quadratic Galois extension of k that admits an involution of the second kind. We will now consider each individual absolutely simple adjoint quasi-split K group G∗ and describe its inner forms by folding the Dynkin diagram D ∗ ∗ according to the image of  f . We will record G by the pair of types of G K and ∗ k , G . For each inner form we will give the relative affine Dynkin diagram D the relative finite Dynkin diagram Dk , and the Tits index. We refer to §2.6(c) for a simple recipe that computes the relative root system of a Tits index, which we have used implicitly. We will also record for each vertex the integer d(v) of Remark 10.7.4, as well as whether the vertex is special (s) or hyperspecial (hs). We will put a circle around vertices corresponding to multipliable roots. We will give the diagrams of the quasi-split groups explicitly only for groups that split over K but not over k. In the other cases, the diagram of the quasi-split K , which we are taking as input, so we form coincides with the diagram of D will not repeat this information.

(a) (1 An−1, 1 An−1 ) K is of type An−1 and is a cycle with n vertices. The affine Dynkin diagram D The group Ξ = Z/nZ acts on it by cyclic permutations. The image of  f is a k of cyclic group of order d |n. Letting r = (n/d) − 1, the affine root system D the form G is of type Ar , where A0 is to be interpreted as the empty affine root system, signifying that G is anisotropic. The group G is PGLr+1 (D) associated to a division algebra of degree d and has Tits index 1 A(d) n−1,r . Each integer d(v) equals d. Each vertex is special, but not absolutely special or superspecial.

10.7 Classification of Connected Reductive k-Groups

407

(b) (1 An−1, 2 An−1 ) We have Aut(DK ) = Z/2Z and the k-structure of G∗ is given by a surjective K ) is the dihedral group homomorphism f : Γ → Z/2Z. Moreover, Aut(D D2n = Z/nZ  Z/2Z and Ξ is the unique cyclic subgroup Z/nZ. The group G is determined by a homomorphism  f : Γ → D2n , well defined up to conjugation by Z/nZ, that lifts f . Before we proceed we recall a few basic facts about the structure of the dihedral group D2n . It is equal to Z/2Z when n = 1, equal to (Z/2Z)2 when n = 2, and non-abelian otherwise. An element of the normal subgroup Z/nZ is customarily called a “rotation” and we will write t for 1 ∈ Z/nZ. Write r for 1 ∈ Z/2Z, so that t and r generate D2n . We let D2n act on Z/nZ where t acts by the “translation” x → x + 1, and r acts by x → −x. If we represent Z/nZ as the vertices of a regular n-gon, then t acts by rotation, and r by reflection. Any element of the form t i r is of order 2, and acts as a reflection of the regular n-gon. When n is odd there is a unique Z/nZ-conjugacy class of reflections, and each reflection fixes exactly one point of Z/nZ; when n is even there are two conjugacy classes of reflections. These are {r, t 2 r, . . . , t n−2 r }, in which each reflection fixes exactly two points of Z/nZ, and {tr, t 3 r, . . . , t n−1 r }, in which none of the reflections fix a point of Z/nZ. We illustrate the reflections r and tr in the cases n = 5 and n = 6:

A subgroup H ⊂ D2n is either contained in Z/nZ and is thus cyclic, or not, in which case it is generated by t l , t i r for some l |n, 1  l  n, and 0  i < l and is thus isomorphic to D2n/l . If n is odd then H is conjugate under Z/nZ to t l , r . The set of orbits of t l , r in Z/nZ consists of the orbit l Z/nZ of size n/l, and the orbits (i + l Z/nZ) ∪ (−i + l Z/nZ) of size 2n/l for i = 1, . . . , (l − 1)/2. If n is even then H is conjugate under Z/nZ to exactly one of t l , r or t l , tr . The set of orbits of t l , r in Z/nZ consists of the orbit l Z/nZ of size n/l, the orbits (i + l Z/nZ) ∪ (−i + l Z/nZ) of size 2n/l for i = 1, . . . , (l − 1)/2, and, if l is even, also the orbit 2l + l Z/nZ of size n/l. The set of orbits of t l , tr in Z/nZ consists of the orbits (i + l Z/nZ) ∪ (1 − i + l Z/nZ) of size 2n/l for i = 1, . . . , l/2, and, if l is odd, also the orbit l+1 2 + l Z/nZ of size n/l. To see this, note that the orbits of l Z/nZ in Z/nZ are the fibers of the natural projection Z/nZ → Z/l Z, and this projection is equivariant under the reflections r : x → −x and tr : x → 1 − x.

Residue Field f of Dimension  1

408

Returning to the study of G, the fact that  f lifts f implies that the image of  f is a dihedral subgroup of Dn of order 2d for some d |n. Let us examine first the case d = 1. Then the image of  f is generated by a single reflection in D2n . Up to Ξ-conjugation this reflection is equal to r when n is odd, or to exactly one of r or tr when n is even. When the reflection is r, K in the cases of n even and n odd are given (in the examples the folding of D of n = 10 and n = 9) by

1

2

2

2

2

hs

1

3

hs

s

2

2

2

1 hs

yielding relative affine root systems of type Cn/2 when n is even and of type (BC(n−1)/2, C(n−1)/2 ) when n is odd. The relative finite root systems have type Cn/2 when n is even and type BC(n−1)/2 when n is odd. The Tits indices are 2 A(1) when n is even and 2 A(1) when n is odd. These foldings n−1, n/2 n−1,(n−1)/2 correspond to the trivial element of H1 (Γ, Ξ) and lead to the quasi-split form G = G∗ . When n is even and the reflection is tr, the folding is given (in the example n = 10) by

3 s

2

2

2

3 s

∨ yielding a relative affine root system of type (C(n/2)−1 , C(n/2)−1 ), hence a relative finite root system of type BC(n/2)−1 . The rank of G is thus one less than that of G∗ . The group G is the projective unitary group of a non-degenerate Hermitian form of index r = (n/2) − 1 relative to the unramified quadratic extension that 2 (2) splits G∗ , and has Tits index 2 A(1) n−1,r . Note that the Tits index A3,1 does not occur, since it would produce a relative finite root system of type A1 rather than BC1 . Consider now the case d > 1. The image of  f is conjugate to either t l , r or t l , tr , with l = n/d. Folding first under the action of t l we obtain as in the case of (1 An−1, 1 An−1 ) the affine root system of type Al−1 , which is empty when l = 1. Assuming l > 1 we fold this affine root system under the action of r or tr, respectively, to obtain as in the case of d = 1 just discussed one of the affine ∨ , C(l/2)−1 ). root systems Cl/2 , (BC(l−1)/2, C(l−1)/2 ), or (C(l/2)−1 The group obtained in this way is the projective unitary group PUr (D, h),

10.7 Classification of Connected Reductive k-Groups

409

where r is the rank of the relative root system (r = l/2, (l − 1)/2, or l/2 − 1, in the respective cases), D is a central division algebra of degree d over the unramified quadratic extension /k with  being the fixed subfield of K of  f −1 ( t l ) ⊂ Γ, equipped with an involution σ of the second kind, and h is a non-degenerate Hermitian form of index r relative to σ. The Tits index of this group is 2 A(d) n,r . Note that, since the group D2d is not cyclic when d > 1, this case cannot occur when Γ is procyclic (for example, when f is finite), but it may occur otherwise. This sheds a new light on Example 10.3.3, in which case f = C(x) and Γ is a free profinite group, so every dihedral subgroup of D2n , in particular the entire D2n , can be a homomorphic image of Γ.

(c) (2 An, 2 An ) K is of type BCn/2 and Ξ = Aut(D K ) = {1}, so H1 (k, G∗ ) = When n is even, D {1}. Thus, only the quasi-split form exists, and its index is 2 A(1) . n, n/2 ∨   When n is odd, DK is of type B and Ξ = Aut(DK ) = Z/2Z. The quasi(n+1)/2

split form has index 2 A(1) . It corresponds to the trivial folding, which we n,(n+1)/2 K corresponding to the non-trivial will not display. The shape of the folding of D

element of H1 (Γ, Ξ) depends on whether n = 3 or n > 3, because the diagram ∨ for m > 2. When n = 3, the of B2∨ = C2∨ has a different shape from that of Bm K is given by folding of D

2

1

s

s

It results in a relative affine root system of type BC1 , a relative finite root system of type BC1 , and Tits index 2 A(2) 3,1 . In the case of n > 3, the folding is illustrated (in the case of n = 9) by

2 s

1

1

1

1 s

and results in a system of type BC(n−1)/2 . The group is PUr (h), where r = (n − 1)/2 and h is a non-degenerate Hermitian form of index r relative to the ramified quadratic extension /k that is the fixed field in k s of ker( f ) ⊂ 2 (2) Gal(k s /k). The Tits index is 2 A(1) n,r . Note that the Tits index A3,1 is again

Residue Field f of Dimension  1

410

shown not to occur, since it would produce a relative finite root system of type A1 rather than BC1 .

(d) (Bn, Bn ), n  3 It is more natural to consider a group of type B2 as a group of type C2 , so we K ) = Z/2Z. will assume in this subsection that n  3. We have Ξ = Aut(D The trivial element of Ξ gives the trivial folding (which we will not display) K under the non-trivial and leads to a split group of type Bn . The folding of D element is given (in the example of n = 5) by

2

1

1

1

1

s

s

∨ and the relative finite The resulting relative affine root system is of type Cn−1 root system is of type Bn−1 . The Tits index is Bn, n−1 .

(e) (Cn, Cn ), n  2 K ) = Z/2Z. The trivial element of Ξ gives the trivial We have Ξ = Aut(D folding (which we will not display) and leads to a split group of type Cn . The K under the non-trivial element depends on the parity of n. We folding of D illustrate the two cases in the examples of n = 9 and n = 8:

3 s

2

2

2

2

1

s

s

2

2

2

2 s

For n odd the relative affine root system of G is of type (BC(n−1)/2, C(n−1)/2 ), the relative finite root system of G is of type BC(n−1)/2 , and the Tits index is (2) Cn,(n−1)/2 . For n even the relative affine root system is of type Cn/2 , the relative

(2) finite root system is of the same type, and the Tits index is Cn, . Note that the n/2

(1) (1) Tits indices Cn,(n−1)/2 , respectively Cn, , can be ruled out because they would n/2 imply the existence of an anisotropic kernel of type C, which is impossible by Theorem 10.3.1. It’s worth emphasizing that in the case of n = 2 we interpret C1 to mean A1 . Thus, in that case, the relative affine and finite root systems are of type A1 and (2) K takes the form . The folding of D the Tits index is C2,1

10.7 Classification of Connected Reductive k-Groups

1

2

s

s

411

(f) (1 Dn, 1 Dn ), n > 4 The trivial element of Ξ gives the trivial folding (which we will not display) and leads to a split group of type Dn . f is of order 2, it leads to one When n is even Ξ = (Z/2Z)2 . If the image of  of the following two possible foldings (in the examples of n = 6 and n = 10):

2

1

1

1

2

s

1

2

2

s

2

2 s 2 s

∨ , the relative In the first case, the relative affine root system is of type Cn−2 (1) finite root system is of type Bn−2 , and the Tits index of G is 1 Dn, n−2 . In the ∨ second case, the relative affine root system is of type Bn/2 , the relative finite

(2) root system is of type Cn/2 , and the Tits index is 1 Dn, . n/2

If n is even and the image of  f is all of Ξ (which is not possible when Γ is procyclic), then the folding is obtained from the second of the above two foldings by additionally folding the two branches on the right, as follows (for n = 10):

1 s

2

2

2

4 s

The relative affine root system of G is of type BC(n/2)−1 and the Tits index is

1 D(2) . n,(n/2)−1

If n is odd, then Ξ = Z/4Z. If the image of  f is of order 2, the folding is the ∨ , the same as in the case of n even, the relative affine root system is of type Cn−2 (1) relative finite root system is of type Bn−2 , and the Tits index of G is 1 Dn, n−2 . If  the image of f is of order 4, the folding is as follows (for n = 9):

412

Residue Field f of Dimension  1

3

2

2

s

4 s

∨ The relative affine root system of G is of type (C(n−3)/2 , BC(n−3)/2 ), the relative

(2) finite root system is of type BC(n−3)/2 , and the Tits index is 1 Dn,(n−3)/2 . Note that a higher degree division algebra will not have an involution of the first kind, cf. Remark 10.7.5.

(g) (1 Dn, 2 Dn ), n > 4 K is of type Dn . Recall from Remark 1.3.76 that The affine Dynkin diagram D K ) as (Z/2Z)2  Z/2Z, where the generators (1, 0, 0), we can represent Aut(D (0, 1, 0), and (0, 0, 1), are given by the following automorphisms: (0, 0, 1) (1, 0, 0)

(0, 1, 0) (0, 0, 1)

When n is even then Ξ = {(0, 0, 0), (0, 0, 1), (1, 1, 0), (1, 1, 1)}  (Z/2Z)2 , and K ) → Aut(DK ) is the map (Z/2Z)2 Z/2Z → Z/2Z the projection map Aut(D sending (1, 0, 0) and (0, 1, 0) to 1 and (0, 0, 1) to 0. When n is odd then Ξ = {(0, 0, 0), (1, 0, 1), (1, 1, 0), (0, 1, 1)}  Z/4Z and the K ) → Aut(DK ) is the map (Z/2Z)2  Z/2Z → Z/2Z projection map Aut(D sending each of (1, 0, 0), (0, 1, 0), (0, 0, 1), to 1. The k-structure of G∗ is given by a surjective homomorphism f : Γ → Aut(DK ) = Z/2Z and the possible k-structures of G are given by homomorK ) → K ) which lift f under the projection map Aut(D phisms  f : Γ → Aut(D K ) of order 2, 4, or 8, and it f is a subgroup of Aut(D Aut(DK ). The image of  is not contained in Ξ. Now consider first the case that the image of  f has order 2. The elements of K ) = (Z/2Z)2  Z/2Z are (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), order 2 in Aut(D (1, 1, 1). Of these, (1, 0, 0) and (0, 1, 0) are conjugate under Ξ and if the image of  f is generated by either of them, then G = G∗ . The corresponding folding of DK takes the form

10.7 Classification of Connected Reductive k-Groups

1 hs 1 hs

1

1

1

413

2

(1) resulting in a relative affine root system of type Bn−1 and Tits index 2 Dn, n−1 . The element (1, 1, 0) lies in Ξ, both when n is even and when n is odd. The elements (0, 0, 1) and (1, 1, 1) lie in Ξ when n is even; when n is odd they do not K , for n odd, that either of lie in Ξ, but are conjugate under Ξ. The folding of D (0, 0, 1) and (1, 1, 1) gives rise to, is shown below for n = 9:

3

2

2 s

2

2 s ∨ resulting in a relative affine root system of type (B(n−1)/2, B(n−1)/2 ), relative

(2) finite root system of type BC(n−1)/2 , and Tits index 2 Dn,(n−1)/2 .  Consider now the case that the image of f has order 4. This image is K ) isomorphic to either Z/4Z or (Z/2Z)2 . There is a unique subgroup of Aut(D isomorphic to Z/4Z and it consists of {(0, 0, 0), (1, 0, 1), (1, 1, 0), (0, 1, 1)}. When n is odd this group is Ξ, but when n is even it is not, and the folding it gives rise to is shown below for n = 10:

1 s

2

2

2

4 s

The relative affine root system is of type BC(n/2)−1 , the relative finite root (2) system is BC(n/2)−1 , and the Tits index is 2 Dn, . n/2−1 Next we consider the case when the image of  f is isomorphic to (Z/2Z)2 (this case requires that Γ is not procyclic). There are two such subgroups of (Z/2Z)2  Z/2Z, namely {(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0)} and {(0, 0, 0), (0, 0, 1), (1, 1, 0), (1, 1, 1)}.

Residue Field f of Dimension  1

414

The first of these subgroups produces a folding, both when n is even and when n is odd. This folding is shown below for n = 6:

2

1

1

1

2

s

s

∨ , the relative finite root system The relative affine root system is of type Cn−2 (1) 2 is of type Bn−2 , and the Tits index is Dn, n−2 .

The subgroup {(0, 0, 0), (0, 0, 1), (1, 1, 0), (1, 1, 1)} is equal to Ξ when n is even and hence cannot be the image of  f . When n is odd, it gives a folding which is shown below for n = 9:

3

2

2

4

s

s

∨ The relative affine root system is of type (C(n−3)/2 , BC(n−3)/2 ), the relative

(2) finite root system is of type BC(n−3)/2 , and the Tits index is 2 Dn,(n−3)/2 . The final remaining case is that of  f being surjective, which again requires K depend on the parity of n that Γ not be procyclic. The resulting foldings of D

and are exemplified in the cases of n = 10 and n = 9 as follows:

1

2

2

s

2

4

3

s

s

2

2

4 s

When n is even the relative affine root system is of type BC(n/2)−1 , the (2) relative finite root system is of type BC(n/2)−1 , and the Tits index is 2 Dn,(n/2)−1 . On the other hand, when n is odd, the relative affine root system is of type ∨ , BC(n−3)/2 ), the relative finite root system is of type BC(n−3)/2 , and the (C(n−3)/2 (2) Tits index is 2 Dn,(n−3)/2 .

(h) (2 Dn, 2 Dn ), n > 4 K is of type C ∨ and Ξ = Aut(D K ) = Z/2Z. The The affine Dynkin diagram D n−1 trivial element of Ξ gives the trivial folding (which we will not display) and

10.7 Classification of Connected Reductive k-Groups

415

leads to a quasi-split group with Tits index 2 Dn,n−1 and relative root system of type Bn−1 . The folding induced by the non-trivial element depends on the parity of n and is exemplified in the cases n = 10 and n = 9 by:

3 s

2

2

2

2

1

s

s

2

2

2

2 s

∨ When n is even the relative affine root system is of type (C(n/2)−1 , BC(n/2)−1 ) and when n is odd the relative affine root system is of type BC(n−1)/2 . Setting r = n/2 − 1 or (n − 1)/2 respectively, the relative finite root system is of type (2) . BCr and the Tits index is 2 Dn,r

(i) D4 We will discuss here all cases in which the absolute root system of G is of type D4 . Let L/K be the splitting extension of the quasi-split group G∗K . If the K is of type G∨ and has no automorphisms, degree of L/K is 3 or 6, then D 2 ∗ 2 or 6 D2 , respectively. hence G = G is a quasi-split group with Tits index 3 D4,2 4,2 K ) = Z/2Z. K is of type C ∨ and Ξ = Aut(D If the degree of L/K is 2, then D 3

The trivial element of Ξ gives the trivial folding, which leads to a quasi-split (1) . The non-trivial element of Ξ leads to the group G = G∗ with Tits index 2 D4,3 folding:

3

2

s

s

This results in a relative affine root system of type (C1∨, BC1 ), a relative finite (2) . root system of type BC1 , and Tits index 2 D4,1 For the rest of our discussion of D4 we assume that the degree of L/K is 1; that K is of type D4 , Aut(DK ) = S3 , and Aut(D K ) = S4 . The is, G∗K is split. Then D group Ξ is the unique 2-Sylow subgroup of A4 and is isomorphic to (Z/2Z)2 . We have the exact sequence 1 → (Z/2Z)2 → S4 → S3 → 1 which we can split by choosing any of the four non-central nodes of the affine

416

Residue Field f of Dimension  1

Dynkin diagram D4 and identifying S3 with the subgroup of S4 that fixes that node. K arising from subgroups Let us first enumerate the possible foldings of D K ). These depend only on the size of the orbits for the action of this of Aut(D subgroup on the set of non-central vertices of DK . If there are four orbits of size 1, we obtain the trivial folding (which we will not display), leading to a split group of type D4 . If there are two orbits of size 2, we obtain the following folding with relative affine root system of type C2∨ and relative finite root system of type B2 :

2

1

2

s

s

If there are two orbits of size 1 and one orbit of size 2, we obtain the following folding with relative affine root system of type B3 and relative finite root system of type B3 :

1 hs 1 hs

1

2

If there is an orbit of size 1 and an orbit of size 3, we obtain the following folding with relative affine root system of type G2 :

1

1

3

hs

If there is a single orbit of size 4, we obtain the following folding with relative affine root system is of type BC1 :

1

4

s

s

K ) = S4 : cyclic group of order We now list all possible subgroups of Aut(D 2 generated by a transposition, cyclic group of order 2 generated by a double

10.7 Classification of Connected Reductive k-Groups

417

transposition (the product of two commuting transpositions), cyclic group of order 3 generated by a 3-cycle, cyclic group of order 4 generated by a 4-cycle, Klein-4 group generated by two commuting transpositions (not normal), Klein4 group generated by two commuting double transpositions (this is the subgroup Ξ, normal), a copy of S3 , a copy of A3 , a copy of A4 , a copy of D8 . In each case, it is clear what the orbit structure is. It remains to compute the image of each such subgroup under the map S4 → S3 . For this, it is enough to compute the intersection of such a subgroup with the normal subgroup Ξ, which is the normal Klein-4 group. A cyclic group generated by a transposition or a 3-cycle has trivial intersection with Ξ and projects isomorphically to S3 . A cyclic group of order 4, as well as a non-normal Klein-4 group, intersects Ξ in a subgroup of order 2, hence maps to a subgroup of order 2 in S3 . A copy of S3 intersects Ξ trivially, and hence is a complement to Ξ in S4 . A copy of A3 also intersects Ξ trivially. A copy of D8 contains Ξ and maps to a subgroup of S3 of order 2. A cyclic subgroup generated by a double transposition is contained in Ξ. From the above we can read off the possible k-structures of G depending on f are the k-structure of G∗ . If G∗ is k-split, then the only possible images of  the subgroups of Ξ, giving the possible orbit structures (1, 1, 1, 1), (2, 2) and (4), (1) (2) , and 1 D4,1 , respectively. with Tits indices 1 D4,4 , 1 D4,2 f are a cyclic If G∗ splits over a quadratic extension, the possible images of  group generated by a transposition or a 4-cycle, a non-normal Klein-4 group, or a dihedral group of order 8. The orbit structures are (1, 1, 2), (4), (2, 2), and (4), (1) 2 (2) 2 (1) (2) , D4,1 , D4,2 , and 2 D4,1 , respectively. respectively. The Tits indices are 2 D4,3 f are the cyclic If G∗ splits over a cubic extension, the possible images of  2 . groups of order 3. The orbit structure is (1, 3) and the Tits index is 3 D4,2 If G∗ splits over an A3 or S3 -extension, the possible images of  f are subgroups isomorphic to A3 , S3 , or S4 . The corresponding orbit structures are (1, 3), (1, 3), 2 , 6 D2 , and 6 D9 , respectively. and (4). The Tits indices are 6 D4,2 4,2 4,1

(j) (1 E6, 1 E6 ) The group Ξ is isomorphic to Z/3Z and acts by cyclically permuting the three branches of the affine Dynkin diagram of type E6 . The group H1 (Γ, Ξ) = Hom(Γ, Ξ) has two non-trivial elements, both of which fold E6 as follows:

418

Residue Field f of Dimension  1

1

3

3 s

The relative affine root system is of type G∨2 , the relative finite root system 16 . On the other hand, the trivial element is of type G2 , and the Tits index is 1 E6,2 of Ξ gives the trivial folding (which we will not display) and leads to the split group of type E6 .

(k) (1 E6, 2 E6 ) K ) = S3 and Aut(DK ) = Z/2Z. The projection Aut(D K ) → We have Aut(D Aut(DK ) is given by the unique projection S3 → Z/2Z. The k-structure of G∗ is given by a surjective homomorphism f : Γ → Aut(DK ) = Z/2Z. The k-structure of G is given by a homomorphism  f : Γ → S3 that lifts f . The image of  f is a subgroup of S3 that is either of order 2 or of order 6. All elements of S3 of order 2 are conjugate under Ξ = Z/3Z. Therefore, there is a K ) whose image unique Ξ-conjugacy class of homomorphisms  f : Γ → Aut(D 1 is of order 2. It corresponds to the trivial element of H (Γ, Ξ) and hence to the K is given by quasi-split form. The folding of D

1

1

1

2

2

hs

2 . resulting in a relative affine root system is of type F4 and Tits index 2 E6,4 A non-quasi-split inner form of that group is obtained when  f is surjective

(not possible when Γ is procyclic), and the folding is as in the case (1 E6, 1 E6 ):

1

3

3 s

resulting in a relative affine root system of type G∨2 , relative finite root system 16 . of type G2 , and Tits index 2 E6,2

10.7 Classification of Connected Reductive k-Groups

419

(l) E7 K ) = Z/2Z. The group H1 (Γ, Ξ) has a unique non-trivial We have Ξ = Aut(D K as follows: element which leads to a non-split inner form and folds D

1

1

2

2

2 s

The relative affine root system of G is of type F4∨ , the relative finite root system 9 . On the other hand, the trivial element is of type F4 , and the Tits index is 1 E7,4 of Ξ gives the trivial folding (which we will not display), leading to the split group of type E7 . Remark 10.7.6 According to Remark 10.6.3, for groups of type (2 E6, 2 E6 ), E8 , F4 and G2 , H1 (k, G∗ ) vanishes. Therefore, every k-group G such that G K is of outer type E6 is uniquely determined by its quasi-split k-form G∗ . Moreover, groups of type E8 , F4 and G2 are split (when the residue field f is of dimension  1.)

11 Component Groups of Integral Models

Let k be a field given with a discrete valuation ω that is normalized so that ω(k × ) = Z. We denote the ring of integers of k by o, the maximal ideal of o by m. We assume that o is Henselian and the residue field f is perfect. Fix a separable closure k s and let Θ = Gal(k s /k), I = Gal(k s /K), and Γ = Γ = Gal(K/k), where K ⊂ k s is the maximal unramified extension. We write O for the ring of integers of K and M for the maximal ideal of O. Then f = O/M is an algebraic closure of f. In this chapter we are going to introduce the Kottwitz homomorphism and use it to describe the groups of connected components of the integral models T ft and T lft of a k-torus T (cf. Appendix B), and more generally of the various integral models of a connected reductive k-group G associated to subsets of an apartment of its Bruhat–Tits building as in §8.3. More precisely, let G be a connected reductive k-group. We will recall the definition due to Borovoi [Bor98] of the finitely generated abelian group π1 (G) with Θ-action, called the algebraic fundamental group of G, and a functorial homomorphism κ = κG : G(k) → π1 (G)ΓI , constructed by Kottwitz [Kot97, §7] which is continuous with respect to the analytic topology on G(k) and the discrete topology on π1 (G)ΓI . This homomorphism is surjective provided dim(f)  1, in particular when k = K. The group G(k)0 of Definition 2.6.23 will be identified with the kernel of κG . This will aid us in the study of G(k)0 and of the component groups of the integral models of G constructed in Chapter 8. 420

11.1 The Kottwitz Homomorphism for Tori

421

11.1 The Kottwitz Homomorphism for Tori For a k-torus T we set π1 (T) = X∗ (T). Recall the valuation homomorphism ωT : T(k) → X∗ (T) ⊗Z Q of (2.5.1) and the fact that it takes image in X∗ (T) when T is split. Proposition 11.1.1 Let T be a K-torus. There exists a unique homomorphism κT : T(K) → X∗ (T)I with the following property: for any finite Galois extension L/K splitting T, the diagram T(L)  T(K)

ωTL

κT

/ X∗ (T)  / X∗ (T)I

commutes, where the left vertical arrow is the norm map, the right vertical arrow is the natural projection, and the top horizontal map is the valuation homomorphism with respect to the valuation on L normalized so that L × maps onto Z. This homomorphism is surjective. If T arises via base change from a k-torus, then κT is Gal(K/k)-equivariant. Proof The norm map T(L) → T(K) is continuous and open, since it coincides with the map on K-points induced by the norm map R L/K (TL ) → T, which is smooth and surjective. The kernel of the norm map R L/K (TL ) → T is a torus. Proposition 2.3.5 implies that the norm map T(L) → T(K) is surjective, and the uniqueness part of the claim is proved. The Gal(K/k)-equivariance follows from this uniqueness and the Gal(L/k)-equivariance of ωTL whenever L is Galois over k. To prove existence, we first note that if we have a map κT that makes the above diagram commutative for one particular Galois extension L/K splitting T, then it also makes it commutative for any other Galois extensions L/K splitting T. This follows from the fact that if L1 /K and L2 /K are two such extensions and L1 ⊂ L2 , then the diagram T(L2 )  T(L1 )

ωT , L2

ωT , L1

/ X∗ (T)  / X∗ (T)

commutes, due to the fact that L2 /L1 is totally ramified. Fixing one finite Galois extension L/K splitting T, we consider the composition of ωT ,L with X∗ (T) → X∗ (T)I . It is I-invariant and hence factors through the quotient

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T(L)I . We have dim(K)  1 according to Corollary 2.3.7. Since any algebraic extension of a dim  1 field is also a dim  1 field, Lemma 2.5.4 implies that for any separable extension F of L and any i > 0 we have Hi (F,T) = 0. Thus, part (i) of [Ser97, Chapter I, Appendix 1, Lemma 1] holds and we infer from part (iii) that the norm provides an isomorphism T(L)I → T(K). We have thus proved that the composition of ωT ,L : T(L) → X∗ (T) with X∗ (T) → X∗ (T)I factors through the norm T(L) → T(K).   and T(L) → T(K) are Remark 11.1.2 Since the norm maps T( L) → T(K)  and T(L) → T( compatible with the inclusions T(K) → T(K) L), cf. Proposition 2.1.6, κT is the restriction to T(K) of κTK+ . We continue with a K-torus T. The natural identification HomZ (X∗ (T), Z) = X∗ (T) induces a homomorphism X∗ (T)I → HomZ (X∗ (T)I , Z) whose kernel is the torsion subgroup of X∗ (T)I . At the same time we have the homomorphism (2.5.1) ωT : T(K) → HomZ (X∗ (T)I , Z),

t → ( χ → −ω( χ(t))).

Recall the definitions of T(k)0 and T(k)1 = T(k)b from Definition 2.5.13 and Proposition 2.5.8. Lemma 11.1.3 The composition of X∗ (T)I → HomZ (X∗ (T)I , Z) with κT coincides with ωT . In particular, κT−1 (X∗ (T)I , tor ) = T(K)b = T(K)1 . Proof The second point is immediate from the first point and Proposition 2.5.8. Therefore it is enough to prove the first point. When T is split the map X∗ (T)I → HomZ (X∗ (T)I , Z) is an isomorphism and identifies κT and ωT , as one sees immediately from the construction of κT . To obtain the statement for general T we consider the maximal split quotient T → A and note that X∗ (A) is the torsion-free quotient of X∗ (T)I , while X∗ (A) = X∗ (T)I , after which the  claim follows from the functoriality of κT and ωT . Lemma 11.1.4 Let E/K be a finite separable extension. Then the following diagrams commute T(E)  T(K)

κTE

κT

/ X∗ (T)I E

T(E) O

 / X∗ (T)I

T(K)

κTE

/ X∗ (T)I O E

κT

/ X∗ (T)I

where T(E) → T(K) is the norm map for the extension E/K, T(K) → T(E) is the natural inclusion induced by the inclusion E ⊂ K, X∗ (T)IE → X∗ (T)I is the

11.1 The Kottwitz Homomorphism for Tori

423

natural projection, induced by the identity on X∗ (T), and X∗ (T)I → X∗ (T)IE is the trace map for the action of the finite group I/IE . Proof Choose a resolution R1 → R0 → T as in Lemma 2.5.3. The maps R0 (K) → T(K) and R0 (E) → T(E) are surjective due to Proposition 2.3.5. This reduces the proof to the case that T is induced, in which case the argument  follows by comparing κT with ωT using Lemma 11.1.3. Proposition 11.1.5 T(K)0 = ker(κT ). Proof In the case when T is split we can reduce to T = Gm . Then κ is identified with the valuation ωk and the claim is immediate. For a general K-torus T choose a finite Galois extension L/K splitting T. By Definition 2.5.13 we have T(K)0 = NL/K (T(L)0 ). Consider the diagram in Proposition 11.1.1. The kernel of T(L) → X∗ (T) is equal to T(L)1 = T(L)0 according to the case of split tori, and the commutativity of the diagram shows that T(K)0 ⊂ ker(κT ). Consider conversely t ∈ T(L) such that NL/K (t) ∈ ker(κT ). Let x = κT ,L (t) ∈ X∗ (T). Then x belongs to the augmentation module for the action of I in X∗ (T), thus x is a sum of elements of the form y − σy for σ ∈ I and y ∈ X∗ (T). If ω ∈ L × is a uniformizer, then κT ,L (y()/σ(y())) = y − σy, while NL/F (y()/σ(y())) = 1. We see that we can modify t without changing NL/F (t) to achieve κT ,L (t) = 1. This shows ker(κT ) ⊂ NL/K (T(L)0 ) =  T(K)0 . We now consider a k-torus T and define κT : T(k) → π1 (T)Gal(K/k) I as the restriction of κTK . The identities T(k)1 = κT−1 (π1 (T)I , tor ) and T(k)0 = ker(κT ) remain valid due to the identities T(k)1 = T(K)1 ∩ T(k) and T(k)0 = T(K)0 ∩ T(k). Corollary 11.1.6

The homomorphism κT induces injective homomorphisms

T(k)1 /T(k)0 → (X∗ (T)I , tor )Γ,

T(k)/T(k)0 → (X∗ (T)I )Γ

which are surjective when k = K. Proof Immediate from Lemma 11.1.3 and Proposition 11.1.5 and the surjectivity of the Kottwitz homomorphism when k = K.  Remark 11.1.7 More generally, the homomorphisms of Corollary 11.1.6 are isomorphisms when dim(f)  1, cf. Corollary 11.7.2. Lemma 11.1.8 The Kottwitz homomorphism is compatible with restriction of

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scalars: If /k is a finite separable extension, S is an -torus, and T = R/k S, then the following diagram commutes T(k)

S()

κT

/ X∗ (T)Γ I

κS

/ X∗ (S)Γ I

Proof Abbreviate Δ = Θ . We have X∗ (T) = IndΘ Δ X∗ (S). Recall for a ΔA) ⊗Z B → IndΘ module A and a Θ-module B the isomorphism (IndΘ Δ Δ (A ⊗Z B) sending f ⊗ b to φ f ,b (σ) = f (σ) ⊗ σb. Via this isomorphism we obtain T(k s ) = X∗ (T) ⊗Z k s× = IndΘ Δ S(k s ). Let  /k be a finite Galois extension containing  and splitting S. Then   T(L ) = T(k s )I  = IndΘ Δ S(L ). Therefore we may replace k s by L and Θ by Γ . This puts us in the situation of considering the subgroups I = Gal(L /K) and Δ = Gal(L /) of Θ = Gal(L /k) and their actions on S(L ) and X∗ (S). We will use the following elementary facts. Given a group A, a finite index subgroup B ⊂ A, a normal subgroup C ⊂ A contained in B, and an A-module M, the natural maps IndBA (M C ) → (IndBA M)C and (IndBA M)C → IndBA (MC ) are isomorphisms. If we now drop the assumption that C is contained in B A BC C IndBC and use the isomorphisms IndBA M = IndBC B M, Ind B M = Ind B∩C M, C C C B∩C , and (IndB∩C M)C = MB∩C , then we obtain the isomor(IndB∩C M) = M phisms A (M B∩C ) (IndBA M)C = IndBC

A and (IndBA M)C = IndBC (MB∩C ),

of which the first one maps f : A → M to f  : A → M B∩C given by f (a) = f (a), and the second maps f : A → M to f  : A → MB∩C given by f (a) = B∩C\C f (ca). These facts allow us to compute  I Θ T(K) = T(L )I = (IndΘ Δ S(L )) = Ind I ·Δ S(L), and Θ X∗ (T)I = (IndΘ Δ X∗ (S)) I = Ind IΔ (X∗ (S) I ∩Δ ).

We claim that the Kottwitz homomorphism T(K) → X∗ (T)I is obtained by applying IndΘI Δ to the Kottwitz homomorphism S(L) → X∗ (S)I∩Δ . Indeed, the norm map T(L ) → T(K) is translated to the map  Θ I ·Δ  Θ IndΘ Δ S(L ) = Ind I ·Δ IndΔ S(L ) → Ind I ·Δ S(L)

obtained by applying the functor IndΘI ·Δ to the map IndΔI ·Δ S(L ) → S(L) sending  f to τ ∈I f (τ). Let s ∈ S(L) and let s ∈ S(L ) be such that NL /L (s) = s. By

11.1 The Kottwitz Homomorphism for Tori

425

construction of κS (s) is the image of κS,L (s) ∈ X∗ (S) under the projection X∗ (S) → X∗ (S)Δ∩I . Let f ∈ IndΔI Δ S(L ) be the function supported on Δ ⊂ IΔ  with f (1) = s. Then τ ∈I f (τ) = s. On the other hand, f is sent via IndΔI Δ κS,L to the element g of IndΔIΔ X∗ (S) supported on Δ and determined by g(1) = κS,L (s). The image of g under the isomorphism (IndΔIΔ X∗ (S))I → X∗ (S)Δ∩I is κS,L (s) = κS,L (s), and the claim is proved. Finally we have T(k) = (IndΘIΔ S(L))Θ = S(L)Δ = S() and X∗ (T)ΓI = (IndΘIΔ X∗ (S)Δ∩I )Γ = (X∗ (S)Δ∩I )I Δ/I ∩Δ . The Kottwitz homomorphism κT is the restriction to the diagonal of κTK =  IndΘI Δ κS L , which recovers κS . Corollary 11.1.9 If T is an induced torus then κ : T(k) → X∗ (T)ΓI is surjective. Proof

Lemma 11.1.8 reduces to the case T = Gm , which is obvious.



Lemma 11.1.10 Let 1 → T1 → T2 → T3 → 1 be an exact sequence of tori. If T1 is induced, then 1 → T1 (k)0 → T2 (k)0 → T3 (k)0 → 1 is exact. Proof We will use Proposition 11.1.5 without mention. Proposition 2.3.5 implies that the sequence of K-points remains exact. We have the commutative diagram 1

/ T1 (K) κT1

 X∗ (T1 )I

/ T2 (K) κT2

 / X∗ (T2 )I

/ T3 (K)

/1

κT3

 / X∗ (T3 )I

/0

with exact rows and surjective vertical maps. We claim that then the homomorphism X∗ (T1 )I → X∗ (T2 )I is injective. Indeed, its kernel is the image of the connecting homomorphism H1 (I, X∗ (T3 )) → X∗ (T1 )I . We can replace I by a finite quotient through which it acts to see that H1 (I, X∗ (T3 )) is finite. Since X∗ (T1 )I is torsion-free, the claim follows. We take invariants under Γ in the above diagram and obtain the commutative diagram 1

/ T1 (k)

0

 / X∗ (T1 )Γ I

κT1

/ T2 (k) κT2

 / X∗ (T2 )Γ I

/ T3 (k)

/1

κT3

 / X∗ (T3 )Γ I

/0

The rows of the above diagram are exact due to the vanishing of H1 (Γ,T1 (K))

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Component Groups of Integral Models

and H1 (Γ, X∗ (T1 )I ), owing to the fact that T1 is induced. Corollary 11.1.9 and the snake lemma complete the proof.  Remark 11.1.11 In the above lemma one can relax the condition that T1 is induced to the condition that it has induced ramification in the sense of Definition 2.5.16 below, provided one also assumes dim(f)  1, cf. Corollary 11.7.4. Lemma 11.1.12 (1) If T is a k-torus with induced ramification, then T(k)0 = T(k)1 . (2) If 1 → A → B → C → 1 is an exact sequence of k-tori and A has induced ramification, then the sequence 0 → X∗ (A)I → X∗ (B)I → X∗ (C)I → 0 is exact. Proof If T has induced ramification, then X∗ (T)I is torsion-free and the first point follows from Corollary 11.1.6. For the second point, apply X∗ to the exact sequence of tori to obtain an exact sequence of finite-rank free Z-modules with Θ-action. We claim that after taking inertial coinvariants the sequence remains exact. The only issue would be the injectivity of X∗ (A)I → X∗ (B)I . We may of course replace I by a suitable finite quotient through which it acts. The kernel of this map is the image of the connecting homomorphism H1 (I, X∗ (C)) → X∗ (A)I . But H1 (I, X∗ (C)) is finite, while by assumption X∗ (A)I is torsion-free, so this connecting homomorphism is zero. This shows the exactness of the second sequence. 

11.2 The Component Groups of T ft and T lft Let T lft and T ft be the Néron lft-model and the Néron ft-model of T, respectively. These are smooth o-group schemes whose generic fiber is T. We have T ft ⊂ T lft and both group schemes share the same relative identity component, which we denote by T 0 . We refer the reader to Appendix B for detailed discussion. Since the generic fibers of these group schemes are connected, their component groups π0 (T lft ) and π0 (T ft ) are étale group schemes over f. As such, they are completely determined by their groups of f-points viewed as modules over Gal(f/f) = Gal(K/k). These can be described explicitly in terms of the cocharacter module X∗ (T) via the Kottwitz homomorphism. For this, we must recall that T 0 (o) = T(k)0 according to Corollary B.8.7.

11.3 The Algebraic Fundamental Group Corollary 11.2.1 modules

427

The homomorphism κT is an isomorphism of Gal(K/k)-

π0 (T lft )(f) → X∗ (T)I and π0 (T ft )(f) → X∗ (T)I , tor .

11.3 The Algebraic Fundamental Group For this section only, k can be any field. Let G be a connected reductive k-group. Construction 11.3.1 Let Gsc be the simply connected cover of the derived subgroup of G. For any maximal torus T ⊂ G its preimage Tsc in Gsc is also a maximal torus. If T is defined over k, then so is Tsc . The map Tsc → T induces an injective map X∗ (Tsc ) → X∗ (T) whose image is the lattice of coroots for the coroot system R∨ (T) ⊂ X∗ (T). The quotient X∗ (T)/X∗ (Tsc ) is a finitely generated abelian group with an action of the absolute Galois group of k. If T  ⊂ G is another maximal k-torus there exists g ∈ Gsc (k s ) such that Tsc = gTsc g −1 and hence T  = gT g −1 . The Weyl group of T acts trivially on the quotient X∗ (T)/X∗ (Tsc ), because the reflection along a root a has the effect of adding an integer multiple of the coroot a∨ ∈ X∗ (Tsc ). Therefore the isomorphism X∗ (T)/X∗ (Tsc ) → X∗ (T)/X∗ (Tsc ) induced by Ad(g) is independent of the choice of g. In particular, it is Galois-equivariant. In this way we obtain a system of finitely generated abelian groups with Galois action, indexed by the maximal k-tori of G, and between any two such there is given a Galois-equivariant isomorphism. What we mean by system is that the composition of two composable isomorphisms in the system is again an isomorphism in the system. Definition 11.3.2 The algebraic fundamental group π1 (G) of G is the limit (equivalently colimit) of that system. Fact 11.3.3 π1 (G) = {0} if and only if G is semi-simple and simply connected. Remark 11.3.4 When the ground field is C, the algebraic fundamental group of G coincides with the topological fundamental group of the complex Lie group G(C), due to the fact that the algebraic and topological notions of simply connected coincide in that case. Fact 11.3.5 If G = T is a torus then π1 (G) = X∗ (T). Lemma 11.3.6 The assignment G → π1 (G) is a functor from the category of connected reductive groups to the category of finitely generated abelian groups with Galois action.

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Component Groups of Integral Models

Proof Let f : H → G be a morphism between connected reductive groups. It lifts uniquely to a morphism fsc : Hsc → Gsc . Choose a maximal torus T ⊂ H. Choose a maximal torus S ⊂ G containing f (T). Then f : T → S and fsc : Tsc → Ssc lead to the desired morphism π1 (H) → π1 (G). 

11.4 z-Extensions Let G be a connected reductive k-group. The construction of κG will employ a useful device, called a z-extension, which is an exact sequence  → G → 1, 1→Z →G  torus and G  is a connected where Z is an induced (necessarily central in G) reductive k-group whose derived subgroup is simply connected. This concept, originally introduced by Langlands and Ono ([Lan79, p. 721], [Ono65, p. 97]) and used to great effect by Kottwitz, is easily constructed as follows. Construction 11.4.1 Let Gsc be the simply connected cover of the derived subgroup of G and Z(G)◦ the identity component of the center of G. The product morphism Z(G)◦ × Gsc → G is surjective with finite kernel, which we call μ. Let /k be a finite Galois extension that splits G, hence also Z(G)◦ and Gsc . The finite Gal(/k)-module X ∗ (μ) can be presented as a quotient of a free Gal(/k)-module M. Let Z be the torus with X ∗ (Z) = M. Thus Z is an  be the pushout of induced torus equipped with an embedding μ → Z. Let G the extension 1 → μ → Z(G)◦ × Gsc → G → 1  = (Z × Z(G)◦ × Gsc )/μ, where along the embedding μ → Z. Explicitly, G μ is embedded into Z(G)◦ × Gsc as the kernel of the morphism to G, and μ is embedded into Z by the inverse of the embedding μ → Z that we just  is a quotient of a connected reductive group, it is itself constructed. Since G  is an connected and reductive. By construction, Z is an induced torus and G  extension of G by Z. The construction of G provides a natural injective map  whose image is normal and whose cokernel is abelian, hence its Gsc → G  image equals the derived subgroup of G. 2 → G are two z-extensions, so is their fiber 1 → G and G Fact 11.4.2 If G product over G.  → G and H  → H be z-extensions. Given a homoLemma 11.4.3 Let G 1 → G such that the following morphism G → H there exists a z-extension G

11.5 The Kottwitz Homomorphism for Reductive Groups

429

diagram commutes o G

1 G

/H 

 G

 G

 /H

1 to be the fiber product of G  → G → H ← H.  The above Proof Define G 1 → G diagram is true by definition, but we need to check that the natural map G is a z-extension.  Since  → H implies the surjectivity of G 1 → G. The surjectivity of H   G → G is surjective, the composition G1 → G is surjective. The kernel of  → G and H  → H, and 1 → G is equal to the product of the kernels of G G 1 is an extension of G by a torus, is therefore an induced torus. In particular, G  with abelian kernel, 1 surjects onto G hence connected and reductive. Since G   the derived subgroup of G1 is simply connected.

11.5 The Kottwitz Homomorphism for Reductive Groups We now extend the definition of the Kottwitz homomorphism to a functorial surjective homomorphism κ = κG : G(K) → π1 (G)I , for any connected reductive K-group. As for tori, this homomorphism will be Gal(K/k)-equivariant when G arises via base change from k. Assume first that the derived subgroup of G is simply connected and let D = G/Gsc . Then the homomorphism G → D induces an isomorphism π1 (G) → π1 (D) and the Kottwitz homomorphism is defined so that the diagram G(K)

/ D(K)

 π1 (G)I

 π1 (D)I

commutes. In the general case we choose a z-extension  → G → 1. 1→Z →G   → π1 (G) are surjective. The Kottwitz The maps G(K) → G(K) and π1 (G)

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Component Groups of Integral Models

 has just been defined and one defines the Kottwitz hohomomorphism for G momorphism for G as the dotted arrow below.  G(K)

/ G(K)

 I π1 (G)

 / π1 (G)I

Lemma 11.5.1 There exists a unique homomorphism G(K) → π1 (G)I that  surmakes the above diagram commute. It is independent of the choice of G, jective, functorial in G, and Gal(K/k)-equivariant.  be its preimage in G.  The Proof Let T ⊂ G be a maximal torus and let T ∨  coroot lattice Q ⊂ X∗ (T) has trivial intersection with X∗ (Z), which leads to the exact sequence  → π1 (G) → 1. 0 → π1 (Z) → π1 (G) We have the commutative diagram with exact rows 1

/ Z(K)

/ G(K) 

/ G(K)

/1

 π1 (Z)I

 / π1 (G) I

/ π1 (G)I

/1

The square commutes due to the functoriality of the Kottwitz homomorphism for tori, applied to the homomorphism of tori Z → D. The composed map  I → π1 (G)I therefore factors through G(K), establishing the  G(K) → π1 (G)  follows from the surjectivity of existence of κG . The uniqueness, given G,  G(K) → G(K). This also implies the Gal(K/k)-equivariance. The indepen follows from Fact 11.4.2, and the functoriality in G follows from dence of G Lemma 11.4.3.  Given a connected reductive k-group G, we define κG : G(k) → π1 (G)Gal(K/k) I as the restriction to G(k) of κG K . Recall the definitions of G(k)1 and G(k)0 from Definitions 2.6.15 and 2.6.23. Lemma 11.5.2 −1 ((π1 (G)I , tor )Γ ). G(k)1 = κG

11.5 The Kottwitz Homomorphism for Reductive Groups

431

Proof Choose a maximal torus T ⊂ G and present π1 (G) = X∗ (T)/X∗ (Tsc ). We have the surjection X∗ (T)/X∗ (Tsc ) → X∗ (T)/X∗ (Tder ) = X∗ (T/Tder ) = X∗ (Gab ) with finite kernel. It induces a surjection with finite kernel on the level of I-coinvariants. The functoriality of κ implies that for g ∈ G(k) with image g ∈ Gab (k) the condition κG (g) ∈ π1 (G)I , tor is equivalent to the condition κGab (g) ∈ X∗ (Gab )I , tor . The claim now follows from Lemma 11.1.3 and the  definition of G(k)1 .  → G → 1 be a z-extension. The maps Proposition 11.5.3 Let 1 → Z → G  → G(k) and ker(κ  ) → ker(κG ) are surjective. G(k) G  → G → 1 remains exact on kProof The exact sequence 1 → Z → G  I → π1 (G)I → 0 is points, since Z is induced. The sequence X∗ (Z)I → π1 (G) exact, since taking I-coinvariants is right-exact. The torsion-freeness of X∗ (Z)I implies, as was argued in the proof of Lemma 2.5.20, that in fact the first map in this sequence is injective. The Gal(K/k)-module X∗ (Z)I is induced, so taking Gal(K/k)-invariants keeps the sequence exact. Now apply the snake lemma and of Corollary 11.1.9 to conclude use the surjectivity of Z(k) → X∗ (Z)Gal(K/k) I  the surjectivity of ker(κG ) → ker(κG ). Proposition 11.5.4

The identity G(k)0 = ker(κG ) holds.

Proof By definition we have ker(κG ) = ker(κG K ) ∩ G(k), while G(k)0 = G(K)0 ∩ G(k) was proved in Proposition 9.3.24. This reduces the proof to the case k = K. In that case, G is quasi-split by Corollary 2.3.8. The functoriality of the Kottwitz homomorphism and its triviality on simply connected groups shows G(k)0 ⊂ ker(κG ), so we need to argue the opposite inclusion. Assume first that G has simply connected derived subgroup and let D = G/Gsc . Then ker(κG ) is the preimage of ker(κD ) under G(k) → D(k). Proposition 11.1.5 states D(k)0 = ker(κD ). Let T ⊂ G be a maximally split maximal torus and let Tsc = T ∩ Gsc . Then we have the exact sequence 1 → Tsc → T → D → 1. Since Tsc is induced, Lemma 2.5.20 implies that T(k)0 → D(k)0 is surjective, hence G(k)0 → D(k)0 is also surjective. Now drop the assumption that the derived subgroup of G is simply connected.  → G → 1. The group G  is also quasi-split. Fix again a z-extension 1 → Z → G According to Proposition 11.5.3, the map ker(κG ) → ker(κG ) is surjective. We  0 . Since the image of G(k)  0 in G(k) is have just shown that ker(κG ) = G(k) 0 0  contained in G(k) , we see that G(k) = ker(κG ) as claimed. Remark 11.5.5

Proposition 11.5.4 gives a new interpretation of the ad hoc

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Definition 2.6.23. It also implies that G(k)0 is functorial with respect to all homomorphisms of reductive groups. Lemma 11.5.6 Let M ⊂ G be a Levi subgroup. Then M(k)1 ∩ G(k)0 = M(k)0 . Proof We have M(k)1 = M(K)1 ∩ M(k), M(k)0 = M(K)0 ∩ M(k), and G(k)0 = G(K)0 ∩ G(k), cf. Proposition 9.3.24. This reduces the proof to the case k = K. In particular, G and M are quasi-split by Corollary 2.3.8. The natural map π1 (M) → π1 (G) is surjective by definition. We claim that its kernel is a torsion-free induced Galois module. To see this, choose a maximally split maximal torus T of M and let TGsc and TMsc be the preimages of T in Gsc and Msc . The kernel of this map is isomorphic to X∗ (TGsc )/X∗ (TMsc ). Choose a parabolic K-subgroup P ⊂ G with Levi factor M and a Borel K-pair (T, B) of G contained in (M, P). Then (T, B ∩ M) is a Borel K-pair for M and the set of simple roots Δ M determined by (T, B ∩ M) is an I-invariant subset of the set of simple roots Δ determined by (T, B). The subset Δ∨M of Δ∨ is I-invariant and so is its complement. But that complement is a basis for X∗ (TGsc )/X∗ (TMsc ). An element of M(K)0 maps trivially to π1 (M)I under κ M , hence trivially to π1 (G)I under κG , so lies in G(K)0 . Conversely consider an element m ∈ M(K)1 ∩ G(K)0 . Then κ M (m) is a torsion element of π1 (M)I whose image in π1 (G)I is zero. Thus it lifts to an element of ker(π1 (M) → π1 (G))I . The kernel of ker(π1 (M) → π1 (G))I → π1 (M)I is finite (replace I by a finite quotient through which it acts, then that kernel is the image of H1 (I, π1 (G)), a finite abelian group). Therefore the lift of κ M (m) in ker(π1 (M) → π1 (G))I is still torsion. But this group is torsion-free by the claim just proved. 

11.6 The Component Groups of Parahoric Integral Models Let G again be a connected reductive k-group, not necessarily quasi-split. Recall that G K is quasi-split according to Corollary 2.3.8. Proposition 11.6.1 Let C be a chamber in B(G K ). The restriction of the homomorphism κ : G(K) → π1 (G)I to G(K)C is still surjective and induces an isomorphism G(K)C /G(K)0C → π1 (G)I . Proof Surjectivity follows from the fact that Gsc (K) acts transitively on the set of chambers of B(G K ) = B(Gsc,K ) and that κ is trivial on Gsc (K). Proposition  7.6.4 states that G(K)0C = G(K)C ∩ G(K)0 , hence the second statement.

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Corollary 11.6.2 The Kottwitz homomorphism induces isomorphisms G(K)/G(K)0 → π1 (G)I , G(K)1 /G(K)0 → π1 (G)I , tor, G(K)/G(K)1 → π1 (G)I , free . Corollary 11.6.3 Let C be a chamber in B(G K ). The component group of GC is identified via κ with π1 (G)I . If C is Θ-invariant then this identification is Gal(K/k)-equivariant. More generally, if Ω is any bounded subset of an apartment of B(G K ), the component group of GΩ is identified via κ with a subgroup of π1 (G)I , and this identification is Θ-invariant provided Ω is Θstable.

11.7 The Case of dim(f)  1 In this section we shall assume that dim(f)  1. Recall that this holds in particular when f is finite. This allows us to derive additional results about the Kottwitz homomorphism. We begin with the case of tori and let T be a k-torus. Lemma 11.7.1 We have H1 (Gal(K/k),T(K)0 ) = {0}. In particular, the Kottwitz homomorphism induces a continuous surjective homomorphism T(k) → . X∗ (T)Gal(K/k) I Proof The second statement follows from the first, and the first statement follows from Corollary B.8.7 and Lemma 8.1.4.  Corollary 11.7.2 The injective homomorphisms T(k)1 /T(k)0 → (X∗ (T)I , tor )Γ,

T(k)/T(k)0 → (X∗ (T)I )Γ

of Corollary 11.1.6 are isomorphisms. Corollary 11.7.3 The image of the valuation homomorphism ωT : T(k) → X∗ (T)Θ ⊗ Q is equal to the image of the renormalized norm map  σ(x) (X∗ (T)I )Gal(K/k) → X∗ (T)Θ ⊗ Q, x → −e(E/K)−1 σ ∈Gal(E/K)

for any finite Galois extension E/K splitting T. Proof According to Lemmas 11.7.1 and 11.1.3 the image of ωT is equal to the image of the composition → HomZ (X∗ (T)I , Z)Gal(K/k) → X∗ (T)Θ ⊗ Q, X∗ (T)Gal(K/k) I

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where X∗ (T)I → HomZ (X∗ (T)I , Z) is the natural map induced by the duality between X∗ (T) and X∗ (T), and the second map is due to Lemma 2.5.5 applied to the base field K. One checks easily that the composition of the two is given by the renormalized norm map for the action of I.  Corollary 11.7.4 If 1 → A → B → C → 1 is an exact sequence of tori and A has induced ramification, then the sequence 1 → A(k)0 → B(k)0 → C(k)0 → 1 is exact. Proof

We consider the commutative diagram of Kottwitz homomorphisms 1

/ A(K)

/ B(K)

/ C(K)

/1

0

 / X∗ (A)I

 / X∗ (B)I

 / X∗ (C)I

/0

The exactness of the top row on the right follows from Proposition 2.3.3, while the exactness of the bottom row follows Lemma 11.1.12. The vertical maps are surjective. The snake lemma implies that the sequence 1 → A(K)0 → B(K)0 → C(K)0 → 1 is exact. Lemma 11.7.1 implies that taking invariants under Γ gives the desired exactness.  We now consider a general connected reductive group G. Corollary 11.7.5 The Kottwitz homomorphism κ : G(k) → π1 (G)Gal(K/k) is I surjective and induces an isomorphism . G(k)C /G(k)0C → π1 (G)Gal(K/k) I Proof Let C be a Γ-invariant chamber of B(G K ), which exists by Proposition 9.10.4). According to Proposition 11.6.1 we have the exact sequence 1 → G(K)0C → G(K)C → π1 (G)I → 1. Theorem 8.3.13 gives a smooth integral model GC0 of G with connected fibers  such that G(K)0C = GC0 (O). The claim follows from Lemma 8.1.4. Corollary 11.7.6

The Kottwitz homomorphism induces isomorphisms G(k)/G(k)0 → π1 (G)Gal(K/k) , I , G(k)1 /G(k)0 → π1 (G)Gal(K/k) I , tor . G(k)/G(k)1 → (π1 (G)I )Gal(K/k) free

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435

We now present a strengthening of Theorem 10.6.8 that is valid for arbitrary connected reductive groups. For this, assume that k is a local field. Theorem 11.7.7 There is a functorial isomorphism H1 (k, G) → π1 (G)Θ,tor . Proof There is a bijection between maximal tori T ⊂ G and maximal tori Tsc of Gsc , given by taking the preimage Tsc of T in Gsc . The morphism [Tsc → T] → [Gsc → G] of crossed modules is a quasi-isomorphism. We obtain the following diagram with exact rows H1 (k, Gsc )

/ H1 (k, G) O H1 (k,T)

(∗)

/ H1 (k, Gsc → G) O / H1 (k,Tsc → T) 

/ H 2 (k,Tsc )



H−1 (k, X∗ (Tsc ) → X∗ (T)) The map (∗) is injective due to the vanishing of H1 (k, Gsc ), cf. Theorem 10.6.4. According to Theorem 10.5.1 we can choose Tsc to be anisotropic. Theorem 10.6.7 then implies that the map (∗) is surjective. The bottom isomorphism is due to the Tate–Nakayama duality. But one sees from the definition that H−1 (k, X∗ (Tsc ) → X∗ (T)) = π1 (G)Θ,tor . The composition of this identification with the two vertical isomorphisms and the map (∗) induces the desired isomorphism H1 (k, G) → π1 (G)Θ,tor , which is easily seen to be independent of the choice of T, and functorial.  Remark 11.7.8 One can show that the composition of the above isomorphism with the inflation map H1 (K/k, G(K)) → H1 (k, G) sends a 1-cocycle c to the image of c(Fr) under the Kottwitz homomorphism. This follows from the explicit description of the Tate–Nakayama isomorphism. We refer to [Kot85, §2.5] and [Kot97, §7].

12 Finite Group Actions and Tamely Ramified Descent

Let O be a discretely valued Henselian local ring with valuation ω. Let K be the field of fractions of O. We assume in this chapter that the residue field of O is algebraically closed, and we will denote it by f. Let H be a connected reductive K-group and Θ be a finite group of Kautomorphisms of H. We assume that the order of Θ is not divisible by the characteristic of the residue field f. Let G = (H Θ )0 . This group is also reductive, see [Ric82, Proposition10.1.5] or [PY02, Theorem2.1]. The goal of this chapter  is to show that the enlarged Bruhat–Tits building B(G) of G(K) can be identified  as a metric space with the subspace of the enlarged Bruhat–Tits building B(H) consisting of points that are fixed under the action of Θ and explicitly relate the Bruhat–Tits group schemes and the parahoric subgroups of G(K) to those of H(K). We will also relate the polysimplicial structure of B(G) to that of B(H). The main result is Theorem 12.7.1. We will follow here the approach developed in [Pra20a]. Using a different argument from the one given below, it   was shown in [PY02] that B(G) is indeed the subspace of B(H) consisting of points that are fixed under Θ. We mention that in [Pra20a], it has only been assumed that the residue field of K is separably closed. Since over such a field, H need not be quasi-split, it has been assumed in [Pra20a] that Bruhat–Tits theory is available for H. The results of this chapter apply in particular to the case of a given connected reductive K-group G and a Galois extension L/K of degree prime to p: Set  L )Gal(L/K) . This = B(G ting H = R L/K (G L ) we obtain an identification B(G) latter result was proved by Guy Rousseau in his unpublished thesis[Rou77, Proposition 5.1.1]. 436

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12.1 Preliminaries In what follows, for any K-split torus S, we will write Hom(Gm, S) for the X∗ (S) and denote by V(S) the R-vector space R ⊗Z X∗ (S). Let C be the maximal K-split torus contained in the center of H. Then there is a natural action of H(K) on V(C) by translations, with Hder (K), as well as every bounded subgroup of H(K), acting trivially (cf.§4.3). The enlarged Bruhat–  Tits building B(H) of H(K) is the product V(C) × B(H). The apartments of this building, as well as that of B(H), are in bijective correspondence with maximal K-split tori of H. Given a maximal K-split torus T of H, the corresponding apartment of the enlarged building is an affine space under V(T) := R ⊗Z X∗ (T).  For more details, see §4.3. By a facet of B(H) we mean the product V(C) × F, for a facet F of B(H). By abuse of language, we will say that a subset of the  enlarged building B(H) = V(C) × B(H) of H(K) is bounded if its projection in B(H) is bounded. Given a K-split torus S of H, let M be the centralizer of S in H. We view   the enlarged building B(M) of M(K) as the union of the apartments of B(H) corresponding to maximal K-split tori of H that contain S. We recall, cf. Axioms 4.1.20–4.1.22 or §8.5, that given a non-empty bounded subset Ω of an apartment of B(H) there is a smooth affine O-group scheme HΩ1 with generic fiber H, associated to Ω, such that HΩ1 (O) is the subgroup H(K)1Ω of H(K)1 consisting of elements that fix Ω pointwise. The relative identity component HΩ0 of HΩ1 is an open affine O-subgroup scheme of the latter; it is by definition the union of the generic fiber H of HΩ1 and the identity component of its special fiber. The group scheme HΩ0 is called the Bruhat–Tits group scheme associated to Ω. The special fiber of HΩ0 will be denoted by H Ω0 .  This can also be phrased in terms of the enlarged building B(H). Recall from 1 §4.3 that H(K) is the subgroup of H(K) consisting of elements that fix one, hence any, point of V(C). Therefore, if Ω is a non-empty bounded subset of an  apartment of B(H), then the stabilizer of Ω in H(K) equals the stabilizer of the projection of Ω to B(H) in the group H(K)1 . We will use HΩ1 and HΩ0 for Ω a  subset of either B(H) or B(H). 12.1.1 Let G be a smooth affine O-group scheme and G be its generic fiber. According to Lemma 2.10.9 its coordinate ring has the following description: O[G ] = { f ∈ K[G] | f (G (O)) ⊂ O}. In fact, the proof of that lemma gives the following slightly stronger statement which we will find useful. If G ⊂ G (O) is a subgroup whose image in G (f) is

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Zariski-dense in the special fiber G , then O[G ] = { f ∈ K[G] | f (G) ⊂ O}, thus G determines the model G . In particular, we obtain the following strengthening of Corollary 2.10.10. If Y is an affine O-scheme of finite type with generic fiber Y , then a morphism G → Y extends to a morphism G → Y if and only if it carries G to Y (O). 12.1.2 Let Ω be a non-empty bounded subset of an apartment of B(H), or of  B(H). As the O-group scheme HΩ1 is smooth and affine and its generic fiber is H, the coordinate ring of HΩ1 has thus the following description: O[HΩ1 ] = { f ∈ K[H] | f (H(K)1Ω ) ⊂ O}. Let P = HΩ0 (O) be the parahoric subgroup corresponding to Ω and let P ⊂ P be a subgroup of finite index. Then the image of P in H Ω0 (f) is Zariski-dense, and therefore by 12.1.1, O[HΩ0 ] = { f ∈ K[H] | f (P) ⊂ O}. Thus the subgroup P determines the group scheme HΩ0 , and P is the unique parahoric subgroup containing P as a subgroup of finite index. Proposition 12.1.3 Let Ω be a non-empty bounded subset of an apartment  of B(H). Let G be a smooth connected K-subgroup of H and G be a smooth affine O-group scheme with generic fiber G and connected special fiber. Let G be a subgroup of G (O) that fixes Ω pointwise and whose image in G (f) is Zariski-dense in the special fiber of G . Then G (O) ⊂ HΩ0 (O) and the inclusion G → H extends uniquely to an O-group scheme homomorphism ϕ : G → HΩ0 . In particular, G (O) fixes Ω pointwise, as well as any facet F of B(H) that meets Ω. Proof According to 12.1.1 the inclusion G → H(K)1Ω induces an O-group scheme homomorphism ϕ : G → HΩ1 that is the natural inclusion G → H on the generic fibers. Since G has connected fibers, the homomorphism ϕ factors through HΩ0 . Any facet F of B(H) that meets Ω is stable under G (O)(⊂ H(K)), so a subgroup of G (O) of finite index fixes it pointwise. Now applying the result of the preceding paragraph, for F in place of Ω, we see that there is an O-group scheme homomorphism G → HF0 that is the natural inclusion G → H on the generic fibers and hence G (O) fixes F pointwise.  Proposition 12.1.4 Let Ω be a non-empty bounded subset of an apartment of  B(H). Let S be a K-split torus of H and S the split O-torus with generic fiber

12.2 Certain Group Schemes Associated to H and G

439

S. If a subgroup of the maximal bounded subgroup S (O) of S(K) of finite index fixes Ω pointwise, then there is a maximal K-split torus T of H containing S such that Ω is contained in the apartment corresponding to T. Proof According to Proposition 12.1.3 there is an O-group scheme homomorphism ι : S → HΩ0 that is the natural inclusion S → H on the generic fibers (ι is a closed immersion, see [SGA3, Exp.IX, Theorem6.8 and Corollary2.5] and also the proof of Lemma B.2.3). Applying Proposition 8.2.1(1) to the centralizer of ι(S ) (in HΩ0 ) in place of G , and O in place of o, we see that there is a closed O-torus T of HΩ0 that commutes with ι(S ) and whose generic fiber T is a maximal K-split torus of H. The torus T clearly contains S, and Proposition 9.3.5(2) implies that Ω is contained in the apartment corresponding to T.  The following is a simple consequence of the preceding propositions, from which we retain the notations G, S, G , and S . Corollary 12.1.5 (1) The set of points of B(H) that are fixed under G (O) is the union of facets pointwise fixed under G (O). (2) Let M be the centralizer in H of a K-split torus S. The set of points  of B(H) that are fixed under a finite-index subgroup S of S(K)b equals  B(M).

12.2 Certain Group Schemes Associated to H and G 12.2.1 Let Θ be a finite group of automorphisms of the reductive K-group H. According to Axiom 4.1.17 there is a natural action of Θ on B(H), as well as on   B(H), by isometries such that for all h in H(K), x in B(H) or B(H), and Θ in Θ, we have Θ(h · x) = Θ(h) · Θ(x). The action of Θ on B(H) is polysimplicial. If a facet of B(H) is stable under the action of Θ, then its barycenter is fixed under Θ. Conversely, if a facet F of B(H) contains a point x fixed under Θ, then being the unique facet containing x, F is stable under the action of Θ. For a subset X of a set given with an action of Θ, X Θ will denote the subset of points of X that are fixed under Θ. 12.2.2 In the following we will assume that the characteristic p of the residue field of K does not divide the order of Θ. Then G := (H Θ )0 is a reductive group, see [Ric82, Proposition10.1.5] or [PY02, Theorem2.1]. The subspace Θ of points of B(H)   B(H) fixed under Θ is closed, convex, and stable under the Θ action of H(K) , in particular under the action of G(K). We will prove that the enlarged Bruhat–Tits building of G(K) can be G(K)-equivariantly identified

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Θ as a metric space. We will describe the polysimplices in B(G) in  with B(H) terms of those of B(H) Θ , and the integral models of G in terms of those of H.

12.2.3 Group schemes associated to Θ-stable subsets of apartments of  B(H) and parahoric subgroups of G(K) (1) Let Ω be a non-empty Θ-stable bounded subset of an apartment of B(H),  or of B(H). Then HΩ1 (O) = H(K)1Ω is stable under the action of Θ on H(K), so the coordinate ring O[HΩ1 ] is stable under the action of Θ on K[H]. This implies that Θ acts on the group scheme HΩ1 by O-group scheme automorphisms. The relative identity component HΩ0 of HΩ1 is of course stable under this action. We define the functor HΩΘ (respectively (HΩ0 ) Θ ) of Θ-fixed points that associates to a commutative O-algebra C the subgroup HΩ1 (C) Θ (respectively HΩ0 (C) Θ ) of HΩ1 (C) (respectively HΩ0 (C)) consisting of elements fixed under Θ. The functors HΩΘ and (HΩ0 ) Θ are represented by closed smooth O-subgroup schemes of HΩ1 and HΩ0 respectively (see Propositions 2.11.1 and 2.11.3, or [Edi92, Propositions 3.1 and 3.4]); we will denote these closed smooth O-subgroup schemes also by HΩΘ and (HΩ0 ) Θ respectively. The generic fiber of both of these subgroup schemes is H Θ , and so the identity component of their generic fibers is G. The relative identity component (HΩΘ )0 of HΩΘ is by definition the union of the identity components of its generic and special fibers; it is an open (so smooth) affine O-subgroup scheme[SGA3, Exp.VIB , 3.4] with generic fiber G. The index of the subgroup (HΩΘ )0 (O) in HΩΘ (O) is finite. To see this, we observe that since G is the identity component of H Θ , G(K) is of finite index in H Θ (K). Moreover, the inverse image I of (HΩΘ )0 (f) under the map HΩΘ (O) → HΩΘ (f) is of finite index in HΩΘ (O) since the index of (HΩΘ )0 (f) in HΩΘ (f) is finite. Now as (HΩΘ )0 (O) = I ∩ G(K), it is of finite index in HΩΘ (O). Since the generic fiber of HΩΘ is H Θ , it follows from Lemma A.2.1(3) that HΩΘ is just the schematic closure of H Θ in HΩ . We will denote (HΩΘ )0 by GΩ0 henceforth. As GΩ0 (O) ⊂ HΩ1 (O), GΩ0 (O) fixes Ω pointwise. Moreover, as GΩ0 (O) is a subgroup of finite index of HΩΘ (O) = HΩ (O) ∩ G(K) and HΩ (O) is open in H(K), we see that GΩ0 (O) is open in G(K). Θ has been   It is also bounded. Once the identification B(G) = B(H) established, we will show that if Ω is pointwise fixed by Θ, then GΩ0 coincides with the Bruhat–Tits O-group scheme associated to G and Ω. (2) Let Ω ≺ Ω  be non-empty Θ-stable bounded subsets of an apartment  of B(H). The O-group scheme homomorphism HΩ1 → HΩ1 of Axiom

12.2 Certain Group Schemes Associated to H and G

441

4.1.22 restricts to a homomorphism ρΩ,Ω : HΩ0 → HΩ0 , and by Proposition 2.11.5, it induces an O-group scheme homomorphism HΩΘ → HΩΘ . The last homomorphism gives an O-group scheme homomorphism G Θ 0 0 0 Θ 0 ρΩ,Ω  : (HΩ  ) = GΩ  → GΩ = (HΩ )

that is the identity homomorphism on the generic fiber G. Θ , we will refer to the subgroup P := G 0 (O) of G(K) as  (3) For x ∈ B(H) x {x } 0 for a the “parahoric” subgroup associated to x. The group scheme G {x } given “parahoric” subgroup Px of G(K) is the unique smooth model of G whose group of O-rational points equals Px , cf. Corollary 2.10.11. Once Θ has been established, we show in §11.5   the identification B(G) = B(H) that Px coincides with the parahoric subgroup associated to x. (4) More generally, if P is a “parahoric” subgroup of G(K), and P is a subgroup of P of finite index, then according to 12.1.1, P determines the group scheme G 0 associated to P, and hence it determines P. (5) Let P and the associated group scheme G 0 be as in the preceding paragraph. Let Ω be a non-empty Θ stable subset of an apartment of B(H), or  of B(H), and GΩ0 be as in (1). We assume that P fixes Ω pointwise. Then the inclusion of P in H(K)1Ω (= HΩ1 (O)) gives an O-group scheme homomorphism G 0 → HΩ0 (Proposition 12.1.3). Since G 0 (O) = P ⊂ HΩ0 (O)Θ , this homomorphism factors through GΩ0 (= (HΩΘ )0 ) to give an O-group scheme homomorphism G 0 → GΩ0 that is the identity on the generic fiber G. Θ that are fixed by P, and [x, y] is the  Suppose x, y are two points of B(H) 0 be as geodesic joining x and y. Then P fixes every point z of [x, y]. Let G[x,y] 0 0 in (1) for Ω = [x, y]. There are O-group scheme homomorphisms G → G[x,y] 0 that are the identity on the common generic fiber G. and G 0 → G {z }

We will eventually show that B(H)Θ is an affine building with apartments described in 12.4.3 below. We begin with the following proposition which has been suggested by Proposition 1.1 of [PY02]. The proof given here is different from that in [PY02]. Proposition 12.2.4 Let Ω be a non-empty Θ-stable bounded subset of an  apartment of B(H). Let H := HΩ0 be the Bruhat–Tits smooth affine O-group scheme with generic fiber H, and connected special fiber H , associated to Ω. Let H := H /Ru (H ) be the maximal reductive quotient of H . Then there exist closed O-tori S ⊂ T in H such that

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(1) the generic fiber T of T is a maximal K-split torus of H and Ω is contained in the apartment corresponding to T; (2) S , and therefore its generic fiber S, is stable under Θ and the special fiber of S maps onto the central torus of H. We first prove the following lemma in which F is any field of characteristic p  0. Lemma 12.2.5 Let H be a smooth connected affine algebraic F-group given with an action by a finite group Θ by F-automorphisms. Let U be a Θ-stable smooth connected unipotent normal F-subgroup of H. We assume that p does not divide the order of Θ. Let S be a Θ-stable F-torus of H := H/U. Then there exists a Θ-stable F-torus S in H that maps isomorphically onto S. In particular, there exists a Θ-stable F-torus in H that maps isomorphically onto the central torus of H. Proof Let T be an F-torus of H that maps isomorphically onto S(⊂ H). Consider the Θ-stable solvable subgroup TU. Using conjugacy under U(F) of maximal F-tori of this solvable group [Bor91, Theorem19.2], we see that for Θ ∈ Θ, Θ(T) = u(Θ)−1 Tu(Θ) for some u(Θ) ∈ U(F). Let U(F) =: U0 ⊃ U1 ⊃ U2 ⊃ · · · ⊃ Un = {1} be the descending central series of the nilpotent group U(F). Each subgroup Ui is Θ-stable and Ui /Ui+1 is a commutative p-group if p  0, and a Q-vector space if p = 0. Now let i  n, be the largest integer such that there exists a torus S in TU that maps isomorphically onto S, and for every Θ ∈ Θ, there is a u(Θ) ∈ Ui such that Θ(S) = u(Θ)−1 Su(Θ). The proof will be complete if we can show i = n. By way of contradiction, assume that i < n. Let Ni be the normalizer of S in Ui . Then, for Θ ∈ Θ, Θ(Ni ) = u(Θ)−1 Ni u(Θ) and hence as Ui /Ui+1 is commutative, we see that Θ(Ni Ui+1 ) = Ni Ui+1 , that is, Ni Ui+1 is Θ-stable. It is easy to see that Θ → u(Θ) mod(Ni Ui+1 ) is a 1-cocycle on Θ with values in Ui /Ni Ui+1 . But H1 (Θ, Ui /Ni Ui+1 ) is trivial since the finite group Θ is of order prime to p if p  0, and Ui /Ni Ui+1 is divisible if p = 0. So there exits a u ∈ Ui such that for all Θ ∈ Θ, u−1 u(Θ)Θ(u) lies in Ni Ui+1 . Now let S = u−1 Su. Then the normalizer of S in Ui is u−1 Ni u and again as Ui /Ui+1 is commutative, u−1 Ni u · Ui+1 = Ni Ui+1 . For Θ ∈ Θ, we choose u (Θ) ∈ Ui+1 such that u−1 u(Θ)Θ(u) ∈ u−1 Ni u · u (Θ). Then Θ(S) = u (Θ)−1 Su (Θ) for all Θ ∈ Θ. This contradicts the maximality of i.  Proof of Proposition 12.2.4 There is a natural action of Θ on HΩ1 by Ogroup scheme automorphisms(12.2.3(1)). This action keeps H = HΩ0 stable and induces an action of Θ on H , as well as on H, by f-group automorphisms. According to the preceding lemma, there exists a Θ-stable torus S in H that

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maps isomorphically onto the central torus of H. Let T be a maximal f-torus of H containing S . We will denote the natural inclusion S → H by ι. The character group X∗ (S ) of S is a free abelian group on which Θ acts by automorphisms. Let S be the split O-torus with coordinate ring the group ring O[X∗ (S )]. The action of Θ on X∗ (S ) induces an action (of Θ) on S by O-group scheme automorphisms. The homomorphism functor HomSpec(O)−gr (S , H ) is representable by a smooth O-scheme X ([SGA3, Exp.XI,4.2]). There is a natural action of Θ on X induced by its action on HomSpec(O)−gr (S , H ) described as follows. For f ∈ HomSpec(O)−gr (S , H ), Θ ∈ Θ, and s ∈ S (C), (Θ · f )(s) = Θ( f (Θ−1 (s))), for any commutative O-algebra C. By Proposition 2.11.5 the subscheme X Θ of Θ-fixed points in X is a closed smooth O-subscheme of X . The inclusion ι : S → H clearly lies in X Θ (f). Now since O is Henselian, the natural map X Θ (O) → X Θ (f) is surjective by Lemma 8.1.3 and hence there is an O-group scheme homomorphism ι : S → H that is Θ-equivariant and lies over ι. As ι is a closed immersion, using [SGA3, Exp.IX, 2.5 and 6.6] we see that ι is a closed immersion. We identify S with a Θ-stable closed O-torus of H in terms of ι. Using Proposition 8.2.1(1) we see that H contains a closed O-torus T that contains S and whose special fiber is T . This T is O-split since its special fiber T is split. The generic fiber T of T is a maximal K-split torus of H since T is a maximal torus of H . Proposition 9.3.5(2) implies that the apartment of  B(H) corresponding to T contains Ω. 

12.3 A Reduction Let ZG be the maximal K-split torus in the center of G. Then M := Z H (ZG )  is a Levi subgroup of H and it contains G. Recalling that we view B(M) as the  union of apartments of B(H) corresponding to the maximal K-split tori of H that contain ZG , we have the following result. Θ ⊂ B(M). Θ = B(M) Θ.     Hence, B(H) (1) B(H) Θ , there exists a maximal K-split torus S of G,  (2) Given points x, y ∈ B(H) G and a maximal K-split torus T of H containing SG , and hence contained  H (SG ))) corresponding to in Z H (SG ), such that the apartment A(⊂ B(Z Θ is the affine subspace  T contains x and y. Moreover, AΘ = A ∩ B(H) V(SG ) + x of A of dimension dim(SG ).

Proposition 12.3.1

Proof

 Let F be a facet of B(H) which contains x in its closure and is maximal

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Θ , and let Ω = F ∪ {y}. Let S ⊂ T be a pair  among the facets that meet B(H) of K-split tori with properties (1) and (2) of Proposition 12.2.4. Let SG and TG be the maximal subtori of S and T respectively contained in G. Let A be the  apartment of B(H) corresponding to T. Then A contains y and the closure of F, and so it also contains x. Moreover, A is an affine space under V(T), and Lemma 9.4.33 applied to H/K, in place of G/k, implies that the affine subspace V(S) + x of A contains Ω. The affine subspaces V(SG ) + x ⊂ V(TG ) + x of Θ . As V(S) Θ = V(S ) and F ⊂ V(S) + x, we  A are clearly contained in B(H) G Θ Θ  is contained in V(SG ) + x. But since the facet F is see that F = F ∩ B(H) Θ , AΘ (= A ∩ B(H) Θ ) is contained   maximal among the facets that meet B(H) Θ Θ in the affine subspace of A spanned by F . Therefore, A = V(SG ) + x. This implies that V(SG ) + x = V(TG ) + x, and hence, SG = TG . We will now show that SG is a maximal K-split torus of G.

Let S  be a maximal K-split torus of G containing SG . The centralizer Z H (S ) of S  in H is a Θ-stable Levi subgroup contained in the Levi subgroup  H (S )) of Z H (S )(K) is identified with the Z H (SG ). The enlarged building B(Z  union of apartments of B(H) corresponding to maximal K-split tori of H that  H (S )) ⊂ B(Z  H ((SG )). Let z be a point of B(Z  H (S )) Θ contain S . Hence, B(Z and T  (⊃ S ) be a maximal K-split torus of H such that the corresponding  H (S )) contains z. Then A = V(T ) + z, and hence, A Θ = apartment A of B(Z  Θ  A ∩ B(H) = V(T ) Θ + z = V(S ) + z is an affine subspace of A of dimension dim(S ). Let F  be a facet of A which contains the point z in its closure and Θ . Then A Θ is contained in  is maximal among the facets of A that meet B(H) the affine subspace of A spanned by F  Θ , so dim(F  Θ ) = dim(S )  dim(SG ). But dim(F Θ ) = dim(SG )  dim(F  Θ ). This implies that dim(SG ) = dim(S ) and hence S  = SG . Thus we have shown that SG is a maximal K-split torus of G. Θ and take y = x. Let S be as above  Now let x be an arbitrary point of B(H) G

and ZG be the maximal K-split central torus of G. Then ZG is contained in SG and hence Z H (SG ) is contained in the centralizer M = Z H (ZG ) of ZG in H. So  H (SG )) is contained in B(M).   B(Z In particular, x ∈ B(M), which implies that Θ   B(H) ⊂ B(M).  12.3.2 Now let Ω be a Θ-stable non-empty bounded subset of an apartment of  B(M). We define (MΩΘ )0 in the same way as (HΩΘ )0 was defined in 12.2.3(1). Now since (H Θ )0 = G = (M Θ )0 , the group schemes (HΩΘ )0 and (MΩΘ )0 have the same generic fiber, namely G. Furthermore, MΩΘ (O) ⊂ HΩΘ (O), so the identity of G extends to a homomorphism (MΩΘ )0 → (HΩΘ )0 . The identity (H Θ )0 = G = (M Θ )0 further implies HΩΘ (O) ∩ G(K) = MΩΘ (O) ∩ G(K), showing that (MΩΘ )0 (O) is of finite index in HΩΘ (O). So 12.1.1 implies that the

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identity of G extends to a homomorphism (HΩΘ )0 → MΩΘ , which necessarily factors through (MΩΘ )0 . We conclude (MΩΘ )0 (O) = (HΩΘ )0 (O) as subgroups of G(K), and Corollary 2.10.11 implies (HΩΘ )0 = (MΩΘ )0 . Θ with B(G),   and the Bruhat–Tits So for the purpose of comparing B(H) parahoric group schemes associated to G, H and Ω, we may (and will) replace H with M and assume until the end of §11.6 that the maximal K-split central torus of G is also central in H. Let Z be the maximal K-split central torus of H and H  be the derived subgroup of H. Then Z is also stable under Θ, and (ZΘ )0 = ZG . The subgroup G  := (H  Θ )0 is contained in G, it contains the derived subgroup of G, and the maximal central torus of G  is K-anisotropic. There is a natural action  = of H(K) on B(H) = B(H ); so G(K)(⊂ H(K)Θ ) acts on B(H)Θ . As B(H) V(Z) × B(H), we see that Θ  B(H) = V(Z)Θ × B(H)Θ = V(ZG ) × B(H)Θ .

We will denote B(H)Θ by B in the rest of this chapter. It is obvious that B is a closed convex subset of B(H)Θ , and as pointed out above there is a natural action of G(K) on B. We will show below that B carries a natural structure of an affine building on which G(K) operates by polysimplicial isometries and it admits a unique G(K)-equivariant identification with B(G).

12.4 Apartments of B Following 12.3.2 we are assuming that the maximal K-split central torus ZG of G is contained in the maximal K-split central torus Z of H. Our goal is to establish an identification of B(G) with B as a metric space, and our strategy will be to use the uniqueness properties of the building described in §4.4, especially Proposition 4.4.3 and Corollary 4.4.5. For this, we will equip the set B with the structure of an affine building for the group G and verify that it satisfies enough of the axioms laid out in §4.1. The following propositions are the first steps of this plan – they describe which subsets of B will be the apartments. Let H, H  and G  be as in 12.3.2. From now on, we will use S to denote a maximal K-split torus of G and write S  for the maximal subtorus of S contained in G . As the centralizer Z H  (S) of S in H  is stable under Θ, the enlarged  H  (S)) of Z H  (S)(K) contains a Θ-fixed point. There Bruhat–Tits building B(Z

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 is a natural action of V(C) (⊃ V(S)) on B(H), where C (⊃ S) is the maximal K-split central torus of Z H  (S). Proposition 12.4.1 Let S and S  be as above. Let x be a Θ-fixed point in  H  (S)). Then B(Z  H  (S)) Θ = V(S ) + x. So B(Z  H  (S)) Θ is an affine space B(Z  under V(S ). Let T be a maximal K-split torus of H containing S such that the apartment A  H  (S)) Θ . of B(H) corresponding to T contains a Θ-fixed point. Then AΘ = B(Z Proof Let Z (⊃ S ) and Z H  (S) be the maximal K-split central torus and the derived subgroup of Z H  (S), respectively. Then Z, and Z H  (S) are stable  H  (S)) under Θ; moreover, (Z Θ )0 = S . There is a natural action of V(Z) on B(Z  Θ  (cf.12.3.2). Hence, there is an action of V(S ) = V(Z) on B(Z H  (S))Θ . It follows from Proposition 9.3.11 applied to Z H  (S) in place of G, and K in  H  (S)) is a union of apartments containing x. For any such place of k, that B(Z apartment A that corresponds to a maximal K-split torus T of H  containing S , we see that A = V(T) + x, hence AΘ = V(T)Θ + x = V(S ) + x. This implies  H  (S))Θ is an affine space under  H  (S))Θ = V(S ) + x. Therefore, B(Z that B(Z V(S ). We will now prove the second assertion of the proposition. Assume that A contains a Θ-fixed point y. Then A = V(T) + y. Hence, AΘ = V(T)Θ + y =  H  (S))Θ , so it V(S ) + y. Thus AΘ is an affine subspace under V(S ) of B(Z coincides with the latter.  Proposition 12.4.2 Let S1 and S2 be maximal K-split tori of G and Ω be  H  (S2 )). Then there exists  H  (S1 )) ∩ B(Z a non-empty Θ-stable subset of B(Z 0 Θ   H  (S2 ))Θ . Any such an element g ∈ GΩ (O) that maps B(Z H  (S1 )) onto B(Z element fixes Ω pointwise. Proof We will use Proposition 8.2.1(2), with O in place of o, and denote GΩ0 by G , and its special fiber by G . Let S1 and S2 be the closed O-tori of G with generic fibers S1 and S2 respectively. The tori S1 and S2 are maximal O-tori of G . The special fibers S1 and S2 of S1 and S2 are maximal tori of G , and hence there is an element g of G (f) that conjugates S1 onto S2 , cf.[Bor91, Corollary 11.3(1)]. Now Proposition 8.2.1(2) implies that there exists a g ∈ G (O) lying over g that conjugates S1 onto S2 . This element  H  (S1 ))Θ onto fixes Ω pointwise and conjugates S1 onto S2 and hence maps B(Z Θ  H  (S2 )) . B(Z   H  (S)) Θ for maximal K-split 12.4.3 We now equip B with the apartments B(Z tori S of G. The apartment for S is an affine space under the R-vector space V(S ), where S  is the maximal subtorus of S contained in G , so the dimension

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of V(S ) is equal to the K-rank of G . Conjugacy of maximal K-split tori of G under G(K) implies that this group acts transitively on the set of apartments of B. Propositions 12.3.1(2) and 12.4.1 imply the following proposition at once: Proposition 12.4.4 Given any two points of B, there is a maximal K-split torus S of G such that the corresponding apartment of B contains these two points. 12.4.5 Let S be a maximal K-split torus of G and S  be the maximal subtorus of S contained in G . Let NG (S) and ZG (S) respectively be the normalizer and the centralizer of S in G. As NG (S) normalizes the centralizer Z H  (S) of S  H  (S)) and B(Z  H  (S))Θ is in H , there is a natural action of NG (S)(K) on B(Z clearly stable under this action. For n ∈ NG (S)(K), the action by n carries an  H  (S)) to the apartment n · A by an affine transformation. apartment A of B(Z  H  (S)) Θ under Proposition 12.4.6 Axiom 4.1.4 holds for the affine space B(Z  V(S ). Proof Let T be a maximal K-split torus of H  that contains S  and such that  H  (S)) contains a Θ-fixed point. the corresponding apartment A := AT of B(Z  H  (gSg −1 )), x → g · x,  For g ∈ G(K), the isomorphism g : B(Z H  (S)) → B(Z carries the apartment A = AT to the apartment AgT g−1 corresponding to the torus gT g −1 containing gSg −1 via an affine isomorphism f (g) : A → AgT g−1 covering the natural isomorphism V(T) → V(gT g −1 ). By restriction, f (g) pro H  (gSg −1 )) Θ  H  (S)) Θ = AΘ → AΘ −1 = B(Z vides an affine isomorphism B(Z T gT g

covering the natural isomorphism V(S ) → V(gS  g −1 ). This proves A 2. In particular, we see that for n ∈ NG (S)(K), the affine automorphism f (n) of A keeps AΘ stable and so ϕ(n) := f (n)| AΘ is an affine automorphism of AΘ . The derivative df (n) : V(T) → V(nT n−1 ) is induced from the map X∗ (T) → X∗ (nT n−1 ),

λ → α(n) · λ,

where α(n) is the restriction to H  of the inner automorphism Int(n). So, the map df (n)|V (S ) : V(S ) → V(S ) is induced from the homomorphism X∗ (S ) → X∗ (S ) given by λ → α(n) · λ. Thus, df (n) is the Weyl group action of n ∈ NG (S)(K) on X∗ (S ). In particular, dϕ = df | AΘ is trivial on ZG (S)(K). The  H  (S)) Θ = AΘ admits a action of the bounded subgroup ZG (S)(K)b on B(Z fixed point by Theorem 1.1.15. We fix such a point x. As dϕ(z) is trivial, for all v ∈ V(S ), and z ∈ ZG (S)(K), z · (v + x) = v + z · x. For z ∈ ZG (S)(K), let z · x = φ(z) + x, where φ is a V(S )-valued function on ZG (S)(K). It is easily

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seen that φ is a homomorphism that is trivial on ZG (S)(K)b . So z ∈ ZG (S)(K)  H  (S)) Θ by translation by φ(z) ∈ V(S ) and ZG (S)(K)b acts trivially. acts on B(Z We will now prove the formula in A 1, by showing that φ = ν. Since the image of S(K) in ZG (S)(K)/ZG (S)(K)b  Zdim(S) is a subgroup of finite index, to show that φ = ν, it suffices to prove this equality on S(K). But for s ∈ S(K) we have sT s−1 = T and we can apply A 1 to the apartment A = AT of B(H) and conclude that f (s) is the translation of the apartment A by an element ν(s) ∈ V(T), where ν(s) is given by χ(ν(s)) = −ω( χ(s)) for all χ ∈ X∗K (T). This implies the equality φ = ν on S(K), since the restriction map X∗K (T) → X∗K (S) is surjective and the image of the restriction map X∗K (ZG (S)) → X∗K (S)  is of finite index in X∗K (S). Proposition 12.4.7 Let A be an apartment of B. Then there is a unique  H  (S)) Θ . So the stabilizer of A maximal K-split torus S of G such that A = B(Z in G(K) is NG (S)(K).  H  (S)) Θ . We Proof We fix a maximal K-split torus S of G such that A = B(Z will show that S is uniquely determined by A. For this purpose, we observe that the subgroup NG (S)(K) of G(K) acts on A and the maximal bounded subgroup ZG (S)(K)b of ZG (S)(K) acts trivially (Proposition 12.4.6). So the subgroup G of G(K) consisting of elements that fix A pointwise is a bounded subgroup of G(K), normalized by NG (S)(K), and it contains ZG (S)(K)b . Therefore, for any g ∈ G, the orbit {sgs−1 | s ∈ S(K)} of S(K) through g, under the conjugation action, is contained in G and hence it is bounded. Now, using Lemma 2.2.8 we see that S(K) commutes with g. This implies that G commutes with S(K) and hence G ⊂ ZG (S)(K). As G contains ZG (S)(K)b , the Zariski closure of G is ZG (S). Since S is the unique maximal K-split torus of G contained in ZG (S), both the assertions follow.  12.4.8 Let A be the apartment of B corresponding to a maximal K-split torus S of G and let Ω be a non-empty bounded subset of A. The apartment A is contained in an apartment A of B(H) that corresponds to a maximal K-split torus T of H containing S and A = A ∩ B = AΘ (Proposition 12.4.1). So Ω is a bounded subset of A. The group scheme HΩ1 contains a closed split O-torus T with generic fiber T, see Axiom 4.1.20. Let S be the closed Osubtorus of T whose generic fiber is S (S is the schematic closure of S in T ). The automorphism group Θ of HΩ1 acts trivially on the O-torus S (since S ⊂ G ⊂ H Θ ) and hence S is contained in GΩ0 . The special fiber S of S is a maximal torus of G Ω0 since S is a maximal K-split torus of G.

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0 (O) be the “parahoric” subgroup of G(K) associated For x ∈ B, let Px := G {x } with the point x. Let S be a maximal K-split torus of G such that x lies in the 0 contains a apartment A of B corresponding to S. Then the group scheme G {x } closed split O-torus S whose generic fiber is S.

12.5 The Polyhedral Structure on B Following 12.3.2 we are assuming that the maximal K-split central torus ZG of G is contained in the maximal K-split central torus Z of H. We will define and study the polyhedral structure on the set B(= B(H)Θ ). The polyhedrons will turn out to be polysimplices, namely the polysimplices of B(G) under the identification B with B(G) of Theorem 12.7.1. 12.5.1 For a Θ-stable non-empty bounded subset Ω of an apartment of B(H), let GΩ0 be the group scheme defined in 12.2.3(1). We will denote the special 0 in what follows. Given a point x ∈ B, for simplicity we will fiber of GΩ0 by G Ω 0 denote G {x } , H {x1 } , H {x0 } and H {xΘ} by Gx0 , Hx1 , Hx0 and HxΘ respectively, and the special fibers of these group schemes will be denoted by Gx0 , H x1 , H x0 and H xΘ respectively. The subgroup of H(K) (respectively G(K)) consisting of elements that fix x will be denoted by H(K)x (respectively G(K)x ). The subgroup Gx0 (O)(⊂ G(K)x ) is of finite index in G(K)x . 12.5.2 Polyhedral structure on B Let P := Px , for x ∈ B, be a “parahoric” subgroup of G(K) and Gx0 be the O-group scheme associated with x. Let B(H) P denote the set of points of B(H) fixed by P. According to Corollary 12.1.5, B(H) P is the union of facets pointwise fixed by P. Let F P := B(H) P ∩ B. This closed convex subset is by definition the closed facet of B associated with the “parahoric” subgroup P. Then x ∈ F P and as Gx0 (O) = P, x is not fixed by any “parahoric” subgroup of G(K) larger than P. The O-group scheme Gx0 contains a closed split O-torus S whose generic fiber S is a maximal K-split torus of G (12.4.8). The subgroup S (O) (of S(K)) is the maximal bounded subgroup of S(K) and it is contained in P(= Gx0 (O)), so, according  H  (S)) of to Corollary 12.1.5, B(H) P is contained in the enlarged building B(Z Z H  (S)(K). This implies that the closed facet F P is contained in the apartment  H  (S)) ∩ B) of B corresponding to the maximal  H  (S)) Θ (= B(Z A := B(Z K-split torus S of G. Given another “parahoric” subgroup Q of G(K), we claim that if F Q = F P , then Q = P. To see this, we choose points x, y ∈ B such that Gx0 (O) = P and Gy0 (O) = Q. Then y ∈ F Q = F P . So P fixes y. Now using 12.2(4) we see that P ⊂ Q. We similarly see that Q ⊂ P.

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We conclude that if Q P, then F Q is properly contained in F P . Let F P be the subset of points of F P that are not fixed by any “parahoric” subgroup  of G(K) larger than P, that is, F P = F P − QP F Q . We call F P the facet of B associated with the “parahoric” subgroup P of G(K). The facet F P is non-empty (recall that x ∈ F P ) and F P = F P if and only if P is a maximal “parahoric” subgroup of G(K). We will show below (Propositions 12.5.6 and 12.5.9) that F P is convex and bounded. By varying P over the set of “parahoric” subgroups of G(K), we obtain all the facets of B. In the following two lemmas (Lemmas 12.5.3 and 12.5.4), F is any perfect field of characteristic p  0 and F is an algebraic closure of F. We will use the notation introduced in §2.11. Lemma 12.5.3 Let H be a smooth connected affine algebraic F-group and Q be a parabolic F-subgroup of H. Let S be an F-torus of Q whose image in the maximal reductive quotient Q/Ru (Q) of Q contains the maximal central torus of Q/Ru (Q). Then any 1-parameter subgroup λ : Gm → H such that Q = PH (λ)Ru (H) has a conjugate under Ru (Q)(F) with image in S. Proof Let λ be a 1-parameter subgroup of H such that Q = PH (λ)Ru (H). The image T of λ is contained in Q and it maps into the central torus of Q/Ru (Q). Therefore, T is contained in the solvable subgroup SRu (Q) of Q. Note that as S is commutative, the derived subgroup of SRu (Q) is contained in Ru (Q), so the maximal F-tori of SRu (Q) are conjugate to each other under Ru (Q)(F)[Bor91, Theorem19.2]. Hence, there is a u ∈ Ru (Q)(F) such that uTu−1 ⊂ S. Then the image of the 1-parameter subgroup μ : Gm → S, defined  as μ(t) = uλ(t)u−1 , is contained in S. Lemma 12.5.4 Let H be a smooth connected affine algebraic F-group given with an action by a finite group Θ. We assume that p does not divide the order of Θ. Let G = (H Θ )0 . (1) Ru (G) = Ru (H) Θ = G ∩ Ru (H) and the induced map G/Ru (G) → ([H/Ru (H)]Θ )0 is an isomorphism. (2) Given a Θ-stable parabolic F-subgroup Q of H, P := G∩Q is a parabolic F-subgroup of G, so P is connected and it equals (Q Θ )0 . (3) Conversely, given a parabolic F-subgroup P of G, and a maximal Fsplit torus S ⊂ P, there is a Θ-stable parabolic F-subgroup Q of H, Q containing the centralizer ZH (S) of S in H, such that P = G ∩ Q = (Q Θ )0 .

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Proof To prove (1), we apply Lemma 8.1.5 to Ru (H)F to conclude that Ru (H) Θ is smooth and connected. Hence Ru (H) Θ is contained in Ru (G). To prove the reverse inclusion we note that the natural homomorphism G → ([H/Ru (H)] Θ )0 is surjective since the induced Lie algebra homomorphism Lie(G) → Lie([H/Ru (H)] Θ ) is surjective as Lie(G) = (Lie(H)) Θ and Lie([H/Ru (H)] Θ ) = (Lie[H/Ru (H)]) Θ . Now it only remains to note that ([H/Ru (H)] Θ )0 is reductive (Proposition 2.11.5) and hence G ∩ Ru (H) = G ∩ Ru (H) Θ contains Ru (G). Since Ru (G) = G ∩ Ru (H) ⊂ G ∩ Q, to prove (2), we can replace H by its reductive quotient H/Ru (H) and assume that H is reductive. Then G is also reductive. The unipotent radical Ru (Q) of Q is Θ-stable. Let S be a Θstable torus in Q that maps isomorphically onto the maximal central torus of the reductive quotient Q := Q/Ru (Q) (Lemma 12.2.5). By Lemma 12.5.3, there exists a 1-parameter subgroup λ : Gm → S such that Q = PH (λ). Let μ = Θ∈Θ Θ · λ. Then μ is invariant under Θ and so it is a 1-parameter subgroup of G. We will now show that Q = PH (μ). Let Φλ (respectively Φμ ) be the set of weights in the Lie algebra of Q (respectively PH (μ)) with respect to the adjoint action of S. Then since Q, PH (μ) and S are Θ-stable, the subsets Φλ and Φμ (of X∗ (S)) are stable under the action of Θ on X∗ (S). Hence, for all a ∈ Φλ , as a, λ  0, we conclude that a, μ  0. Therefore, Φλ ⊂ Φμ . On the other hand, for b ∈ Φμ , b, μ  0. If b(∈ Φμ ) does not belong to Φλ , then for Θ ∈ Θ, Θ· b  Φλ , so for all Θ ∈ Θ, Θ· b, λ < 0, which implies that b, μ < 0. This is a contradiction. Therefore, Φλ = Φμ and so Q = PH (μ). Now observe that (Q Θ )0 ⊂ G ∩ Q ⊂ Q Θ . As Q Θ is a smooth subgroup (Proposition 2.11.5), G ∩ Q is a smooth subgroup, and since it contains the parabolic F-subgroup PG (μ) of G, it is a parabolic F-subgroup of G [Bor91, Corollary 11.2]. Hence, in particular, it is connected. Therefore, G ∩ Q = (Q Θ )0 . Now we will prove (3). Let λ : Gm → S be a 1-parameter subgroup such that P = PG (λ)Ru (G). Then Q := PH (λ)Ru (H) is a parabolic subgroup of H that is Θ-stable (since λ is Θ-invariant) and it contains P as well as ZH (S). According to (2), G ∩ Q = (Q Θ )0 is a parabolic F-subgroup of G containing P. The Lie algebras of P and (Q Θ )0 are clearly equal. This implies that P = G ∩ Q = (Q Θ )0 and we have proved (3).  12.5.5 We recall that if F and F  are facets of B(H) with F ≺ F , or equivalently, HF0 (O) ⊂ HF0 (O), we say that F is a face of F . In a collection

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C of facets, a facet is maximal if it is not a proper face of any facet belonging to C, and a facet is minimal if no proper face of it belongs to C. Now let X be a convex subset of B(H) and C be the set of facets of B(H), or facets lying in a given apartment A, that meet X. Recall the following from 9.3.1. (1) All maximal facets in C are of equal dimension and a facet F ∈ C is maximal if and only if dim(F ∩ X) is maximal. (2) Let F be a facet lying in an apartment A. Assume that F is maximal among the facets of A that meet X, and let AF be the affine subspace of A spanned by F. Then every facet of A that meets X is contained in AF and A ∩ X is contained in the affine subspace of A spanned by F ∩ X. The subset B(= B(H) Θ ) of B(H) is closed and convex. Hence the assertions of the preceding paragraph hold for B in place of X. Proposition 12.5.6 F P be as in 12.5.2.

Let P be a “parahoric” subgroup of G(K) and F P and

(1) Given x ∈ F P and y ∈ F P , for every point z of the geodesic [x, y], except possibly for z = y, Gz0 (O) = P. Thus, in particular, Gx0 = P for all x ∈ FP . (2) Let F be a facet of B(H) that meets F P and is maximal among such facets. Then GF0 (O) = P. Thus F Θ ⊂ F P . Proof To prove the first assertion, let F0, F1, . . . , Fn be all the facets of B(H) that contain a segment of positive length of the geodesic [x, y]. Then each Fi is Θ-stable and is fixed pointwise by P, hence P ⊂ GF0i (O), cf.12.2.3(4),  and moreover [x, y] ⊂ i F i . Using 12.5.5 we see that the Fi s are of equal dimension. We assume the facets {Fi } to be indexed so that x lies in F 0 , y lies in F n , and for each i < n, F i ∩ F i+1 is non-empty. Let z 0 = x. For every positive integer i( n), F i−1 ∩ F i contains a unique point of [x, y]; we will denote this point by z i . To prove the second assertion of the proposition along with the first, we take x to be a point of F P and y to be any point of F Θ . Let [x, y], and for i  n, Fi and z i be as in the preceding paragraph. Then Fn = F. Since z0 = x ∈ F 0 , there is an O-group scheme homomorphism GF00 → Gz00 which is the identity on the generic fiber G. Thus, GF00 (O) ⊂ P. But P ⊂ GF00 (O), so Gz00 (O) = P = GF00 (O). Let j ( n) be a positive integer such that for all i < j,

Gz0i (O) = P = GF0i (O). As z j ∈ F j−1 ∩ F j , there are Θ-equivariant O-group

12.5 The Polyhedral Structure on B σj

453

ρj

scheme homomorphisms HF1j−1 −→ Hz1j ←− HF1j that are the identity on the generic fiber H. The images of the induced homomorphisms σj

ρj

H F0j−1 −→ H z0j ←− H F0j are parabolic subgroups of H z0j (Axiom 4.1.22(2)). Theorem 8.4.19(1) applied



 for H in place of G shows that the kernels K σ j and K ρ j of σ j and ρ j respectively are smooth subgroups, hence, 









 



Lie σ j H F0j−1 = σ j Lie H F0j−1 and Lie ρ j H F0j ) = ρ j Lie H F0j .

Θ

Θ Moreover, as K σ j and K ρ j are also smooth (Proposition 2.11.5), we see that











 Lie σ j GF0j−1 = σ j Lie GF0j−1 and Lie ρ j GF0j = ρ j Lie GF0j . Since the characteristic p of f does not divide the order of Θ, every representation of Θ on a finite-dimensional f-vector space is completely reducible. Hence, given two such representations of Θ on f-vector spaces V and W and a Θequivariant surjective linear map V → W, the induced map V Θ → W Θ is also surjective. Now 







  Θ



Lie σ j GF0j−1 = σ j Lie GF0j−1 = σ j (Lie H F0j−1 Θ 0 Θ  Θ





. = Lie σ j H Fj−1 = Lie σ j H F0j−1 = σ j Lie H F0j−1 As





Θ 



 Θ σ j GF0j−1 ⊂ σ j H F0j−1 and Lie σ j GF0j−1 = Lie σ j H F0j−1 ,



 Θ0 . In an analogous way we see we conclude that σ j GF0j−1 = (σ j H Fj−1

0



 Θ0 . Lemma 12.5.4(2) implies that both of these that ρ j GFj = ρ j H Fj subgroups are parabolic subgroups of G z j . As GF0j−1 (O) = P, whereas, P ⊂



 GF0j (O)(⊂ Gz0j (O)), we see that σ j GF0j−1 is contained in ρ j GF0j . Let Q and Q 



 respectively be the images of σ j GF0j−1 and ρ j GF0j in the maximal reductive

quotient G z j := G z0j /Ru (G z0j ) of G z0j . Then Q ⊂ Q , and both of them are parabolic subgroups of G z j .

Now let S be a maximal K-split torus of G such that the apartment of B(H)Θ corresponding to S contains the geodesic [x, y] (Proposition 12.4.4). Let v ∈ V(S) such that v + x = y. Then for all sufficiently small positive real numbers , − v + z j ∈ Fj−1 and  v + z j ∈ Fj . Using Axiom 4.1.22(3) we infer



 that the images of the parabolic subgroups σ j H F0j−1 and ρ j H F0j (of H z0j )

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in the maximal reductive quotient H z j := H z0j /Ru (H z0j ) of H z0j are opposite 



 parabolic subgroups. Therefore, the image H of σ j H F0j−1 ∩ ρ j H F0j in H z j is reductive. The natural homomorphism π : G z j → H z j is injective (Lemma 12.5.4(1)). It is obvious that the image of Q = Q ∩ Q  under π is (H Θ )0 . As (HΘ )0 is reductive, we see that Q is a reductive subgroup of G z j . But since Q is a parabolic subgroup of the latter, we must have Q = G z j , and hence, Q  = G z j . So, σ j (GF0j−1 ) = Gz0j = ρ j (GF0j ). Since the natural homomorphism GF0j−1 (O) → GF0j−1 (f) is surjective, and σ j (GF0j−1 ) = Gz0j , the image of GF0j−1 (O)(⊂ Gz0j (O)) in Gz0j (f) is all of Gz0j (f). From this and 12.1.1 we see that O[Gz0j ] = { f ∈ K[G] | f (GF0j−1 (O)) ⊂ O} = O[GF0j−1 ]. Therefore, σj | G 0

F j−1

: GF0j−1 → Gz0j is an O-group scheme isomorphism. We

similarly see that ρ j | G 0 : GF0j → Gz0j is an O-group scheme isomorphism. Now Fj

since GF0j−1 (O) = P, we conclude that P = Gz0j (O) = GF0j (O). By induction it

follows that P = Gz0i (O) = GF0i (O) for all i  n. In particular, for all z ∈ [x, y],  except possibly for z = y, Gz0 (O) = P, and GF0n (O) = P. Corollary 12.5.7

Let P and Q be “parahoric” subgroups of G(K). Then

(1) F P is convex. (2) F P is an open-dense subset of F P , hence the closure of F P is F P . Moreover, F P  F P unless P is a maximal “parahoric” subgroup of G(K). (3) If F P ∩ FQ is non-empty, then for any z in this intersection, P = Gz0 (O) = Q. Thus every point of B(H)Θ is contained in a unique facet. For a “parahoric” subgroup Q of G(K) containing P, obviously, FQ ⊂ F Q ⊂ F P , thus FQ ≺ F P and hence F P is a maximal facet if and only if P is a minimal “parahoric” subgroup of G(K). The maximal facets of B are called the chambers of B. It is easily seen using the observations contained in 12.5.5 that all the chambers are of equal dimension. We say that a facet F  of B is a face of a facet F if F  ≺ F, that is, if F  is contained in the closure of F. We will use the following simple lemma in the proof of the next proposition. Lemma 12.5.8 Let S be a maximal K-split torus of G, A the corresponding apartment of B(H)Θ , and C a non-compact closed convex subset of A. Then for any point x ∈ C, there is an infinite ray originating at x and contained in C. Proof Recall that A is an affine space under V(S) = R⊗Z X∗ (S). We identify A with V(S) using translations by elements in the latter, with x identified with the

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origin 0, and use a positive definite inner product on V(S) to get a norm on A. With this identification, C is a closed convex subset of V(S) containing 0. Since C is non-compact, there exist unit vectors vi ∈ V(S), i  1, and positive real numbers si → ∞ such that si vi lies in C. After replacing {vi } by a subsequence, we may (and do) assume that the sequence {vi } converges to a unit vector v. We will now show that for every non-negative real number t, tv lies in C. This will prove the lemma. To see that tv lies in C, it suffices to observe that for a given t, the sequence {tvi } converges to tv, and for all sufficiently large i (so  that si  t), tvi lies in C. Proposition 12.5.9 For any “parahoric” subgroup P of G(K), the associated closed facet F P of B, and so also the associated facet F P (⊂ F P ), is bounded. Proof Let S be a maximal K-split torus of G such that the corresponding apartment A of B contains F P (12.5.2). Assume to the contrary that F P is non-compact and fix a point x of F P . Then, according to Lemma 12.5.8, there is an infinite ray R := {tv + x | t ∈ R0 }, for some v ∈ V(S), originating at x and contained in F P . As the central torus of G has been assumed to be K-anisotropic, there is a non-multipliable root a of G, with respect to S, such that a(v) < 0. Let Ua be the root group of G corresponding to the root a. Since P is an open subgroup of G(K) there exists a non-trivial element u ∈ P ∩ Ua (K). Let ψ := ψau be as in Proposition 9.4.3(3). Then since u fixes every point of R, ψ(tv + x) = ta(v) + ψ(x)  0 for all t  0 (Proposition9.4.3(4)). But as a(v) < 0, this is impossible.  We state the following proposition for future reference. Proposition 12.5.10 Let P be a “parahoric” subgroup of G(K) and F := F P be the facet of B associated to P in 12.5.2. Then for any x ∈ F, since P ⊂ GF0 (O) ⊂ Gx0 (O) = P (12.5.6(1)), GF0 (O) = P, and hence the natural O-group scheme homomorphism GF0 → Gx0 is an isomorphism. In particular, for any facet F of B(H) that meets F, P = GF0 = GF0 , and hence F Θ ⊂ F. 12.5.11 Let F and F  be two facets of B(H)Θ , with F ≺ F . As in 12.2.3(2) G 0 0 be the : GF0  → GF0 . Let ρF,F we obtain a homomorphism ρG : G  → G F F F,F  G  induced homomorphism. We will denote the image of ρF,F  by pF (F ). The following proposition shows that this image is a parabolic subgroup of GF0 . Let P and Q be two “parahoric” subgroups of G(K) and F P and FQ be the corresponding facets of B. Then F P ⊂ F Q if and only if P ⊃ Q. Thus F P ≺ FQ if and only if P ⊃ Q. Given a “parahoric” subgroup P of G(K),

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the following proposition provides a recipe for determining all “parahoric” subgroups Q contained in P. Let F and F  be two facets of B(H)Θ with F ≺ F . Fix an x ∈ F and let F be a facet of B(H) that contains x. Then the union of all facets of B(H) that contain F in their closure is a neighborhood of x in B(H). Since x belongs to the closure of F , there is a facet F  of B(H) that meets F  and contains F in its closure, that is, F ≺ F . Proposition 12.5.12 Let F and F  be two facets of B(H)Θ , with F ≺ F . (1) If F and F  are facets of B(H), F ≺ F , that meet F and F  respectively, then pF (F ) = (Q Θ )0 , where Q is the image of ρF , F  : H F0 → H F0 . Moreover, pF (F ) is a parabolic subgroup of GF0 . (2) Given a parabolic subgroup P of GF0 , there is a facet F  of B(H)Θ with G 0 0 F ≺ F  such that the image of the homomorphism ρF,F : G  → G F F equals P. Proof As the image of the Lie algebra homomorphism Lie(GF0 ) → Lie(GF0 ) G G Θ  0 Θ 0 induced by ρF,F  is Lie(Q) , the containment pF (F ) = ρF,F  (G  ) ⊂ (Q ) F is equality. According to Lemma 12.5.4(2), (Q Θ )0 is a parabolic subgroup of GF0 . This proves (1). To prove assertion (2), we fix a facet F of B(H) that meets F. Then GF0 = GF0 (Proposition 12.5.10). Using Lemma 12.5.4(3) for H F0 in place of H, we find a Θ-stable parabolic subgroup Q of H F0 such that P = (Q Θ )0 . Let (F ≺) F  be the facet of B(H) corresponding to the parabolic subgroup Q of H F0 . Then F  is stable under the Θ-action on B(H). As F ≺ F , the O-group scheme homomorphism ρF , F  : HF0 → HF0 restricts to an O-group scheme homomorphism ρFG, F  : GF0  → GF0 . The image of ρF , F  : H F0 → H F0

Θ 0 is Q. Then the image of ρG F , F  is (Q ) = P (see the proof of (1)). Let Q := GF0  (O) ⊂ GF0 (O) = P, and F  = FQ . Then Q ⊂ P are “parahoric subgroups” of G(K), F  = F Q ⊃ F P ⊃ F, thus F ≺ F . As F and F  meet F and F  respectively, GF0 = GF0 and GF0  = GF0  (Proposition 12.5.10), the image G 0 0 equals P. of the homomorphism ρF,F  : G  → G F F

12.6 Identification of Parahoric Subgroups Following 12.3.2 we are assuming that the maximal K-split central torus ZG of G is contained in the maximal K-split central torus Z of H. In fact, the

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457

conclusions of this section hold without this assumption, due to the identity (MΩΘ )0 = (HΩΘ )0 proved in 12.3.2. We also recall from 12.3.2 that there is a natural action of G(K) on B. Proposition 12.6.1 (1) A subgroup of G(K) is “parahoric” in the sense of 12.2 if and only if it is a parahoric subgroup of G(K) in the sense of Bruhat–Tits, that is, the stabilizer in G(K)0 of a point of B(G). (2) A smooth model of G is “parahoric” if and only if it is parahoric. Proof (1) Let P be a maximal “parahoric” subgroup and F := F P be the corresponding facet of B. We apply Corollary 4.2.12 to the bounded subgroup P of G(K) acting on B(G) to obtain a point z ∈ B(G) fixed by P. Thus P ⊂ G(K)1z . According to Corollary 2.10.10 the identity of G extends to a homomorphism of group schemes P → Gz1 , where P is the smooth affine O-group scheme constructed in §12.2, and Gz1 is the group scheme constructed in §8.3. Since P has connected fibers, this homomorphism factors through a homomorphism P → Gz0 , from which we conclude P ⊂ G(K)0z . Let y be a vertex of the facet of B(G) that contains z. Then P ⊂ G(K)z0 ⊂ G(K)y0 . The subgroup G(K)0y is bounded and its action on B(H) preserves the closed convex subset B. By Corollary 4.2.12 this subgroup fixes a point x in B. But then x is also fixed by P. Hence, x lies in F. Since P is maximal, F is closed, so x lies in F, and hence G(K)0y is contained in P. Therefore, G(K)0y = P. Thus P is a Bruhat–Tits maximal parahoric subgroup. By a similar argument one can show that every Bruhat–Tits maximal parahoric subgroup of G(K) is a maximal “parahoric” subgroup. From the above results and Corollary 2.10.11 we conclude that the maximal “parahoric” group schemes of §12.2 are same as the Bruhat–Tits group schemes associated to maximal parahoric subgroups of G(K). Proposition 12.5.12 and Theorem 8.4.19(4) provide identical recipes for determining the “parahoric” subgroups of G(K) contained in a maximal “parahoric” subgroup and parahoric subgroups of G(K) contained in a maximal parahoric subgroup. Therefore “parahoric” subgroups of G(K) are parahoric subgroups in the sense of Bruhat– Tits and vice versa. (2) This follows at once from (1) and Corollary 2.10.11. 

12.7 The Main Theorem We now drop the assumption that the maximal K-split central torus ZG of G is contained in the maximal K-split central torus Z of H.

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Recall from §12.2 the notions of “parahoric” subgroups of G(K) and “parahoric group schemes.” Theorem 12.7.1 Let H be a connected reductive K-group and Θ be a finite group of K-automorphisms of H. We assume that the characteristic of the residue field of K does not divide the order of Θ. Then G := (H Θ )0 is a reductive K-subgroup. We have the following. Θ with the fol  (1) There exists a G(K)-equivariant bijection B(G) → B(H) lowing property: for each maximal K-split torus S of G, the apartment Θ.  of B(G)   H (S))Θ of B(H)  A(S) is identified with the apartment B(Z  (2) Under this bijection, the facets of B(G) are identified with the facets of Θ as defined in §12.5.  B(H) Θ.   Let F be a facet of B(G) and let F  be the corresponding facet of B(H)

(3) The parahoric subgroup of G(K) associated to F equals the “parahoric” subgroup associated to F . In other words, G(K)0F = H(K)0F  ∩ G(K)0 . (4) The identity automorphism of G extends to an isomorphism between the parahoric group scheme associated to F and the “parahoric group scheme” associated to F . (5) On the level of maximal reductive quotients of special fibers, we have Θ 0 GF = (HF ) .

Proof As explained in 12.3.2, to prove this theorem we may (and will) assume that the maximal K-split central torus of G is central in H. Under this assumption, we will show that there is a unique G(K)-equivariant polysimplicial isomorphism B → B(G). Let S ⊂ G be a maximal K-split torus and A(S) be the apartment of B(G) corresponding to S. Propositions 12.4.6 and 4.4.3 provide a unique NG (S)(K) H  (S))Θ covering the equivariant isomorphism of affine spaces A(S) → B(Z identity on V(S). We will now show that the equality of maximal parahoric subgroups with maximal “parahoric” subgroups given by Proposition 12.6.1 is compatible with this isomorphism. For this, let y be a vertex in A(S). Then the maximal Bruhat–Tits parahoric subgroup P := G(K)0y is a maximal “parahoric” subgroup. Let F := F P be the corresponding facet in B and x be a point of F. Since y is a vertex, Corollary 7.4.8 implies that y is the unique fixed point for the action of NG (S)(K) ∩ P on A(S). But as P fixes F pointwise, we infer that F is the singleton {x}. Therefore, x corresponds to y under the bijection  H  (S))Θ . A(S) → B(Z

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459

 H  (S))Θ maps vertices to We have thus shown that the bijection A(S) → B(Z vertices and identifies the Bruhat–Tits maximal parahoric subgroups with the maximal “parahoric” subgroups. Finally, we show that these bijections, for the various maximal split tori S ⊂ G, glue to a bijection B(G) → B = B(H)Θ . For this we recall the presentation of B(G) of Proposition 4.4.4. It will be enough to establish the same presentation for B. Proposition 12.4.4 shows that the map  H  (S))Θ → B sending (g, x) to gx G(K) × B(Z  H  (S))Θ and g, h ∈ G(K) with gx = hy, we is surjective. Given x, y ∈ B(Z  H  (S))Θ ,  H  (S))Θ and h−1 g B(Z apply Proposition 12.4.2 to the apartments B(Z 0 −1  both of which contain y, to obtain j ∈ G(K)y such that j h g B(Z H  (S))Θ =  H  (S))Θ . By Proposition 12.4.7 we have n := j h−1 g ∈ NG (S)(K). Then B(Z nx = y and g −1 hn ∈ G(K)0x . Now assertions (3) and (4) follow from Proposition 12.6.1, and assertion (5) follows from Lemma 12.5.4(1).  Remark 12.7.2 Recall from §4.2 that there exists a G(K)-equivariant metric on B(G) whose restriction to each apartment is a Euclidean metric, and that any two such metrics are related to each other by rescaling on each K-simple factor of G. Since the metric on B(H)Θ is G(K)-equivariant and restricts to a Euclidean metric on each apartment of B(H)Θ , the metric on B(G) can be Θ becomes an isometry.   chosen so that the bijection B(G) → B(H)

12.8 The Case of a Finite Cyclic Group We begin by proving the following proposition in which F is any field of characteristic  0. Proposition 12.8.1 Let H be a reductive F-group, Θ an F-automorphism of H of finite order not divisible by the characteristic of F and G := (HΘ )0 . (1) If H is non-commutative, then G is non-trivial and no maximal torus of G is central in H. (2) The centralizer T := ZH (S) in H of any maximal torus S of G is a Θ-stable maximal torus of H. (3) Every maximal Θ-stable F-torus of H is a maximal torus (of H). (4) If F is separably closed, then H contains a Θ-stable Borel F-subgroup. Proof To prove assertion (1), we can (and will) replace H by its derived subgroup and assume that H is semi-simple. We fix an algebraic closure F of

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F. According to a theorem of Steinberg[Ste68, Theorem7.5], HF contains a Θ-

stable Borel subgroup B, and this Borel subgroup contains a Θ-stable maximal torus T. We endow the root system of HF with respect to the maximal torus T with the ordering determined by the Borel subgroup B. Let b be the sum of all positive roots. Then as B is Θ-stable, b is fixed under Θ acting on the character group X∗ (T) of T. Therefore, X∗ (T) admits a non-trivial torsion-free quotient on which Θ acts trivially. This implies that T contains a non-trivial subtorus S that is fixed pointwise under Θ. The subtorus S is therefore contained in GF . Since the center of the semi-simple group H does not contain a non-trivial smooth connected subgroup, we infer that S is not central in HF . Now by conjugacy of maximal tori in GF , we see that no maximal torus of this group can be central in HF . This proves (1). The centralizer ZH (S) is a Θ-stable reductive subgroup of H, and (ZH (S)Θ )0 = ZG (S) = S. So to prove (2), it would suffice to show that ZH (S) is commutative. As S is central in ZH (S), if ZH (S) were non-commutative, we could apply (1) to this subgroup in place of H to get a contradiction. To prove (3), let T be a maximal Θ-stable F-torus. To show that T is a maximal torus, it would suffice to show that its centralizer Z is a torus, since Z is also stable under Θ. Let us assume that Z is a non-commutative reductive group. Then according to (1), (ZΘ )0 is non-trivial. Let S be a maximal F-torus of this group. Then (1) implies that S is not contained in the center of Z. So, in particular, S is not contained in T. Therefore, S · T is a Θ-stable F-torus that properly contains T. This contradicts the maximality of T. To prove (4), we assume now that F is separably closed and let S be a maximal torus of G. Then S, and its centralizer T in H, are F-split. In view of (2), there exists a 1-parameter subgroup λ : Gm → S such that for every root c of H with respect to the split maximal torus T, c, λ  0. Then B := PH (λ) is a Θ-stable  Borel F-subgroup of H. In the following proposition we will use the notation introduced in §12.4. We will assume that H is K-isotropic, Θ is a finite cyclic group of automorphisms of H, and the characteristic of the residue field of K does not divide the order of Θ. Proposition 12.8.2 (1) The Bruhat–Tits building B(H) of H(K) contains a Θ-stable chamber. (2) Given a Θ-stable chamber C in B(H), there is a Θ-stable maximal Ksplit torus T of H that contains a maximal K-split torus of G and the apartment corresponding to T contains C. Proof Let C be a Θ-stable facet of B(H) that is maximal among the Θ-stable

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facets. Let H := HC0 be the Bruhat–Tits smooth affine O-group scheme corresponding to C. The generic fiber of H is H, and its special fiber H is connected. Let H := H /Ru (H ) be the maximal reductive quotient of H . In case H is commutative, H does not contain a proper parabolic subgroup and so C is a chamber of B(H). We assume, if possible, that H is not commutative. As C is stable under the action of Θ, there is a natural action of this finite cyclic group on H by O-group scheme automorphisms (12.2.3(1)). This action induces an action of Θ on H , and so also on its maximal reductive quotient H. Now taking Θ to be a generator of Θ, and using Proposition 12.8.1(4), we conclude that H contains a Θ-stable Borel subgroup. The inverse image B in H of any such Borel subgroup of H is a Θ-stable Borel subgroup of H . The facet C corresponding to B is Θ-stable and C ≺ C. This contradicts the maximality of C. Hence, H is commutative (so it is a torus) and C is a chamber. Now applying Proposition 12.2.4 for Ω = C, we see that since H is a torus, H contains a maximal K-split torus T that is stable under Θ and the apartment of B(H) corresponding to T contains C. Arguing as in the proof of Proposition 12.3.1, we see that the maximal subtorus of T contained in G is a maximal K-split torus of G.  Proposition 12.8.3 Let Θ be a finite cyclic group of automorphisms of the reductive K-group H. We assume that the order of Θ is not divisible by the characteristic of f, and let G = (H Θ )0 be as above. Let S be a maximal K-split torus of G and T be a Θ-stable maximalK-split torus of H that contains S, cf.Proposition 12.8.2(2). Let A be the apartment in the Bruhat–Tits building B(G) = B(H)Θ corresponding to S and A(⊃ A) be the Θ-stable apartment of B(H) corresponding to T. Then every affine root of G with respect to S is the restriction to A of an affine root of H with respect to T. Proof Let ψ be an affine root of G with respect to S with derivative a ∈ Φ(S, G). We fix a point z ∈ A where ψ vanishes and let Gz0 and Hz0 be the O-group schemes associated to z; let Gz0 and H z0 respectively be their special fibers and Gz and Hz be the maximal reductive quotients of these special fibers. We will denote the maximal torus of Gz (respectively Hz ) corresponding to S (respectivelyT) by S (respectively T (⊃ S)). As ψ vanishes at z, Theorem 8.4.10 (applied to Ω = {z}) implies that a is a root of Gz with respect to S. 0 Using Lemma 12.5.4(1), we see that Gz = (HΘ z ) . So there exists a root, say b, of Hz with respect to T whose restriction to S is a. Applying Theorem 8.4.10 (to Hz and Ω = {z}) we see that there exists an affine root ψb of H with respect to T whose derivative is b and which vanishes at z. Now the restriction of ψb to A is an affine function with derivative a and as it vanishes at z, this restriction is equal to ψ. 

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Remark 12.8.4 (1) It is not true in general that the restriction of every affine root of H with respect to T is an affine root of G with respect to S. To see an example, let G be a non-trivial K-split semi-simple group, L a totally ramified Galois extension of K of finite degree n which is not divisible by the characteristic of f. It is known that Θ := Gal(L/K) is cyclic. Let H = R L/K (G L ). Then Θ acts on H by K-automorphisms and H Θ = G. Let Φ = Φ(S, G) be the root system of G with respect to S, then its affine root system is Φ × Z, whereas the affine root system of H with respect to T is Φ × n1 Z. (2) We can use Proposition 9.4.28 to obtain an analog of the above proposition over k. Proposition 12.8.5 Let S be a maximal K-split torus of G and let T be a Θinvariant maximal K-split torus of H containing S, cf. Proposition 12.8.2(2). Assume that the restriction map X∗ (T) → X∗ (S) maps Φ(T, H) onto Φ(S, G) and that the fibers of this map are orbits under Θ. Assume that for r = f (0), (Z H (T)(K)r )Θ = ZG (S)(K)r .  be a concave function such that f (0) > 0 and let  G) → R Let f : Φ(S,  be the composition of f with the restriction map Φ(T,  G) → R  H) → F : Φ(T,  Φ(S, G). Let x ∈ A(S, G) ⊂ A(T, H). Then the group H(K)x, F is Θ-invariant, G(K)x, f = (H(K)x, F )Θ and Gx, f = (Hx, F )Θ . Proof By assumption we have F ◦ Θ = F and Θ(x) = x. It follows that for H H every b ∈ Φ(T, H) and Θ ∈ Θ, we have Θ(Ub, x, F ) = UΘ(b), x, F . Since H is quasi-split, Z H (T) is a Θ-stable maximal torus of H. The functoriality of the filtration Z H (T)(K)r shows that Θ(Z H (T)(K)r ) = Z H (T)(K)r . We conclude from Definition 7.3.3 that H(K)x, F is Θ-invariant. The identity Gx, f = (Hx, F )Θ would follow from Corollary 2.10.11 and the identity G(K)x, f = (H(K)x, F )Θ , whose proof we now turn to. Choose a system of positive roots Φ(S, G)+ for Φ(S, G) and let Φ(T, H)+ be its preimage under the surjective map Φ(T, H) → Φ(S, G). Then Φ(T, H)+ is Θ-stable. Applying Proposition 7.3.12 we obtain the following set-theoretic direct product decomposition   H H U−b, Ub, H(K)x, F = x, F × Z H (T)(K)r × x, F . b ∈Φ(T ,H)+

b ∈Φ(T ,H)+

Each of the three factors is invariant under Θ. Therefore, Θ    H Θ U−b, × (Z (T)(K) ) × (H(K)x, F )Θ = H r x, F b ∈Φ(T ,H)+

 b ∈Φ(T ,H)+

H Ub, x, F

Θ

.

By assumption, (Z H (T)(K)r )Θ = ZG (S)(K)r . Now consider b ∈ Φ(T, H)+ .

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Even though Φ(T, H) may be non-reduced, the considerations of [Ste68, §8.2] remain valid, since the unique irreducible non-reduced root system BCn does not have outer automorphisms. Therefore, For any b ∈ Φ(T, H) exactly one of the following cases occurs: (1) the Θ-orbit of b consists of strongly orthogonal roots {b1, . . . , bn }, that is, no pair of roots adds up to a root, (2) the irreducible component containing b is of type A2n and there is an element Θ ∈ Θ that preserves that component and acts non-trivially on it, and Θ · b = {b1, Θ(b1 ), b2, Θ(b2 ), . . . , bn, Θ(bn )}. Thus bi + Θ(bi ) ∈ R(T, H). In case (1), the root groups UbH1 , . . . , UbHn commute with each other and their direct product UBH is a Θ-stable subgroup of H. Writing a = b|S , the root group Ua ⊂ G is given by Ua = (UBH )Θ . H . Then In case (2), let UiH be the root group generated by UbHi and UΘ(b i) H the groups Ui again commute with each other and their direct product UBH is Θ-stable. Writing a = b|S we see that both a and 2a belong to Φ(S, G) and again Ua = (UBH )Θ . To complete the proof it suffices to show that, for any r ∈ R, Ua, x,r = H Θ (UB, x,r ) . This reduces to the case r = 0 by choosing y ∈ A(S, G) such that x − y, a = r and replacing x by y. Now Proposition 7.3.12 shows that Ua, x,0 = Ua (K) ∩ G(K)0x . The latter equals (UBH (K) ∩ H(K)0x )Θ according to Theorem 12.7.1 and the identity Ua = (UBH )Θ . Proposition 7.3.12 shows  H H  UBH (K) ∩ H(K)0x = UB, b Ub, x,0 . x,0 = Remark 12.8.6 Proposition 12.8.5 complements Theorem 12.7.1, in particular part (3). In fact, more can be said about the Θ-action on Hx . Beyond taking fixed points under Θ, one can also study the eigenspaces for the action of Θ as modules under Gx = (HΘx )0 and establish connections to Vinberg’s theory. We refer the reader to [RY14].

12.9 Tamely Ramified Descent 12.9.1 Let k be a field endowed with a non-archimedean discrete valuation ω. We assume that the valuation ring o of k is Henselian and its residue field f is perfect. Then the valuation ω extends uniquely to any algebraic extension of k; we will denote all such extensions by ω. For a finite extension  of k, its ramification index denoted e(/k) is [ω( × ) : ω(k × )]. The extension  is said to be totally ramified if e(/k) equals the degree [ : k] of  over k, and  is

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a tamely ramified extension if the characteristic of the residue field of k does not divide e(/k). Thus if the residue field is of characteristic zero, then every finite algebraic extension of k is tamely ramified. Let G be a connected reductive k-group. Let  be a tamely ramified Galois extension of k. Let K and L be the maximal unramified extensions of k and , respectively, contained in a fixed separable closure k s of k. Theorem 12.9.2 The Bruhat–Tits building of G(k) has a natural identification with B(G ) Gal(/k) as a metric space. This theorem was first proved by Guy Rousseau by a completely different method than used here. Proof Let k  be the maximal unramified extension of k contained in . Then L = K ⊗ k  . We will denote the Galois groups of K/k  and /k  by Γ and Θ respectively. Then the Galois group of L/k  is canonically isomorphic to the direct product Γ × Θ and the Galois group of L/K is canonically isomorphic to Θ. Let H = R/k  (G ). Then H(k ) = G(), so B(H) = B(G ). Moreover, HK = R L/K (G L ); Θ operates on HK by K-rational automorphisms and HKΘ = G K . Theorem 12.7.1 provides the equality B(HK ) Θ = B(G K ) of metric spaces. Γ

So, B(HK ) Θ×Γ = B(HK ) Θ = B(G K ) Γ . By unramified descent, B(G K ) Γ = B(G k  ). Therefore, B(HK ) Θ×Γ = B(G k  ). On the other hand, once again by unramified descent, B(HK ) Γ = B(H) = B(G ). Hence, B(HK ) Γ×Θ = B(G ) Θ and we conclude that B(G ) Θ = B(G k  ). Now since Θ is a normal subgroup of Gal(/k) and Gal(/k)/Θ = Gal(k /k), we see that

 Gal(k  /k)  = B(G k  ) Gal(k /k) . B(G ) Gal(/k) = B(G ) Θ As k  is an unramified extension of k, again by unramified descent we see that   B(G k  ) Gal(k /k) = B(G). Thus B(G ) Gal(/k) = B(G). The following proposition now follows at once from Proposition 12.5.10. Proposition 12.9.3 F Gal(/k) ⊂ F.

If a facet F of B(G ) meets a facet F of B(G), then

Proposition 12.9.4 Let L/K be a finite tamely ramified Galois extension. Let S be a maximal K-split torus of G and T be the K-torus of G containing S such  be a concave  G) → R that TL is a maximal L-split torus of G L . Let f : Φ(S,   function such that f (0) > 0 and let F : Φ(T, G) → R be the composition of f  G) → Φ(S,  G). Let x ∈ A(S) ⊂ A(TL ). Then the with the restriction map Φ(T,

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group G(L)x, F is Θ-invariant, G(K)x, f = (G(L)x, F )Θ , and Gx, f = (Hx, F )Θ , where H = R L/K (G L ) and Θ = Gal(L/K). Proof We will apply Proposition 12.8.5. Let Z be the centralizer of S in G. Since G is quasi-split, Z is a maximal torus defined over K, and T is the maximal L-split subtorus of Z. The restriction map X∗ (Z) → X∗ (S) maps Φ(Z, G) surjectively onto Φ(S, G). The fibers of the surjection Φ(Z, G) → Φ(S, G) are orbits under Gal(Ks /K). This surjection factors through Φ(T, G). The map Φ(Z, G) → Φ(T, G) is surjective and its fibers are orbits under Gal(Ks /L). Therefore the map Φ(T, G) → Φ(S, G) is surjective and its fibers are orbits under Gal(L/K). Under the embedding G → H the image of Z is identified with Z H  (S)Θ . Thus Z H  (S)Θ = ZG (S). The assumptions of Proposition 12.8.5 are thus verified and the proof is complete. 

13 Moy–Prasad Filtrations

Let k be a field endowed with a discrete valuation ω : k × → R. Let o be its ring of integers and m be the maximal ideal of o. We assume that o is Henselian and the residue field f = o/m is perfect. We further assume that ω(k × ) = Z. We will denote the unique extension of the valuation ω to any algebraic field extension /k also by ω. Let K be a maximal unramified extension with ring of integers O and let Γ = Gal(K/k). Let x ∈ B(G). The papers [MP94] and [MP96] introduced a decreasing separated filtration G(k)x,r of the bounded open subgroup G(k)0x indexed by r ∈ R0 , as well as decreasing separated filtrations gx,r and g∗x,r of the Lie algebra g of G and of its dual g∗ , indexed by r ∈ R. In this chapter we will review the construction and properties of these filtrations and discuss certain generalizations. We note that analogous filtrations were defined independently in [SS97]. We will use the notation established in §1.6. In particular, for any  decreasing filtration {Xr } of a group X we define Xr+ = s>r Xs .

13.1 Filtrations of Tori The filtrations for a general connected reductive group G are a combination of filtrations for the various root groups, which are provided by the concept of a valuation of the root datum (cf. Definition 6.1.2), and a filtration on the centralizer of a maximal split torus of G. Such a filtration was constructed in §9.8 under the assumption that an appropriate filtration has been defined on all tori. Which is the right filtration for a torus can be a subtle question in general. This question is discussed in some detail in Appendix B, particularly §B.5, §B.6 and §B.10. To briefly summarize, the question about the correct filtration on a torus T is easy to resolve when T is weakly induced (cf. Definition B.6.2), which 466

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means that there exists a finite tamely ramified field extension /k such that T is an induced torus. In applications, the main examples of this situation are when T is a maximally split maximal torus of a quasi-split connected reductive group G that either splits over a tame extension, or is simply connected, or adjoint. For example, [MP94] dealt exclusively with simply connected groups. In the situation when T is weakly induced, the correct filtration on T is the so-called standard filtration described in §B.5. It is defined as follows, cf. Definition B.5.1. T(k)0 = T(k)0 and for r > 0, T(k)r = {t ∈ T(k)0 | ω( χ(t) − 1)  r for all χ ∈ X∗ (T)}.

(13.1.1)

There is also the corresponding filtration on the Lie algebra, defined for all r ∈ R by t(k)r = {X ∈ t(k) | ω(dχ(X))  r for all χ ∈ X∗ (T)}

(13.1.2)

and one has a functorial isomorphism T(k)r /T(k)s → t(k)r /t(k)s

(13.1.3)

for all 0 < r < s  2r, called the Moy–Prasad isomorphism, cf. Proposition B.6.9. More generally, one would like to have a well-behaved filtration defined on all k-tori, not just those which are weakly induced. What well-behaved might mean is of course a matter of debate, but in this book we take it to include at least the primary properties of being functorial (cf. Definition 7.2.2) and admissible (cf. Definition B.10.2), and ideally the secondary properties of being schematic, connected, and congruent (cf. Definitions 2.10.15 and B.5.3), and satisfying the Moy–Prasad isomorphism. The standard filtration is functorial and schematic (cf. Proposition B.10.1). Its restriction to the full subcategory consisting of tori which are weakly induced is connected and congruent (cf. Proposition B.6.4), and admissible (cf. Proposition B.9.1). Moreover, on that subcategory it is the unique admissible filtration (cf. Proposition B.10.5). However, on the whole category of k-tori the standard filtration is not connected and congruent and does not satisfy the Moy–Prasad isomorphism. A better behaved filtration is the minimal congruent filtration introduced by Yu and defined in §B.10. It is a schematic functorial filtration which is admissible (hence coincides with the standard filtration on any torus which is weakly induced) and also connected and congruent (cf. Proposition B.10.10, Corollary B.10.13). It also satisfies the Moy–Prasad isomorphism, although its functoriality only holds in depth greater than or equal to 1 (cf. Propositions B.10.16 and B.10.17). The minimal congruent filtration has the drawback of being less explicit. For many purposes the particular filtration of the torus is not too important.

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Therefore, we will now fix any functorial admissible filtration that satisfies (1) and (2) of §7.2, and is hence commutator-friendly according to Lemma 7.2.3. This will be our arrangement until §13.5, where we will need a filtration that satisfies the Moy–Prasad isomorphism. The material of §9.8 then provides a commutator friendly filtration on the centralizer of a maximal k-split torus of a connected reductive group G.

13.2 Filtrations of Parahoric Subgroups Let S ⊂ G be a maximal k-split torus and let Z ⊂ G the centralizer of S. Let A ⊂ B(G) the apartment for S. Let Φ be the root system for G relative to S and  = Φ ∪ {0}. Φ  sending 0 the constant function Φ →R Consider x ∈ A. For every r ∈ R  each element of Φ to r is concave. We will denote this function also by r. 0 the Moy–Prasad filtration subgroup Definition 13.2.1 For x ∈ A and r ∈ R Px,r ⊂ G(k) is defined as the subgroup G(k)x,r of Definition 7.3.3 Recalling the construction of G(k)x,r , we see that it is the open bounded subgroup of G(k) generated by Z(k)r and Ua, x,r for all a ∈ Φ, equivalently the open bounded subgroup generated by Z(k)r and Uψ for all ψ ∈ Ψx,r = {ψ ∈ Ψ | ψ(x)  r }. In §9.8 we used this construction in the special case of the group Z. Remark 13.2.2 As we have already remarked in the general case of G(k)x, f , the group Px,r depends on the choice of filtration of Z(k), which in turn depends on the choice of functorial filtration on tori. Note further that, while the parahoric subgroup Px,0 and its subgroup Px,0+ depend only on the facet containing x, in general this is not true for Px,r , which may well depend on the particular point x within that facet. Remark 13.2.3 We can consider x as an element of an apartment A(TK ) of B(G K ) for a special k-torus T containing S, cf. Proposition 9.3.4. Applying this construction over K we obtain Moy–Prasad filtration groups Px,r of G(K). It follows from Proposition 9.8.3(2) that Px,r = (Px,r )Γ . Definition 13.2.4 Let M be a Levi k-subgroup of G and AM be its central k-split torus. Let P = M N be a parabolic subgroup of G whose unipotent radical is N, and K ⊂ G(k) a compact open subgroup. Let N+ = K ∩ N(k),

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N− = K ∩ N − (k), where P− = M N − is the parabolic k-subgroup that is Mopposite to P, and M = K∩M(k). Then K is said to have Iwahori decomposition with respect to P if (1) the product map N− × M × N+ → K is a homeomorphism (but not necessarily a homomorphism); (2) given a ∈ AM (k) such that ω(α(a))  0 for all α ∈ R(AM , N) one has aN+ a−1 ⊂ N and a−1 N− a ⊂ N− . Proposition 13.2.5 The filtration (Px,r )r ∈R 0 is a decreasing separated filtration of G(k) by bounded open subgroups, with the following properties. (1) The group Px,r depends only on x and r, but not on S. (2) For g ∈ G(k), gPx,r g −1 = Pgx,r . In particular, Px,r is normal in the stabilizer G(k)x of x. (3) If r > 0 then the product map   Ua, x,r × Z(k)r × Ua, x,r → Px,r a ∈Φ−

a ∈Φ+

is a bijection, where Φ+ is any system of positive roots in Φ, and the factors in each product are taken with respect to an arbitrary fixed order. (4) [Px,r , Px, s ] ⊂ Px,r+s . Let now P = M N be a parabolic subgroup, and assume that x ∈ B(G) M as in §9.7. (5) If r > 0, Px,r has an Iwahori decomposition with respect to P. (6) Px,r ∩ M(k) = PxMM ,r , where x M ∈ B(M) is the image of x under (9.7.1). Proof Propositions 7.3.17 and 7.6.4 imply (1). Lemma 7.7.8(1) implies (2). Proposition 7.3.12(5) implies (3). If r = 0 or s = 0 then (4) follows from (2). Assume now that r > 0 and s > 0. Using (2) and (3) it is enough to show that zwz −1 w −1 ∈ Px,r+s whenever z ∈ Ua, x,r and w ∈ Ub, x, s for any a, b ∈ Φ. The cases where a is not a negative rational multiple of b is covered by Axiom V 3 of Definition 6.1.2, while the case where a is a negative rational multiple of b is covered by Lemma 7.1.2, where we are using in addition Definition 7.1.3 and the fact that the filtration on Z(k) is commutator friendly. Assume that x ∈ B(G) M and choose an apartment A corresponding to a maximal split torus S of M. For (5) we choose Φ+ so that it contains a system ⊂ Φ+,nd , Φ−,nd ⊂ Φ−,nd , and the of simple roots for M. Then we have Φ+,nd M M claim follows from (3). In fact, this same argument also shows that (6) holds whenever r > 0. The validity of (6) when r = 0 was already established in Corollary 9.7.3. 

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13.2.6 Reductive quotient Let x ∈ B(G) be as above, and Gx0 be the associated Bruhat–Tits parahoric group scheme with connected fibers. Let Gx0 be the special fiber of Gx0 and G x the maximal reductive quotient of Gx0 . Then the 0

kernel of the natural surjective homomorphism Px,0 = Gx (o) → G x (f) equals Px,0+ according to Corollary 8.4.12, and hence Px,0 /Px,0+ can be identified with G x (f). The same is true over K, that is, the reduction map Gx0 (O) → G x (f) identifies Px,0 /Px,0+ with G x (f). More generally, we have an identity of f-groups G x = cok(G x,0+ → G x,0 ), where Gx,0 (= Gx0 ) and Gx,0+ are the integral models associated to the constant concave functions 0 and 0+ by Theorem 8.5.2, overline denotes their special fibers, and the homomorphism G x,0+ → G x,0 is defined in 8.5.17.

13.3 Filtrations of the Lie Algebra and its Dual 13.3.1 Definition of the filtrations Let x ∈ B. For each r ∈ R0 we have the group scheme Gx,r associated to the constant concave function f = r in §8.5. Let gx,r be the Lie algebra of this group scheme, it is an o-lattice in the kvector space g(k). Let m denote the maximal ideal of o. For n ∈ N the identity gx,r+n = m n · gx,r holds by Proposition 8.5.16. Therefore, we may extend the definition of gx,r to all R via this relation. The sequence {gx,r } of open subgroups is a separated descending filtration of the vector space g(k). Recalling  by gx,r+ = s>r gx, s . the convention of §1.6, this filtration extends to R For any real number r, we define the lattice g∗x,r in g∗ as follows: g∗x,r = {X ∈ g∗ | X(Y ) ∈ m for all Y ∈ gx,(−r)+ }. 13.3.2 The quotient gx,0 /gx,0+ Let G x denote the maximal reductive quotient of the special fiber of the parahoric group scheme Gx0 . According to 13.2.6, G x = cok(G x,0+ → G x,0 ). From the discussion of §2.4(a) and the smoothness of ker(G x,0+ → G x,0 ) due to Proposition 8.5.18, we see that Lie(G x ) = cok(gx,0+ ⊗o f → gx,0 ⊗o f) = gx,0 /gx,0+, where the last identity comes from the inclusions m · gx,0 ⊂ gx,0+ ⊂ gx,0 of o-lattices in the k-vector space g = Lie(G). 13.3.3 Description of the lattice gx,r The lattice gx,r can be described more explicitly as follows. Let g(K)x,r be the O-lattice in g(K) which is the Lie algebra of the group scheme (Gx,r )O . Then gx,r = (g(K)x,r )Γ . The O-lattice g(K)x,r can be described explicitly in terms of a Chevalley–Steinberg basis

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of g(K) relative to a fixed maximally split maximal torus T of G K . This was done in 8.5.20 for an arbitrary concave function group scheme. In fact, this description also works over o provided G is quasi-split. One can also give the following less explicit description. Let S be a maximal k-split torus whose apartment contains x. Given an affine root ψ with derivative a ∈ Φ := Φ(S, G), we have the group scheme Uψ with generic fiber Ua defined in 9.4.5, characterized by the property that Uψ (O) is the subgroup of Ua (K) that pointwise fixes the affine root hyperplane Hψ . Let uψ be the Lie algebra of Uψ . Then uψ = gz ∩ u a (k), where gz is the Lie algebra of the parahoric group scheme Gz0 associated to any point z ∈ Hψ . Let S be the standard integral model of S. According to Proposition 9.8.3 there is a natural action S × Gx,r → Gx,r . Let Zr be the fixed point subgroup scheme of Gx,r . Its generic fiber is Z := ZG (S). Let zr be the Lie algebra of Zr , a lattice in z. Then gx,r is the o-span of zr and uψ for all ψ ∈ Ψx,r . In what follows, for X ∈ g (respectively X ∈ g∗ ) and g ∈ G(k), we will denote Adg(X) (respectively Ad∗ g(X)) simply by gX or g(X). 13.3.4 Let x ∈ B(G). As G(K)x,r is a normal subgroup of G(K)0x by Proposition 13.2.5, the conjugation action of G on itself extends (see Corollary 2.10.10) to an action of the group scheme Gx0 on the group scheme Gx,r , and hence also on the Lie algebra gx,r and its dual g∗x,r , for all r ∈ R. This action descends to 0

a k-rational action of G x on gx,r ⊗o f, which induces an f-rational action of the reductive quotient G x on gx,r /gx,r+ , as well as on g∗x,r /g∗x,r+ .

13.4 Optimal Points These points were defined in [MP94, §6.1] for g∗ and optimal points for g were later defined in [AD02, §2.3]. It has been observed by Adler–DeBacker [AD02, Lemma 2.3.1], and also independently by Shu-Yen Pan, that the set of optimal points for g∗ is a subset of the set of optimal points for g. We will content ourselves by just defining the optimal points for g here and refer the reader to [MP94] for the analogous definition of optimal points for g∗ . These points have played an important role in the proof of existence of “unrefined minimal K-types” in admissible representation of reductive p-adic groups. They have been used in defining the important notion of “depth” of such representations, and in showing that the depth of any admissible representation is a non-negative rational number, see [MP94] and [MP96]. In this subsection, Ψ will denote the set of affine roots of G with respect to

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a fixed maximal k- split torus S of G. Let  be the smallest positive rational number such that for any affine root ψ, ψ +  is also an affine root. Fix a chamber C in A. We say that an affine root ψ is positive (respectively negative), and write ψ > 0 (resp.ψ < 0), if it takes positive (respectively negative) values on C. A positive affine root is said to be a simple affine root if it is not the sum of two positive affine roots. Let Δ be the set of simple affine roots; Δ is a basis of the affine root system Ψ. The closure C of C equals the subset {x ∈ A | ψ(x)  0 for allψ ∈ Δ}. The derivatives of any #Δ − 1 simple affine roots constitute a basis of the Q-linear span of the k-root system Φ(S, G). In what follows, we will denote by Σ the finite set of affine roots ψ such that ψ > 0 and ψ −  < 0. Definition 13.4.1 A real valued affine functional on A is called rational if it is a linear combination of simple affine roots with rational coefficients. A point x ∈ A is called rational, if every affine root takes a rational value at x. Definition 13.4.2 Consider a non-empty subset S of Σ. An optimal point for S is any point x of C satisfying the following conditions: (1) minψ ∈S ψ(x)  minψ ∈S ψ(y) for all y ∈ C, (2) x is a rational point, that is, ψ(x) ∈ Q for every affine root ψ. A point x of B is optimal if there exists an apartment A of B, a chamber C ⊂ A, and a non-empty finite subset S of the Σ associated to C, such that x ∈ C is optimal for S. We will now show that optimal points for S exist. Finding them explicitly is a problem of linear programming. For integral linear programming, we recommend the book [Sch86]. Enumerate the affine roots in S as ψ1, . . . , ψm . We define the affine functions ψi j := ψ j − ψi . For a given i  m, consider the following compact polyhedron Ci = {x ∈ C | ψi j (x)  0 for all j  m}.  It is obvious that i Ci = C and for x ∈ Ci , ψi (x) = min{ψ j (x) | j  m}. From the definition of Ci , it is also obvious that every element of Δ takes rational values at every vertex of Ci . In other words, the vertices of Ci are rational points of A. Let ci be the maximum of the values of ψi on Ci . Consider the intersection with Ci of the parallel half-apartments Aψi , c = {x ∈ A | ψi (x)  c} for all c ∈ R,. It is clear that Ci ∩ Aψi , ci contains a vertex of Ci , but Ci ∩ Aψi , c = ∅ for all c > ci . As ci is the value taken by ψi at a vertex of Ci , it is a rational number.

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473

We fix a j such that c j is the maximum of {c1, . . . , cm }. We define the function φ on C as φ(x) = min{ψi (x) | x ∈ C, i  m}. The maximum value taken by φ on C is c j , and there is a vertex of C j on which this value is taken. It is obvious that this vertex is an optimal point for S. Readers interested in more details, and in determining the maximum value c j of φ, are referred to [Sch86]. It can happen that there are infinitely many optimal points for a given apartment A, chamber C, and the finite set S. They are all equally useful to us. So, we fix A and C, and for each S ⊂ Σ, we make an arbitrary choice of one such point in C and denote it by xS . Let O be the subset of B(G) consisting of those points that are G(k)-conjugate to one of the xS . A rational number r is called an optimal number if for an optimal point x, gx,r  gx,r+ . Since O is a locally finite set, we see that the set of optimal numbers is a discrete subset of Q. Example 13.4.3 Consider the split group SL3 and its diagonal maximal torus. The standard integral structure corresponds to a special vertex x ∈ A according to Theorem 9.9.3. Using this vertex we can identify Ψ with Φ × Z. The standard upper triangular Borel subgroup leads to a Weyl chamber with corresponding set of simple roots {a, b}, where a(t1, t2, t3 ) = t1 /t2 and b(t1, t2, t3 ) = t2 /t3 and (t1, t2, t3 ) denotes the diagonal matrix with the given entries. The Weyl chamber together with the chosen vertex distinguishes a chamber C in A. This chamber corresponds to the set of affine simple roots {a, b, 1 − a − b}. Let y, z ∈ C be the vertices determined by a(y) = 1 and b(z) = 1. The set Σ corresponding to C is given by {a, b, a + b, 1 − a, 1 − b, 1 − a − b}. The set of optimal points for {a} is {y}, the set of optimal points for {b} is {z}, and the set of optimal points for {a + b} is the entire geodesic [y, z]. The set of optimal points for {a, b} is the midpoint of the geodesic [y, z]. The set of optimal points for {a, b, 1 − a − b} is the barycenter of C. All in all, if we take the vertices x, y, z, the midpoints of [x, y], [y, z], [x, z], and the barycenter of C, we obtain a set containing an optimal point for each subset of Σ. The following lemma gives a tool for reducing problems involving arbitrary points of B(G) to optimal points. Lemma 13.4.4 Given a point z ∈ B(G) and a real number s, there exists a point y ∈ O such that gy, s ⊃ gz, s . Proof We may (and do) assume that z ∈ C. Let S  = {ψ ∈ Ψ | ψ(z)  s}. Let n be the smallest integer such that for all ψ ∈ S , ψ + n is a positive affine root, and let S = {ψ + n | ψ ∈ S  } ∩ Σ. Note that S is non-empty. Let y := xS (∈ C). Now to prove the lemma, it suffices to show that for all ψ ∈ S , ψ(y)  s. We consider a ψ ∈ S , such that ψ + n is in S. For any such ψ, ψ + (n − 1) < 0

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so ψ(z) + (n − 1)  0 and as ψ(z)  s, we infer that (1 − n)  s. Also, ψ(y) + n  min ϕ(z)  s + n, ϕ ∈S

and hence ψ(y)  s. On the other hand if ψ ∈ S  is such that ψ + n is not in S, then ψ + (n − 1) > 0, so ψ(y) + (n − 1)  0, which implies that ψ(y)  (1 − n)  s. 

13.5 The Moy–Prasad Isomorphism We continue with a connected reductive k-group G. In this section we will prove the Moy–Prasad isomorphism relating the filtration of the group to the filtration of the Lie algebra. For this, it is necessary that the filtration chosen on the category of tori satisfies this isomorphism. Thus, we now equip the category of tori with the minimal congruent filtration. Fix positive real numbers 0 < r  s and x ∈ B(G). We consider the following conditions. (1) The maximally split maximal torus of G K becomes induced after a tame base change (i.e. it is weakly induced in the sense of Definition B.6.2) and s  2r. (2) r = r0 + n with 0  r0 < 1 and n a positive integer, and s  r0 + 2n. (3) 0 < r  s  2r  r + 1. These conditions are not mutually exclusive. Note that the maximally split maximal torus of G K is automatically induced if G is simply connected, or adjoint, or split over K. Theorem 13.5.1 Assume that at least one of conditions (1)–(3) holds. (1) There exists an isomorphism of abstract abelian groups MPx,r , s : G(k)x,r /G(k)x, s → g(k)x,r /g(k)x, s compatible with unramified algebraic extensions of k. (2) If either (1) or (2) of the above conditions holds, then such an isomorphism can be chosen with the following property: for any k-rational automorphism θ of G, dθ ◦ MPx,r , s ◦ θ −1 = MPθ(x),r , s . (3) When both (1) and (2) hold simultaneously, the isomorphisms they lead to coincide.

13.5 The Moy–Prasad Isomorphism

475

Proof Assume that condition (2) holds. We apply Proposition A.5.19 to the group scheme Gx,r0 to obtain a functorial isomorphism G(k)x,r0 +n /G(k)x,r0 +2n → g(k)r0 +n /g(k)r0 +2n . Setting s0 = s − n > 0 and using this functoriality for the homomorphism Gs0 → Gr0 one obtains the isomorphism G(k)x,r0 +n /G(k)x, s0 +n → g(k)r0 +n /g(k)s0 +n, as desired. Furthermore, this isomorphism is still functorial, from which the second claim follows. We now assume that at least one of conditions (1) and (3) holds. We will first construct the isomorphism over K so we assume for a moment that k = K. In particular, G is quasi-split by Corollary 2.3.8. Choose a maximally split maximal torus T ⊂ G whose apartment contains x. The group G(k)x,r is generated by T(k)r and Ua, x,r for all a ∈ Φ = Φ(T, G). Parts (3) and (4) of Proposition 13.2.5 imply that the natural map   (Ua, x,r /Ua, x, s ) × (T(k)r /T(k)s ) × (Ua, x,r /Ua, x, s ) → Px,r /Px, s a ∈Φ−, nd

a ∈Φ+, nd

is an isomorphism of abstract abelian groups. At the same time, the natural map * u a,x,r /u a,x,s → g(k)x,r /g(k)x, s t(k)r /t(k)s ⊕ a ∈Φnd

is also an isomorphism of groups. The filtration Ua, x,r of each root group is indexed by all real numbers, and according to Proposition C.5.1 the corresponding integral models Ua, x,r satisfy the congruence property (Ua, x,r )n = Ua, x,r+n . Therefore Proposition A.5.19 applied to the group scheme Ua, x,r−n produces a functorial isomorphism Ua, x,r /Ua, x, s → u a, x,r /u a, x, s . On the other hand, Proposition B.6.9 produces a functorial isomorphism T(k)r /T(k)s → t(k)r /t(k)s under the assumption that condition (1) holds, and Proposition B.10.17 provides a possibly non-functorial such isomorphism under the assumption that condition (3) holds. Putting all of these together we obtain an isomorphism Px,r /Px, s → g(k)x,r /g(k)x, s . If we assume condition (1), we see that this isomorphism does not depend on the choice of T by applying the functoriality of all isomorphisms that were involved in its construction and the fact that any two choices of T are conjugate under G(k)x,0 according to Proposition 7.7.5. The same argument also implies

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the equivariance of this isomorphism under automorphisms θ, thus proving claim (2). We now drop the assumption that k = K. We have thus produced an isomorphism G(K)x,r /G(K)x, s → g(K)s,r /g(K)x, s . It is Gal(K/k)-equivariant. It is enough to show that the Gal(K/k)-fixed points in G(K)x,r /G(K)x, s are equal to G(k)x,r /G(k)x, s and the Gal(K/k)-fixed points in g(K)x,r /g(K)x, s are equal to g(k)x,r /g(k)x, s . This would follow from the vanishing of the cohomology sets H1 (Gal(K/k), G(K)x, s ) and H1 (Gal(K/k), g(K)x, s ). Consider the smooth model Gx, s . The vanishing of H1 (Gal(K/k), G(K)x, s ) is reduced by Lemma 8.1.4 to the vanishing of H1 (Gal(K/k), Gx, s (f)). Since Gx, s ⊗o f is a smooth connected unipotent group by Theorem 8.5.14 and f is perfect, this group has a filtration with successive subquotients isomorphic to Ga , and the desired vanishing follows from Lemma 2.4.2. The vanishing of H1 (Gal(K/k), Lie(Gx, s )(f)) is analogous. Note finally that when conditions (1) and (2) are both satisfied then the two constructions produce the same isomorphism: this follows from the functoriality of the isomorphism of Proposition A.5.19 applied to the various inclusions Ua → G and T → G, and the compatibility of the isomorphisms T(k)r /T(k)s → t(k)r /t(k)s provided by Proposition B.6.9 and Proposition A.5.19, as discussed in Proposition B.10.16.  The following characterization of Iwahori subgroups when k is a local field (i.e. k is complete and its residue field f is finite of characteristic p) was given by Hideya Matsumoto. We will give its proof in our set-up using Moy–Prasad filtrations. Proposition 13.5.2

Assume that k is a local field.

(1) For any x ∈ B(G) the group Px,0+ is an open pro-p subgroup of G(k). It is maximal if and only if x lies in a chamber, that is, Px,0 is an Iwahori subgroup. (2) Every maximal open pro-p subgroup of G(k)0 equals Px,0+ for an Iwahori subgroup Px,0 , and Px,0 is the normalizer of Px,0+ in G(k)0 . Proof (1) Since k has been assumed to be a local (i.e., locally compact) field, the topological space G(k) is locally compact. As Px,0+ is a bounded open subgroup of G(k), it is compact. Therefore Px,0+ = lim Px,0+ /Px,r . The ←−− quotient Px,0+ /Px,r has a finite filtration whose successive quotients are vector groups according to Theorem 13.5.1. This shows that Px,0+ is a pro-p group. If Px,0 is not an Iwahori subgroup, the reductive quotient of G x,0 will contain a proper parabolic subgroup. The preimage H ⊂ G(k)x,0 of the group of f-points of its unipotent radical is an extension of the pro-p group (Px,0+ =) G(k)x,0+

13.5 The Moy–Prasad Isomorphism

477

by the group of f-points of that unipotent radical. The latter is a finite p-group, so H is an open pro-p subgroup of G(k)0 that contains G(k)x,0+ properly. To complete the proof of (1) it remains to show that G(k)x,0+ is a maximal pro-p open subgroup if x lies in a chamber. This will be done in the course of proving (2). (2) Consider a maximal pro-p subgroup H of G(k)0 . It is compact, hence bounded, so fixes a point in B(G) due to Corollary 4.2.14. Any facet of B(G) invariant under H is pointwise fixed by H by Proposition 1.5.13(3). Let F be a facet that is maximal among the facets fixed by H. Thus H ⊂ G(k)0F . Since G(k)F,0+ is an open pro-p subgroup by (1), normalized by H, the product G(k)F,0+ ·H is an open pro-p subgroup of G(k)0 , thus equal to H by maximality of H. Thus G(k)F,0+ ⊂ H. The image of H in G(k)F,0 /G(k)F,0+ is a maximal p-subgroup of the group of f-points of the quasi-split connected reductive fgroup which is the maximal reductive quotient of Gx0 . Therefore, this image is the group of f-points of the unipotent radical of a Borel f-subgroup of that quasi-split reductive group. If C is the chamber such that F ≺ C and the image of ρF,C : G C0 → G F0 modulo the unipotent radical of G F0 equals that Borel subgroup, we see that H ⊂ G(k)C,0 .

The maximality of F implies F = C. The root system of the reductive quotient is then empty according to Theorem 8.4.10, so this reductive quotient is a torus and has no non-trivial unipotent subgroups. We conclude that H = G(k)C,0+ . This proves that every maximal pro-p subgroup of G(k)0 is the pro-unipotent radical of an Iwahori subgroup. Since all Iwahori subgroups are conjugate under G(k)0 , we see that the pro-unipotent radical of every Iwahori subgroup is a maximal pro-p subgroup of G(k)0 . This completes the proof of (1). To complete the proof of (2) it remains to show that I = G(k)C,0 is the normalizer of G(k)C,0+ in G(k)0 . Let N be that normalizer. According to Proposition 2.2.16, N is a compact open subgroup of G(k)0 . As it contains I, it is a parahoric subgroup, since the parahoric subgroups of G(k) are precisely the parabolic subgroups of the Iwahori–Tits system of G(k)0 , that is, those subgroups of G(k)0 that contain an Iwahori subgroup. Let F be the corresponding facet of B(G). It is contained in the closure of C. Then G(k)F,0+ ⊂ G(k)C,0+ by 6.3.21. The image of G(k)C,0+ in G(k)F,0 /G(k)F,0+ is the group of f-points of the unipotent radical of a Borel subgroup of the maximal reductive quotient of G F0 . But it is also normal in G(k)F,0 /G(k)F,0+ . This is impossible unless  G(k)F,0 /G(k)F,0+ is (the group of f-points of a) torus. Thus F = C.

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13.6 Semi-stability Theorem 13.5.1 identifies Px,r /Px,r+ with the f-vector space gx,r /gx,r+ and the action of G x (f) = Px /P+x on Px,r /Px,r+ arises from a rational representation of G x on that f-vector space, cf. 13.3.4. Semi-stable vectors in this representation space have been studied in [RY14], [FR17], and [Fin21], beginning with an initial work in [MP94]. These semi-stable vectors play an important role in the construction and study of supercuspidal representations of reductive p-adic groups. Recall from Theorem 9.4.23 that the absolute and relative root systems of G x are the sets {ψ | ψ ∈ ΨK , ψ(x) = 0} and {ψ | ψ ∈ Ψk , ψ(x) = 0}, respectively, where here, and henceforth, for an affine root ψ, we write ψ for its derivative. Definition 13.6.1 We will say that a 1-parameter subgroup λ : Gm → G defined over k is compatible with a point x of B(G) if λ maps the maximal bounded subgroup Gm (O) of Gm (K) into Gx1 (O) (⊂ G(K)). It follows from Corollary 2.10.10 that λ is compatible with x if and only if it extends to an o-group scheme homomorphism Gm → Gx0 . If such an extension exists, we will denote it again by λ. 13.6.2 Let S be a maximal k-split torus of G, A be the corresponding apartment in B(G). We fix a chamber C in A and a point x in the closure of C. Let Gx0 be the Bruhat–Tits parahoric o-group scheme associated to x, Px = Gx0 (o), and G x be the maximal reductive quotient of the special fiber of Gx0 . Let Ψ = Ψ(S, G) be the relative affine root system of G with respect to S and let Δ be the basis of Ψ determined by the chamber C, cf. Proposition 1.3.22. Then the derivatives of affine roots in Δx := {ψ ∈ Δ|ψ(x) = 0} constitute a basis of the relative root system of G x with respect to the maximal f-split torus S (of G x ) determined by S, cf. Proposition 1.3.35. Proposition 13.6.3 ([MP94, Proposition 4.3]) Let ρ : G → GL(V) be a krational representation of G on a finite-dimensional k-vector space V. Let X be a vector in V. If there is a 1-parameter k-subgroup λ : Gm → G such that limt→0 λ(t) X = 0, then there is a 1-parameter k-subgroup μ : Gm → G, compatible with x, such that limt→0 μ(t) X = 0. Proof Let y be a special vertex of C and G := Gy0 be the corresponding Bruhat–Tits parahoric o-group scheme. Let P = G (o). Let P := PG (λ) = {g ∈ G | lim λ(t)gλ(t)−1 exists in G}. t→0

Then P is a parabolic k-subgroup of G which contains the image of λ. By the Iwasawa decomposition (Theorem 5.3.4), G(k) = P · P(k). By conjugacy of

13.6 Semi-stability

479

maximal k-split tori of G under G(k), there exists an element pg ∈ G(k), with p ∈ P and g ∈ P(k), such that the conjugate of λ under pg takes values in S. Now lim gλ(t)g −1 X = lim g · λ(t)g −1 λ(t)−1 · λ(t) X = 0,

t→0

t→0

since limt→0 λ(t)g −1 λ(t)−1 exists. Hence we may (and do) replace λ by its conjugate under g ∈ P(k) to assume that the 1-parameter subgroup t → pλ(t)p−1 takes values in S. So λ takes values in the conjugate S  := p−1 Sp of S. As the maximal bounded subgroup of S(K) is contained in G (O) and p ∈ P(= G (o)), the maximal bounded subgroup of S (K) is also contained in G (O). Hence G contains a closed o-split torus S  with generic fiber S . The 1-parameter subgroup λ : Gm → S  extends to an o-group scheme homomorphism Gm → S  (⊂ G ) that we will denote again by λ. Let G be the special fiber of G and λf : Gm → G be the 1-parameter subgroup of G induced by λ. Let P := PG (λ); then P is a closed smooth o-subgroup scheme of G (cf. Propositions 2.11.1 and 2.11.3), and its generic fiber is P. (Note that P(O) = G (O) ∩ P(K) and this equality determines P.) Let P be the special fiber of P; it equals the closed connected f-subgroup PG (λf ) of G and contains the special fiber S  of S . Let Ru (G ) be the unipotent radical of G and G := G /Ru (G ) be the maximal reductive quotient of G . We denote the image of the special fiber of S  in G by S  . Let λ : Gm → (S)  (⊂ G) be the 1-parameter subgroup of G induced by the 1 parameter subgroup λf of G . Let P be the image of P in G. Then P = PG (λ), hence it is a parabolic f-subgroup of the reductive group G. Moreover, the kernel K of the natural homomorphism P → P equals Ru (G ) ∩ P = Ru (G ) ∩ PG (λf ) = PRu (G ) (λf ), and hence this kernel is a smooth connected

unipotent f-subgroup of P (Propositions 2.11.1 and 2.11.3). Since we have assumed f to be perfect, K is f-split, Therefore, H1 (f, K ) is trivial and so the natural homomorphism P(f) → P(f) is surjective. As o is Henselian, the natural homomorphism P(o) → P(f) is also surjective. Hence the composite homomorphism P(o) → P(f) is surjective. Let [x, y] be the geodesic joining x to y. This geodesic lies in the closure of the chamber C. Moreover, if x  y, there is a face F of C that contains 0 = GF0 . There is a natural o-group scheme (x, y) := [x, y]− {x, y}, and then G[x,y] 0 → Gy0 = G that is the identity on the homomorphism ρ := ρ {y },[x,y] : G[x,y] 0 (O)(⊂ G (O)) contains the common generic fiber G (9.3.2). We know that G[x,y]

kernel of the composite π of the natural homomorphisms G (O) → G (f) → 0 0 G(f), and the image of the special fiber G [x,y] of G[x,y] in G is a parabolic

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f-subgroup Q (Theorem 8.4.19). Let Q be the image of Q in G. Then Q is a parabolic f-subgroup of G. The intersection P ∩ Q of two parabolic f-subgroups of G contains a maximal f-split torus S of G ([Bor91, Proposition20.7(i)]). By conjugacy of maximal f-split tori of P, there exists an element h ∈ P(f) such that h S h−1 = S. Now as P(o) → P(f) is surjective, there exists an h ∈ P(o)(⊂ G (o)) that lies over h. Let S  := hS  h−1 ; S  is a closed o-split torus of G and its special fiber maps isomorphically onto S. Let μ be the conjugate of λ under h. Then μ is a 1-parameter subgroup of S . It is obvious that under the homomorphism π : G (O) → G(f), S (O) is mapped onto S(f)(⊂ Q(f)). Now as the kernel 0 (O), we conclude that S  (O), so also μ(G (O)), of π is contained in G[x,y] m 0 (O)(⊂ G 0 (O)), and hence the 1-parameter subgroup μ is is contained in G[x,y] x compatible with x. Since h ∈ P(o)(⊂ P(k)), limt→0 μ(t) X exists and is 0.  Proposition 13.6.4 ([MP94, Proposition 4.4]) We will use the notation introduced in 13.6.2 and Proposition 13.6.3. There is a 1-parameter k-subgroup ν : Gm → S(⊂ G) compatible with x and an element p ∈ Px such that: (1) limt→0 ν(t)(pX) = 0,   0. (2) for all ψ ∈ Δx , ψ(ν) Proof Let the 1-parameter k-subgroup μ : Gm → G be as in the preceding proposition. By the conjugacy of maximal f-split tori in G x under G x (f) and the fact that the f-Weyl group W(G x , S) acts transitively on the set of W(G x , S)Weyl chambers in X∗ (S), we conclude using Proposition 8.2.1(4) that there exists a p ∈ Px such that the conjugate ν of the 1-parameter subgroup μ, under p, takes values in S and its image lies in the positive Weyl chamber of X∗ (S),  i.e., ψ(ν)  0 for all ψ ∈ Δx . Since ν(t)(pX) = pμ(t)(X), we see that limt→0 ν(t)(pX) = 0.  Definition 13.6.5 We say that an element X of g or g∗ is nilpotent if there is a 1-parameter subgroup λ : Gm → G defined over k such that limt→0 λ(t) X = 0. Let x be a point of B(G). We say that a coset Ξ in gx,r /gx,r+ or g∗x,r /g∗x,r+ is degenerate if it contains a nilpotent element. Given a degenerate coset Ξ in gx,r /gx,r+ (respectively g∗x,r /g∗x,r+ ) we can write it as X + gx,r+ (respectively X + g∗x,r+ ) with X nilpotent. A coset Ξ as above is called non-degenerate if it is not degenerate. Remark 13.6.6 An element X ∈ g is nilpotent in the sense of Definition 13.6.5 if and only if it belongs to the nilpotent radical of a parabolic k-subalgebra. Hence, such an element is nilpotent in the sense of Jordan decomposition. The converse can fail when the characteristic of k is too small for G, because a

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481

nilpotent element need not belong to the nilpotent radical of any parabolic k-subalgebra. For example, if k is a non-perfect field of characteristic 2 (i.e. the local function field F2 ((t))), and x ∈ k is a non-square, then the elements " # " # 1 x 0 1 and x −1 1 0 0 of the Lie algebra of SL2 (k) are nilpotent, but not SL2 (k)-conjugate to each other. Elements that are brought to 0 by a 1-parameter k-subgroup of SL2 are always conjugate to the second element, as one sees by a direct matrix computation. Proposition 13.6.7 ([MP94, Proposition 6.3]) Let Ξ = X + gx,r+ be a degenerate coset in gx,r /gx,r+ with X nilpotent. Then there is an optimal point y such that for some s > r, gy, s ⊃ Ξ. Moreover, Ξ considered as an element of gx,r /gx,r+ is “unstable” under the action of G x , i.e., the closure of the G x –orbit through it contains 0. Proof According to Proposition 13.6.4, there are p ∈ Px and a 1-parameter k-subgroup ν : Gm → S, compatible with x, such that limt→0 ν(t)(pX) = 0. Let pX = Y0 + Yψ , with Y0 ∈ zr and Yψ ∈ uψ , and the summation is over a set of absolute affine roots ψ with distinct derivatives such that ψ(x)  r. Let S be the set of absolute affine roots ψ such that ψ(x) = r and Yψ  muψ . Then  > 0. since limt→0 ν(t)(pX) = 0, we see that Y0 = 0 and for all ψ ∈ S, ψ(ν)  For any affine root ψ and a real number , ψ( ν + x) =  ψ(ν) + ψ(x). So, for all positive  and all affine roots ψ ∈ S, ψ( ν + x) > r. On the other hand, if ψ(x) > r, then for all sufficiently small positive real , ψ( ν + x) > r. Thus we see that if  is chosen to be a suitably small positive real number and we set x  =  ν + x, then pΞ = pX + gx,r+ is contained in gx , s for some s > r. We set y  = p−1 x , then gy , s ⊃ Ξ. Now let y be an optimal point such that gy, s ⊃ gy , s (Lemma 13.4.4). We will now prove the second assertion. For this, we use Proposition 13.6.3 to find a 1-parameter k-subgroup μ : Gm → G, compatible with x, such that limt→0 μ(t) X = 0. This μ determines a 1-parameter f-subgroup μ : Gm → G x such that limt→0 μ(t)Ξ = 0. This implies that the closure of the G x -orbit through Ξ contains 0. 

13.7 G-Domains in the Lie Algebra g In the following we will use the notation introduced in Chapters 5 and 9. Thus S will denote a maximal k-split torus of G, Φ = Φ(S, G) and Φ+ a positive

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system of roots in Φ. Moreover, z is a special vertex in the apartment A in B(G) corresponding to S and Pz is the parahoric subgroup of G(k) determined by z. Let M = ZG (S) be the centralizer of S in G and let P be the minimal parabolic k-subgroup with Levi factor M determined by Φ+ . Let U be the unipotent radical of P. Then P = M  U. Let P− be the opposite parabolic subgroup containing M, and U − be its unipotent radical. Let g, m, u and u − be the Lie algebras of G, M, U and U − respectively. Then g = u ⊕ m ⊕ u − . Let N denote the set of nilpotent elements in g (see Definitions 13.6.5). Let N  denote the set of elements X ∈ g such that the closure of G(k) · X, in the Hausdorff topology on g induced by the valuation topology on k, contains 0. It is obvious that N ⊂ N . Lemma 13.7.1 Let {pi } be a sequence in Pz and {Xi } be a sequence in g. Then {pi · Xi } converges to 0 if and only if {Xi } converges to 0. Proof As Pz is bounded, it is obvious that if {Xi } converges to 0, then so does {pi · Xi }. To prove the converse, let us assume that {pi · Xi } converges to 0. Then as Xi = p−1 i · pi · Xi and {pi · Xi } converges to 0, we see that the sequence  {Xi } converges to 0. Lemma 13.7.2 [AD02, Lemma 2.5.1] If either k is perfect or it is locally compact (i.e., it is complete and its residue field f is finite), then N = N . Moreover, if k is locally compact, then N is closed in g. Proof We first assume that k is perfect. Let X ∈ N . Then the Zariski closure of G(k) · X contains 0 and then by a well-known result of Kempf [Kem78, Corollary 4.3] there exists a 1-parameter subgroup λ ∈ Homk (Gm, G) such that limt→0 λ(t) · X = 0, and hence, by the definition of N, X belongs to it. This shows that if k is perfect then N = N . Now let us assume that k is locally compact. Then Pz is compact. We will show that N  = Pz · u − . As u − is closed in g, this would imply that N  = Pz · u − is closed. Moreover, since u − is contained in N, this would also imply that N  = N. Let X be an element of N , that is, the subset G(k) · X of g contains 0 in its closure. Then there exists a sequence {gi } in G(k) such that the sequence {gi · X } converges to 0. Using the Cartan decomposition (Theorem 5.2.1) we write gi = pi mi qi , with pi , qi ∈ Pz and mi ∈ Z, where Z is as in (5.2.1). Then the sequence {mi qi · X } converges to 0 (Lemma 13.7.1). Let Xi := qi · X = Yi + Zi + Yi− , with Yi ∈ u, Yi− ∈ u − and Zi ∈ m. Note that the adjoint action of the elements of Z on u is “non-contracting” and the adjoint action of M(k) on m factors through the compact quotient M(k)/S(k). Hence, the sequence {mi · Xi } can converge to 0, only if Yi → 0 and Zi → 0. Now since Pz is compact, there exists a subsequence of {qi } that converges, say to

13.7 G-Domains in the Lie Algebra g

483

the element q ∈ Pz . Then a subsequence of {Xi } converges to q · X. On the other hand, any convergent subsequence of {Xi = Yi + Zi + Yi− } converges to an element of u − since {Yi− } ⊂ u − . Hence, q · X ∈ u − , and so X ∈ q−1 · u − .  Lemma 13.7.3 [AD02, Lemma 3.2.1] Let x, y ∈ B(G). Then for all r ∈ R, gy,r ⊂ gx,r + N. Proof Without loss of generality we can assume that both x and y lie in A. Let V = R ⊗Z X∗ (S ), where S  is the maximal subtorus of S contained in the derived group G  of G. Then the apartment A is an affine space under V. Let v ∈ V be the vector such that x = v + y. Let n (respectively n − ) be the span of root spaces corresponding to roots a such that a(v) > 0 (respectively a(v) < 0) and z be the span of the Lie algebra of the centralizer of S and the root spaces corresponding to roots a such that a(v) = 0. Then n and n − are the Lie algebras of unipotent radicals of (opposite) parabolic k-subgroups, and hence both are contained in N. As g = n ⊕ z ⊕ n −, we have gy,r = (n ∩ gy,r ) ⊕ (z ∩ gy,r ) ⊕ (n − ∩ gy,r ) ⊂ (n ∩ gy,r ) ⊕ (z ∩ gy,r ) + N. It is obvious that (n ∩ gy,r ) ⊕ (z ∩ gy,r ) ⊂ gx,r , hence gy,r ⊂ gx,r + N.



In the following proposition, x is a point of the apartment A of B(G) corresponding to the maximal k-split torus S of G. The o-split torus of Gx0 with generic fiber S will be denoted by S . The special fiber of S will be denoted by 0

S ; S is a maximal f-split torus of G x . The image S of S in G x is a maximal f-split torus of the latter. 0

We recall that the natural homomorphisms Px = Gx0 (o) → G x (f) and

0 G x (f)

→ G x (f) are surjective.

Proposition 13.7.4 [MP94, Proposition 6.4] Let Ξ be a coset in gx,r /gx,r+ . (1) If for some y ∈ B(G) and s > r, Ξ ∩ gy, s  ∅, then Ξ is degenerate. (2) Ξ considered as an element of gx,r /gx,r+ is “unstable” under the action of G x , that is, the G x -orbit through Ξ contains 0 in its closure, if and only if Ξ is degenerate. Proof (1) Let us assume that for some s > r and y ∈ B(G), Ξ ∩ gy, s  ∅. Since s > r, by the preceding lemma, gy, s ⊂ gx, s + N ⊂ gx,r+ + N. So since Ξ ∩ gy, s  ∅, we see that Ξ is contained in gx,r+ + N. This implies that Ξ contains a nilpotent element and hence it is degenerate. (2) We have already shown (Proposition 13.6.7) that if Ξ is degenerate, then it is unstable under the action of G x on gx,r /gx,r+ . To prove the converse, let us assume that Ξ is unstable. Then using [Kem78, Corollary 4.3] we find

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a 1-parameter f-subgroup λ : Gm → G x such that limt→0 λ(t)Ξ = 0. Using the conjugacy of maximal f-split tori in G x under G x (f), and the surjectivity of the homomorphism Px → G x (f), we see that there is an o-group scheme homomorphism μ : Gm → S (i.e., a 1-parameter subgroup of S ) and p ∈ Px such that the conjugate of μ under p−1 lies over λ. Arguing as in the proof of Proposition 13.6.7, we see that for a sufficiently small positive real , if we take y = p−1 ( μ + x), then for some s > r, Ξ ⊂ gy, s . Assertion (1) now implies that Ξ is degenerate.   Definition 13.7.5 For r ∈ R, we define gr = x ∈B(G) gx,r , and also gr+ =    ∗ ∗ ∗ ∗ x ∈B(G) g x,r+ , g r = x ∈B(G) g x,r , and g r+ = x ∈B(G) g x,r+ . It is clear that the first two are open subsets of g and both are stable under the adjoint action of G(k) on g, and the last two are open subsets of g∗ and they are stable under the coadjoint action of G(k) on g∗ .  By Lemma 13.4.4, we see that gr is gx,r where the union is over all optimal points x as defined in §13.4. It also follows then that gr = gr+ unless r is an optimal number (as in §13.4). Lemma 13.7.6

Assume that k is locally compact.

(1) For every r ∈ R and x ∈ B(G), gr ⊂ gx,r + N. (2) N = ∩r ∈R gr . Proof (1) We know from Lemma 13.7.3 that for all y ∈ B(G), gy,r ⊂ gx,r +N.  Hence, gr = y ∈B(G) gy,r ⊂ gx,r + N. (2) Suppose X ∈ N(= N ), then the closure of the G(k)-orbit of X under the adjoint action on g contains 0, so the orbit intersects gx,r for all x ∈ B(G) and all r ∈ R. This implies that X lies in ∩r ∈R gr ; thus we have shown that N ⊂ ∩r ∈R gr . On the other hand, since gr ⊂ gx,r + N for a given x ∈ B(G) and all r ∈ R, we see that ∩r ∈R gr ⊂ ∩r ∈R (gx,r + N). But since N is closed (Lemma 13.7.2), and the gx,r constitute a neighborhood base at 0, we obtain  that ∩r ∈R (gx,r + N) = N. Hence, N ⊃ ∩r ∈R gr . Thus the lattices in the collection {gr |r ∈ R} form a neighborhood base of N consisting of open sets that are stable under the adjoint action of G(k). Following [AD02], we now define the depth d(X) of X ∈ g to be ∞ if X is nilpotent, otherwise d(X) is the optimal number r such that X ∈ gr − gr+ . Before we prove the next proposition, we observe that if k is locally compact, for all x ∈ B(G) and r ∈ R, gx,r + N is closed, since gx,r is compact and N

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485

is closed (Lemma 13.7.2). Now the following proposition implies that for all r ∈ R, gr is closed. Proposition 13.7.7 Assume that k is locally compact. For all r ∈ R, gr = ∩x ∈B(G) (gx,r + N), so gr is closed. Proof Lemma 13.7.6(1) implies that the left-hand side is contained in the right-hand side. Suppose X belongs to the right-hand side but not to gr , then d(X) < r. Hence, there is a point x ∈ B(G) such that X ∈ gx,d(X) . But X also belongs to gx,r + N(⊂ gx,d(X)+ + N), so we see that the coset X + gx,d(X)+ is a degenerate coset. According to Proposition 13.6.7, there exists a point y ∈ B(G) and a real number s > d(X) such that the coset X + gx,d(X)+ is contained in gy, s .  This implies that X ∈ gy, s , so s  d(X); a contradiction. Definition 13.7.8 A non-empty subset of g (respectively g∗ ) is called a Gdomain if it is open, closed, and invariant under the adjoint action (respectively coadjoint action) of G(k). As we have shown above, for every r ∈ R, gr is a G-domain. For r ∈ R, gr∗ is also a G-domain. In fact the analog of all the results proved above for g hold also for g∗ , with almost identical proofs. In the same vein as above, we say that a non-empty subset of G(k) is a G-domain if it is open and closed and is stable under the conjugation action of G(k). It is shown in [AD02] and [DeB02] that for every non-negative real  number r, G(k)r := x ∈B(G) Px,r is a G-domain.

13.8 Vanishing of Cohomology Let x ∈ B(G) and r ∈ R0 . Given a Galois extension /k we can use the embedding B(G) → B(G ) to form the Moy–Prasad filtration subgroup G()x,r . The action of Gal(/k) on B(G ) stabilizes x, and hence also G()x,r . It is often useful to have information about H1 (Gal(/k), G()x,r ), and as well as about Hn (Gal(/k),T()r ) for a k-torus T. In this section we will collect some results on the vanishing of such cohomology groups. The case of a torus and an unramified Galois extension /k is treated in Corollary B.10.14. Proposition 13.8.1 Let k /k be an unramified algebraic extension, x ∈ B(G). Then H1 (Gal(k /k), G(k )x,r ) vanishes for r > 0. If in addition dim(f)  1, then H1 (Gal(k /k), G(k )x,0 ) vanishes.

486

Moy–Prasad Filtrations

Proof Write Gx,r for the smooth integral model with Gx,r (O) = G(K)x,r . According to Lemma 8.1.4, the vanishing of H1 (Gal(K/k), G(K)x,r ) reduces to the vanishing of H1 (Gal(f/f), Gx,r (f)). The latter is automatic when dim(f)  1 due to Lang’s theorem. If r > 0, then the special fiber of Gx,r is a smooth connected unipotent group by Corollary 8.5.12 and the desired vanishing follows from Lemma 2.4.2. The vanishing of H1 (Gal(k /k), G(k )x,r ) follows from that of H1 (Gal(K/k), G(K)x,r ) via the inflation–restriction sequence [Ser79, Chapter VII, §6, Proposition 4].  Before we discuss extensions that are not unramified, as well as higher degrees of cohomology, we prove some preparatory lemmas. Lemma 13.8.2 Let F be a field, V a finite-dimensional F-vector space, and Θ a finite group whose order is not divisible by the characteristic of F, acting linearly on V. Then Hn (Θ,V) = {0} for all n > 0. Proof Let r be the order of Θ. It is known that multiplication by r annihilates Hn (Θ,V) for all n > 0, cf. [Ser97, Chapter I, §2.4, Proposition 9] applied to H = {1}. On the other hand, multiplication by r is an automorphism of V, since it is invertible in F.  Lemma 13.8.3 Let F be a field, F /F a Galois extension, and V a finitedimensional F -vector space with a semi-linear action of Gal(F /F). Then Hn (Gal(F /F),V) = {0} for all n > 0. Proof The semi-linear action of Gal(F /F) on V leads, by Galois descent, to an F-structure, that is, an F-subspace W of V such that W ⊗F F  → V is an isomorphism. Induction on dim(W) reduces to studying Hn (Gal(F /F), F ), which vanishes by the normal basis theorem.  Recall that an inverse system of sets { An } is said to satisfy the Mittag-Leffler condition if for any n there exists m0  n so that for all m > m0 the image of Am → An is independent of m. If the transition maps are all surjective, then the Mittag-Leffler condition is trivially satisfied. One sees at once that if the inverse system satisfies the Mittag-Leffler condition and each An is non-empty, then lim An is non-empty. ←−− The following lemma is well known for abelian coefficients, cf. [NSW08, Theorem 2.7.5]. The case of non-abelian coefficients is proved similarly, and we give a simple proof below for the convenience of the reader. Lemma 13.8.4 Let Θ be a finite group and let An be an inverse system of (not necessarily abelian) groups given with Θ-action and surjective transition maps. Let A = lim An . For each x ∈ Z 1 (Θ, A) let x An and x A be the groups An ←−−n

13.8 Vanishing of Cohomology

487

and A with the Θ-action twisted by x = (xn ), so that x A = lim x An . Assume ←−−n that, for each such x, the inverse system H0 (Θ, x An ) satisfies the Mittag-Leffler condition. Then the natural map of sets H1 (Θ, A) → lim H1 (Θ, An ) ←−− is bijective. Proof Assume given zn ∈ Z 1 (Θ, An ) for each n such that the image in H1 (Θ, An ) of the class [zn+1 ] of zn+1 equals the class [zn ] of zn . Starting with n = 1, we can successively modify zn+1 by a coboundary so that the image of zn+1 in Z 1 (Θ, An ) equals zn . We are using here the surjectivity of An+1 → An . Then θ → (zn (θ)) is an element of Z 1 (Θ, A) and [(zn )] ∈ H1 (Θ, A) maps to ([zn ]) ∈ lim H1 (Θ, An ). This proves the surjectivity of the map. ←−− To prove injectivity, assume given x, y ∈ Z 1 (Θ, A) such that their classes [x] and [y] map to the same element of lim H1 (Θ, An ). Replacing An and A with ←−− the groups x An and x A respectively, and y with yx −1 , we may (and do) assume that x = 1. We need to show then that the class of y in H1 (Θ, A) is trivial. Write yn for the composition of y with A → An . We are thus given a sequence of elements hn ∈ An such that yn (θ) = hn−1 θ(hn ). We would like to find an element g = (gn ) ∈ lim An = A such that yn (θ) = gn−1 θ(gn ). For such a putative element ←−− g = (gn ) consider the difference δn = hn gn−1 . It satisfies the two properties: • δn ∈ H0 (Θ, An ); • pn+1, n (δn+1 )δn−1 = pn+1, n (hn+1 )hn−1 , where pm, n : Am → An , are the transition maps in the inverse system. Conversely, a sequence δn ∈ An with these properties would determine g = (gn ) ∈ A = lim An with the desired property by setting gn = δn−1 hn . To find ←−− the sequence (δn ), we consider for each n the set Pn of all truncated sequences (δ1, . . . , δn ), with δr ∈ H 0 (Θ, Ar ) for r  n, that satisfy pr+1,r (δr+1 )δr−1 = pr+1,r (hr+1 )hr−1 for r  n − 1. The map (δ1, . . . , δn ) → δn is a bijection of sets Pn → H0 (Θ, An ). In particular, Pn is non-empty. The projection map Pn → Pn−1 that sends (δ1, . . . , δn ) to (δ1, . . . , δn−1 ) translates to the map H0 (Θ, An ) → H0 (Θ, An−1 ) sending a to γn−1 · pn, n−1 (a), where γn−1 = hn−1 · pn, n−1 (hn−1 ). Therefore, for m > n, the map Pm → Pn translates to the map qm, n : H0 (Θ, Am ) → H0 (Θ, An ) that sends a to γn · pn+1, n (γn+1 ) · pn+2, n (γn+2 ) · · · pm−1, n (γm−1 ) · pm, n (a). We see that if for a fixed n the image of pm, n : H0 (Θ, Am ) → H0 (Θ, An ) stabilizes for all m  m0 , then the image of qm, n stabilizes for all m > m0 . Therefore, the inverse system Pn also satisfies the Mittag-Leffler condition. It

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follows that lim Pn is non-empty. An element of lim Pn is a sequence (δn ) of ←−− ←−− the kind we want. This completes the proof of injectivity.  We now return to the study of Hn (Gal(/k), G()x,r ). Theorem 13.8.5 Assume that k is complete. Let /k be a (not necessarily finite) Galois extension whose ramification degree is finite and not divisible by the characteristic p of f. Let G be a connected reductive k-group, x ∈ B(G), and r ∈ R. Assume that either r  1, or r > 0 and the maximally split maximal torus of G K is weakly induced. Let s > r. (1) H1 (Gal(/k), G()x,r ) and H1 (Gal(/k), G()x,r /G()x, s ) vanish. (2) If G is a torus, then Hn (Gal(/k), G()r ) and H1 (Gal(/k), G()x,r /G()x, s ) vanish for all n > 0. Proof There exists a finite Galois extension   of k in  such that /  is unramified. Let k /k be the maximal unramified subextension of  /k. We will employ the inflation–restriction sequence ([Ser79, Chapter VII, §6, Proposition 4]) multiple times to reduce the proof to two basic cases. When n > 1 we employ induction on n which allows us to assume the vanishing of Hn−1 in order to use the higher inflation–restriction sequence ([Ser79, Chapter VII, §6, Proposition 5]). Let us write Hn (Gal(/k), G()x,r ) with the understanding that n = 1 or G is a torus. First, the vanishing of Hn (Gal(/k), G()x,r ) is reduced to the van

  ishing of Hn (Gal(/ ), G()x,r ) and that of Hn Gal( /k), (G()x,r )Gal(/ ) .

  Now Hn (Gal(/ ), G()x,r ) = lim Hn Gal( / ), (G()x,r )Gal(/ ) as   −−→ traverses the finite Galois subextensions of / . Since /  is unramified   we have (G()x,r )Gal(/ ) = G( )x,r and (G()x,r )Gal(/ ) = G( )x,r from Proposition 9.8.3(2). The vanishing of Hn (Gal( /k), G( )x,r ) is reduced to the

   vanishing of Hn (Gal( /k ), G( )x,r ) and Hn Gal(k /k), (G( )x,r )Gal( /k ) .   According to Proposition 12.9.4, (G( )x,r )Gal( /k ) = G(k )x,r . Therefore, we have reduced the proof of the theorem to the two cases where /k is finite and either unramified or totally tamely ramified. The completeness of k implies the completeness of , which in turn im >0 − {∞} the group plies G()x,r = lim G()x,r /G()x,t . For any r, s ∈ R ←−−t G()x,r /G()x, s , which we will abbreviate by G()x,r:s , has a filtration indexed by r < t < s with subsequent quotients G()x,t:t+ that are non-trivial only for finitely many values of t. According to Theorem 13.5.1 the quotient G()x,t:t+ is functorially isomorphic to a finite-dimensional f -vector space. Fix any r < s. For the finitely many r < t < s for which the quotient

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489

G()x,t:t+ is non-trivial we apply repeatedly Lemma 13.8.2 when /k is totally ramified, and Lemma 13.8.3 when /k is unramified, to conclude that Hn (Gal(/k), G()x,r:s ) vanishes. For a fixed r < t < s consider now the commutative diagram, where we abbreviate G() to G and Hn (Gal(/k), −) to Hn (−): Hn−1 (Gx,t ) O

/ Hn−1 (Gx,r )

/ Hn−1 (Gx,r:t ) O

/ Hn (Gx,t ) O

/ Hn (Gx,r )

Hn−1 (Gx, s )

/ Hn−1 (Gx,r )

/ Hn−1 (Gx,r:s )

/ Hn (Gx, s )

/ Hn (Gx,r ).

We have just argued that Hn (G()x,r:s ) vanishes, from which we conclude that Hn (G()x, s ) → Hn (G()x,r ) is surjective. A simple diagram chase as in the proof of the 5-lemma shows that the map Hn−1 (G()x,r:s ) → Hn−1 (G()x,r:t ) is surjective. In the case that G is not a torus, so that n = 1, we apply this reasoning not just to G, but to any twist of G by an element of Z 1 (Gal(/k), G()x,r ). This allows us to apply Lemma 13.8.4 and its abelian analog to conclude that Hn (G()x,r ) = lim Hn (G()x,r:s ). ←−− s

We have thus proved the vanishing of H1 (Gal(/k), G()x,r ) for any /k as in the statement of the theorem. For the vanishing of Hn (Gal(/k), G()x,r:s ) we use again the same reduction procedure, except now we need to identify with G(k)x,r:s . This identification holds due to the vanishing of G()Gal(/k) x,r:s 1  H (Gal(/k), G()x, s ) just proved.

14 Functorial Properties

Let k be a discretely valued Henselian field with perfect residue field. Let f : G → H be a homomorphism of connected reductive k-groups. Recall the   enlarged buildings B(G) and B(H) introduced in §4.3. Definition 14.0.1

  A map ϕ : B(G) → B(H) is called f -admissible if

 (1) it is f -equivariant, that is, it translates the action of g ∈ G(k) on B(G) to  the action of f (g) ∈ H(k) on B(H); (2) for every maximal k-split torus S ⊂ G there exists a maximal k-split torus  H), and the resulting map  G)) ⊂ A(T, T ⊂ H such that f (S) ⊂ T, ϕ(A(S,   ϕ : A(S, G) → A(T, H) is equivariant for the map V(S) → V(T) induced by f . A map ϕ : B(G) → B(H) is called f -admissible if it satisfies the analogous conditions, “with the tildes removed.” Fact 14.0.2 If f1 : G1 → G2 and f2 : G2 → G3 are two homomorphisms  1 ) → B(G  2 ) is f1 -admissible, between connected reductive k-groups, ϕ1 : B(G  2 ) → B(G  3 ) is f2 -admissible, then ϕ2 ◦ ϕ1 is f2 ◦ f1 -admissible. and ϕ2 : B(G The same holds for maps between the reduced buildings. In this chapter we study the set of f -admissible maps B(G) → B(H) as   well as B(G) → B(H). Results about this problem were originally obtained by Landvogt in [Lan00]. Our treatment is influenced by his ideas, but is somewhat different. Despite the title of this chapter, we do not claim that the Bruhat–Tits building is a functor. What we will show is that an f -admissible map always exists. However, in general there can be many such maps, a phenomenon already encountered in our discussion of Levi subgroups in §9.7. The question of 490

Functorial Properties

491

uniqueness of such maps is not fully understood; we offer some thoughts on this matter in §14.9. As the following example shows, in general one cannot expect an f -admissible map between the reduced buildings, which is why we have brought in the enlarged buildings. We will see however that an f -admissible map between the reduced buildings does exist when f (AG ) ⊂ AH , where AG and AH are the maximal split central tori of G and H, respectively, and that one can reduce the general case to this case. Example 14.0.3 Consider H = SL2 and G the diagonal split torus, with f being the inclusion. Then B(G) is a single point, while B(H) is a regular tree. Giving an f -equivariant map B(G) → B(H) is then equivalent to giving a point of B(H) fixed under G(k). There does not exist such a point.   On the other hand, B(G) = R, while the apartment in B(H) = B(H) corresponding to G is an affine space over R. The choice of a point of the apartment of  G identifies this apartment with R, and hence with B(G), in a G(k)-equivariant way. Lemma 14.0.4 Let f1 : G1 → H1 and f2 : G2 → H2 be two homomorphisms of connected reductive k-groups and let f : G1 × G2 → H1 × H2 be their  1 × H2 ) equals the  1 × G2 ) → B(H product. Each f -admissible map ϕ : B(G   product (ϕ1, ϕ2 ) of fi -admissible maps ϕi : B(Gi ) → B(Hi ). The same holds for maps between the reduced buildings. Proof Let S ⊂ G and T ⊂ H be maximal k-split tori such that f (S) ⊂ T, f (A(S, G)) ⊂ A(T, H), and the map A(S, G) → A(T, H) is equivariant for V(S) → V(T). Then S = S1 × S2 and T = T1 × T2 for maximal k-split tori  G) = A(S  1, G1 ) × A(S  2, G 2 ) Si ⊂ Gi and Ti ⊂ Hi and fi (Si ) ⊂ Ti . We have A(S,  2, H2 ) and ϕ induces an affine map A(S  1, G 1 ) ×  H) = A(T  1, H1 ) × A(T and A(T,    A(S2, G2 ) → A(T1, H1 ) × A(T2, H2 ) whose derivative V(S1 ) × V(S2 ) → V(T1 ) ×

V(T2 ) is induced by f and therefore decomposes as the product of V(Si ) → V(Ti ). This implies that the affine map itself decomposes as the product of affine  i , Hi ) with derivative V(Si ) → V(Ti ). Since this holds  i , Gi ) → A(T maps A(S for all S and T, we see that ϕ decomposes as required.  Lemma 14.0.5 If f is an isomorphism then there exists a unique f -admissible map B(G) → B(H), namely the map induced by transport of structure. Its product with the isomorphism V(AG ) → V(AH ) induced by f is an f -admissible   map B(G) → B(H), and any other such is a translate by V(AH ). Proof It is clear that the map induced by transport of structure is f -admissible. Conversely, if ϕ : B(G) → B(H) is f -admissible, then its restriction to each

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Functorial Properties

apartment is an isomorphism onto its image, because it is affine map whose derivative is an isomorphism. Therefore ϕ is an isomorphism of buildings. If ϕ  is another such, then ϕ  ◦ ϕ−1 is a G(k)-equivariant automorphism of B(H) whose restriction to each apartment has derivative equal to the identity. Therefore ϕ  ◦ ϕ−1 is an isometry for any metric on B(H). Corollary 4.2.21 shows that ϕ  ◦ ϕ−1 is the identity.   Let ϕ : B(G) → B(H) be f -admissible. For x ∈ B(G) and v ∈ V(AG )  2 (x, v)) ∈ B(H) × V(AH ) = B(H). By considering write ϕ (x, v) = ( ϕ1 (x, v), ϕ  individual apartments in B(G) one sees that ϕ is equivariant for the isomorphism V(AG ) → V(AH ) induced by f , and that the image of B(G) × {0} projects to 1 (x, v) = ϕ 1 (x, 0) and ϕ 2 (x, v) = a single point w ∈ V(AH ). This means that ϕ  descends to a map B(G) = ϕ 2 (x, 0) + v = w + v. It further implies that ϕ   B(G)/V(A G ) → B(H)/V(AH ) = B(H) that is still f -admissible and therefore the unique such map ϕ. We conclude ϕ (x, v) = (ϕ(x), w + v).  Recall that the image of the map of topological spaces underlying the map f of group schemes is a closed subspace of the topological space of H, and the image of f is defined as this closed subspace with the reduced scheme structure. We call this image f (G). Then f (G) is a connected reductive k-subgroup of H, in particular it is smooth. The map f factors as the composition of G → f (G) and f (G) → H. The map G → f (G) is a quotient map, that is, faithfully flat, in particular surjective, and the map f (G) → H is a closed immersion, in particular injective. It is therefore enough to treat these two cases separately. The case of a quotient map is rather simple. Most of the work will be involved in the treatment of closed immersions. We impose the following condition on the homomorphism f : let f  : Gsc → f (G)sc be the unique homomorphism lifting G → f (G). For each k s -simple factor (equivalently for each k-simple factor) G  of Gsc , we demand that the restriction of f  to G  is either trivial or an isomorphism onto its image. An example of a map f that fails this property is the Frobenius isogeny SLn → SLn in characteristic p > 0, or the exceptional isogeny SO2n+1 → Sp2n in characteristic 2. On the other hand, if k has characteristic zero, the condition on f is automatically satisfied: the kernel of f  |G is reduced and, since G  is simple, either equal to G  or contained in the center of G ; in the latter case f  induces a central isogeny G  → f (G ) ⊂ f (G)sc , but f (G ) is a smooth closed normal subgroup of the simply connected group f (G)sc , hence itself simply connected, and we deduce that f  |G : G  → f (G ) is an isomorphism.

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14.1 Quotient Maps We consider in this section a quotient map f : G → H, that is, a faithfully flat homomorphism of k-groups. Proposition 14.1.1 There exists a unique f -admissible map B(G) → B(H). It is surjective. Its product with the map V(AG ) → V(AH ) induced by f is an   f -admissible map B(G) → B(H) and any other such is a translate of this by V(AH ). The remainder of this section is devoted to the proof of this proposition. Lemma 14.1.2

The multiplication map induces a central isogeny Z(G)0 × Gsc → G.

Proof Since Z(Gder ) is finite and G = Z(G) · Gder , the natural map Z(G)0 → G/Gder is a homomorphism between tori of the same rank, with finite kernel, and therefore an isogeny, in particular surjective. We conclude that the multiplication map Z(G)0 ×Gsc → G is surjective. Its kernel being finite and central, it is a central isogeny.  Combining the lift fsc : Gsc → Hsc with the map Z(G)0 → Z(H)0 obtained from f by restriction, and the decomposition of both Gsc and Hsc as products of k-simple factors, we obtain the commutative diagram Z(G)0 × Gsc,1 × · · · × Gsc, n

/G

 Z(H)0 × Hsc,1 × · · · × Hsc, m

 /H

Since the bottom horizontal map is an isogeny (as well as the top one) the left vertical map must be surjective, for otherwise its image will have a strictly smaller dimension than that of the bottom left corner, hence also smaller than the dimension of H. Consider now fsc . Its kernel is a closed normal subgroup of Gsc . By our assumption on f this kernel is a product of some of the simple factors of Gsc . Let us arrange the enumeration of these factors so that the kernel of fsc is the product of the factors with indices d + 1, . . . , n for some d. Then fsc induces an isomorphism Gsc,1 × · · · × Gsc, d → Hsc . Since the decomposition of Hsc into simple factors is unique ([Bor91, 14.10]), we see that d = m and, up to rearrangement of the factors of Hsc , fsc induces an isomorphism Gsc,i → Hsc,i for all i = 1, . . . , d.

494

Functorial Properties m n B(Gsc,i ) and B(H) = i=1 B(Hsc,i ). The map fsc We have B(G) = i=1 induces the map B(G) → B(H) that first projects onto the first d factors of B(G), and then, on a factor with index 1  i  d is given by the isomorphism Gsc,i → Hsc,i . That this map is equivariant for f follows from the fact that f induces the map Gad,1 × · · · × Gad, n = Gad → Had = Had,1 × · · · × Had, m that is again given by the projection onto the first d factors of Gad , and then, on a factor with index 1  i  d is given by the isomorphism Gad,i → Had,i induced by the isomorphism Gsc,i → Hsc,i . Since the action of G(k) on B(G) factors through Gad (k), and the analogous statement holds for H, the f -equivariance of B(G) → B(H) follows.   = Turning to enlarged buildings, we have B(G) = B(G)×V(Z(G)0 ) and B(H) 0   → B(H) that on the B(H) × V(Z(H) ). The map f induces the map B(G) factor B(G) is the map just constructed, and on the factor V(Z(G)0 ) is the map V(Z(G)0 ) → V(Z(H)0 ) induced by the map Z(G)0 → Z(H)0 . Recall that the action of G(k) on V(Z(G)0 ) comes from the isomorphism V(Z(G)0 ) → V(G/Gder ) induced by the isogeny Z(G)0 → G/Gder . The map Z(G)0 → Z(H)0 induced by f is translated under these isomorphisms for G and H to the map G/Gder → H/Hder again induced by f . The f -equivariance of V(Z(G)0 ) →   → B(H), follows. V(Z(H)0 ), and hence of B(G) In both cases, part (2) of Definition 14.0.1 follows immediately from the construction. We have thus proved existence of an f -admissible map, and turn to uniqueness. If ϕ : B(G) → B(H) is f -admissible, it is also admissible for fsc : Gsc → Hsc . Lemma 14.0.4 implies that ϕ decomposes as the product of ϕi , where ϕi is admissible for fsc(i) : Gsc,i → Hsc,i and we have arranged the k-simple factors of Gsc and Hsc so that fsc(i) is an isomorphism for i = 1, . . . , d and Hsc,i = {1} for i = d + 1, . . . , n. Lemma 14.0.5 implies the uniqueness of ϕi and we con  clude the uniqueness of ϕ. The claim about f -admissible maps B(G) → B(H) follows from the same argument as in the proof of Lemma 14.0.5.

14.2 Embeddings: Isometric Properties We now come to the case of a closed immersion f : G → H. Thus let H be a connected reductive k-group and G ⊂ H a connected reductive k-subgroup. In   Definition 14.0.1 we defined the notion of an f -admissible map B(G) → B(H). Since f is now the natural inclusion, we will drop it from the notation. In fact,

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495

as the next lemma shows, such a map is always injective. To emphasize this fact, we will refer to an f -admissible map as an admissible embedding. Lemma 14.2.1

  Any admissible map ϕ : B(G) → B(H) is injective.

 Choose a maximal k-split torus S ⊂ G such that Proof Let x1  x2 ∈ B(G).  x1, x2 ∈ A(S, G) and choose a maximal k-split torus T ⊂ H as in Definition  G)) ⊂ A(T,  H) and the 14.0.1(2). Then 0  x2 − x1 ∈ V(S). Since ϕ(A(S,  G) → A(T,  H) is equivariant for the inclusion V(S) → resulting map ϕ : A(S, V(T), we have ϕ(x2 ) = ϕ(x1 + (x2 − x1 )) = ϕ(x1 ) + (x2 − x1 )  ϕ(x1 ).



We now discuss the isometric properties of admissible embeddings. Recall from §4.3 that a compatible system of metrics on V(S) for all maximal k-split tori S ⊂ G is a choice of a Euclidean metric on V(S) for each S such that for each g ∈ G(k) the isomorphism V(S) → V(gSg −1 ) provided by conjugation by g is an isometry. Such a compatible system is specified by fixing a Weyl groupinvariant Euclidean metric on V(S) for one single S ⊂ G and then transporting it to V(S1 ) for all other S1 ⊂ G. Recall further that such a compatible system  endows B(G) with a metric, and the compatible system can be recovered from  the metric on B(G). Proposition 14.2.2

 Assume chosen a metric on B(H).

(1) Let S and T be maximal k-split tori in G and H, respectively, such that S ⊂ T. The restriction to V(S) of the metric on V(T) depends only on S, and not on T. (2) The resulting metrics on V(S) for all maximal k-split tori S ⊂ G form a compatible system.  (3) If we endow B(G) with the corresponding metric, then any admissible   embedding B(G) → B(H) is an isometry. Proof (1) Let T1 be another maximal k-split torus of H that contains S. Since S is contained in both T and T1 , there exists h ∈ ZS (H)(k) such that T1 = hT h−1 . Conjugation by h is an isometry V(T) → V(T1 ) that commutes with the inclusions of V(S) into V(T) and V(T1 ). (2) Let S and T be as in (1) and let g ∈ G(k). Then Ad(g) : V(S) → V(gSg −1 ) is the restriction of Ad(g) : V(T) → V(gT g −1 ), but the latter is an isometry.   (3) Let ϕ : B(G) → B(H) be an admissible embedding. To check that it is an isometry it is enough to check that the restriction of ϕ to a given apartment  G) is an isometry, because any two points of B(G)  A(S, are contained in a common apartment. By Definition 14.0.1(2) there exists a maximal k-split torus  G)) ⊂ T ⊂ H containing the maximal k-split torus S ⊂ G such that ϕ(A(S,    A(T, H) and the resulting map A(S, G) → A(T, H) is an affine map with

496

Functorial Properties

derivative map the natural inclusion V(S) → V(T). Since this inclusion is  G) and an isometry by construction and the metrics on the affine spaces A(S,  A(T, H) are obtained from the metrics on their translation spaces, the claim follows.    It may be useful to refer to the metrics on B(G) and B(H) as in Proposition 14.2.2 as compatible.

14.3 Embeddings: Factorization through a Levi Subgroup Let AG and AH be the maximal k-split tori in the centers of G and H. In general AG is not a subgroup of AH and this precludes the existence of admissible embeddings B(G) → B(H), as we saw in Example 14.0.3, necessitating the use of the enlarged buildings. In this section we will see that the general case reduces to the case in which AG ⊂ AH , and to the construction of admissible embeddings B(G) → B(H) in that case. Let M = Z H (AG ). Then M is a Levi subgroup of H that contains G. In particular, we have a the natural inclusion ι : AG → AM . Given an admissible embedding ϕ : B(G) → B(M) we can compose   ϕ × ι : B(G) = B(G) × V(AG ) → B(M) × V(AM ) = B(M)   with an admissible embedding ψ : B(M) → B(H). Recall that the latter were classified in §9.7, see especially 9.7.5.   Proposition 14.3.1 Let ϕ : B(G) → B(M) and ψ : B(M) → B(H) be admissible embeddings. The composition of ϕ × ι with ψ is an admissible embedding     B(G) → B(H). Every admissible embedding B(G) → B(H) arises this way from a unique pair (ϕ, ψ). For the proof we will need the following lemma.   Lemma 14.3.2 Let ϕ : B(G) → B(M) be an admissible embedding. The   and the projection B(M) → composition of ϕ with the inclusion B(G) → B(G) V(AM ) is constant.  Proof The embedding ϕ is G(k)-equivariant, the projection B(M) → V(AM )  is M(k)-equivariant, and the inclusion B(G) → B(G) is Gder (k)-equivariant. The action of M(k) on V(AM ) is via the negative of the valuation homomorphism for M. The latter is functorial (Lemma 2.6.16) and we conclude that restricting this action to G(k) we obtain the action via the negative of the valuation homomorphism ωG : G(k) → V(AG ) composed with the embedding

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497

V(AG ) → V(AM ). This homomorphism restricts trivially to Gder (k). Therefore, the map B(G) → V(AM ) is Gder (k)-invariant. Since all apartments of B(G) are conjugate under Gder (k), it is enough to show that the image in V(AM ) of a single apartment of B(G) is a point. Let S ⊂ G be a maximal k-split torus and let A(S, G) ⊂ B(G) be the  G) is a single orbit unassociated apartment. The subset A(S, G) ⊂ A(S,   der V(S ), where S is the largest torus contained in S ∩ Gder . Let T ⊂ M  G)) ⊂ A(T,  M) = be a maximal k-split torus containing S such that ϕ(A(S,  G) is translated via ϕ to the A(T, M) × V(AM ). The action of V(S ) on A(S,  M). Recall that the decomporestriction to V(S ) of the action of V(T) on A(T,  sition V(T) = V(T ) ⊕ V(AM ) comes from the isogeny T  × AM → T, where T  is the maximal torus in T ∩ Mder . The projection V(T) → V(AM ) is effected by the projection T → T/T  and the isomorphism V(AM ) → V(T/T ) induced by the isogeny AM → T/T . The inclusion S → T maps S  into T  and therefore induces the trivial map V(S ) → V(T) → V(AM ). We conclude that the  G) projects the subset A(S, G) onto a single point of restriction of ϕ to A(S,  V(AM ). Proof of Proposition 14.3.1 That ψ ◦ϕ is admissible follows from Fact 14.0.2.  Since the images of the subset B(M) of B(M) under the various choices of ψ are all disjoint, cf. 9.7.5, we see that ψ ◦ ϕ  ψ  ◦ ϕ if ψ  ψ . Since ψ is injective, we see that ψ ◦ ϕ  ψ ◦ ϕ  if ϕ  ϕ .   Given an admissible embedding η : B(G) → B(H), Definition 14.0.1(2)  implies that the image of every apartment of B(G) is contained in an apartment  of B(H) corresponding to a maximal k-split torus of H that contains AG . Thus    the image of B(G) under η is contained in B(H) AG . Note that B(H) AG =  B(H) A M , because any maximal torus of H that contains AG is contained  in M and hence contains AM . According to 9.7.5, B(H) A M is the common   image of all admissible embeddings B(M) → B(H), so we may choose one   such ψ and factor η as ψ ◦ ϕ, where ϕ : B(G) → B(M) is an admissible embedding. Applying Lemma 14.3.2 we find v ∈ V(AM /AG ) such that ϕ + v is  an admissible embedding mapping B(G) into the subset B(M) of B(M). Then η = (ψ − v) ◦ (ϕ + v). 

14.4 Embeddings of Apartments We continue with the notation established in the first paragraph of §14.3. Following Proposition 14.3.1 we will seek to construct an admissible embedding

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Functorial Properties

B(G) → B(M). In this section we take the first step, by considering a single apartment in B(G). Let S be a maximal k-split torus of G and let A(S, G) be the corresponding apartment in B(G). Recall that A(S, G) is an affine space under the vector space V(S/AG ). We have (S ∩ AM )0 = AG . Let B(M)S be the union of those apartments that correspond to maximal ksplit tori of M that contain S. Recall from Proposition 9.7.2(1) that there is an action of V(S/AG ) on B(M)S whose restriction to an apartment A(T, M) equals the restriction to V(S/AG ) of the natural translation action of V(T/AM ) on A(T, M). The action of NM (S)(k) on B(M) preserves B(M)S and is compatible with the action of V(S/AG ), in the sense that the two actions glue to an action of V(S/AG )  NM (S)(k). In particular, NM (S)(k) acts on B(M)S /V(S/AG ) and the subgroup S(k) acts trivially. Restricting this action to NG (S)(k) we obtain an action of NG (S)(k)/S(k) on B(M)S /V(S/AG ). Recall from 9.7.5 that B(M)S /V(S/AG ) can be identified with the enlarged building of Z M (S)/(S · AM ). We consider embeddings ϕ : A(S, G) → B(M)S that are V(S)  NG (S)(k)equivariant. Explicitly, this means that there exists a maximal k-split torus T of M such that S ⊂ T and ϕ(A(S, G)) is contained in A(T, M), and the resulting embedding ϕ : A(S, G) → A(T, M) is an affine map whose derivative is the embedding V(S/AG ) → V(T/AM ). Proposition 14.4.1 (1) The map that sends a V(S/AG )  NG (S)(k)-equivariant embedding ϕ : A(S, G) → B(M)S to its image is a bijection between the set of such embeddings and the set [B(M)S /V(S/AG )] NG (S)(k) . (2) Given y ∈ [B(M)S /V(S/AG )] NG (S)(k) let ϕS, y : A(S, G) → B(M)S denote the corresponding embedding. The assignment (S, y) → ϕS, y is G(k)-equivariant. Proof (1) Given ϕ, its image is contained in B(M)S and is a single orbit under V(S/AG ), and therefore projects to a point of B(M)S /V(S/AG ). Since the image of ϕ is invariant under NG (S)(k), the corresponding point is fixed by this action. Conversely, assume given a point y ∈ B(M)S /V(S/AG ) that is fixed by NG (S)(k). Let Y ⊂ B(M) be its preimage. There exists a maximal k-split torus T ⊂ Z M (S) such that the apartment of T/(S AM ) in B(M)S /V(S/AG ) =  M (S)/S AM ) contains y. Therefore the apartment of T in B(M) contains Y . B(Z We now check that Y satisfies the axiomatic properties of the apartment A(S, G) as laid out in Axiom 4.1.4. This will provide a canonical isomorphism of affine spaces A(S, G) → Y , whose composition with the inclusion Y → B(M)

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499

will be the map ϕ. By construction Y is an affine space under V(S/AG ). The action of NG (S)(k) on B(M) preserves Y . We claim that this action is by affine transformations, and its derivative is the Weyl action of NG (S)(k) on V(S/AG ). To see this, let n ∈ NG (S)(k) and choose a maximal k-split torus T of M containing S such that Y ⊂ A(T, M). Then n : A(T, M) → A(nT n−1, M) is an affine transformation whose derivative is Ad(n) : V(T/AM ) → V(nT n−1 /AM ). The point y¯ is fixed by n and hence contained in both A(T, M)/V(S/AG ) and A(nT n−1, M)/V(S/AG ). Therefore Y is contained in both A(T, M) and A(nT n−1, M), and n : A(T, M) → A(nT n−1, M) preserves Y . At the same time, we have the commutative diagram

V(T) O

/ V(nT n−1 ) O

Ad(n)

ι S , nT n−1

ιS , T

V(S)

Ad(n)

/ V(S)

where ιS,T is the map V(S) → V(T) induced by the inclusion S → T, and ιS, nT n−1 is the analogous map. This proves the claim. The claim implies that the action of ZG (S)(k) on Y has trivial derivative, and is thus given by translations. Let ν : ZG (S)(k) → V(S/AG ) be the corresponding homomorphism. The subgroup ZG (S)(k)1 acts on Y and the Bruhat– Tits fixed point theorem for B(M) implies that this group fixes a point of Y . Since the actions of ZG (S)(k) and V(S/AG ) on B(M)S commute, we conclude that ZG (S)(k)1 fixes every point of Y . Therefore, ν induces a homomorphism ZG (S)(k)/ZG (S)(k)1 → V(S/AG ). The image of S(k) in the quotient ZG (S)(k)/ZG (S)(k)1 is of finite index. The action of S(k) on A(T, M) is via translations by −ωS : S(k) → V(S/AG ) → V(T/AM ). The functoriality of the valuation homomorphism shows that ν = −ω ZG (S) . We have thus established that Y satisfies the axiomatic properties of the apartment A(S, G). (2) Let g ∈ G(k). Given y ∈ [B(M)S /V(S/AG )] NG (S)(k) let again Y ⊂ B(M)S be its preimage, so that ϕS, y : A(S, G) → Y is the unique NG (S)(k)equivariant isomorphism of affine spaces over V(S/AG ). It is clear that Ad(g) ◦ ϕS, y ◦ Ad(g)−1 is an NG (gSg −1 )(k)-equivariant isomorphism A(gSg −1, G) → gY of affine spaces over V(gSg −1 /AG ). But so is ϕgSg−1 ,gy and the uniqueness property of this isomorphism implies that the two isomorphisms are equal. 

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14.5 Adapted Points: Definition and Properties We continue with the notation established in the first paragraph of §14.3. Recall   that there is an embedding B(G) → B(G) and a retraction B(G) → B(G).  Given  x ∈ B(G) we will write x ∈ B(G) for the image of  x under this retraction. Then G(k)x = G(k)1x . We employ the analogous notation for H. The following notion is the key to the study of admissible embeddings   B(G) → B(H). We will spend this section studying its properties, and will apply it to the study of admissible embeddings in §14.6. Definition 14.5.1 A pair (x, y) of points x ∈ B(G) and y ∈ B(M) is called adapted if for each maximal k-split torus S ⊂ G with x ∈ A(S, G) (1) y ∈ B(M)S and the image y ∈ B(M)S /V(S/AG ) is NG (S)(k)-fixed, (2) ϕS, y¯ (x) = y, where ϕS, y is as in Proposition 14.4.1, (3) G(k)1x+v ⊂ M(k)1y+v for all v ∈ V(S/AG ).   A pair ( x,  y ) of points  x ∈ B(G) and  y ∈ B(H) is called adapted if y ∈ B(H) AG and the pair (x, y M ) is adapted, where y M ∈ B(M) is the image of y under the projection B(H) AG = B(H) A M → B(M). Remark 14.5.2 While the definition of ( x,  y ) being adapted is in terms of (x, y M ), one can rephrase conditions (1) and (3) more directly in terms of  ( x,  y ). Indeed, using the M(k)-equivariant isomorphism B(H) S /V(S · A M ) → y in B(M)S of §9.7, we see that condition (1) is equivalent to the image of   B(H) /V(S · A ) being fixed by N (S)(k). In turn, this is equivalent to the S M G  /V(S) is fixed by seemingly stronger condition that the image of  y in B(H) S  NG (S)(k). To see this, note that the set of all points of B(H)S /V(S) having the  same image in B(H) S /V(S · A M ) is a torsor for the action of V(A M /AG ) on  B(H) /V(S). If that image is fixed, then this torsor is invariant under NG (S)(k). S The subgroup S(k) acts trivially, and NG (S)(k)/S(k) is bounded, so Corollary 4.3.6 implies that this group has a fixed point in this torsor. Since the action of V(AM /AG ) commutes with the action of NG (S)(k)/S(k), we conclude that every point in this torsor is fixed. Condition (3) is equivalent to G(k)x +v ⊂ H(k)y +v for all v ∈ V(S), according to Corollary 9.7.3 and Lemma 4.3.3. We will see in Lemma 14.5.4 that condition (2) is implied by conditions (1) and (3) when x is special. Lemma 14.5.3 Let x ∈ B(G) and y ∈ B(M). Assume that there exists a maximal k-split torus S ⊂ G with x ∈ A(S, G) so that (1)–(3) of Definition 14.5.1 are satisfied. Then (x, y) is a pair of adapted points.

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501

Proof We need to show that (1)–(3) of Definition 14.5.1 are satisfied for any other maximal k-split torus S1 ⊂ G with x ∈ A(S1, G). Fix h ∈ G(k)1x such that S1 = hSh−1 . (1) Then y = hy lies in hB(M)S = B(M)S1 . The action of h on B(M) induces a bijection B(M)S /V(S/AG ) → B(M)S1 /V(S1 /AG ) that is equivariant for the bijection Ad(h) : NG (S)(k) → NG (S1 )(k) and maps the image of y in B(M)S /V(S/AG ) to the image of y in B(M)S1 /V(S1 /AG ). Therefore the latter is NG (S1 )(k)-fixed. (2) According to Proposition 14.4.1(2) we have hϕS, y (z) = ϕhSh−1 ,hy (hz). But hSh−1 = S1 and we have just argued that hy = y. Therefore ϕS1 ,y (x) = y due to G(k)1x ⊂ M(k)1y . (3) We have V(S1 /AG ) = hV(S/AG ). For any v ∈ V(S/AG ) we have that  G(k)1x+hv = hG(k)1x+v h−1 ⊂ hM(k)1y+v h−1 = M(k)1y+hv . Lemma 14.5.4 Let x ∈ B(G) and y ∈ B(M) be such that x is special, G(k)1x ⊂ M(k)1y and for some S ⊂ G with x ∈ A(S, G), y belongs to B(M)S and its image y ∈ B(M)S /V(S/AG ) is fixed by ZG (S)(k). Then y is fixed by NG (S)(k) and ϕS, y (x) = y. Proof Since x is special, every element of NG (S)(k) is the product of an element of ZG (S)(k) and an element of NG (S)(k) ∩ G(k)1x . Therefore, every element of NG (S)(k) preserves the subset y + V(S/AG ) of B(H)S , that is, y is fixed under NG (S)(k). Again since x is special, it is the unique point of A(S, G) fixed by the group NG (S)(k) ∩ G(k)1x . Therefore ϕS, y (x) is the unique point of the image of ϕS, y that is fixed by NG (S)(k) ∩ G(k)1x . By assumption y is fixed by all of G(k)1x , so  we conclude y = ϕS, y (x). Corollary 14.5.5 Let x ∈ B(G) and y ∈ B(M) be a pair of adapted points and let S ⊂ G be a maximal k-split torus such that x ∈ A(S, G). For v ∈ V(S/AG ), the pair of points (x + v, y + v) is adapted, where y + v is the translation of y by v within B(M)S . Proof Write x  = x + v and y  = y + v. We will check the parts of Definition 14.5.1 for (x , y ) in terms of that for (x, y). According to Lemma 14.5.3 it is enough to check (1)–(3) with respect to the chosen maximal k-split torus S. But (1) is the same for y and y , since y = y , (2) follows from the V(S/AG ) equivariance of ϕS, y , and (3) is again the same for (x, y) and (x , y ).   Lemma 14.5.6 Fix  x ∈ B(G). The set of  y ∈ B(H) such that the pair ( x,  y ) is  adapted is a closed convex subset of B(H). Proof

Write x ∈ B(G) for the projection of  x . The set of  y for which ( x,  y ) is

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Functorial Properties

adapted is the preimage of the set of y M ∈ B(M) for which (x, y M ) is adapted  under the projection B(H) AG → B(H) AG → B(M). Therefore it is enough to study the latter set. To ease notation, we will write y in place of y M . Choose a maximal k-split torus S ⊂ G such that x ∈ A(S, G). Then the set of y such that (x, y) is adapted to x lies in B(M)S . According to Corollary 14.5.5, we may assume, after translating this set within B(M)S by a suitable element of V(S/AG ), that x is special. We prove convexity first. Consider y1, y2 ∈ B(M)S such that (x, y1 ) and (x, y2 ) are adapted. Let y ∈ [y1, y2 ]. Since NG (S)(k)/S(k) fixes y1 and y2 , it  M (S)/S), fixes the geodesic [y1, y2 ] in the building B(M)S /V(S/AG ) = B(Z and hence also its point y. Lemma 14.5.4 implies ϕS, y (x) = y. From ϕS, y1 (x) = y1 , ϕS, y2 (x) = y2 , we have ϕS, y1 (x + v) = y1 + v and ϕS, y2 (x + v) = y2 + v for v ∈ V(S/AG ). By assumption G(k)1x+v fixes both y1 + v and y2 + v. Therefore it fixes every point of [y1 + v, y2 + v]. Since translation by v is an isometry of B(H)S we see [y1 + v, y2 + v] = [y1, y2 ] + v. In particular, G(k)1x+v fixes y + v. The proof of convexity is complete. We now prove closedness. Consider a sequence (yn ) ∈ B(M) converging to y ∈ B(M) such that (x, yn ) is adapted for all n. We check the parts of Definition 14.5.1 hold for y. Since B(M)S is closed in B(M) (cf. Proposition 9.3.11 and the paragraph preceding it) and [B(M)S /V(S)] NG (S)(k) is closed in B(M)S /V(S), we conclude that (1) holds. For (2) and (3) we use Lemma 4.2.17 together with the claim that ϕS, yn (z) converges to ϕS, y (z) for all z ∈ A(S, G). To see the latter, the V(S/AG )-equivariance of ϕS, yn and ϕS, y reduces the claim to the case of a single z ∈ A(S, G) that we may freely choose. We choose z = x and the claim follows. 

14.6 Embeddings of Buildings via Adapted Points   Proposition 14.6.1 Let ( x,  y ) be a pair of adapted points in B(G) × B(H). There exists a unique admissible embedding   → B(H) ϕx ,y : B(G) mapping  x to  y . If (ϕ, ψ) is the corresponding pair as in Proposition 14.3.1, the restriction of ϕ to any apartment A(S, G) equals the map ϕS, y . Proof We first construct ϕx ,y by specifying the pair (ϕ, ψ) as in Proposition 14.3.1.

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503

Choose a maximal k-split torus S ⊂ G with x ∈ A(S, G). Define a map ϕ S, y : G(k) × A(S, G) → B(M) by (g, z) → g · ϕS, y (z). For any n ∈ NG (S)(k) S, y (gpn−1, nz) = ϕ S, y (g, z) because ϕS, y is equivariand p ∈ G(k)1z we have ϕ ant under NG (S)(k) and because G(k)1z ⊂ M(k)1ϕ (z) according to Definition S, y 14.5.1(3). Therefore, ϕ S, y is constant on the equivalence classes of the equivalence relation (4.4.2), and Proposition 4.4.4 implies that the map ϕ S, y factors through a map ϕ : B(G) → B(M). We now check that ϕ does not depend on the choice of S. If S1 is another choice, there exists h ∈ G(k)1x such that S1 = hSh−1 . It is enough to S1 ,y (gh−1, hz) for all g ∈ G(k) and z ∈ A(S, G). show that ϕ S, y (g, z) = ϕ The point y belongs to both B(M)S and B(M)S1 and equals hy. Therefore y ∈ B(M)S1 /V(S1 ) is equal to the image of y ∈ B(M)S /V(S) under the map B(M)S /V(S) → B(M)S1 /V(S1 ) induced by the action of h on B(M). Therefore ϕS1 , y = ϕhSh−1 ,hy = h ◦ ϕS, y ◦ h−1 , the last identity by Proposition 14.4.1(2). S1 , y (gh−1, hz) follows. From this the identity ϕ S, y (g, z) = ϕ It is clear from the construction that ϕ is an admissible embedding B(G) → B(M) and that ϕ(x) = y M , where y M ∈ B(M) is the image of y under the x = (x, v) ∈ B(G) × V(AG ). projection B(H) AG = B(H) A M → B(M). Write    There exists a unique admissible embedding ψ : B(M) → B(H) that maps   to  y ∈ B(H), cf. §9.7.5. The admissible (y M , v) ∈ B(M) × V(AM ) = B(M) embedding ϕx ,y corresponding to the pair (ϕ, ψ) as in Proposition 14.3.1 has the desired properties.   We now check uniqueness. Any admissible embedding B(G) → B(H) takes  and is equivariant for the action of V(A ). Therefore we image in B(H) AG G  may freely assume that  x = (x, 0) ∈ B(G) = B(G) × V(AG ). Let (ϕ1, ψ1 ) x to  y . Since ψ1 is an section of be another pair such that ψ1 ◦ ϕ1 sends   y. the projection B(H) A M → B(M) we see that ϕ1 (x) = y M and ψ1 (y M ) =  This already implies ψ1 = ψ. To see ϕ = ϕ1 it is enough to restrict both to the apartment A(S, G). These restrictions are V(S)  NG (S)(k)-equivariant embeddings A(S, G) → B(M) and by Proposition 14.4.1 uniquely determined by their images. But their images are both equal to y M + V(S) ⊂ B(M)S .  Proposition 14.6.2 (1) Let ( x1 ,  y1 ) and ( x2,  y2 ) be two pairs of adapted points. Then ϕx1 ,y1 = ϕx2 ,y2 if and only if for one, hence any, maximal x2 ∈ A(S, G), the points  y1 ,  y2 ∈ B(H)S k-split torus S ⊂ G such that  x1 ,  are in the same V(S)-orbit and elements  x2 −  x1 and  y2 −  y1 of V(S) are equal.   (2) Let ϕ : B(G) → B(H) be an admissible embedding. Then ( x, ϕ ( x )) is a  pair of adapted points for any  x ∈ B(G). Proof

(1) According to Proposition 14.6.1, we have ϕx1 ,y1 = ϕx2 ,y2 if and

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Functorial Properties

x2 ) =  y2 . Choose a maximal k-split torus of G such that  x1 ,  x2 ∈ only if ϕx1 ,y1 (    y2 ∈ B(H)S and the restriction of ϕx1 ,y1 to A(S, G) is a V(S) A(S, G). Then  y1 ,   G) → B(H)  x2 ) = NG (S)(k)-equivariant embedding A(S, S . Therefore, ϕ x1 , y1 ( ϕx1 ,y1 ( x1 + ( x2 −  x1 )) = ϕx1 ,y1 ( x1 ) + ( x2 −  x1 ) =  y1 + ( x2 −  x1 ), showing that x2 ) =  y2 is indeed equivalent to the equality of  x2 −  x1 and  y2 −  y1 in ϕx1 ,y1 ( V(S).   (2) Fix  x ∈ B(G) and let  y = ϕ( x ). Then  y ∈ B(H) AG and since ϕ is V(AG )-equivariant and the notion of adaptedness is also V(AG )-equivariant by  definition, we may assume that  x = (x, 0) in B(G) = B(G) × V(AG ). Use Proposition 14.3.1 to assign to ϕ  a pair (ϕ, ψ). Letting y M ∈ B(M) be the  image of  y under B(H) AG → B(H) AG → B(M) we have ϕ(x) = y M . Since adaptedness of ( x,  y ) is equivalent to adaptedness of (x, y M ) by definition, we focus on ϕ.  G). The restriction of Let S ⊂ G be a maximal k-split torus such that  x ∈ A(S, ϕ to A(S, G) is an embedding A(S, G) → B(M)S as in Proposition 14.4.1, and that Proposition implies that the image of that embedding in B(M)S projects to a single point of B(M)S /V(S/AG ) fixed by NG (S)(k). Since this image contains y M , we see that Definition 14.5.1(1) is satisfied, while (2) and (3) follow from ϕ|A(S,G) = ϕS, y M , the identity ϕ(x) = y M noted above, and the G(k)-equivariance of ϕ. 

14.7 The Space of Embeddings and Galois Descent We continue with the notation established in the first paragraph of §14.3. Let /k be a Galois extension of finite ramification degree.   ) with respect   ) → B(H Let M be the set of all admissible embeddings B(G to the Henselian discretely valued field . The group Θ = Gal(/k) acts on M   ).   ) and B(H by its actions on B(G We define a metric dM on M by setting x ), g( x )) dM ( f , g) = d( f (   ).   ) is arbitrary and on the right we are using the metric of B(H where  x ∈ B(G Proposition 14.7.1 The above definition does not depend on the choice of   ) and defines a metric on M. With respect to this metric, M is a  x ∈ B(G complete metric space of non-positive curvature. The group Θ acts on M with finite orbits. Proof

  ). Choose a maximal -split torus Consider another point  x  ∈ B(G

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 G). The restrictions of f and g to A(S,  G) take S ⊂ G such that  x,  x  ∈ A(S,  image in B(H )S and are V(S)-equivariant, therefore d( f ( x ), g( x )) = d( f ( x ) + ( x −  x ), g( x ) + ( x −  x )) = d( f ( x ), g( x )),   )S is isometric. as the action of V(S) on B(H x) Fix now  x ∈ B(G ). According to Proposition 14.6.2, the map f → f (   ) such that establishes an isometry between M and the set of points  y ∈ B(H ( x,  y ) is adapted. By Lemma 14.5.6 the latter set is a closed convex subset of   ), and thus inherits the properties of being complete and non-positively B(H curved.   ), we see that it acts   ) and B(H Since Θ acts with finite orbits on both B(G with finite orbits on M.   ) → B(H   ), then there Corollary 14.7.2 If there exists an admissible B(G, exists a Θ-equivariant such embedding. Proof

This follows from Proposition 14.7.1 and Corollary 1.1.19.



 K ) → B(H  K ), Corollary 14.7.3 If there exists an admissible embedding B(G   then there exists an admissible embedding B(G) → B(H). Proof This follows from Corollary 14.7.2 applied to  = K and the identities   K )Gal(K/k) and B(H)   K )Gal(K/k) , cf. §9.5. B(G) = B(G = B(H 

14.8 Existence of Adapted Points We continue with the notation established in the first paragraph of §14.3. Corol lary 14.7.3 has reduced the existence of an admissible embedding B(G) →  B(H) to the case where k is strictly Henselian. We now assume that this is the case. Propositions 14.6.1 and 14.6.2 show that such embeddings are given by pairs of adapted points. In this section we prove that pairs of adapted points exist under the assumption that k is strictly Henselian. Lemma 14.8.1 Let x ∈ B(G) and y ∈ B(H) be such that G(k)1x ⊂ H(k)1y . For every maximal k-split torus S ⊂ G whose apartment contains x there exists a maximal k-split torus T ⊂ H that contains S and whose apartment contains y. Proof Consider the Bruhat–Tits group scheme Hy1 . We have the embeddings S(k)0 ⊂ G(k)1x ⊂ H(k)1y = Hy1 (o). According to Proposition B.2.4 the schematic closure S of S in Hy1 is an o-torus. Proposition 8.2.1 applied to any maximal f-torus of H

1 y

that contains S produces a maximal o-torus T

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Functorial Properties

containing S . The generic fiber T of T is a maximal k-split torus of H that contains S. Proposition 9.3.5(2) shows that y is contained in the apartment A(T, H).  Lemma 14.8.2 Assume that G and H are split. Let x ∈ B(G) be special. If y ∈ B(M) satisfies G(k)1x ⊂ M(k)1y , then (x, y) is a pair of adapted points. In particular, pairs of adapted points ( x,  y ) for G and H exist. Proof According to Definition 14.5.1, the existence of a pair of adapted points for G and H follows directly from the existence for G and M. Choose a k-split maximal torus S ⊂ G such that x ∈ A(S, G). Lemma 14.8.1 shows y ∈ B(M)S . We now verify Definition 14.5.1(1)(2). For any maximal k-split torus T ⊂ M containing S, the action of S(k) on A(T, M) is the restriction of the action of T(k), which is by translations via the homomorphism ν : T(k) → V(T/AM ). The restriction of ν to S(k) is the homomorphism ν : S(k) → V(S/AG ). Hence S(k) acts trivially on A(T, M)/V(S/AG ), therefore on B(M)S /V(S/AG ), and thus fixes y. Since G is split, ZG (S) = S and Lemma 14.5.4 shows that NG (S)(k) also fixes y and that ϕS, y (x) = y. We now show that G(k)1x+v ⊂ M(k)1y+v for each v ∈ V(S/AG ). First, recall that G(k)1x+v = G(k)0x+v · NG (S)(k)1x+v . The V(S/AG )  NG (S)(k)-equivariance of ϕS, y shows that NG (S)(k)1x+v ⊂ NG (S)(k)1y+v ⊂ M(k)1y+v . Therefore it is enough to show that G(k)0x+v ⊂ M(k)0y+v . Let X ⊂ A(S, G) be the set of all points z ∈ A(S, G) that satisfy G(k)0z ⊂ M(k)0ϕ (z) . We claim that if X contains a chamber of A(S, G), then X = S, y A(S, G). To see this, assume that X contains a chamber of A(S, G). The NG (S)(k)-equivariance of ϕS, y implies that X is stable under NG (S)(k), thus X contains all chambers of A(S, G). Consider any x  ∈ A(S, G) and let C1, . . . , Cn be the chambers that contain x  in their closure. Then G(k)0x is generated by G(k)0Ci for i = 1, . . . , n. For a fixed 1  i  n let (xm ) be a sequence of points in Ci converging to x . Then G(k)0Ci = G(k)0xm ⊂ M(k)0ym for all m, where ym = ϕS, y (xm ). Since the sequence (ym ) converges to y  = ϕS, y (x ), applying Lemma 4.2.17 to B(M) we see that G(k)0Ci ⊂ M(k)0y . Hence G(k)0x ⊂ M(k)0y , showing x  ∈ A(S, G). The claim is proved. Therefore, it is enough to show that X contains some chamber of A(S, G). Choose a chamber C whose closure contains x and consider z ∈ C. Recall G for all that G(k)0z is generated by S(k)0 and the root filtration groups Ua,z,0 G 0 a ∈ Φ(S, G). So we need to show that Ua, z,0 ⊂ M(k)ϕ (z) for all a ∈ Φ(S, G), S, y and all z ∈ C. Fix a ∈ Φ(S, G). Fix a k-split maximal torus T ⊂ M that contains S and let Φ(T, M)a = {b ∈ Φ(T, M) | b|S ∈ Z>0 · a}.

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Choose an arbitrary order on this finite set. The root group UaM for the root a ∈ Φ(S, M) is the direct product (as an affine variety) of the root groups {UbM | b ∈ Φ(T, M)b } in the chosen order. We have UaG ⊂ UaM . According to Proposition 7.3.12(3), applied to a positive system of roots that contains M Φ(T, M)a , UaM (k) ∩ M(k)ϕS , y (z) is the direct product of Ub,ϕ for all S , y (z),0 b ∈ Φ(T, M)a , again in the chosen order. This holds for all z ∈ A(S, G), in particular for z = x. Fix z ∈ C and write v = z − x ∈ V(S). Then ϕS, y (z) = y + v according G to Lemma 14.5.4 and the V(S)-equivariance of ϕS, y . We have Ua, x+v,0 = G M M Ua, and U = U . Let m be the positive integer such that b, y+v,0 b, y,−b(v) x,−a(v) b|S = ma. Then b(v) = ma(v). Since x is special and z ∈ C we have either G G = Ua, 0 < a(v) < 1, in which case Ua, x,0 , or −1 < a(v) < 0, in which x,−a(v) G G G G = Ua, case Ua, x,−a(v) = Ua, x,1 . Thus, when a(v) > 0 we obtain Ua, x,0 ⊂ x,−a(v)    M M M U ⊂ U = U . b ∈Φ(T , M) a b, y,0 b ∈Φ(T , M) a b, y,−b(v) b ∈Φ(T , M) a b, y+v,0 Consider now the case a(v) < 0. For any b ∈ Φ(T, M)a write pa, b for the composition of the inclusion UaG → UaM with the projection UaM → UbM given by the product structure on the variety UaM coming from the chosen order of Φ(T, M)a . Then pa, b is a morphism of affine varieties UaG → UbM . It is enough G M ) ⊂ Ub, . We have −b(v) = −ma(v) < m, hence to show pa, b (Ua, y,−b(v) x,−a(v) G M M M Ub, y, m ⊂ Ub, y,−b(v) and it is enough to show pa, b (Ua, x,1 ) ⊂ Ub, y, m . Choose an isomorphism τa : UaG → Ga that is equivariant for a : S → Gm , and an isomorphism τb : UbM → Ga that is equivariant for b: T → Gm . Then pa, b = τb ◦ pa, b ◦ τa−1 is an endomorphism of the affine variety Ga . Since pa, b is S(k)equivariant we have pa, b (su) = s m pa, b (u) for all u ∈ k and all those s ∈ k × that lie in the image of a : S(k) → k × . Since there are infinitely many such s and pa, b is a polynomial map, we see that pa, b (su) = s m pa, b (u) holds for all G u ∈ k and all s ∈ k × . If La, Lb ⊂ k are the o-lattices such that τa (Ua, x,0 ) = L a G M M and τb (Ub, y,0 ) = Lb , then τa (Ua, x,1 ) = m · La and τb (Ub, y, m ) = m m · Lb . G M The assumption G(k)0x ⊂ M(k)0y implies pa, b (Ua, x,0 ) ⊂ Ub, y,0 , which in turn G −1 pa, b (mLa )) ⊂ τb−1 (m m Lb ) = implies pa, b (La ) ⊂ Lb . Then pa, b (Ua, x,1 ) = τb ( M  Ub, y, m .  Recall that, if /k is a finite Galois extension, then B(H) is a closed convex   y1 ∈ B(H ), Proposition 1.1.20 guarantees the subset of B(H ) and, given  existence and uniqueness of a point  y ∈ B(H) closest to  y1 . Lemma 14.8.3 Let  be a finite Galois extension splitting G and H. Let   ) be such that (  x,  y1 ) is a pair  x ∈ B(G) be absolutely special and let  y1 ∈ B(H

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y ∈ B(H) be the point closest to  y1 . Then of adapted points for G and H . Let  ( x,  y ) is a pair of adapted points for G and H.  G). Since k Proof Let S ⊂ G be a maximal k-split torus such that  x ∈ A(S, is strictly Henselian, G is quasi-split according to Corollary 2.3.8. Therefore Z = ZG (S) is a maximal torus of G (which may properly contain S) and   )Z ,  G ). By Definition 14.5.1 and Remark 14.5.2 we have y1 ∈ B(H x ∈ A(Z,  y1 in B(H ) Z /V(Z) is fixed by NG (Z)(). G()x ⊂ H()y1 , and the image of  Consider the group Z = {(z, v) ∈ Z() × V(Z)|ω Z (z) = v}. Since both Z() and V(Z) act on B(H ) Z by isometries, so does the group Z. This action fixes every point of B(H ) Z , in particular y1 . We now consider the embeddings

B(H ) Z

o B(H O )

B(H) O

/ B(H )S o

B(H)S

 Proposition 1.1.20 guarantees that the unique point  y ∈ B(H) that is closest  to  y1 is fixed by G(k)x . Lemma 14.8.1 shows that this point lies in B(H) S.   )S , and we conclude that  y is the The latter is a closed convex subset of B(H  y1 ∈ B(H )S . Since the action of unique point of B(H) S that is closest to  Gal(/k)   )S fixes  = {(z, v) ∈ Z(k) × V(S) | ν(z) = −v} on B(H y1 and preserves Z  B(H)S , Proposition 1.1.20 implies that this action fixes  y . Thus, the image y  of  y in B(H) S /V(S · A M )  B(M)S is fixed by Z(k). Lemma 14.5.4 shows that y is fixed by NG (S)(k). Finally we show that G(k)1x+v ⊂ H(k)1y+v for all v ∈ V(S). We know that ϕz, y1 (x + v) = y1 + v and hence G(k)1x+v ⊂ G()1x+v ⊂ H()1y1 +v . Since the action of V(S) on B(H )S is by isometries and preserves B(H)S , y + v ∈ B(H)S is the point closest to y1 + v ∈ B(H )S . Therefore, y + v is fixed by G(k)1x+v .  Proposition 14.8.4 Pairs ( x,  y ) of adapted points exist. More precisely, let   ) a point satisfying   x ∈ B(G) be an absolutely special point,  y1 ∈ B(H  y ∈ B(H) the unique point closest to  y1 . Then ( x,  y) G()x ⊂ H()y , and  1

is a pair of adapted points. Proof According to Proposition 7.11.4 there exists an absolutely special point   x ∈ B(G). If /k is a finite Galois extension that splits both G and H,  x remains   ). Lemma 14.8.2 shows that ( x,  y1 ) is a pair of adapted points special in B(G x,  y ) is a pair of adapted points for G and H , and Lemma 14.8.3 shows that ( for G and H. 

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14.9 Uniqueness of Admissible Embeddings In this section we present some positive and negative results about the uniqueness of admissible embeddings. A complete solution to this problem is at the moment not available. We continue with a connected reductive k-group H and a connected reductive subgroup G. Proposition 14.3.1 reduces the classification of admissible embed  dings B(G) → B(H) to that of the admissible embeddings B(G) → B(M), where M is the centralizer of AG in H. Thus, we will assume from now on that H = M, that is, that AG ⊂ AH . We will reuse the letter M for other purposes.   Lemma 14.9.1 Fix  x ∈ B(G). The set of admissible embeddings ϕ : B(G) →   B(H) is in bijection with the set of points  y ∈ B(H) for which ( x,  y ) is an adapted pair; the bijection is given by  y = ϕ( x ). Proof

This follows at once from Propositions 14.6.1 and 14.6.2(2).



The map M → AM is an inclusion-reversing bijection between the set of Levi subgroups of H that contain G and the set of split tori in the connected centralizer Z H (G)0 of G in H. Therefore, Z H (G)0 (F) acts transitively on the set of minimal such Levi subgroups. We fix one such M and consider the set of admissible embeddings B(G) → B(M). Then AG ⊂ AM and (AM ∩G)0 = AG . Let M = M/AM and G = G/(AM ∩ G). Then G ⊂ M and both of these groups have anisotropic centers. In fact, the following stronger statement is true. Lemma 14.9.2

The centralizer of G in M does not contain a split torus.

Proof It is enough to show that Z M (G) = Z M (G)/AM . Given g ∈ G and m ∈ M so that the image m ∈ M centralizes G, a(m, g) := gmg −1 m−1 ∈ AM . The element a(m, g) depends only on the image g ∈ G ⊂ M of g and the image m ∈ M of m. A direct computation shows that a(m, g) is multiplicative in both variables. The multiplicativity in g shows that for each fixed m the map g → a(m, g) is a homomorphism G → AM . This homomorphism factors through the maximal abelian quotient of G, which is an anisotropic torus due to AG ⊂ AM ∩ G. Thus this homomorphism is trivial, and we conclude that  Z M (G) = Z M (G)/AM . Proposition 14.9.3 Assume that K = f((t)) for an algebraically closed field f of characteristic zero, and that G is split. There exists a unique admissible embedding B(G K ) → B(MK ). It maps every special point of B(G K ) to a vertex in B(MK ).

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Proof We replace G and M by G and M, thereby assuming that G is semisimple. Choose a special point x ∈ B(G K ). Let z ∈ B(MK ) be a point fixed by G(K)x . The embedding G(K)x → M(K)z extends to a homomorphism of O-group schemes Gx → Mz . Therefore the inclusion G(K)x → M(K)x maps the kernel G(K)x,1 of the reduction map Gx (O) → Gx (f) to the kernel M(K)x,1 of the reduction map Mz (O) → Mz (f). Since G is split and x is special, G(K)x, 0+ = G(K)x, 1 . In particular, G(K)x, 0+ ⊂ M(K)z, 0+ . The group Z M (G)0 is reductive since K has characteristic zero. Since K is strictly Henselian, this reductive group is quasi-split. Lemma 14.9.2 implies that it does not contain a split torus. We conclude that Z M (G)0 is an anisotropic torus. Let Mz be the reductive quotient of the special fiber of Mz and let S be the maximal central torus of Mz . Let Z be the smoothening of the schematic closure in Mz of Z M (G)0 . Then Z is an integral model of the anisotropic torus Z M (G)0 and has the property Z (O) = M(K)z ∩ Z M (G)0 (K). Consider the map Z → M z on special fibers and compose it with the reduction map M z → Mz . The image of Z in Mz is a smooth closed subgroup. Since Z M (G)0 is an anisotropic torus, Z has unipotent identity component. The same then holds for its image in Mz . We claim that this image contains a finite index subgroup of S. Granting the claim, we conclude S = {1}, which shows that z is a vertex. Since this is true for any point z ∈ B(MK ) fixed by G(k)x , we conclude that there is a unique such point, for if z  is another such point, then the entire geodesic [z, z ] is fixed by G(k)x , but all its points are vertices, which is only possible if z = z  is a vertex. Proposition 14.6.1 now completes the proof, modulo the outstanding claim. We now prove the claim that the image of Z in Mz contains a finite index subgroup of S. More precisely, we will prove that every s ∈ S(f) has a lift in Z M (G)(K) ∩ M(K)z,0 . Let s0 ∈ M(K)z,0 be an arbitrary lift of s. Let 0 = r0 < r1 < · · · be the set of jumps of the filtration M(K)z,r . Consider the map cs0 : G(f) ⊂ G(K)x,0 → H(K)z,r0 ,

cs0 (g) = s0 gs0−1 g −1 .

We are using here the fact that the integral model Gx is reductive (Proposition 8.4.14) as well as the embedding f → O to identify Gx with the base change of its special fiber, which we again denote by G, since it has the same root datum. Since s is central in Mz the image of cs0 is trivial in Mz (f) = M(K)z,r0 /M(K)z,r1 , thus cs0 takes values in M(K)z,r1 . We consider V1 = M(K)z,r1 /M(K)z,r2 . This is an f-vector space with an algebraic action of G(f). The map cs0 is an element of Z 1 (G(f), V1 ). By vanishing of rational cohomology there exists h1 ∈ M(K)z,r1

14.9 Uniqueness of Admissible Embeddings

511

such that 1 = h1 (s0 gs0−1 g −1 )(gh1−1 g −1 ) = (h1 s0 )g(h1 s0 )−1 g −1 holds modulo M(K)z,r2 . Set s1 = h1 s0 . Then s1 is another lift of s and maps to s0 modulo M(K)z,r1 . The map cs1 is now an element of Z 1 (G(f), V2 ), where V2 = M(K)z,r2 /M(K)z,r3 . Repeating this procedure inductively we obtain a sequence s0, s1, s2, . . . of lifts of s. The completeness of K implies that this sequence converges to an element s ∈ M(K)z,0 . This element has the property that the map cs (g) is trivial for all g ∈ G(f). In other words, s commutes with the subgroup G(f) of G(K)x,0 . Since G(f) is Zariski dense in G we conclude  that s ∈ Z M (G). Example 14.9.4 In this example we show that the analog of Proposition 14.9.3 is not always true when the residue field of k has positive characteristic, even if k itself has characteristic zero. Let k be a finite extension of Q p . Consider the adjoint representation G = PGL2 → SL3 = H. Since k has characteristic zero, this representation is irreducible, hence G is not contained in a Levi subgroup of H. The Lie algebra of G is the quotient of M2 (k) by the subspace of diagonal matrices. Taking the basis " # " # " # 0 −1 1 0 0 0 , , 0 0 0 0 1 0 for it we obtain the formula "

a c

⎡ a2 # 1 ⎢⎢ b  → 2ac d det ⎢⎢ 2 ⎣c

ab ad + bc cd

b2 ⎤⎥ 2bd ⎥⎥ d 2 ⎥⎦

for the embedding G → H. Let x ∈ B(G) and y ∈ B(H) be the standard vertices, corresponding to the standard integral structures on PGL2 and SL3 . One checks that (x, y) is a pair of adapted points. When p  2 the adjoint representation PGL2 (f) → SL3 (f) is irreducible. Since this is the map between special fibers Gx0 → H y0 , we see that the image of this map is not contained in a proper parabolic subgroup, which precludes the existence of another y  ∈ B(H) such that G(k)x ⊂ H(k)y (indeed, if such a y  existed, the entire geodesic [y, y ] would be fixed by G(k)x , but this geodesic will contain points from at least one facet whose closure contains y). We conclude using Lemma 14.9.1 that there is a unique admissible embedding B(G) → B(H). The situation is different when p = 2. The image of Gx0 → H y0 is contained

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in the parabolic subgroup ⎡∗ ⎢ ⎢0 ⎢ ⎢∗ ⎣

∗ ∗⎤⎥ ∗ 0⎥⎥ . ∗ ∗⎥⎦

This is a reflection of the reducibility of the adjoint representation of PGL2 in characteristic 2. We recall that the natural map Lie(SL2 ) → Lie(PGL2 ) is an isomorphism in odd characteristic, while in characteristic 2 it fits into the exact sequence 0 → f → Lie(SL2 )(f) → Lie(PGL2 )(f) → f → 0, where f is embedded into Lie(SL2 )(f) as the group of diagonal matrices, which is the center of that Lie algebra, and the map Lie(PGL2 )(f) → f is given by the trace. This sequence is equivariant for the action of PGL2 (f) and endows Lie(PGL2 )(f) with a filtration whose two successive subquotients are irreducible representations of PGL2 (f). In particular, the image of PGL2 → SL3 does not lie in any smaller parabolic f-subgroup. Let T ⊂ H be the diagonal maximal torus and let λ ∈ X∗ (Tad ) be the 1-parameter subgroup into the adjoint group PGL3 sending t ∈ Gm to the diagonal matrix with entries (1, t, 1). Let e be the ramification degree of k/Q2 . All points of the geodesic [y, y − eλ] are fixed by G(k)x . This geodesic contains e + 1 vertices, namely {y, y − λ, . . . , y − eλ}. When e = 1 the geodesic is the face of a chamber in the apartment A(T, H), and this is the full fixed point set of 0  SL3,f G(k)x in B(H). When e > 1 and we consider the image of Gx0 → H y−λ we see that it lies in the subgroup ⎡∗ 0 ∗⎤ ⎥ ⎢ ⎢0 ∗ 0⎥ . ⎢ ⎥ ⎢∗ 0 ∗⎥ ⎣ ⎦ This is a Levi subgroup of SL3,f that is contained in exactly two (opposite) parabolic subgroups. These correspond to the adjacent vertices y and y − 2λ, and we conclude that no other vertex of B(H) adjacent to y−λ is fixed by G(k)0x . This analysis can be applied to the vertices y − kλ, k = 1, . . . , e − 1, to conclude that the geodesic [y, y − eλ] is the full set of points of B(H) fixed by G(k)0x . Since G(k)0x = G(k)x (cf. Lemma 7.7.10 and Proposition 7.7.11), according to Lemma 14.8.2 (x, y ) is a pair of adapted points for each y  ∈ [y, y − eλ]. Each such pair leads to a distinct admissible embedding by Lemma 14.9.1.

15 The Buildings of Classical Groups via Lattice Chains

In this chapter we will give explicit descriptions of the buildings of classical groups in terms of additive norms and lattice chains. The enlarged building of the group GLn was described by additive norms by Goldman–Iwahori in [GI63] when the base field k is locally compact, in particular complete. The case of more general discretely valued field was treated in [BT84b]. The buildings of classical groups were discussed in [BT87b]. One can also give explicit descriptions of the buildings of some exceptional groups, cf. [GY03], [GY05]. We will treat here the case of classical groups over a field k endowed with a discrete valuation ω : k → R ∪ {∞} normalized so that ω(k × ) = Z. We denote by o the ring of integers of k and by m the maximal ideal of o. Let f = o/m be the residue field. We assume that o is Henselian but do not assume that k is complete. We will further assume that the residual characteristic p is not 2 when discussing orthogonal and ramified unitary groups. The reason for this restriction is that it simplifies the exposition. Readers interested specifically in orthogonal or ramified unitary groups in residual characteristic 2 are referred to [BT87b].

15.1 The Special and General Linear Groups Let X be an n-dimensional vector space over k. The algebraic group G = SL(X) is defined by the functor R → SL(R ⊗k X) where R is any k-algebra. Choosing a basis of X identifies X with k n and hence SL(X) with SLn . One defines analogously the group GL(X). The map det R : GL(X)(R) → R× is functorial in R and leads to a homomorphism det : GL(X) → Gm of algebraic groups, whose kernel is SL(X). The center 513

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The Buildings of Classical Groups via Lattice Chains

of GL(X) is identified with Gm . The group PGL(X) is defined as the quotient GL(X)/Gm . We will describe the building of the algebraic group SL(X), and the enlarged building of the algebraic group GL(X), explicitly. Definition 15.1.1 An additive norm on X is a function α : X → R ∪ {∞} satisfying the following. AN 1 α(v + w)  min(α(v), α(w)), for v, w ∈ X. AN 2 α(x v) = ω(x) + α(v) for x ∈ k, v ∈ X. AN 3 α(v) = ∞ ⇔ v = 0. Fact 15.1.2 If α(v)  α(w) then α(v + w) = min(α(v), α(w)). The function α : X − {0} → R is continuous. Proof The first two items in the definition of α imply α(v)  min(α(v + w), α(w)), which combined with α(v + w)  min(α(v), α(w)) implies the first claim. The second claim follows from the first.  Example 15.1.3 If X is 1-dimensional and 0  v ∈ X, any additive norm α : X → R ∪ {∞} satisfies α(x v) = ω(x) + α(v). Construction 15.1.4 Let X1 and X2 be finite-dimensional k-vector spaces and let αi an additive norm on Xi . Define X = X1 ⊕ X2 and α : X → R ∪ {∞} by α(v1 + v2 ) = min(α1 (v1 ), α2 (v2 )). Then α is an additive norm, called the direct sum of α1 and α2 . We now discuss the converse of this process. Definition 15.1.5

Let α be an additive norm on X.

(1) The norm α splits over a direct sum decomposition X = X1 ⊕ X2 if α(v1 + v2 ) = min{α(v1 ), α(v2 )} for all v1 ∈ X1 and v2 ∈ X2 . (2) A set v1, . . . , vn of non-zero vectors in X is called splitting for α if for all a1, . . . , an ∈ k one has   α ai vi = min{α(ai vi )}. (3) A basis v1, . . . , vn of X is a splitting basis for α if it is a splitting set for α, equivalently if α splits over X = l1 ⊕ · · · ⊕ ln , where li ⊂ X is the line spanned by vi . (4) The norm α is called splitting if it admits a splitting basis. Lemma 15.1.6

A splitting set for α is linearly independent.

15.1 The Special and General Linear Groups

515

Proof Let v1, . . . , vn be a splitting set. Then for any a1, . . . , an ∈ k we have α( i ai vi ) = mini {ai vi } = mini {ω(ai ) + α(vi )}. Thus i ai vi is zero if and  only if each ai is zero. Corollary 15.1.7 If {v1, . . . , vn } is a subset of X − {0} such that α(vi )  α(v j ) mod Z for i  j, then this set is linearly independent. In particular, α(X − {0}) is a closed and discrete subset of R. Proof Fact 15.1.2 implies {v1, . . . , vn } is a splitting set and the first claim follows from Lemma 15.1.6. Since X is finite dimensional, the first claim implies that α has finitely many values modulo Z, hence the second claim.  Example 15.1.8 Let v1, . . . , vn be a basis of X and a1, . . . , an ∈ R. Then - n .  α xi vi = min {ω(xi ) + ai } i=1

i

is an additive norm on X admitting v1, . . . , vn as a splitting basis. Every additive norm admitting v1, . . . , vn as a splitting basis is of this form. Thus the set of additive norms admitting v1, . . . , vn as a splitting basis is in bijection with Rn ; the bijection sends α to (α(v1 ), . . . , α(vn )). Lemma 15.1.9 If x, y ∈ X are such that α(x)  α(x + y) then α(x + y) = min{α(x), α(y)}. Proof If we assume that α(x + y)  min{α(x), α(y)}, then Definition 15.1.1 implies α(x + y) > min{α(x), α(y)}, while Fact 15.1.2 implies α(x) = α(y).  Together we obtain α(x + y) > α(x), contradicting the assumption. For an additive norm α we define the ball Lα,r = {v ∈ X | α(v)  r }.  This is an o-submodule of X. It is also useful to define Lα,r+ = s>r Lα, s = {v ∈ X | α(v) > r }. Since α(X − {0}) is discrete, there exists some s > r such that Lα,r+ = Lα, s . Recall that a lattice in X is a finitely generated o-submodule of X that generates X over k. Since o is noetherian, an o-submodule of X that is contained in a lattice and contains a lattice is itself a lattice. Since any basis of X can be rescaled to lie in Lα,r , the latter contains a lattice. However, it need not be contained in a lattice, that is, it need not be finitely generated. Lemma 15.1.10 (1) An additive norm α is splitting if and only if for one, hence all, r ∈ R the ball Lα,r is a lattice in X.

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The Buildings of Classical Groups via Lattice Chains

(2) Let α be a splitting additive norm and let Y ⊂ X be a subspace. Then α|Y is a splitting additive norm on Y and there exists Y  ⊂ X such that X = Y ⊕ Y  and α splits over Y ⊕ Y . Proof We induct on dim(X). When dim(X) = 1 and 0  v ∈ X then every norm α satisfies α(xv) = ω(x)+α(v). In particular, α is splitting. All statements are trivial. Consider now a general X. If α is splitting and (v1, . . . , vn ) is a splitting basis, then * m ai · vi , Lα,r = where ai is the smallest integer greater than or equal to r − α(vi ). Thus Lα,r is a lattice. Conversely assume that for some r ∈ R the ball Lα,r is a lattice. If Y ⊂ X is a proper subspace, we have Lα |Y ,r = Lα,r ∩ Y . Since Lα,r is a lattice, so is Lα |Y ,r and the inductive hypothesis implies that α|Y is splitting. Furthermore, Lα,r+n = m n · Lα,r is a lattice for all n ∈ Z, hence the same holds for Lα, s for all s ∈ R. Let Y ⊂ X be a hyperplane. We claim that there is Y  ⊂ X such that X = Y ⊕ Y  and α splits over Y ⊕ Y . Let x0 ∈ X − Y . Since Lα,r is a lattice, the set {m n · Lα,r | n ∈ Z} forms a basis of neighborhoods of 0 in X. Since x0 + Y is a closed subset of X there exists n ∈ Z such that Lα,r+n = m n · Lα,r is disjoint from x0 + Y . This implies that α(x0 + Y ) ⊂ R is bounded above. Since this set is also closed by Corollary 15.1.7, it has a maximum. Thus there is x ∈ x0 + Y such that α(x)  α(x + y) for all y ∈ Y . Lemma 15.1.9 implies α(x + y) = min{α(x), α(y)}. Setting Y  = k · x the claim is proved. The claim just proved and the fact that α|Y is splitting imply that α is splitting. This completes the inductive argument. Consider again a general proper subspace Y ⊂ X. Writing Y = Y0 ⊂ Y1 ⊂ · · · ⊂ Yn = X with dim(Yi /Yi−1 ) = 1 and applying successively the above claim  produces a splitting complement Y . Proposition 15.1.11 splitting basis.

If k is complete then every additive norm admits a

Proof According to Corollary 15.1.7 we can choose a finite set γ1, . . . , γn ∈ [0, 1) of representatives for the distinct values of α modulo Z. For each γi let (vi, j ) be a basis for the f-vector space Lα,γi /Lα,γi + , where j runs over a set Ji which will soon be seen to be finite. We claim that every finite subset of {vi, j } is splitting. To see this, consider a set {ai, j } of elements of k, of which a positive but finite number are non-zero.

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517

The indices (i, j) for which α(ai, j vi, j ) attains the minimal value must all have the same first component, call it i1 . Moreover, the corresponding ai, j must all have the same valuation r = ω(ai, j ). Thus r + γi1 is the minimal value of α(ai, j γi, j ). Let Ji1 be the subset consisting of those j ∈ Ji1 for which α(ai1 , j vi1 , j ) = r + γi1 . Now j ∈Ji ai, j vi, j has non-zero image in Lα,r+γi /Lα,r+γi + and therefore 1 α( j ∈Ji ai, j vi, j ) = r + γi . Since every other ai, j vi, j has larger value under α 1 we see α( i, j ai, j vi, j ) = α( j ∈Ji ai, j vi, j ) = r + γi . This completes the proof 1 of the claim. With the claim proved, Lemma 15.1.6 implies that every finite subset of {vi, j } is linearly independent, hence the whole set {vi, j } is linearly independent, hence finite since dim(X) is finite. We conclude that {vi, j } is splitting. It remains to show that {vi, j } is a basis. Let v ∈ X and let r1 = α(v). There exists a linear combination of elements of {vi, j }, call it v1 , such that v − v1 ∈ Lα,r1 + . In other words, α(v − v1 ) > α(v). Proceeding inductively we obtain a sequence vn of linear combinations of the set {vi, j } such that α(v − vn+1 ) > α(v − vn ). Since α(X − {0}) is discrete, this implies that α(v − vn ) tends to +∞. Induction and Fact 15.1.2 show that if m > n then α(vn − vm ) = α(v − vn ), which tends to +∞ as n tends to ∞. Writing vn = i, j ai,(n)j we see that for each index (i, j) the sequence ω(ai,(n)j − ai,(m) j ) tends to ∞ as n tends to ∞ and n < m. In other words, ai,(n)j is a Cauchy sequence. The completeness assumption on k implies that this sequence converges to some ai, j ∈ k. Writing v  = i, j ai, j vi, j we see that the sequence vn converges to v . Now α(v − v )  α(v − vn ) + α(v  − vn )  for any n, hence α(v − v ) = ∞, showing v = v .

ω of X with respect If k is not complete, we can consider the completion X ω  = k ⊗k X. Consider an additive to the natural topology induced by ω. Thus X ω  α norm α and let  α : X → R ∪ {∞} be the continuous extension of α. Then  automatically satisfies AN 1 and AN 2, but may fail to satisfy AN 3. Thus, it is a semi-norm. Lemma 15.1.12 The additive norm α is splitting if and only if  α satisfies AN 3 and is hence a norm. Proof If α is splitting let (v1, . . . , vn ) be a splitting basis. Thus α( xi vi ) = α(  xi vi ) = mini {ω( xi ) + α(vi )}, where we denote mini {ω(xi ) + α(vi )}. Then  again by ω the unique continuous extension to  k of the valuation ω. In particular,  α is an additive norm. Conversely, assume  α is an additive norm. By Proposition 15.1.11 it admits ω for any r. a splitting basis. By Lemma 15.1.10 the ball Lα,r is a lattice in X ω   Fix r and a lattice L ⊂ X. Then L is a lattice in X . Since Lα,r is a lattice

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The Buildings of Classical Groups via Lattice Chains

L. Intersecting with X we obtain there exist a, b ∈ k such that a  L ⊂ Lα,r ⊂ b  a L ⊂ Lα,r ⊂ bL and conclude that Lα,r is a lattice, hence by Lemma 15.1.10 that α is splitting.  α be the completion of We also have the following characterization. Let X X for the topology induced by α, that is, the topology for which the balls Lα,r form a basis of neighborhoods of 0. This topology is Hausdorff by AN 3, α is injective. The scalar multiplication of k therefore the natural map X → X α α is naturally  k of k with respect to ω; thus X on X extends to the completion  α . The ω → X a topological  k-vector space. We thus have the natural map f : X α , which we also denote by α. norm α extends uniquely to a norm on X α is surjective and satω → X Lemma 15.1.13 (1) The natural map f : X isfies α( f (x)) =  α(x). ω |  α(x) = ∞}. (2) ker( f ) = {x ∈ X (3) α is splitting if and only if f is injective, hence bijective.  that contains α , so any  k-subspace of X Proof By construction X is dense in X α  X must equal X . The image of f has this property, so f is surjective. The identity α( f (x)) =  α(x) holds trivially for x ∈ X and this implies that it holds ω . This proves (1) and (2) follows from it, while (3) follows from for all x ∈ X (2) and Lemma 15.1.12.  Example 15.1.14 We now give an example of an additive norm that is not splitting. Assume that k is not complete. Let V be a 2-dimensional k-vector subspace of  k that contains k. Let α be the restriction to V of the natural ω =  α =  k, while V k ⊗k V is a 2-dimensional extension of ω to  k. Then V  k-vector space.   li ) be the Given a direct sum decomposition X = l1 ⊕ · · · ⊕ ln , let A( set of all additive norms that split over this decomposition. The group Rn acts   on A( li ) as follows. Fix a non-zero vector vi ∈ li for each i = 1, . . . , n. As   discussed in Example 15.1.8, this choice gives a bijection A( li ) → Rn . We n transport the natural translation action of R on itself via this bijection. One   li ) obtained this way does not checks immediately that the action of Rn in A(   li ) does not depend depend on the choice of x1, . . . , xn . Note that the set A( n on the ordering of the lines l1, . . . , ln , but the action of R on this set does, since the standard basis of Rn comes equipped with an ordering.  Define B(X) to be the set of all splitting additive norms. There is an action  of R on B(X): if α is an additive norm and r ∈ R we can define the translated  norm α + r by (α + r)(v) = α(v) + r. This action preserves every subset of B(X)   of the form A( li ) and on that subset it equals the restriction of the natural

15.1 The Special and General Linear Groups

519

action of Rn constructed above to the diagonal copy of R. We will call an orbit  of R in B(X) a homothety class.     li ) and B(X) by the action Let A( li ) and B(X) be the quotients of A( of R.  is an affine space under Rn , and the set A is an Fact 15.1.15 (1) The set A affine space under the vector space Rn /R, where R is embedded into Rn diagonally. (2) The choice of a basis (v1, . . . , vn ) of X with vi ∈ li specifies an element of  A( li ), namely the unique α that satisfies α(vi ) = 0 for i = 1, . . . , n.  is the union of A,  and B is the union of A, the unions being taken over (3) B all direct sum decompositions X = l1 ⊕ · · · ⊕ ln . We can define a simplicial structure on the space B by interpreting splitting additive norms in terms of graded periodic lattice chains. Definition 15.1.16 (1) A periodic lattice chain in X is a set L of lattices of X, totally ordered by inclusion, such that L ∈ L implies x · L ∈ L for all x ∈ k ×. (2) A grading of L is a strictly decreasing function c : L → R having the property c(m · L) = c(L) + 1. Remark 15.1.17 Given a periodic lattice chain L, fix a lattice L0 ∈ L and consider the set of elements of L contained in L0 and containing mL0 . Taking the quotient modulo mL0 maps this set bijectively to the set of elements of a partial flag in the finite-dimensional f-vector space L0 /mL0 . In particular, this set is finite. We can enumerate its members as mL0  Lr  · · ·  L0 . We may call this finite lattice chain a segment of L, and the number r + 1 the rank of L. It is clear that the segment is uniquely determined by L up to rescaling by k × and cyclic permutation of the indexing, and conversely that L is uniquely determined by any of its segments. Definition 15.1.18 Let L be a periodic lattice chain. A basis (v1, . . . , vn ) of X is called adapted to L, if for every L ∈ L there exist x1, . . . , xn ∈ k × such that (x1 v1, . . . , xn vn ) is a basis of L. Lemma 15.1.19

Every periodic lattice chain admits a basis adapted to it.

Proof Let L be a periodic lattice chain and let mL0  Lm−1  Lm−2  · · ·  L0 be a segment as in Remark 15.1.17. Let L i = Li /mL0 . There exists a basis (l¯1, . . . , l¯n ) of L 0 that is adapted to the flag L m−1  · · ·  L 0 . This means that

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for each 0  i  m − 1 there is 1  k i < n such that (l¯1, . . . , l¯ki ) is a basis of L m−i . Nakayama’s lemma implies that there is a basis (l1, . . . , ln ) of L0 lifting (l¯1, . . . , l¯n ). If π ∈ m is a uniformizer, then the elements (l1, . . . , lki , πlki +1, . . . , πln ) are still linearly independent and generate a submodule of L0 that contains πL0 and maps surjectively onto L m−i , hence must equal Lm−i .  Construction 15.1.20 There are maps in both directions between the set of graded periodic lattice chains and the set of splitting additive norms constructed as follows. From a splitting additive norm α we obtain the periodic lattice chain L = {Lα,r | r ∈ R} and the grading c(L) = inf{α(v) | v ∈ L}. Conversely, given a periodic lattice chain L with grading c we can define α(v) = sup{c(L) | L ∈ L, v ∈ L}. If the norm α corresponds to (L, c), then the norm α + r corresponds to (L, c − r), where (c − r)(L) = c(L) − r. Thus a homothety class of additive norms α + R corresponds to a periodic lattice chain L endowed with a grading well defined up to a shift, which we denote by c +R. The latter notation signifies the set of gradings c + r : L → R, where c : L → R is one grading and r ∈ R is arbitrary. Proposition 15.1.21 The above construction gives mutually inverse bijections between the set of splitting additive norms and the set of periodic lattice chains. A basis of X is a splitting basis of an additive norm α if and only if it is adapted to the lattice chain corresponding to α. Proof It is clear that a splitting norm α gives rise to a graded lattice chain.  a m i vi , where ai is the If (v1, . . . , vn ) is a splitting basis for α, then Lα,r = smallest integer greater than or equal to r − α(vi ). Thus (v1, . . . , vn ) is adapted to L = {Lα,r }. Conversely let (L, c) be a graded periodic lattice chain. It is again immediate that α(v) = sup{c(L) | L ∈ L, v ∈ L} is an additive norm. Lemma 15.1.19 gives a basis (v1, . . . , vn ) of X adapted to L. Write v = xi vi . Since (v1, . . . , vn ) is adapted to L, we have for L ∈ L v ∈ L ⇔ for every i, xi vi ∈ L. It follows that α( xi vi ) = mini {α(xi vi )}, that is, that (v1, . . . , vn ) is a splitting basis for α. That the two maps are mutual inverses is left to the reader.  Fact 15.1.22 Let α be an additive norm and (L, c) the corresponding lattice chain with grading. A basis v1, . . . , vn of X is a splitting basis for α if and only if for each L ∈ L there are x1, . . . , xn ∈ k such that x1 v1, . . . , xn vn is a basis of L.

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521

15.1.23 We declare two points of B to belong to the same simplex if they correspond to the same lattice chain. We say that the simplex containing x ∈ B is a face of the simplex containing y ∈ B if the lattice chain corresponding to x is a subset of the lattice chain corresponding to y. With respect to this simplicial structure the facets of dimension r are given by the lattice chains of rank r + 1 as in Remark 15.1.17. In particular, the vertices are the lattice chains whose segments are single lattices, while the chambers are the lattice chains whose segments have successive quotients that are 1-dimensional f-vector spaces. Each periodic lattice chain L has a natural grading, well defined up to homothety, which is obtained by choosing a segment mL0  Lr  Lr−1  · · ·  L0 of L and setting c(Li ) = i/(r + 1). The corresponding point of B is the barycenter of the facet determined by L. 15.1.24 Let (v1, . . . , vn ) be a basis of X and identify X with k n using that basis. This specifies the lattices Li = m ⊕ · · · ⊕ m ⊕ o ⊕ · · · ⊕ o . /012 /012 i−times

(n−i)−times

Each of these lattices specifies a vertex in B and these are the vertices of a chamber specified by the segment mL0  Ln−1  · · ·  L0 as in Remark 15.1.17. The faces of this chamber correspond bijectively to the subsets of {0, . . . , n−1}, with each such subset S corresponding to the segment containing the lattices {Li | i ∈ S}. Note here that the set {0, . . . , n − 1} is also in bijection with the set of affine simple roots in the affine root system Ψ(An−1 ). In this way we identify the subset of {0, . . . , n − 1} corresponding to a face of the chamber mL0  Ln−1  · · ·  L0 with the type of that face as in Remark 1.5.15. This discussion can be applied to an arbitrary chamber in B using Lemma 15.1.19. For a description of the parahoric subgroups in terms of periodic lattice chains, cf. 15.1.33. 15.1.25 For lattices L1, L2 ⊂ X we define the relative volume vol(L1 /L2 ) = ω(det(g)) for any g ∈ GL(X) such that gL2 = L1 . This is independent of the choice of g. It is clear that vol(L1 /L3 ) = vol(L1 /L2 ) + vol(L2 /L3 ). Lemma 15.1.26 Let L be a lattice in X and α an additive norm on X. There exists a unique real number vol(α/L) such that for any splitting basis v1, . . . , vn for α the equality holds  vol(α/L) = vol( v1, . . . , vn /L) − α(vi ).

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For any g ∈ GL(X) we have vol(gα/L) = vol(α/L) + ω(det(g)). For any c ∈ R we have vol(α + c/L) = vol(α/L) − nc. Proof Let v1, . . . , vn be a splitting basis for α. We are free to change the lattice L to any other lattice M and then define vol(α/L) = vol(α/M) + vol(M/L). We choose M to belong to the lattice chain L determined by α. Then there exist x1, . . . , xn ∈ k such that x1 v1, . . . , xn vn is a basis of M. Consider the lattices in L that lie between M and mM. Since M/mM is an n-dimensional k-vector space, there are only finitely many such lattices mM = Mr  · · ·  M0 = M. Let ni = dimk (Mi−1 /Mi ) and let mi = n1 + · · · + ni and m0 = 0. Then 0 = m0 < m1 < · · · < mr = n. We can reindex the basis v1, . . . , vn in such a way that for each i the elements πx1 v1, . . . , πxmi vmi , xmi +1 vmi +1, . . . , xn vn form a basis of Mi , where π ∈ k is a fixed uniformizer. Then α(xi vi ) = c(M j ) n α(xi vi ) = for the unique 0  j < r satisfying m j < i  m j+1 . Therefore i=1 r−1 c(M )n . It follows that j j+1 j=0 .    vol( v1, . . . , vn /M) − α(vi ) = − ω(xi ) + α(vi ) i

i

=−



i

α(xi vi )

i

=−

r−1 

c(M j )n j+1 .

j=0

We see that this is independent of the choice of splitting basis, as claimed. For the second equation note that gv1, . . . , gvn is a splitting basis for gα.  15.1.27 Given a graded lattice chain (L, c) in X we obtain a graded lattice 3 3 3 chain n (L, c) in ∧n X by taking the lattice chain n L := { n L | L ∈ L} and 3 3 3n 3n c( L) = c(L). Note that vol(L1 /L2 ) = vol( n L1 / n L2 ). In the grading 3n 3n L1  L2 . particular, for L1 ⊂ L2 , L1  L2 is equivalent to 3   n X) sending (L, c) to Corollary 15.1.28 The map B(X) → B(X) × B(

 3n (L, c + R), (L, c) is bijective and GL(X)-equivariant. 3  3n X) = Remark 15.1.29 Since n X is a 1-dimensional vector space, B(  3n X) is a 1-dimensional affine space. Note however that it does not have a A( 3 distinguished point, unless a basis for n X has been chosen. Fact 15.1.30 Consider G = GL(X) or G = SL(X). Let v1, . . . , vn be a basis of X. Let li ⊂ X be the line spanned by vi and Xi = l1 ⊕ · · · ⊕ li .

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(1) The stabilizer T of the direct sum decomposition X = l1 ⊕ · · · ⊕ ln is a maximal torus of G. (2) The stabilizer B of the flag X1 ⊂ X2 ⊂ · · · ⊂ Xn is a Borel subgroup containing T. (3) For i = 1, . . . , n − 1, the morphisms Ga → G sending x to the linear map that sends vi+1 to x vi + vi+1 and fixes v j for j  i + 1, parameterize the simple root subgroups. Thus, the basis v1, . . . , vn leads to a pinning of G. Let β = (v1, . . . , vn ) be a basis of X. From Fact 15.1.30 we obtain a Chevalley  oβ ∈ B(GL(X)) lifting point oβ ∈ B(GL(X)), as discussed in §6.1. Choose any  oβ . On the other hand, we obtain the norm α for which β is a splitting basis and  xβ ∈ B(X), whose image in B(X) we call α(vi ) = 0, and hence obtain a vertex  xβ . The graded lattice chain corresponding to α is L = {m n L | n ∈ Z}, where L is the lattice spanned by β, and c(m n L) = n. Proposition 15.1.31 Let L be the lattice spanned by β. (1) The group GL(L) equals the stabilizer of  oβ for the action of GL(X) on   B(GL(X)), as well as the stabilizer of  xβ for the action of GL(X) on B(X). It is a maximal bounded subgroup of GL(X). (2) The group SL(L) equals the stabilizer of oβ for the action of SL(X) on B(SL(X)), as well as the stabilizer of xβ for the action of SL(X) on B(X). It is a maximal bounded subgroup of SL(X). (3) Every maximal bounded subgroup of GL(X) respectively SL(X) is of this form. Proof Use the basis β to identify X with k n and therefore GL(X) with GLn . xβ is equal to the stabilizer of In these coordinates L = o n . The stabilizer of  L, which is GL(L). For an element g ∈ SL(X) we have vol(L/gL) = 0 and therefore if gL ⊂ L or L ⊂ gL then L = gL. Thus the stabilizer of xβ in SL(X) is again SL(L). Let G = SL(X). Since G is simply connected we have G(k) = G(k)0 . By Proposition 7.6.4 the stabilizer of oβ in G(k) is G(k)0x . This is the group generated by the integral points of the diagonal maximal torus and all standard root subgroups, which evidently is SLn (o). Let G = GL(X). Then G(k)0 = G(k)1 = {g ∈ G(k) | ω(det g) = 0}. By Lemma 4.3.3, the stabilizer of  xβ in G(k) is the stabilizer of xβ in G(k)1 , which by the same argument as for SL(X) is the stabilizer of L in G(k)1 . This equals GL(L). That SL(L) and GL(L) are maximal bounded subgroups, and that every

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maximal bounded subgroup is of this form, follows from Theorem 4.2.15. Alternatively, one can use Proposition 2.2.13.  Remark 15.1.32 The Proposition 15.1.31 implies that the correspondence xβ ↔ oβ induces an SL(X)-equivariant isomorphism B(X)  B(SL(X)). It is independent of the choice of β, and hence canonical.  oβ induces a GL(X)-equivariant isomorphism B(X)  Furthermore,  xβ ↔   B(GL(X)). This isomorphism is however not quite canonical. The issue is  that of Remark 15.1.29. Namely, B(GL(X)) = B(SL(X)) × X∗ (Gm )R , while 3n  B(X) = B(X) × A( X). When there is a chosen basis of X, for example 3 when X = k n so that GL(X) = GLn , then n X = k and A(k) = R = X∗ (Gm )R and the isomorphism is canonical. More generally, it is enough to choose a 3 3 3 basis of n X. But when there is no chosen basis of n X, then A( n X) is a 1-dimensional affine space without distinguished point, that is, without a natural isomorphism to R. 15.1.33 Consider the group G = GL(X). We can describe the groups G(k)0x ⊂ G(k)x and Gx0 (f) as follows. Let L be a periodic lattice chain. It specifies a facet F in B(X) = B(G). The barycenter of this facet, which we will denote by x, corresponds to the “equispaced” grading of L, which is the function c : L → R for which the value c(L ) − c(L) is constant for any pair of consecutive lattices L  L  in L. The group G(k)x is the stabilizer in G(k) of the lattice chain L. We have G(k)0 = G(k)1 and this is the subgroup of GL(X) consisting of elements whose determinant lies in o× . Thus G(k)0F = G(k)0x = G(k)1x = G(k)†x . This group is equal to the subgroup that stabilizes each individual lattice in the lattice chain. Let L0 ∈ L, so that G(k)0F ⊂ GL(L0 ). Consider the segment mL0  Lr  Lr−1  · · ·  L0 of L as in Remark 15.1.17. Its image in V = L0 /mL0 is a flag {0}  Vr  · · ·  V. The image of G(k)0F in GL(V) is the stabilizer of this flag, and hence a parabolic subgroup. Its reductive quotient  is identified with ri=1 GL(Vi−1 /Vi ). An analogous description holds for G = SL(X).

15.2 Symplectic, Orthogonal, and Unitary groups Let α be an additive norm on X. Let X ∗ be the dual space of X. For λ ∈ X ∗ − {0} define α∗ (λ) =

inf

x ∈X−{0}

{ω(λ(x)) − α(x)} ∈ R ∪ {−∞}.

Note that the function ω(λ(x)) − α(x) is invariant under rescaling. Therefore the infimum can be taken over x ∈ L − mL, where L is an arbitrary lattice in X.

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The infimum is always achieved due to Corollary 15.1.7, but it may well equal −∞. In fact, we have the following result. Lemma 15.2.1 λ ∈ X ∗ − {0}.

The norm α is splitting if and only if α∗ (λ)  −∞ for all

Proof If α is splitting, then the ball Lα,0 is a lattice due to Lemma 15.1.10. On Lα,0 − mLα,0 the norm α takes the value 0. The functional λ is not identically zero on any lattice, and maps lattices in X to lattices in k. Therefore ω(λ(x)) is bounded below on L, so the infimum is not equal to −∞. Assume now that α is not splitting. According to Lemma 15.1.13 there exists ω with  α(x0 ) = ∞. There exists λ ∈ X ∗ whose continuous extension to x0 ∈ X ω  satisfies λ(x0 )  0. Let {xn } ⊂ X be a sequence converging to x0 . Then X lim λ(xn ) = λ(x0 )  0. This sequence shows that inf{ω(λ(x)) − α(x) | x ∈ X − {0}} = −∞.  Recall that given a basis (v1, . . . , vn ) there exists a unique basis (v1∗, . . . , vn∗ ) such that v j∗ (vi ) = δi, j , where δi, j is the Kronecker delta function. Lemma 15.2.2

Let α be a splitting additive norm.

(1) The map α∗ is a splitting additive norm. (2) If (v1, . . . , vn ) is a splitting basis for α, the dual basis (v1∗, . . . , vn∗ ) is a splitting basis for α∗ and α∗ (vi∗ ) = −α(vi ). (3) Under the identification X ∗∗  X the identity (α∗ )∗ = α holds. (4) Given c ∈ R, (α + c)∗ = α∗ − c. (5) If β is another splitting additive norm and α  β, then β∗  α∗ . Proof Consider x = j x j v j with x j ∈ k. Then ω(λ(x))  min{ω(x j ) + ω(λ(v j ))}, j

while α(x) = min j {ω(x j ) + α(v j )}. Therefore, ω(λ(x)) − α(x)  min{ω(λ(v j )) − α(v j )}. j

This minimum is obtained for some j = j0 , which means that the above inequality becomes an equality when x = v j0 , showing α∗ (λ) = min j {ω(λ(v j ))− α(v j )}. Taking λ = vi∗ we obtain α∗ (vi∗ ) = −α(vi ). With this the previous equation can be rewritten as α∗ (λ) = min j {ω(λ(v j )) + α∗ (v j∗ )}. This confirms that α∗ is an additive norm and that (v1∗, . . . , vn∗ ) is a splitting basis for α∗ . The equation (α∗ )∗ = α follows immediately from (α∗ )∗ (vi∗∗ ) = −α∗ (vi∗ ) = α(vi ),

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since the identification X ∗∗  X identifies vi∗∗ with vi . The equation (α + c)∗ = α∗ − c follows as well. The inequality β∗  α∗ follows immediately from α  β  and the definition of α∗ . Definition 15.2.3 The norm α∗ is called the dual norm of α. Remark 15.2.4 The definition of α∗ is a special case of a more general construction. If X and W are finite-dimensional k-vector spaces and α : X → R ∪ {∞} and β : W → R ∪ {∞} are additive norms and α is splitting, we can define the additive norm Hom(α, β) by Hom(α, β) : Homk (X, W) → R ∪ {∞},

u →

inf

x ∈X−{0}

(β(u(x)) − α(x)).

Then α∗ = Hom(α, ω). We will not have use for this more general definition. For the rest of this section we assume char(f)  2. See Remark 15.2.12 for a brief discussion of this assumption. Let σ be an automorphism of k of order 1 or 2 that preserves ω and let k 0 ⊂ k be its fixed field. Let ε ∈ {±1}. Let X be a finite-dimensional k-vector space and let ·, · : X × X → k be an ε-Hermitian form, that is, (1) v1 + v2, w = v1, w + v2, w , v, w1 + w2 = v, w1 + v, w2 . (2) av, w = a v, w , v, aw = σ(a) v, w . (3) v, w = εσ( w, v ). We remark that our condition (2) is different from the convention of [BT87b], cf. page 144 of loc. cit., which results in differences in some formulas, especially the parameterization of root groups. We assume that ·, · is non-degenerate, which means that the map X σ → X∗ defined by w → −, w is an isomorphism of k-vector spaces. Here X σ = X ⊗k k where the second factor is equipped with the k-module structure given by the map σ : k → k. Let G = U(X, ·, · ) be the unitary group associated to this data. The possible cases of this set-up are the following: (1) σ = id, ε = 1. Then ·, · is a symmetric bilinear form and U(X, ·, · ) is the corresponding orthogonal group. (2) σ = id, ε = −1. Then ·, · is an alternating bilinear form and U(X, ·, · ) is the corresponding symplectic group. (3) σ  id, ε = 1. Then ·, · is a Hermitian form and U(X, ·, · ) is the corresponding unitary group. (4) σ  id, ε = −1. If η ∈ k is any element with η + σ(η) = 0, then η · ·, · is a Hermitian form and U(X, ·, · ) = U(X, η · ·, · ) is again a unitary group, so this case produces the same groups as the previous case.

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Note that the set of additive norms on X σ is canonically identified with the set of additive norms on X. Definition 15.2.5 (1) Given a splitting additive norm α : X → R ∪ {∞} we set α∨ : X → R ∪ {∞} to be the splitting additive norm obtained by transporting α∗ under the isomorphism X σ → X ∗ . (2) A splitting additive norm α is called self-dual if α = α∨ . In particular, a self-dual norm α satisfies the following inequality α(x) + α(y)  ω( x, y ).

(15.2.1)

Lemma 15.2.6 An additive norm that satisfies (15.2.1) is automatically splitting. ω =  k ⊗k X of X with respect to the topology Proof Consider the completion X given by ω. We extend continuously the ε-Hermitian form ·, · to a  k-valued ω still satisfies ω . The continuous extension  α of α to X ε-Hermitian form on X ω . The claim follows (15.2.1). In particular, it satisfies  α(x)  ∞ for 0  x ∈ X from Lemma 15.1.12.



Remark 15.2.7 Given a basis (v1, . . . , vn ), let (v1∨, . . . , vn∨ ) be the transport of the dual basis (v1∗, . . . , vn∗ ) of X ∗ under the isomorphism X σ → X ∗ . In other words, vi , v j∨ = δi, j . Consider a basis (v1, . . . , vn ) that is self-dual, that is, vi∨ = vi . Thus the matrix of ·, · with respect to this matrix is the identity matrix. If α is a self-dual norm admitting (v1, . . . , vn ) as a splitting basis, then we must have α(vi ) = 0 for all i = 1, . . . , n. Thus, there is a unique self-dual norm admitting a self-dual basis as a splitting basis. Consider now the opposite case: a basis (v1, . . . , vn ) for which vi∨ = xi vn+1−i for some xi ∈ o× . For an additive norm α with splitting basis (v1, . . . , vn ) we have α∨ (vi ) = −α(vn+1−i ). Thus α is self-dual if and only if α(vn+1−i ) = −α(vi ). There is a unique self-dual norm α for which (v1, . . . , vn ) is a splitting basis and α(vi ) = 0. Any other self-dual norm admitting (v1, . . . , vn ) as a splitting basis has the form α + c for c = (c1, . . . , cn ) ∈ Rn with cn+1−i = −ci . A basis as above exists if and only if X is split. More generally, any X admits a Witt decomposition X = X + ⊕ X 0 ⊕ X − , where X 0 is anisotropic, X + and X − are totally isotropic, and ·, · induces an isomorphism X +,σ → X −,∗ . Let α be a splitting additive norm that splits over this decomposition. If we choose a splitting basis (x1+, . . . , xr+ ) for α|X + and (x10, . . . , xa0 ) for α|X 0 , and let (x1−, . . . , xr− ) be the basis of X − dual to (x1+, . . . , xr+ ) with respect to the isomorphism X +,σ → X −,∗ , then (x1+, . . . , xr+, x10, . . . , xa0 , xr−, . . . , x1− ) is a splitting basis for α.

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Lemma 15.2.8 If X is anisotropic, then there is a unique self-dual norm. It is given by α(x) = ω( x, x )/2. Proof The main part of the proof is the following inequality: for every x, y ∈ X : 2ω( x, y )  ω( x, x ) + ω( y, y ).

(15.2.2)

Admitting that, we check that α is a norm. The validity of AN 3 is immediate from the assumption that X is anisotropic, the validity of AN 2 is immediate from the definition of α, and the validity of AN 1 follows from ω( x + y, x + y )/2 = ω( x, x + y, y + x, y + y, x )/2 

 min ω( x, x ), ω( y, y ) /2, where for the last inequality we have used

 ω( y, x ) = ω( x, y )  min ω( x, x ), ω( y, y ) , which in turn is implied by (15.2.2). We have thus seen that α is an additive norm. That it is self-dual is immediate from (15.2.2) and the fact that the infimum in Definition 15.2.3 is attained for x = y. Let now β be any self-dual norm on X. Then (15.2.1) implies β  α. On the other hand, Lemma 15.2.2(5) and the self-duality of both β and α implies α  β, hence α = β. It remains to prove (15.2.2). Consider first the case that σ = 1. Then necessarily ε = 1. Choose an orthogonal basis (v1, . . . , vn ) of X and let ai = vi , vi . Upon rescaling vi we may assume ω(ai ) ∈ {0, 1}. Reorder so that ω(a1 ) = · · · = ω(am ) = 0 and ω(am+1 ) = · · · = ω(an ) = 1. Identifying X n ai xi yi . with k n using that basis we have x, y = i=1 Choose a uniformizer π ∈ k and consider the quadratic forms q1 (x) = m n 2 −1 2 m and k n−m . The orthogonal i=1 ai xi and q2 (x) = i=m+1 π ai xi on k direct sum of q1 and q2 is the quadratic form x, x . Since the latter is assumed anisotropic, so are q1 and q2 . Let q1 and q2 be the quadratic forms on f m and f n−m induced by q1 and q2 . Hensel’s lemma implies that q1 and q2 are also anisotropic (this is part of what is known as Springer’s theorem). When proving (15.2.2) we are free to rescale x and y. We may thus assume that all coordinates of x are integral and at least one of them is a unit. We assume the same for y. Write x = (x1, x2 ) with x1 ∈ o m , x2 ∈ o n−m and let x1 ∈ f m and x2 ∈ f n−m be their reductions. At least one of x1 and x2 is non-zero. Therefore at least one of q1 (x1 ) or q2 (x2 ) is a unit. We have ω( x, x ) = 0 if and only if q1 (x1 ) is a unit. Write analogously y = (y1, y2 ), y1 ∈ f m , y2 ∈ f n−m and conclude that at least one of q1 (y1 ) or q2 (y2 ) is a unit, and that ω( y, y ) = 0 if and only if q1 (y1 ) is a unit. Since x, y is integral, there is nothing to

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prove if ω( x, x ) = 0 = ω( y, y ). If ω( x, x ) = 1 then q1 (x1 ) = 0, thus x1, . . . , xm ∈ m, thus x, y ∈ m. The same conclusion holds if ω( y, y ) = 1. This completes the proof. The case of σ  1 is analogous and uses the εHermitian analog of Springer’s theorem, which is proved in [Lar06].  The following simple lemma will be used in the proof of the next next proposition. Lemma 15.2.9 Let α be an additive norm and X = l ⊕ H a decomposition over which α splits with dim(l) = 1. If v ∈ l and w ∈ H satisfy α(v)  α(w) and l  = v + w , then α splits over X = l  ⊕ H. Proof

That α splits over l ⊕ H is equivalent to the equality α(v + z) = min(α(v), α(z)),

for all z ∈ H. That in turn is equivalent to α(v) = sup{α(v + H)}, using Fact 15.1.2. But we check that sup{α(v + w + H)} = sup{α(v + H)} = α(v) = min{α(v), α(w)} = α(v + w).  Proposition 15.2.10 If α is a self-dual additive norm on X there exists a Witt decomposition of X over which α splits. Proof Let X = X + ⊕ X 0 ⊕ X − be an arbitrary Witt decomposition. Let Y − ⊂ X be a splitting complement of X + ⊕ X 0 for α. Since α is self-dual, we have for any x + ∈ X + α(x + ) = inf{ω( x +, x ) − α(x) | x ∈ X } = inf{ω( x +, y ) − α(y) | y ∈ Y − },

(15.2.3)

where the second equality follows from writing x = x1 + y with x1 ∈ X + ⊕ X 0 , y ∈ Y − and noting that x +, x1 + y = x +, y , because X + is orthogonal to X + ⊕ X 0 , while α(x1 + y) = min{α(x1 ), α(y)}  α(y), because Y − is a splitting complement of X + ⊕ X 0 . Restricting the isomorphism X σ → X∗ given by ·, · to X +,σ induces an isomorphism X +,σ → (X/X + ⊕ X 0 )∗ . Since the inclusion Y − → X induces an isomorphism Y − → X/X + ⊕ X 0 we see that ·, · restricts to a non-degenerate sesquilinear pairing X + ⊗ Y − → k. Let (y1−, . . . , yr− ) be a splitting basis for α|Y − and let (x1+, . . . , xr+ ) be the basis of X + dual to it with respect to ·, · . Equation (15.2.3) implies that (x1+, . . . , xr+ ) is a splitting basis for α|X + . The subspace Y − may not be totally isotropic and we will now modify it to ensure that it is, while maintaining its property of being a splitting complement of X + ⊕ X 0 for

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α. Let ai, j = yi−, y −j for 1  i < j  r, ai, j = 0 for 1  j < i  r, and ai,i = yi−, yi− /2. Define yi = yi− − rj=1 ai, j x +j . Clearly (y1 , . . . , yr ) is still dual to (x1+, . . . , xr+ ). Let us check that the span Y  of (y1 , . . . , yr ) is totally isotropic. We have yi, y j = yi−, y −j − (ai, j + εσ(a j,i )). If i < j then ai, j = yi−, y −j and a j,i = 0. If i = j then (ai,i + εσ(ai,i )) = yi−, yi− . Either way we obtain yi, y j = 0 and conclude that Y  is a totally isotropic space. Next we check that (y1 , . . . , yr ) is still a splitting basis for α|Y  . The selfduality of α implies ω( x, y )  α(x) + α(y) for all x, y ∈ X. By assumption, for i  j we have ω(ai, j ) = ω( yi−, y −j )  α(yi− ) + α(y −j ) = α(yi− ) − α(x +j ). Hence α( rj=1 ai, j x +j )  α(yi− ). The claim follows from Lemma 15.2.9. Let Y 0 ⊂ X + ⊕ X 0 be a splitting complement of X + for α|X + ⊕X 0 . Since X + is the kernel of the restriction of ·, · to X + ⊕ X 0 , the space Y 0 is anisotropic. Thus X = X + ⊕ Y 0 ⊕ Y  is a Witt decomposition and the norm α splits over it.  Let B(X, ·, · ) be the set of all self-dual additive norms on X. Given a Witt decomposition X = X + ⊕ X 0 ⊕ X − and a decomposition X + = l1+ ⊕ · · · ⊕ lr+ into lines, we obtain dually the decomposition X − = l1− ⊕ · · · ⊕ lr− . Define  A( li , ·, · ) to be set of all self-dual additive norms that split over l1+ ⊕ · · · ⊕ lr+ ⊕ X 0 ⊕ lr− ⊕ · · · ⊕ l1− .

 Lemma 15.2.11 (1) The set B(X, ·, · ) is the union of A( li , ·, · ) over all Witt decompositions X = X + ⊕ X 0 ⊕ X − and all decomposition X + = l1+ ⊕ · · · ⊕ lr+ into lines.  (2) The set A( li , ·, · ) is an affine space under the vector space Rr , where c = (c1, . . . , cr ) ∈ Rr sends α to α + (c1, . . . , cr , 0, . . . , 0, −cr , . . . , −c1 ).  (3) The choice of 0  xi+ ∈ li+ specifies a unique α ∈ A( li , ·, · ) by α(xi+ ) = 0 = α(xi− ), where (x1−, . . . , xr− ) is the basis dual to (x1+, . . . , xr+ ). Proof The first claim follows from Proposition 15.2.10, for we may obtain a splitting basis from the decomposition X = X + ⊕ X 0 ⊕ X − as described in Remark 15.2.7. For the second and third points we apply Lemma 15.2.8 and follow Remark 15.2.7 to see that if (x1+, . . . , xr+, x10, . . . , xa0 , xr−, . . . , x1− ) is a Witt basis, then α(x1+ ) = · · · = α(xr+ ) = 0 specifies a unique self-dual additive norm that splits

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over X + ⊕ X 0 ⊕ X − , and any other such is given by α + c for c ∈ Rn given by  (c1, . . . , cr , 0, . . . , 0, −cr , . . . , −c1 ). Remark 15.2.12 We briefly discuss our assumption char(f)  2. The only place this assumption was used was in the proof of Proposition 15.2.10, in the choice of ai,i . What is required from ai,i ∈ k is that ai,i + εσ(ai,i ) = yi−, yi− and ω(ai,i ) = ω( yi−, yi− ). It is easy to see that in that proof, and hence within this section, the assumption can be dropped when either σ = 1 and ε = −1 (the symplectic case) or when σ  1 and k/k0 is unramified (the unramified unitary case). Indeed, in the first case we have yi−, yi− = 0 so we can simply take ai,i = 0. In the second case there exists λ ∈ k with λ + σ(λ) = 1 and ω(λ) = 0, see Lemma 2.8.1. Thus we can take ai,i = λ yi−, yi− . In the other case however this assumption cannot be dropped. The reason is that the notion of a self-dual norm is no longer the correct notion to use. Instead, one must work with the more general notion of a norm that is maxi-minorant for a pair ( ·, · , q) consisting of an ε-Hermitian form and a pseudo-quadratic form [BT87b, Definition 2.1]. A norm is minorant for ·, · if it satisfies (15.2.1), and maxi-minorant if it is maximal among the minorant norms. It is shown in [BT87b, §2.5] that being maxi-minorant for ·, · is equivalent to being self-dual when ·, · is non-degenerate. It is further discussed in [BT87b, Remark 2.2(3)] that when either char(f)  2 or k/k 0 is an unramified quadratic extension, then being maxi-minorant for ·, · is equivalent to being maxi-minorant for ( ·, · , q). Construction 15.2.13 Let X = X + ⊕ X 0 ⊕ X − be a Witt decomposition, let (v1, . . . , vr ) be a basis for X + and let (v−1, . . . , v−r ) be the corresponding dual basis of X − , such that vi , v j = δi,−j . Let li be the line spanned by vi . The stabilizer of the lines l1, . . . , lr , l−r , . . . , l−1 is a minimal Levi subgroup M of G. The subgroup of M that fixes X 0 pointwise is a torus T, equal to (Rk/k0 Gm )r . The maximal split torus S in this torus is a maximal split torus in M and in G. In particular, the k0 -rank of G equals r. We have M = T × M 0 , where M 0 is the identity component of U(X 0, ·, · |X 0 ). The group G is split if S = T = M. This is the case when σ = 1 and X 0 is at most 1-dimensional. The group G is quasi-split if M, equivalently M 0 , is a torus. This is the case when X 0 is at most 2-dimensional and σ = 1, or X 0 is at most 1-dimensional and σ  1. We list the possible cases that can occur. Let d0 = dim(X 0 ). (1) Symplectic: σ = 1 and ε = −1. Then d0 = 0 and G is split. (2) Odd orthogonal: σ = 1, ε = 1, a odd. Then G is split when d0 = 1 and G is not quasi-split when d0 > 1.

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(3) Even orthogonal: σ = 1, ε = 1, d0 even. Then G is split when d0 = 0, quasi-split when d0 = 2, and not quasi-split when d0 > 2. When d0 = 2 there exists a separable quadratic extension /k and ζ ∈ k × such that X 0 =  with x1, x2 = ζ(x1 · x2 + x1 · x2 ). (4) Odd unitary: σ  1, d0 odd. Then G is quasi-split (but not split) when d0 = 1 and G is not quasi-split when d0 > 1. (5) Even unitary: σ  1, d0 even. Then G is quasi-split (but not split) when d0 = 0, and G is not quasi-split when d0 > 0. Let ai : M → Gm be the character specified by mvi = ai (m)vi . Then a−i = −σ(ai ). By restriction to S we obtain a character of S which we also denote by ai . Then a−i = −ai in X∗ (S). The root system of G relative to S consists of the following characters. (1) ai + a j for i  j. (2) ai whenever a > 0, that is, in the odd orthogonal, even non-split orthogonal, odd unitary, and non-quasi-split even unitary cases. (3) 2ai in the symplectic and unitary cases. Thus the type of Φ is Cr in the symplectic and quasi-split even unitary cases, Br in the odd orthogonal and even non-split orthogonal cases, Dr in the even split orthogonal case, and BCr in the odd unitary and even non-quasi-split unitary cases. We can parameterize the relative root groups as follows. Note that these formulas are slightly different from [BT87b, (23),(24)], because our convention for ε-Hermitian forms is different as remarked earlier, and also our convention mvi = ai (m)vi is different from the convention on page 157 of loc. cit. If G is symplectic, we set ε(i) = 1 if i > 0 and ε(i) = −1 if i < 0. The root group for the root ai, j is parameterized by ui, j : Ga → G,

⎧ v , ⎪ ⎪ ⎨ k ⎪ ui, j (ξ)vk = v−i + ε(− j)ξ v j , ⎪ ⎪ ⎪ v − ε(i)ξ v , i ⎩ −j

while the root group for the root 2ai is parameterized by ) vk ui : Ga → G, ui (ξ)vk = v−i − ε(−i)ξ vi

k  −i, − j k = −i k = − j,

k  −i k = −i.

Note that we get in particular a pinning for G. Consider now G orthogonal or unitary. In the unitary case we assume, as we may, that ε = 1. The root group for ai, j is isomorphic to Ga in the orthogonal

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case and to Rk/k0 Ga in the unitary case. Taking ξ in Ga or Rk/k0 Ga , respectively, we map it to the element ui, j (ξ) ∈ G specified by ⎧ v , ⎪ ⎪ ⎨ k ⎪ ui, j (ξ)vk = v−i + σ(ξ)v j , ⎪ ⎪ ⎪v − ξ v , i ⎩ −j

k  −i, − j k = −i k = − j.

The root group for 2ai occurs only in the unitary case and is isomorphic to R0k/k Ga (cf. (2.7.1)). An element ξ ∈ R0k/k Ga is mapped to the element 0 0 ui (ξ) ∈ G acting trivially on X 0 and specified on X + ⊕ X − by ) k  −i vk , ui (ξ)vk = v−i − ξ vi , k = −i. The k 0 -points of the root group for ai are parameterized by pairs (x, ξ) with x ∈ X 0 and ξ ∈ k satisfying ξ + σ(ξ) = x, x . This pair corresponds to the element ui (x, ξ) ∈ G that acts on X0 by ui (x, ξ)v = v − v, x vi and on X + ⊕ X − by

) ui (x, ξ)vk =

vk ,

k  −i

v−i + x − ξ vi ,

k = −i.

When G is quasi-split we can be a bit more explicit about this last parameterization. For an odd orthogonal group X 0 is a 1-dimensional space and we can fix a basis v0 with the property v0, v0 = 1. This gives the isomorphism A1 → X 0 sending x to xv0 . We obtain a parameterization ui : Ga → G of the root group as follows ui : Ga → G,

⎧ v , ⎪ ⎪ ⎨ k ⎪ ui (x)vk = v−i + x v0 − ⎪ ⎪ ⎪v − x v , i ⎩ 0

k  −i, 0 x2 2 vi ,

k = −i k = 0.

The basis element v0 can be replaced only by −v0 , which changes ui (x) to ui (−x). When G is a non-split but quasi-split even orthogonal group, X 0 is a nonsplit 2-dimensional space that splits over a separable quadratic extension /k. It is well known that such a space is isomorphic to R/k A1 equipped with the symmetric bilinear form (x, y) = a · tr/k (x y) for some a ∈ k × , well defined up to N/k ( × ), where y¯ denotes the Galois conjugate of y. We briefly recall the argument. Choose a Witt basis (x+, x− ) of  ⊗k X 0 . Then (x+, x− ) is also a

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Witt basis and must therefore be equal to either (a x+, a−1 x− ) or (ax−, a−1 x+ ) for some a ∈  × . In the first case we see aa = 1, so by Hilbert’s Theorem 90 there exists b ∈  × with a = b−1 b, and then (bx+, b−1 x− ) is a Witt basis of X 0 , contradicting the assumption that X 0 is not split. Therefore we must have x+ = a x− , and we see a−1 a = 1, that is, a ∈ k × . The map l → tr/k (l x+ ) gives the desired isomorphism R/k A1 → X 0 . The Witt basis (x+, x− ) can only be replaced by (z x+, z−1 x− ) or (z x−, z−1 x+ ) and in both cases we see that a is multiplied by a norm from . We now identify X 0 with R/k A1 with the symmetric bilinear form a · tr/k (x y) for some a ∈ k × . As a basis for X 0 we choose v0 = 1 and v0 = η, where η is an arbitrary element of  with η+η = 0. We obtain a parameterization ui : R/k Ga → G of the root group as follows ⎧ ⎪ vk , ⎪ ⎪ ⎪ ⎪ ⎪ v + x − a x x · ei , ⎪ ⎪ ⎨ −i ⎪ ui (x)vk = v0 − a(x + x)vi , ⎪ ⎪ ⎪ ⎪ ⎪ v0 + a, ⎪ ⎪ ⎪ ⎪ eta(x − x)vi , ⎩

k  −i, 0, 0 k = −i k=0 k = 0 .

The choice of η can be changed only by multiplying η by an element of k × , but this has no influence on ui (x). When G is an odd unitary group, then X 0 is a 1-dimensional Hermitian space. Choose v0 ∈ X 0 with v0, v0 = 1. Then we obtain the isomorphism k → X 0 sending x to x v0 . The pairs (x, ξ) such that x ∈ k and ξ ∈ k satisfy ξ +σ(ξ) = x v0, x v0 = x σ(x) are precisely the elements of the group Uk/k0 (k0 ) of (2.7.2). We obtain the parameterization of the root group for the root ai by ui : Uk/k0 → G given by ⎧ v , ⎪ ⎪ ⎨ k ⎪ ui ((x, ξ))vk = v−i + x v0 − ξ vi , ⎪ ⎪ ⎪ v − σ(x)v , i ⎩ 0

k  −i, 0 k = −i k=0

We can replace v0 only by zv0 with z ∈ k satisfying zσ(z) = 1. Then ui ((a, ξ)) is replaced by ui ((za, ξ)). Note that in all of the above cases, the basis (v1, . . . , vr ) leads to a k0 -pinning of the quasi-split group G that is well defined up to units. We continue to assume that G is quasi-split, X = X + ⊕ X 0 ⊕ X − is a Witt decomposition, and X + = l1 ⊕ · · · ⊕ lr is a decomposition into lines. As just argued this specifies a maximal split torus S ⊂ G. Its centralizer T is a maximally split maximal torus. We have the associated apartment A(T)

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in the Bruhat–Tits building B(G). Let 0  vi ∈ li . Let α ∈ A(⊕i li , ·, · ) be the element of Lemma 15.2.11. Let p ∈ A(T) be the Chevalley valuation associated to a pinning obtained from the basis (v1, . . . , vr ) as above. Since that pinning is well defined up to units, p depends only on (v1, . . . , vr ) but not on the choice of pinning. The basis (v1, . . . , vr ) further gives the basis (a1, . . . , ar ) of X∗ (S) ⊗Z R, hence dually an isomorphism VT → Rr . Since both A(T)  and A( i li , ·, · ) are affine spaces over Rr , the points p and α specify an  isomorphism A(T) → A( i li , ·, · ) of affine spaces. Proposition 15.2.14

 (1) The isomorphism A(T) → A( i li , ·, · ) of affine spaces carrying p to α is independent of the choice of (v1, . . . , vr ). (2) The stabilizers in G(k0 ) of α and p coincide. (3) There is a unique G(k0 )-equivariant bijection B(G) → B(X, ·, · ) re stricting to the isomorphism A(T) → A( i li , ·, · ). It is independent of the choice of Witt decomposition.

Proof Since the assignments of p and α to X = X + ⊕ X 0 ⊕ X − and (v1, . . . , vr ) are G(k 0 )-equivariant, and any two choices of (v1, . . . , vr ) in l1 ⊕ · · · ⊕ lr are conjugate under S(k 0 ), the first point follows. The third point follows as well from the same argument, the fact that both B(G) is the union of all apartments  A(T), the fact that B(X, ·, · ) is the union of all apartments A( i li , ·, · ) (cf. Lemma 15.2.11), and the second point. Note that any two Witt decompositions are G(k 0 )-conjugate. It remains to prove the second point. In all cases the center of G is anisotropic and therefore G(k0 ) = G(k 0 )1 . The stabilizer of p in G(k 0 ) is a maximal bounded subgroup according to Theorem 4.2.15. The stabilizer of α in G(k 0 ) is a bounded subgroup: indeed, under the standard representation G → GL(X) the stabilizer of α is identified with the intersection with G(k 0 ) of the stabilizer of α in GL(X)(k), which is a bounded subgroup of GL(X)(k) according to Remark 15.1.32, Lemma 4.3.3, and Proposition 7.7.1. Thus it is enough to show that the stabilizer of p in G(k0 ) is contained in the stabilizer of α in G(k0 ). But Proposition 7.7.5 states that the stabilizer of p in G(k 0 ) is the subgroup generated by T(k 0 )1 = a1 (o× ) · · · ar (o× ) · U(X 0, ·, · |X 0 )0 and the root groups  uα (o). Each of these groups fixes the norm α and the proof is complete. As in the case of the general linear group, the simplices of B(G) can be described in B(X, ·, · ) using lattice chains. We now introduce a notion of duality for graded lattice chains that will correspond to the duality for norms under the correspondence of Proposition 15.1.21.

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Definition 15.2.15

Let L ⊂ X be a lattice. Its dual lattice is defined by

L ∨ = {x ∈ X | for every y ∈ L, x, y ∈ o}. We record the obvious relation L1 ⊂ L2 ⇒ L2∨ ⊂ L1∨ .

(15.2.4)

The natural pairing L ⊗ L ∨ → o descends to a perfect pairing L/mL ⊗ L ∨ /mL ∨ → f of f-vector spaces. Fact 15.2.16

If we define

L ∼ = mL ∨ = {x ∈ X | for every y ∈ L, x, y ∈ m} then for an additive norm α and r ∈ R the following equalities hold (Lα,r )∼ = Lα∨ ,(−r)+,

(Lα,r+ )∼ = Lα∨ ,−r .

Proof The first equality is an immediate computation using a splitting basis (v1, . . . , vn ) for α and the relation α∨ (vi∨ ) = −α(vi ). The second equality follows from the first and (15.2.4).  Definition 15.2.17 Let (L, c) be a graded periodic lattice chain. We define the dual graded periodic lattice chain (L∨, c∨ ) as follows. (1) L∨ = {L ∨ | L ∈ L} = {L ∼ | L ∈ L}. (2) c∨ (L ∼ ) = −c(L − ), where L − is the smallest member of L that properly contains L. Fact 15.2.18 Under the correspondence of Proposition 15.1.21, if the additive norm α corresponds to the graded periodic lattice chain (L, c), then the dual norm α∨ corresponds to the dual graded periodic lattice chain (L∨, c∨ ). In particular, B(X, ·, · ) is identified with the set of self-dual graded periodic lattice chains. Remark 15.2.19 We see that (L, c) → (L, c)∨ is an involution on B(X) and that B(X, ·, · ) is the set of fixed points of that involution. This is a special case of Theorem 12.7.1. 15.2.20 The simplicial structure of B(X, ·, · ) is defined just as in 15.1.23, but one now uses only self-dual graded periodic lattice chains. Unlike the situation of 15.1.23, it is no longer true that vertices correspond to lattice chains whose

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segments are individual lattices. Indeed, while some vertices are of this form, others are not. More precisely, let L be a self-dual periodic lattice chain. Given two lattices M ⊂ L ⊂ X we have L ∨ ⊂ M ∨ and (m · L)∨ = m−1 · (L ∨ ). From this one concludes that there exists a unique lattice L0 ∈ L such that L0∨ ⊂ L0 and there exists no L ∈ L such that L0∨  L  L0 . The simplex of B(X, ·, · ) corresponding to L is a vertex if and only if L = {m n · L0 | n ∈ Z} ∪ {m n · L0∨ | n ∈ Z}. The following three mutually exclusive cases occur. (1) L0∨ = L0 . (2) L0∨ = m · L0 . (3) m · L0  L0∨  L0 . In the first two cases, L = {m n · L0 | n ∈ Z}, and L is also a vertex in B(X). In the third case, L  {m n · L0 | n ∈ Z}, and L is not a vertex of B(X), but rather a simplex of dimension 1, whose two vertices in B(X) are the non-self-dual periodic lattice chains L1 = {m n · L0 | n ∈ Z} and L2 = {m n · L0∨ | n ∈ Z} that satisfy L∨1 = L2 . 15.2.21 We now describe the groups G(k0 )0x ⊂ G(k0 )x and Gx0 (f0 ), in a manner similar to 15.1.33. Since the center of G is anisotropic we have G(k 0 )1 = G(k0 ). However, G(k 0 )0 may be a proper subgroup. More precisely, G(k 0 )0 = G(k 0 ) in the following cases: (1) ε = −1 and σ = 1, that is, G is a symplectic group; (2) k/k 0 is an unramified quadratic extension, that is, G is an unramified unitary group. When ε = 1 and σ = 1, that is, when G is a special orthogonal group, then G(k0 )0 is the subgroup of index 2 of G(k0 ) consisting of those elements whose spinor norm is a unit. When k/k 0 is a ramified quadratic extension, then G(k0 )0 is the subgroup of index 2 consisting of elements whose determinant, an element of k × with trivial k/k0 norm, and hence automatically a unit, is a principal unit. Let L be a self-dual periodic lattice chain and let F be the corresponding facet of B(X, ·, · ). Let c : L → R be the equispaced grading of L, so that the point x ∈ B(X, ·, · ) corresponding to the graded periodic self-dual lattice chain (L, c) is the barycenter of F. Then G(k 0 )x is the stabilizer of the lattice chain L in G(k0 ). It automatically stabilizes each individual lattice. Assume first that L corresponds to a vertex and let L0 ∈ L be the unique lattice as in 15.2.20, so that m · L0 ⊆ L0∨ ⊆ L0 . Consider the two f-vector spaces V0 = L0 /L0∨ and V1 = L0∨ /mL0 . Choose a uniformizer π ∈ k such that σ(π) = π

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if k/k 0 is unramified or trivial, and σ(π) = −π if k/k0 is ramified (recall we are assuming p  2). The restriction of the form ·, · to L0∨ × L0∨ induces a non-degenerate form ·, · 1 : V1 × V1 → f which is ε-Hermitian with respect to the extension f/f0 . The restriction of the form π · ·, · to L0 × L0 induces a non-degenerate form ·, · 0 : V0 × V0 → f that is ε-Hermitian with respect to the extension f/f0 unless k/k 0 is ramified, in which case it is (−ε)-Hermitian (with respect to the trivial extension f = f0 ). The reductive special fiber of Gxb injects into U(V0, ·, · 0 ) × U(V1, ·, · 1 ), while the reductive special fiber of Gx0 is isomorphic to U(V0, ·, · 0 )◦ × U(V1, ·, · 1 )◦ . Given a general self-dual periodic lattice chain L, again let L0 ∈ L be the unique lattice as in 15.2.20, so that m · L0 ⊆ L0∨ ⊆ L0 . The self-dual periodic lattice chain generated by L0 corresponds to a vertex y of the facet F corresponding to L. We have just described the reductive special fiber of Gy0 . The lattices in L between m · L0 and L0∨ project onto a flag in V1 that is self-dual with respect to ·, · 1 , while the lattices between L0∨ and L0 project onto a flag in V0 that is self-dual with respect to ·, · 0 . These flags determine a parabolic subgroup in U(V0, ·, · 0 )◦ ×U(V1, ·, · 1 )◦ , and this parabolic subgroup is the image of the special fiber of GF0 in the reductive special fiber of Gy0 . The reductive special fiber of GF0 is the Levi quotient of this parabolic subgroup. We now give a few examples of self-dual lattice chains and the simplices that they describe, as well as the reductive special fibers of some parahoric subgroups. Example 15.2.22 Consider the group Sp(2n). Thus X = k 2n and the standard basis is a Witt basis. Let us denote by (a1, . . . , a2n ) ∈ Z2n the lattice m a1 ⊕ m a2 ⊕ · · · ⊕m a2n . Then (a1, . . . , a2n )∨ = (−a2n, −a2n−1, . . . , −a2, −a1 ). Self-dual periodic lattice chains comprised of such lattices are the facets of the standard apartment. As an illustration, consider the case n = 2. The lattice L = (0, 0, 0, 0) satisfies L ∨ = L, that is, it is a self-dual lattice. In particular, the periodic lattice chain {mi L | i ∈ Z} it generates is also self-dual. This lattice chain is a vertex in B(X, ·, · ). It is also a vertex in B(X). The lattice L = (1, 1, 0, 0) satisfies L ∨ = m−1 L. It is not self-dual, but the periodic lattice chain it generates is self-dual. This lattice chain is a vertex in B(X, ·, · ). It is also a vertex in B(X). The lattice L = (1, 0, 0, 0) satisfies L ∨ = (0, 0, 0, −1). Therefore the segment L ⊂ L ∨ specifies a self-dual periodic lattice chain. It is a simplex in B(X, ·, · ) and also in B(X). In B(X) this simplex has non-trivial faces, namely the periodic lattice chain generated by L and the one generated by L ∨ , so this simplex is not

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a vertex, but rather has dimension 1. In B(X, ·, · ) this simplex does not have a non-trivial face and is thus a vertex. The periodic lattice chain with segment (1, 1, 0, 0) ⊂ (1, 0, 0, 0) ⊂ (0, 0, 0, 0) ⊂ (0, 0, 0, −1) is self-dual. It is a simplex of dimension 3 in B(X), that is, a chamber. It is a simplex of dimension 2 in B(X, ·, · ), hence also a chamber there. Returning to the case of a general n, let us describe the reductive special fiber of Gx0 for the various vertices x of the standard apartment following 15.2.21. Given a periodic lattice chain L that corresponds to a vertex, we choose L0 ∈ L as in 15.2.20, so that m · L0 ⊆ L0∨ ⊆ L0 . From the description of the duality operation we see that both dim(V0 ) and dim(V1 ) are even; let us denote them by 2n0 and 2n1 . We have n = n0 + n1 . The forms ·, · 0 and ·, · 1 are symplectic, so the reductive special fiber of Gx0 equals Sp(n0 ) × Sp(n1 ). This agrees with Remark 4.1.23(6): the affine root system of G is of type Cn , and removing a vertex from its affine Dynkin diagram leads to a Dynkin diagram of type Cn0 × Cn1 . Example 15.2.23 We now consider the split orthogonal groups. Thus X = k m for some natural number m, and the standard basis is a Witt basis. We use the same notation (a1, . . . , am ) ∈ Zm as in the symplectic case to denote the lattice m a1 ⊕ · · · ⊕ m am , and the duality operation is again given by (a1, . . . , am )∨ = (−am, . . . , −a1 ). As an illustration, consider the case of the 5-dimensional split quadratic space (the case of the 4-dimensional split quadratic space has the same description as that of the 4-dimensional symplectic space). Thus X = k 5 and the standard basis is again a Witt basis. Write again (a1, a2, a3, a4, a5 ) for the lattice m a1 ⊕ m a2 ⊕ m a3 ⊕ m a4 ⊕ m a5 . The duality operation is again (a1, a2, a3, a4, a5 )∨ = (−a5, −a4, −a3, −a2, −a1 ). In this example we will present the analysis in the opposite way from the previous example, by beginning with a periodic lattice chain L of highest rank, namely the one with segment (1, 1, 1, 0, 0) ⊂ (1, 1, 0, 0, 0) ⊂ (1, 0, 0, 0, 0) ⊂ (0, 0, 0, 0, 0) ⊂ (0, 0, 0, 0, −1). This is a chamber in B(X), of dimension 4. Since it is self-dual, it lies in B(X, ·, · ) and is a chamber there as well, of dimension 2. The vertices of that latter chamber are the minimal self-dual periodic lattice chains contained in L. It is immediate that these are given by the following segments: (0, 0, 0, 0, 0)

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is a segment of a self-dual periodic lattice chain that is a vertex in B(X) as well as in B(X, ·, · ); (1, 0, 0, 0, 0) ⊂ (0, 0, 0, 0, −1) is a segment of a self-dual periodic lattice chain that is of rank 2 in B(X), but a vertex in B(X, ·, · ); (1, 1, 1, 0, 0) ⊂ (1, 1, 0, 0, 0) is a segment of a self-dual periodic lattice chain that is of rank 2 in B(X), but a vertex in B(X, ·, · ). Returning to the case of a general m, let us describe the reductive special fiber of Gx0 for the various vertices x of the standard apartment following 15.2.21. Given a periodic lattice chain L that corresponds to a vertex, we choose L0 ∈ L as in 15.2.20, so that m · L0 ⊆ L0∨ ⊆ L0 . Let m = 2n + 1. If L0 corresponds to (a1, . . . , an, . . . , a2n+1 ), then an = 0. In particular, (V0, ·, · 0 ) is an even-dimensional split quadratic space, while (V1, ·, · 1 ) is an odd-dimensional split quadratic space. Let us write 2n0 and 2n1 + 1 for these dimensions. Then n = n0 + n1 and the reductive special fiber of Gx0 equals SO(V0 ) × SO(V1 ). The Dynkin diagram is of type Dn0 × Bn1 , and is obtained by removing a vertex from the affine Dynkin diagram of G, which is of type Bn . Let m = 2n. Then both (V0, ·, · 0 ) and (V1, ·, · 1 ) are even-dimensional split quadratic spaces. Let us write 2n0 and 2n1 for their dimensions. Then n = n0 + n1 and the reductive special fiber of Gx0 equals SO(V0 ) × SO(V1 ). The Dynkin diagram is of type Dn0 × Dn1 , and is obtained by removing a vertex from the affine Dynkin diagram of G, which is of type Dn . Example 15.2.24 Consider now a non-split quasi-split orthogonal group. Thus X = k 2n ⊕ , where k 2n is the split quadratic space for which the standard basis is a Witt basis and /k is a separable quadratic extension with scalar product x1, x2 = x1 x2 + x1 x2 . Let L1 ⊂ k 2n and L2 ⊂  be o-lattices and let L = L1 ⊕ L2 . Then L lies in a self-dual periodic lattice chain in X if and only if L1 lies in a self-dual periodic lattice chain in k 2n and L2 lies in a self-dual periodic lattice chain in . Lemma 15.2.8 states that there is a unique self-dual norm on , namely α(x) = ω( x, x )/2 = ω(2 x x)/2 = ω(x). Therefore by Fact 15.2.18 there is a unique self-dual periodic lattice chain in , namely the one whose members are the balls of α, that is, the powers of the maximal ideal of o . Let us denote by (a1, a2, . . . , an, b, an+1, . . . , a2n ) the lattice m a1 ⊕ · · · ⊕ m a2n mbe,

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where ai ∈ Z, b ∈ e−1 Z, and e ∈ {1, 2} is the ramification degree of /k. Duality sends (a1, . . . , an, b, an+1, . . . , a2n ) to (−a2n, . . . , −an+1,

−b−(e−1) , e

−an, . . . , −a1 ).

Indeed, we recall that by definition (o )∨ is the inverse of the different ideal −1 · m −be . It is known that D/k and therefore more generally (mbe )∨ = D/k  e−1 D/k = m , cf. [Ser79, Chapter III, §6, Proposition 13] and recall we are assuming p  2. If /k is unramified, the duality operation sends (a1, . . . , an, b, an+1, . . . , a2n ) to (−a2n, . . . , −an+1, −b, −an, . . . , −a1 ); if /k is ramified, the duality operation sends 1 (a1, . . . , an, b, an+1, . . . , a2n ) to (−a2n, . . . , −an+1, −b − , −an, . . . , −a1 ). 2 As an illustration, consider the case n = 2 and /k unramified. The analysis proceeds just like for the split group SO(3, 2). The periodic lattice chain L with segment (1, 1, 1, 0, 0) ⊂ (1, 1, 0, 0, 0) ⊂ (1, 0, 0, 0, 0) ⊂ (0, 0, 0, 0, 0) ⊂ (0, 0, 0, 0, −1). is self-dual and corresponds to a chamber in B(X, ·, · ), of dimension 2. Its vertices are given by the following three self-dual periodic lattice chains: the one with segment (0, 0, 0, 0, 0) (a vertex also in B(X)), the one with segment (1, 0, 0, 0, 0) ⊂ (0, 0, 0, 0, −1) (a 1-dimensional simplex in B(X)), and the one with segment (1, 1, 1, 0, 0) ⊂ (1, 1, 0, 0, 0) (again a 1-dimensional simplex in B(X)). One difference to the case of SO(3, 2) is that the periodic lattice chain L does not specify a chamber in B(X). Indeed, the quotient of the lattice (1, 1, 0, 0, 0) by the lattice (1, 1, 1, 0, 0) is a 1-dimensional fl -vector space, hence a 2-dimensional fk -vector space. The lines in this 2-dimensional fk -vector space correspond to the ok -lattices strictly between (1, 1, 0, 0, 0) and (1, 1, 1, 0, 0), hence to the periodic lattice chains in X that strictly contain L, hence to the chambers of B(X) that have the facet corresponding to L as a face. None of these larger periodic lattice chains is self-dual. Consider now the case n = 2 and /k ramified. The periodic lattice chain with segment (1, 1, 0, 0, 0) ⊂ (1, 0, 0, 0, 0) ⊂ (0, 0, 0, 0, 0) ⊂ (0, 0, − 21 , 0, 0) ⊂ (0, 0, − 21 , 0, −1) ⊂ (0, 0, − 12 , −1, −1) is a chamber in B(X), of dimension 5. It is also self-dual, so lies in B(X, ·, · ) and is a chamber there, of dimension 2. Its vertices are the self-dual periodic

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lattice chains with segments (0, 0, 0, 0, 0) ⊂ (0, 0, − 12 , 0, 0), (1, 0, 0, 0, 0) ⊂ (0, 0, − 12 , 0, −1) and (1, 1, 0, 0, 0) ⊂ (0, 0, − 12 , −1, −1). All of these are 1-dimensional simplices in B(X). Returning to the case of a general n, let us describe the reductive special fiber of Gx0 for the various vertices x of the standard apartment following 15.2.21. Given a periodic lattice chain L that corresponds to a vertex, we choose L0 ∈ L as in 15.2.20, so that m · L0 ⊆ L0∨ ⊆ L0 . Consider first /k unramified. If L0 = (a1, . . . , an, b, an+1, . . . , a2n ) then b = 0 and we see that (V0, ·, · ) is a split even-dimensional quadratic space, while (V1, ·, · ) is a quasi-split but non-split even-dimensional quadratic space. Denote these dimensions again by 2n0 and 2n1 +2. Then n = n0 +n1 . The reductive special fiber of Gx0 equals SO(V0 ) × SO(V1 ). The relative Dynkin diagram of this group is Dn0 × Bn1 , and is obtained by removing a vertex from the relative affine Dynkin diagram of the group G, which is of type Bn . Consider next /k ramified. If L0 = (a1, . . . , an, b, an+1, . . . , a2n ) then b = − 21 and we see that (V0, ·, · ) is a split odd-dimensional quadratic space, hence (V1, ·, · ) is also a split odd-dimensional quadratic space. Writing 2n0 + 1 and 2n1 + 1 for their dimensions, we have n = n0 + n1 . The reductive special fiber of Gx0 equals SO(V0 ) × SO(V1 ). The relative Dynkin diagram of this group is Bn0 × Bn1 , and is obtained by removing a vertex from the relative affine Dynkin diagram of the group G, which is of type Cn∨ . Example 15.2.25 Consider the quasi-split unitary group associated to an even-dimensional Hermitian space. This space is necessarily split, hence of the form X = k 2n and the standard basis is a Witt basis. If (a1, . . . , a2n ) denotes the lattice m a1 ⊕ · · · ⊕ m a2n , the duality operation is given by (a1, . . . , a2n )∨ = (−a2n, . . . , −a1 ). We will not give examples of lattice chains, as these are essentially the same as for the case of even-dimensional split quadratic or symplectic spaces. We will only discuss the reductive special fibers of the parahoric group schemes associated to vertices. For this, consider again a periodic self-dual lattice chain L corresponding to a vertex and choose a lattice L0 ∈ L as in 15.2.20, so that m · L0 ⊆ L0∨ ⊆ L0 . We see that both V0 and V1 are even-dimensional. Write 2n0 and 2n1 for their dimensions, so that n = n0 + n1 . When /k is unramified then both (V0, ·, · 0 ) and (V1, ·, · 1 ) are Hermitian spaces. The reductive special fiber of Gx0 is equal to U(V0 ) × U(V1 ).

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Its relative Dynkin diagram is of type Cn0 × Cn1 , and is obtained by removing a vertex from the relative affine Dynkin diagram of G, which is of type Cn . When /k is ramified then (V0, ·, · 0 ) is a symplectic space, while (V1, ·, · 1 ) is an quadratic space. The reductive special fiber of Gx0 is equal to Sp(V0 ) × SO(V1 ). Its relative Dynkin diagram is of type Cn0 × Dn1 , and is obtained by removing a vertex from the relative affine Dynkin diagram of G, which is of type Bn∨ . Example 15.2.26 Consider the quasi-split unitary group associated to an odddimensional Hermitian space. This space is of the form X = k 2n ⊕ k. On k 2n the standard basis is a Witt basis. On k the form is given by (x, y) → axσ(y) for some a ∈ k0× well defined up to norms from k × . We will abbreviate by (a1, . . . , an, b, an+1, . . . , a2n ) ∈ Z2n+1 the o-lattice a m 1 ⊕ · · · ⊕ m a2n ⊕ m b . The duality operation is given by (a1, . . . , an, b, an+1, . . . , a2n )∨ = (−a2n, . . . , −an+1, −b − ω(a), −an, . . . , −a1 ). Up to modifying a by norms from k × we may assume that ω(a) = 0 when k/k0 is ramified, or ω(a) ∈ {0, 1} when k/k0 is unramified. As an illustration, consider the case n = 2. In the case ω(a) = 0 the periodic lattice chain L with segment (1, 1, 0, 0, 0) ⊂ (1, 0, 0, 0, 0) ⊂ (0, 0, 0, 0, 0) ⊂ (0, 0, 0, 0, −1) ⊂ (0, 0, 0, −1, −1) is self-dual. It is a chamber in B(X) of dimension 4, as well as a chamber in B(X, ·, · ) of dimension 2. The vertices of the latter chamber are given by the self-dual periodic lattice chains with the following segments: (0, 0, 0, 0), which is a vertex in both B(X) and B(X, ·, · ); (1, 0, 0, 0, 0) ⊂ (0, 0, 0, 0, −1), which is a 1-dimensional simplex in B(X) but a vertex in B(X, ·, · ); (1, 1, 0, 0, 0) ⊂ (0, 0, 0, −1, −1), which is a 1-dimensional simplex in B(X) but a vertex in B(X, ·, · ). In the case ω(a) = 1 the periodic lattice chain L with segment (1, 1, 0, 0, 0) ⊂ (1, 0, 0, 0, 0) ⊂ (0, 0, 0, 0, 0) ⊂ (0, 0, −1, 0, 0) ⊂ (0, 0, −1, 0, −1) is a chamber in B(X) of dimension 4, as well as a chamber in B(X, ·, · ) of dimension 2. The vertices of the latter chamber are given by the self-dual periodic lattice chains with the following segments: (0, 0, 0, 0, 0) ⊂ (0, 0, −1, 0, 0),

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and (1, 1, 0, 0, 0) ⊂ (0, 0, −1, −1, −1), the first two being simplices of B(X) of dimension 1, but being vertices in B(X, ·, · ), while the third one is a vertex both in B(X) and in B(X, ·, · ). Returning to the case of a general n, let us describe the reductive special fiber of Gx0 for the various vertices x of the standard apartment following 15.2.21. Given a periodic lattice chain L that corresponds to a vertex, we choose L0 ∈ L as in 15.2.20, so that m · L0 ⊆ L0∨ ⊆ L0 . Let L0 correspond to the tuple (a1, . . . , an, b, an+1, . . . , a2n ). Consider first /k unramified. Then both (V0, ·, · ) and (V1, ·, · ) are Hermitian spaces. When ω(a) = 0 then b = 0, implying that the dimension of V0 is even and that of V1 is odd, while when ω(a) = 1 then b = −1, implying that the dimension of V0 is odd and that of V1 is even. Let us assume ω(a) = 0, the other case being entirely analogous. Let 2n0 + 1 and 2n1 be the dimensions of V0 and V1 , respectively. Then n = n0 + n1 . The reductive special fiber of Gx0 equals U(V0 ) × U(V1 ). The relative Dynkin diagram of this group is BCn0 × Cn1 , and is obtained by removing a vertex from the relative affine Dynkin diagram of the group G, which is of type (BCn, Cn ). Consider now /k ramified. Then ω(a) = 0, so b = 0, implying that the dimension of V0 is even and that of V1 is odd. In fact, (V0, ·, · 0 ) is a symplectic space, while (V1, ·, · 1 ) is a quadratic space. Write 2n0 and 2n1 + 1 for their dimensions. Then n = n0 + n1 . The reductive special fiber of Gx0 equals Sp(V0 )× SO(V1 ). The relative Dynkin diagram of this group is Cn0 × Bn1 , and is obtained by removing a vertex from the relative affine Dynkin diagram of the group G, which is of type BCn .

P A R T F OUR APPLICATIONS

16 Classification of Maximal Unramified Tori (d’après DeBacker)

Let k be a field given with a discrete valuation ω : k × → R, with ring of integers o and maximal ideal m of o. We assume that ω(k × ) = Z, o is Henselian and the residue field f = o/m is perfect. We will denote the unique extension of ω to any algebraic field extension of k also by ω. We fix a maximal unramified extension K of k and denote its ring of integers by O. Let Γ = Gal(K/k). Definition 16.1 A k-torus T is said to be unramified if it splits over the maximal unramified extension K of k. The aim of this chapter is to review the results of [DeB06] that classify the G(k)-conjugacy classes of maximal unramified k-tori of a connected reductive group G using Bruhat–Tits theory. Lemma 16.2 Let T be a k-torus of G. Then T is a maximal unramified torus if and only if TK is a maximal K-split torus of G K . Proof It is obviously the case that if TK is a maximal K-split torus of G K , then T is a maximal unramified torus of G. To establish the converse, let T be a maximal unramified k-torus of G. Then considering the centralizer Z(T) of T in G, which is a connected reductive k-subgroup, we see using Proposition 9.3.4 that Z(T) contains a k-torus T  such that TK is a maximal K-split torus of Z(T)K and hence also of G K . Since T is contained in the center of Z(T), T  contains T, and the maximality of T implies T  = T.  Note that a maximal unramified torus T of G need not be a maximal torus. It is if and only if G K is split. Since TK is a maximal split torus of G K , there is an associated apartment A(TK ) of B(G K ). The action of Γ = Gal(K/k) on B(G K ) stabilizes A(TK ). Define A(T) = A(TK )Γ . This is a subset of B(G) = B(G K )Γ . In general A(T) is not an apartment of B(G). The properties of A(T) and 547

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its interactions with the apartments of B(G) are described by the following proposition. If S is a torus of G we denote by S  the maximal subtorus of S contained in the derived subgroup Gder , and V(S ) := R ⊗Z X∗ (S ). Let C be the maximal k-split torus of T, and V(T ) := R ⊗Z X∗ (TK). Then V(T )Γ = V(C ). Proposition 16.3 (1) The subset A(T) of B(G) is non-empty, and it is a union of facets. Given any point x ∈ A(T), A(T) = V(C ) · x (2) There exists an apartment of B(G) containing A(T), and A(T) is an affine subspace of this apartment under the vector space V(C ). (3) If an apartment A of B(G) contains A(T), then the maximal k-split torus S of G corresponding to A contains C. (4) If F is a facet of B(G) contained in A(T) and maximal with this property, and A is an apartment of B(G) containing A(T), then A(T) is the affine subspace of A spanned by F. Proof It is convenient to replace G by its derived subgroup; this does not change A(T) or B(G) and has the effect C  = C and S  = S, simplifying notation. (1) We recall from 9.2.1 that Γ acts on A(TK ) through a finite quotient and hence has a fixed point, showing that A(T) is non-empty. If a facet F of B(G) meets A(T), then the unique facet F of B(G K ) that contains F also meets A(T), hence A(TK ), so F is contained in A(TK ) and also stable under Γ, so F = F Γ is contained in A(T), cf. 9.2.4. Let x be any point of A(T). Then A(TK ) = V(T) · x, and hence A(T) = A(T)Γ = V(T)Γ · x = V(C) · x. (2) Let x be a point of A(T). Every point of the apartment A(TK ) is fixed under the maximal bounded subgroup T(K)b of T(K). So, in particular, x is fixed under the maximal bounded subgroup C(K)b of C(K). Now applying Proposition 9.3.11, for C in place of S and y = x, we see that there is an apartment A of B(G) that contains x and the corresponding maximal k-split torus S of G contains C. Since S contains C, V(C) operates on A by translations. As A(T) = V(C) · x, we conclude that A(T) is an affine subspace of A under the vector space V(C). Assertion (3) follows at once from Corollary 4.2.26. (4) The affine subspace AF of A spanned by F is contained in A(T) by (3). Since A(T) is a union of facets by (1), if AF  A(T) there exists a facet F  contained in A(T) but not in AF , whose closure contains F, contradicting the maximality of F.  Remark 16.4 Let T be a maximal unramified k-torus of G and let Z be the centralizer of T in G. Then Z is a maximal k-torus of G that is a K-Levi

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 subgroup. From the point of view of §9.7, A(T) is the image in B(G) = B(G)    of the enlarged building of Z, where G = G/CG , Z = Z/CG , and CG is the maximal k-split central torus in G. 16.5 The map (T, F) → (F, T) We construct a map from the set of pairs consisting of a maximal unramified k-torus T and a facet F of B(G) contained in A(T), and the set of pairs consisting of a facet F of B(G) and a maximal f-torus T of the reductive quotient G F of the special fiber G F0 . of the Bruhat–Tits o-group scheme GF0 associated to F, cf.§§8.3,8.4. Given (T, F), let T be the closed o-torus of GF0 with generic fiber T. The natural homomorphism π : G F0 → G F is injective on the special fiber T of T . We denote by T the isomorphic image of T in G F . Definition 16.6 Let F be a field. An F-torus T of a reductive F-group H will be called minisotropic (or elliptic) if the maximal torus contained in T ∩ Hder is F-anisotropic. It is easily seen that a maximal F-torus T of H is minisotropic if and only if H does not contain a proper parabolic F-subgroup that contains T. Remark 16.7 If T is a minisotropic maximal unramified k-torus of G, then Proposition 16.3(1) implies that A(T) is a vertex of B(G). Proposition 16.8

The map constructed in 16.5 has the following properties.

It is equivariant with respect to any k-automorphism of G. It is surjective, . The fiber over (F, T) is an orbit of the action of G(k)0+ F Given (T, F) → (F, T), the facet F is maximal among the facets of B(G) contained in A(T) if and only if the f-torus T of G F is minisotropic. (5) If F is maximal, then dim(F) = dim(C) − dim(CG ) = dim(C ), where C and CG are the maximal k-split tori in T and the center of G, respectively, and C  is the maximal k-split torus in T ∩ Gder .

(1) (2) (3) (4)

Proof (1) is immediate from the construction. (2) Consider a facet F of B(G) and a maximal f-torus T of G F . Using Proposition 8.2.1(1) for G = GF0 , we see that there exists a closed o-torus T in GF0 whose special fiber T maps onto T. The generic fiber T of T is a maximal unramified k-torus of G. Let F be the unique facet of B(G K ) containing F. Then as GF0 = GF0 contains T , the apartment A(TK ) of B(G K ) corresponding to TK contains F (Proposition 9.3.5(2)), hence A(T) contains F. (3) If (T1, F) and (T2, F) map to the same (F, T), Proposition 8.2.1(5) implies the existence of an element of G(k)0F that conjugates T1 to T2 and has trivial . image in G F (f), hence it lies in G(k)0+ F

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(4) Consider a pair (T, F) such that F is contained in A(T). There is a facet contained in A(T) that contains F in the boundary of its closure if and only if the reductive group G F contains a proper parabolic f-subgroup containing T. However, as mentioned in Definition 16.6, T is minisotropic if and only if G F does not contain a proper parabolic f-subgroup containing T. These two observations imply (4). (5) Corollary 9.4.24 and (4) show that dim(C) − dim(CG ) = dim(F). Since X∗ (T) = X∗ (T) as Γ-modules, the dimension of C is equal to the dimension of C.  F

Proposition 16.8 implies that the map constructed in 16.5 descends to a bijection between the set of G(k)-conjugacy classes of pairs (T, F) and the set of G(k)-conjugacy classes of pairs (F, T). In order to relate this to the set of G(k)-conjugacy classes of unramified maximal k-tori, we use the following notion. Lemma 16.9 equivalent.

Let F1, F2 be two facets of B(G). The following statements are

(1) There exists an apartment A of B(G) containing F1, F2 such that the affine subspaces of A spanned by F1 and F2 are equal. (2) For every apartment A of B(G) containing F1, F2 the affine subspaces of A spanned by F1 and F2 are equal. Proof Let A1, A2 be two apartments that contain F1, F2 . According to Proposition 4.2.24 there exists g ∈ G(k)0 such that gA1 = A2 and gx = x for all x ∈ A1 ∩ A2 . Since g is an affine transformation A1 → A2 , it maps the affine subspace of A1 spanned by Fi to the affine subspace of A2 spanned by  gFi = Fi , for i = 1, 2. Definition 16.10 Two facets F1, F2 are called strongly associated if they satisfy the equivalent conditions of the above lemma. Recall from 8.4.17 that, given two strongly associated facets F1, F2 we have the isomorphisms G F1 ← G F1 ∪F2 → G F2 . In this way we obtain an isomorphism ϕF2 ,F1 : G F1 → G F2 . The following lemma shows that strong association is close to being an equivalence relation. Lemma 16.11 (1) F1, F1 are strongly associated and ϕF1 ,F1 = idGF1 . Let F1 , F2 and F3 be facets of B(G) and g ∈ G(k). Assume that F1, F2 are strongly associated. −1 . (2) F2, F1 are strongly associated and ϕF1 ,F2 = ϕF 2 ,F1 (3) gF1, gF2 are strongly associated and ϕgF2 ,gF1 = Int(g)◦ϕF2 ,F1 ◦Int(g)−1 .

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(4) If F2, F3 are strongly associated and there exists an apartment containing F1, F2, F3 , then F1, F3 are strongly associated, and ϕF3 ,F1 = ϕF3 ,F2 ◦ ϕF2 ,F1 . Proof (1), (2), and (3) are immediate from the construction, while (4) follows from Corollary 8.4.18.  Lemma 16.12 Let T be a maximal unramified torus of G and F1, F2 be two facets of B(G) contained in A(T) and maximal with this property. Let (F1, T1 ) and (F2, T2 ) be the images of (T, F1 ) and (T, F2 ) under 16.5. Then F1, F2 are strongly associated and ϕF2 ,F1 (T1 ) = T2 . Proof That F1, F2 are strongly associated follows from Proposition 16.3(2). The claim ϕF2 ,F1 (T1 ) = T2 follows from the fact that T1 and T2 are the images  of the special fiber of the schematic closure of T in GF0 1 ∪F2 . Proposition 16.13 The following defines an equivalence relation on the set of pairs consisting of a facet of B(G) and a maximal f-torus of G F : (F1, T1 ) ∼ (F2, T2 ) if and only if there exists g ∈ G(k) such that F1, gF2 are strongly associated and ϕgF2 ,F1 (T1 ) = Int(g)(T2 ). Proof Reflexivity follows from Lemma 16.11(1). Assume (F1, T1 ) ∼ (F2, T2 ) and choose g2 ∈ G(k) so that g2 F2, F1 are strongly associated and Int(g2 )(T2 ) = ϕg2 F2 ,F1 (T1 ). Lemma 16.11(2),(3) implies that g2−1 F1, F2 are strongly associated and Int(g2 )−1 (T1 ) = ϕg−1 F1 ,F2 (T2 ), 2 showing symmetry. To prove transitivity, assume in addition (F2, T2 ) ∼ (F3, T3 ) and choose g3 ∈ G(k) so that F2, g3 F3 are strongly associated and Int(g3 )(T3 ) = ϕg3 F3 ,F2 (T2 ). We use Lemma 16.11(3) repeatedly. Choose apartments A12 , A23 of B(G) such that F1, g2 F2 are contained in A12 and g2 F2, g2 g3 F3 are contained in A23 . Choose h ∈ G(k)0g2 F2 such that hA23 = A12 . Thus F1, g2 F2, hg2 g3 F3 are contained in A12 . Then g2 F2, hg2 g3 F3 are strongly associated. Applying Proposition 8.4.16 to Ω := g2 F2 and Ω  := F1 ∪ g2 F2 ∪ hg2 g3 F3 we obtain h  ∈ G(k)0F1 ∪g2 F2 ∪hg2 g3 F3 such that ϕF1 ∪g2 F2 ∪hg2 g3 F3 ,g2 F2 (h) = (h )−1 . Replacing . Lemma 16.11(3),(4) shows h by h  h we achieve that h ∈ G(k)0+ g2 F2 ϕhg2 g3 F3 ,F1 (T1 ) = ϕhg2 g3 F3 ,hg2 F2 ◦ ϕhg2 F2 ,F1 (T1 ) = Int(hg2 ) ◦ ϕg3 F3 ,F2 ◦ Int(hg2 )−1 ◦ ϕg2 F2 ,F1 (T1 ) = Int(hg2 g3 )(T3 ).



Theorem 16.14 The map constructed in 16.5 and the projection (T, F) → T give rise to a bijection between the set of G(k)-conjugacy classes of maximal

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unramified k-tori of G and the set of equivalence classes of pairs (F, T) consisting of a facet F of B(G) and a minisotropic maximal torus T of G F , for the equivalence relation of Proposition 16.13. This bijection is equivariant with respect to any k-automorphism of G. Proof Proposition 16.8 implies that the map constructed in 16.5 descends to a bijection between the set of G(k)-conjugacy classes of pairs (T, F) and the set of G(k)-conjugacy classes of pairs (F, T). It is enough to show that two G(k)conjugacy classes of pairs (T, F) have the same image under the projection (T, F) → T if and only if their images are equivalent. Given two G(k)-conjugacy classes of pairs that have the same image under the projection onto the first factor, we can choose representatives (T, F1 ) and (T, F2 ) and let (F1, T1 ) and (F2, T2 ) be their images under 16.5. Then F1 and F2 are facets of B(G) contained in A(T) and maximal with this property by Proposition 16.8(4). Lemma 16.12 shows that F1, F2 are strongly associated and ϕF2 ,F1 (T1 ) = T2 . Conversely, assume that (F1, T1 ) and (F2, T2 ) are equivalent. Let (T1, F1 ) and (T2, F2 ) be preimages of (F1, T1 ) and (F2, T2 ) under 16.5. Choose g ∈ G(k) such that F1, gF2 are strongly associated and ϕgF2 ,F1 (T1 ) = Int(g)(T2 ). Conjugating (T2, F2 ) by g we may assume that g = 1. Thus F1, F2 are strongly associated and ϕF2 ,F1 (T1 ) = T2 . Choose an apartment A of B(G) containing F1, F2 , and using Proposition 16.3 choose another apartment A of B(G) that contains A(T1 ) as an affine subspace, and hence F1 . Let h ∈ G(k)0F1 be such that hA = A. Conjugating (T1, F1 ) by h we may assume h = 1, thus A contains A(T1 ) and F2 . Since F1 and F2 are strongly associated, they span the same affine subspace of A. By Proposition 16.3 this affine space equals A(T1 ), and we conclude that F2 belongs to A(T1 ) and is a maximal facet of A(T1 ). Let (F2, T1 ) be the image of (T1, F2 ) under the map 16.5. Lemma 16.12 shows that ϕF2 ,F1 (T1 ) = T1 . By assumption, ϕF2 ,F1 (T1 ) = T2 , so T1 = T2 . Proposition .  16.8(3) implies that T1 and T2 are conjugate under G(k)0+ F2

17 Classification of Tamely Ramified Maximal Tori

Let k be a field endowed with a discrete valuation ω. We assume that k is complete, so it is Henselian. Then the valuation ω extends uniquely to any algebraic extension of k; we will denote all such extensions by ω. Throughout this chapter, G will denote a reductive k-group. Definition 17.1 A k-torus is said to be tamely ramified if it splits over a tamely ramified Galois extension of k. If the residue field of k is of characteristic zero, then every k-torus is tamely ramified. 17.2 Let  be a tamely ramified finite Galois extension of k. We will denote the ring of integers of k (respectively,) by o (respectively,o ), the residue field of k (respectively,) by f (respectively,f ), and the Galois group of /k by Θ. We assume in this chapter that f is perfect. Let k  be the maximal unramified extension of k contained in ; then the residue field of k  is f . We will denote the ring of integers of k  by o . Write I = Gal(/k ) and Γ = Gal(k /k). Then I is a cyclic normal subgroup of Θ and Θ/I = Γ. 17.3 There is a natural action of Θ on the Bruhat–Tits building B(G ) of G(), induced by the action of Θ on . According to Theorem 12.9.2 we have a canonical identification, as metric spaces, of B(G) with B(G )Θ . For a facet F of B(G ), if F Θ is non-empty, then F is stable under the action of Θ on B(G ). If /k is unramified, then k  = , Θ = Γ, and by unramified descent, F Θ is a facet in the building B(G) of G(k). Thus in this case, if F Θ = {x}, then x is a vertex of B(G). 17.4 Let T be a maximal k-torus of G that splits over , and A(T ) be the apartment corresponding to the split maximal torus T of G in the building B(G ). This apartment is stable under the action of Θ on B(G ) and we set A(T) = A(T )Θ . Let F be a facet of A(T ) that is stable under the action of Θ. 553

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Let GF0 be the Bruhat–Tits o -group scheme associated to F, with generic fiber G , and GF be the maximal reductive quotient of the special fiber of GF0 . 17.5 Every θ ∈ Θ(= Gal(/k)) induces an isomorphism GF → GF that covers the induced isomorphism f  → f . In particular, the inertia subgroup I of Θ acts on GF by f -rational automorphisms. There isn’t a natural f-structure on GF in general. However, we can make sense of the concept of an I-stable maximal f-torus T of GF as follows: the group Γ = Θ/I acts naturally on the set of I-stable maximal f -tori of GF , and we require T to be fixed under this action of Γ. The maximal f -torus of GF corresponding to T is obviously an I-stable maximal f-torus of GF . Proposition 17.6 Let F be a Θ-stable facet of B(G ). The map T → T establishes a surjective map from the set of maximal k-tori T of G that split over  and F ∩ A(T)  ∅, to the set of I-stable maximal f-tori of GF . Each ; where F is the facet of B(G) that fiber of this map is an orbit under G(k)0+ F contains F Θ . Proof To prove the statement about fibers, let T1,T2 be two maximal k-tori of G which split over  and assume that both A(T1 ) and A(T2 ) intersect F and that T1 = T2 . Proposition 8.2.1(5) implies the existence of an element −1 = (T ) . For θ ∈ Θ, we have g −1 θ(g) ∈ g ∈ G()0+ 2  F such that g(T1 ) g 0+ −1 N(T1 )() ∩ G()0+ F = T1 () . Thus the 1-cocycle θ → g θ(g) defines a class 1 0+ in H (Θ,T1 () ). According to Theorem 13.8.5, this group vanishes, and we conclude that g can be multiplied on the right by an element of T1 ()0+ to 0+ ensure that it lies in G()0+ F ∩ G(k) = G(k)F , where the last equality is due to Propositions 12.9.4 and 9.8.3. To prove surjectivity, let T be an I-stable maximal f-torus of GF . Let T be a split maximal -torus of G such that A(T) contains F and T is the image in GF of the special fiber of the closed o -torus of GF0 whose generic fiber is T. For each θ ∈ Θ, θ T is another such torus, so Proposition 8.2.1(5) implies θ the existence of gθ ∈ G()0+ F such that Int(gθ )( T) = T. The differential c of the 1-cochain θ → gθ is defined by c : (θ, τ) → gθ · θ(gτ ) · (gθτ )−1 (∈ G()0+ F ), for θ, τ in Θ. It is easy to see that c(θ, τ) normalizes T and hence it belongs 0+ 2 0+ to G()0+ F ∩ N(T)() = T() . Thus c ∈ Z (Θ,T() ). According to Theorem 2 0+ 13.8.5, H (Θ,T() ) = {1}, so for each θ ∈ Θ, there exists tθ ∈ T()0+ such that 1 0+ θ → tθ · gθ is a 1-cocycle on Θ valued in G()0+ F . But since H (Θ, G()F ) = {1} −1 · θ(h) for all (Theorem 13.8.5), there exists h ∈ G()0+ F such that tθ · gθ = h −1 θ ∈ Θ. Now it is easily seen that hT h is stable under the action of Θ, i.e.,

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hT h−1 is defined over k, and T is still the torus of GF corresponding to hT h−1 −1 since h ∈ G()0+  F . Moreover, F ∩ A(hT h )  ∅. Lemma 17.7 Let T be a maximal k-torus of G that splits over . Let F be a facet of B(G ) that meets A(T)(A(T )Θ ) and let TF be the I-stable maximal f-torus of GF associated to T in 17.4. (1) The dimension of the maximal K-split subtorus of TK equals the dimension of (TFI )0 . (2) The dimension of the maximal k-split subtorus of T equals the dimension of the maximal f-split torus of (TFI )0 . Proof The dimension of the maximal K-split subtorus of TK is equal to the rank of X∗ (T)I . But X∗ (T) = X∗ (TF ) as I-modules, and X∗ (TF )I = X∗ ((TFI )0 ). The dimension of the maximal k-split subtorus of T equals the rank of X∗ (T) Θ , which by the previous argument equals the rank of X∗ ((TFI )0 )Γ , which  in turn is the rank of the maximal f-split subtorus of (TFI )0 . 17.8 Let M be a Levi k-subgroup of G. We turn our attention to minisotropic (i.e. elliptic) maximal k-tori T of M which split over . For such a torus T, we denote by AM (T ) the apartment corresponding to the split maximal torus T in the reduced Bruhat–Tits building B(M ) of M(). Let A M (T) := AM (T )Θ . Then A M (T) consists of a single point, since any two points of A M (T) differ by an element of V(T)Θ = R ⊗Z X∗ (T)Θ = {0}, where T  = T ∩ Mder . The following lemma provides some information about A M (T) Lemma 17.9 Let T be a minisotropic maximal k-torus of M which splits over  and let A M (T) = {x}. Then x is the unique Θ-fixed point of the facet F of B(M ) that contains it. (Hence, x is the barycenter of F.) In case /k is unramified, x is a vertex of B(M), and in case G is k-split, F = {x}, so x is a vertex of B(M ). Proof Since x ∈ A M (T) ⊂ AM (T ), the facet F is contained in AM (T ), hence F Θ ⊂ AM (T )Θ = A M (T) = {x}. This proves the first assertion. We assume now that /k is unramified. Then F = F Θ = {x} which implies that x is a vertex of B(M). We assume now that G is k-split and let S be a k-split maximal torus of M such that the apartment A M (S) of B(M) corresponding to S contains x. As S is an -split maximal torus of M , AM (S ) is an apartment of B(M ) which is fixed pointwise under the action of Θ. The facet F that

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contains x lies in the apartment AM (S ). Hence, F is also fixed pointwise under the action of Θ. There exists g ∈ M()0x such that g · AM (S ) = AM (T ), or, equivalently, gS g −1 = T . As x is the unique point of AM (T ) that is fixed under Θ, we see that it is the unique point of AM (S ) fixed by g −1 θ(g) ◦ θ for every θ ∈ Θ. The facet F lies in AM (S ) and is preserved by g −1 θ(g) ◦ θ. Since θ acts trivially on AM (S ) and any element of G()0 that fixes x must fix F pointwise, we see that F is fixed pointwise by g −1 θ(g) ◦ θ. This implies F = {x}, thus x is a vertex.  17.10 Let T be a minisotropic maximal k-torus of a Levi k-subgroup M of G. Let A M (T) = {x}. We denote the Bruhat–Tits o -group scheme associated to x by M,0 x . The maximal reductive quotient of the special fiber of M,0 x will be denoted by M , x . The image in M , x of the special fiber of the o -split torus of M, x with generic fiber T will be denoted by T. Now the following theorem, which follows at once from Proposition 17.6 applied to M in place of G, and the unique facet F of B(M ) that contains x, classifies such minisotropic maximal tori of M. Theorem 17.11 The map sending a minisotropic maximal k-torus T of M that splits over  and A M (T) = {x}, to T ⊂ M , x is a surjection between the set of minisotropic maximal k-tori T of M that split over  and A M (T) = {x}, and the set of I-stable maximal f-tori T of M , x such that (T I ∩ (M , x )der )0 is anisotropic. Each fiber of this map is an orbit under M(k)0+ x . 17.12 Determination of G(k)-conjugacy classes of maximal k-tori of G which split over . We now fix a maximal k-split torus S of G. Given a maximal k-torus T of G which splits over , let CT be the maximal k-split subtorus of T. Since we now wish to classify the G(k)-conjugacy classes of maximal k-tori of G which split over , we may conjugate T by a suitable element of G(k) to assume that CT ⊂ S. Now let M := MT be the centralizer of CT ; M is a Levi k-subgroup of G containing S, and T is a minisotropic maximal k-torus of M. Moreover, the central torus of M is contained in T. Let X be the finite set of Levi k-subgroups of G that contain S. There is an action of NG (S)(k) on X by conjugation. We fix a subset M of X that contains exactly one Levi subgroup in each NG (S)(k)-conjugacy class in X. Now given a maximal k-torus T of G, there is a unique Levi k-subgroup MT ∈ M that contains a conjugate of T under an element of G(k) as a minisotropic maximal torus. It is obvious that it would be enough to classify for each M ∈ M, M(k)conjugacy classes of maximal minisotropic k-tori of M which split over . Let A M (S) be the apartment corresponding to S in the reduced building

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557

B(M) of M(k). We fix a chamber C lying in A M (S) and consider the facets F of the reduced building B(M ) of M() that are stable under Θ and F Θ consists of a single point, say xF , and this point belongs to the closure C of C. The set P M of these points xF , as F varies, is stable under the action of the stabilizer M(k)C of C in M(k). Note that if  is an unramified extension of k, then P M is the set of vertices of C. We choose a subset P M of P M that contains exactly one point of each M(k)C -orbit in P M . Thus to determine G(k)-conjugacy classes of maximal k-tori of G which split over , it is sufficient to determine minisotropic maximal k-tori T of M ∈ M such that A M (T) consists of a single point, say xT , belonging to P M .

18 The Volume Formula

Let k be a global field (i.e., k is either a number field or it is the function field of a curve over a finite field). Unlike in the earlier chapters, in this chapter S and T will denote non-empty finite sets of places of k containing all the archimedean ones. For a place v of k, we will denote by k v the completion of k at v. Given an algebraic k-group G, we will denote the locally compact topological group  v ∈S G(k v ) by G S . In this chapter we will describe Gopal Prasad’s formula for the covolume of “principal” S-arithmetic subgroups of an absolutely simple simply connected algebraic k-group G given in [Pra89]. This formula is derived using a considerable amount of Bruhat–Tits theory of reductive groups over local fields. This theory is needed even in the case that S consists only of archimedean places,  that is when G S := v ∈S G(k v ) is a connected real semi-simple Lie group. Prasad’s formula has had numerous applications. It was originally used in [BP89] to prove finiteness assertions about S-arithmetic subgroups of semisimple groups. It was later used by Alireza Salehi Golsefidy to study Sarithmetic subgroups of minimum covolume in groups over local function fields, and by Amir Mohammadi and Golsefidy to enumerate maximal discrete subgroups that act transitively on vertices of Bruhat–Tits buildings. Recently, it was used by Prasad and Sai-Kee Yeung [PY07] (with computational inputs from Donald Cartwright and Tim Steger) to determine fake projective planes 1 . Even more recently, it was used by François Thilmany [Thi19] to prove that for n  3, up to conjugacy, SLn (Z) is the unique discrete subgroup of SLn (R) of minimal covolume. Besides the results of C.L. Siegel for certain special classical groups the only general results about the volumes of S-arithmetic quotients which were known 1

A fake projective plane is a smooth complex projective algebraic surface with the same Betti-numbers as the complex projective plane (that is, (1, 0, 1, 0, 1)), but which is not isomorphic to the latter.

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before the publication of [Pra89] were concerned with Chevalley groups (that is, the semi-simple groups which split over k); see [Har71]. For the most part, in this chapter we have reproduced Prasad’s paper [Pra89] with some simplifications.

18.1 Remarks on Arithmetic Subgroups 18.1.1 Notation. As usual, the fields of rational numbers, real numbers and complex numbers will be denote by Q, R and C respectively. Throughout this chapter, k will denote a global field and V will denote the set of places of k and Vf (respctively, V∞ ) denote the set of finite (respectively, the set of archimedean) places. For a place v of k, k v will denote the completion of k at v; k v is a local field (i.e., a non-discrete locally compact field). We will denote the normalized absolute value on k v by | |v , and v will denote the additive valuation such that v(k v× ) = Z. The absolute value | |v , and the additive valuation v, have unique extensions to any algebraic extension of k v , to be denoted in the same way. For v ∈ Vf , ov denotes the ring of integers of k v , fv the (finite) residue field and qv the order of fv . We recall that for x ∈ k v× , |x| v = [ov : xov ]−1 = qv−v(x) if x ∈ ov , |x| v = [xov : ov ] = qv−v(x) if x  ov . For v ∈ V∞ , and x ∈ k v , the normalized absolute value is defined to be |x|v = |x| if v is real, that is, if k v = R, and if v is complex, that is, if k v = C, then |x|v = |x| 2 .  Note the product formula: for all x ∈ k × , we have v ∈V |x| v = 1. In what follows, G will denote an absolutely almost simple simply connected k-group, ι : G → G  will denote a central isogeny to a k-group G . Then ι(G(K)) is a normal subgroup of G (K) for any field extension K of k. We will denote the adjoint group of G  by G, and ϕ  : G  → G will denote the natural central isogeny. As no confusion is likely, given a field extension K of k we  simply by G and G  respectively. will sometimes denote G K and G K In this chapter, Xv will denote the Bruhat–Tits building of G(k v ). For the convenience of the reader, we will recall the definition of S-arithmetic subgroups below and prove some well-known properties of these subgroups in our framework. We begin by proving the following proposition. Proposition 18.1.2 Let v be a non-archimedean place of k. Let H be an open

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subgroup of G(k v ). Then the subset F of points of the Bruhat–Tits building Xv of G(k v ) that are fixed under H is compact. Proof H acts continuously on the compactification Xv of Xv constructed in [BS76]. If F were not compact, H would have a fixed point in Xv − Xv . But by construction of the compactification, the isotropy subgroup of a boundary point is of the form P(k v ), where P is a proper parabolic k v -subgroup of G, and P(k v ) does not contain any open subgroups of G(k v ). Since we have not described compactifications of Bruhat–Tits buildings in this book, we will now give a different proof that avoids it. Let A be an apartment in the building Xv . Let I be an Iwahori subgroup of G(k v ) that fixes a chamber in A. Replacing H with H ∩ I we assume, as we may, that H is contained in I, so it is compact and [I : H] is finite. Fix finitely many coset representatives {gi } of the subgroup H in I. Then as  Xv = I · A = i Hgi · A, and F is (pointwise) fixed under H, we see that  F ⊂ i gi · A. Let us assume that F is non-compact. Then one of the gi · A contains a non-compact subset of F. Thus replacing A by a translate, we may (and do) assume that the set C of points of A fixed by H is non-compact. Note that C is a closed convex subset of A and hence it contains an infinite ray R originating at a point c of C (Lemma 12.5.8). The subgroup of G(k v ) that fixes R pointwise contains H and hence it is an open subgroup. Let Sv be the maximal k v -split torus of G corresponding to the apartment A and V = R ⊗Z Homkv (Gm, Sv ). Then A is an affine space under V. Fix a non-zero x ∈ V such that for all non-negative real numbers t, t x + c ∈ R. Let a be a root of G with respect to Sv such that a(x) < 0, and Ua be the root group corresponding to a. Let u be a non-trivial element of H ∩ Ua (k v ) and ψ := ψau be the largest real valued affine function on A, with derivative a, such that u ∈ Uψ (see Proposition 9.4.3(3)). Then as u fixes t x + c for all t  0, we see that ψ(t x + c) = ta(x) + ψ(c)  0. But this is impossible since by choice a(x) < 0.  18.1.3 Let S be a non-empty finite set of places of k containing all the   archimedean ones. Let G S = v ∈S G(k v ) and G S = v ∈S G (k v ). We assume in what follows that G S is non-compact. We denote by A the k-algebra of adèles of k, and by AS the k-algebra of S-adèles that is the restricted direct product of k v for v ∈ V − S. Both A and AS are endowed with the usual locally compact Hausdorff topologies. The natural embeddings of k in the k-algebras A and AS provide embeddings of G(k) in G(A) and G(AS ); G(k), viewed as a subgroup of G(A), is a discrete subgroup. It

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561

is well known ([Bor63] and [Har69]) that in the measure on G(A)/G(k) induced by any Haar measure on G(A), the volume of G(A)/G(k) is finite. 18.1.4 Let K be an arbitrary compact open subgroup of G(AS ) and Λ = G(k) ∩ K. We view G(k), and hence also Λ, embedded in G S diagonally. Then any subgroup of G S which is commensurable with Λ is called an Sarithmetic subgroup of G S . A subgroup of G S is said to be S-arithmetic if it is commensurable with ι(Λ). Since G S has been assumed to be non-compact, being of finite covolume, S-arithmetic subgroups of G S and G S are infinite. If K is a compact open subgroup of G(AS ), then, by strong approximation property ([Pra77] and [Mar91, ChapterII, §6]), the closure of Λ = G(k) ∩ K embedded diagonally in G(AS ) is K. Proposition 18.1.5 [BP89, Prop.1.2] Let Γ  be an S-arithmetic subgroup of G S . Then ϕ (Γ ) is contained in G(k) (G(k) embedded in G(k)S diagonally) and is Zariski-dense. The subgroups Γ  ∩ G (k) and Γ  ∩ ι(G(k)) are normal subgroups of Γ . Proof From reduction theory of arithmetic subgroups we know that the subgroup Γ  ∩ ι(G(k)) is of finite index in Γ , hence it contains a subgroup Γ0 which is normal and of finite index in Γ . Since G (k) is contained in the commensurability group of Γ0 , the latter is Zariski-dense in G  and hence ϕ (Γ0 ) is a Zariski-dense subgroup of G. For γ  ∈ Γ , the element ϕ (γ ) normalizes ϕ (Γ0 ), so it is a k-automorphism of G. This implies that ϕ (Γ ) ⊂ G(k) and that Γ  normalizes G (k) and ι(G(k)), hence also Γ  ∩ G (k) and Γ  ∩ ι(G(k)).  18.1.6 For v ∈ Vf , Aut(G(k v )), and so in particular G(k v ), acts on the building Xv by simplicial automorphisms. In view of the previous proposition, this allows us to define an action of any S-arithmetic subgroup of G S on Xv , with v ∈ Vf . This action will be used from now on without further reference. 18.1.7 For a given finite extension  of k, let S  denote the set of places of  that lie over a place in S. The ring of S-integers of , to be written O (S), is, by definition, the subring {x ∈  | |x|v  1 for every place v  of  not in S  }. For simplicity, we will write Ok (S) for O(S). For a place v of , we will denote by v the completion of  at v, and for non-archimedean v, we denote by ov the ring of integers of v . We fix a k-embedding G → SLn . We consider SLn as the Chevalley O(S)-group scheme determined by the inclusion of O(S)n in k n . Let G be the schematic closure of G in SLn . Then G is a flat O(S)-group scheme of finite

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The Volume Formula

type with generic fiber G. For a ring R containing O(S) and contained in a field extension K of k, we set G(R) := G(K) ∩ SLn (R) = G(R). Now we fix a finite Galois extension  of k over which G splits, and let G be the Chevalley O (S)-group scheme with generic fiber G . Since the group schemes G ×O(S) O (S) and G have the same generic fiber (= G ), they coincide at all but finitely many places of  ([EGAIV3 , Lemma 8.8.2.1]). In particular, for all but finitely many places v of , G splits over v and G (ov ) = G(ov ) = G(v ) ∩ SLn (ov ), where ov denotes the ring of integers of the completion v . As G (ov ) is a hyperspecial parahoric subgroup of G (v ) = G(v ), for all but finitely many non-archimedean places v of , we see that G(v ) ∩ SLn (ov ) is a hyperspecial parahoric subgroup of G(v ). Let k v be the closure of k in v and ov be the ring of integers of k v . Then, as is well known, for all but finitely many non-archimedean v, v is a finite unramified extension of k v . Hence, by unramified descent, we conclude that, for all but finitely many non-archimedean v,

 Gal(v /kv ) G(k v ) ∩ SLn (ov ) = G(v ) ∩ SLn (ov ) is a hyperspecial parahoric subgroup of G(k v ). Now Proposition 10.2.1 implies the following. Proposition 18.1.8 For any k-embedding G → SLn , for all but finitely many non-archimedean places v of k, G(k v ) ∩ SLn (ov ) is a hyperspecial parahoric subgroup of G(k v ). Hence, for all but finitely many non-archimedean places v of k, G is quasi-split over k v and it splits over an unramified extension of k v . 18.1.9 Fix a k-embedding G → SLn . Since any open subgroup of SLn (AS ) contains SLn (ov ) for all but finitely many v  S, we see that given an open subgroup of G(AS ), it contains the hyperspecial maximal compact subgroup G(k v ) ∩ SLn (ov ) of G(k v ) for all but finitely many v. Therefore, a compact open  subgroup K of G(AS ) contains the direct product vS Kv as a subgroup of finite index, where for v  S, Kv is a compact open subgroup of G(k v ) which is hyperspecial for all but finitely many v. Definition 18.1.10 A collection (Pv )vS of parahoric subgroups Pv of G(k v ) is  called a coherent collection if the product vS Pv is a compact open subgroup of G(AS ). From the observation in 18.1.9, it follows that in a coherent collection

18.1 Remarks on Arithmetic Subgroups

563

(Pv )vS of parahoric subgroups, Pv is hyperspecial for all but finitely many v. An S-arithmetic subgroup Λ of G S that is contained in G(k) is said to be a principal S-arithmetic subgroup if there exists a coherent collection (Pv )vS of  parahoric subgroups such that Λ := G(k) ∩ vS Pv ; this Λ will be called the principal S-arithmetic subgroup determined by the coherent collection (Pv )vS of parahoric subgroups. We shall also say that Λ and the normalizer Γ  of ι(Λ) in G S are associated to (Pv )vS . According to the following proposition, any maximal S-arithmetic subgroup of G S is the normalizer of ι(Λ), where Λ is a principal S-arithmetic subgroup of G S . If Λ is a principal S-arithmetic subgroup of G S associated with (Pv )vS , then for v  S, the closure of Λ in G(k v ) is the parahoric subgroup Pv . Proposition 18.1.11 [BP89, Proposition1.4] Let Γ  be an S-arithmetic subgroup of G S and Λ be the inverse image in G(k) of Γ  ∩ ι(G(k)) under ι. (1) The fixed point set of Γ  in Xv (v  S) is compact, not empty. (2) For any field extension K of k, the normalizer of ϕ (Γ ) in G(K)S is contained in G(k) (G(k) embedded in G(K)S diagonally), ϕ (Γ ) is of finite index in its normalizer, and the normalizer N(Γ ) of Γ  in G S is S-arithmetic. (3) Γ  is contained in only finitely many S-arithmetic subgroups of G S .

(4) If Γ  is maximal, then for v  S, the closure Pv of Λ in G(k v ) is a  parahoric subgroup of G(k v ), vS Pv is a compact open subgroup of  G(AS ), Λ = G(k) ∩ v Pv , and Γ  is the normalizer of ι(Λ) in G S .

Proof By strong approximation, the projection of Λ in G(AS ) is dense in a compact open subgroup. Therefore, for v  S, its fixed point set Fv in Xv is compact (Proposition 18.1.2), non-empty (by the fixed-point theorem of Bruhat–Tits 1.1.15), and reduced to the unique fixed point of a hyperspecial parahoric subgroup Pv for v ∈ V − T, where T is a suitable finite subset of V containing S (18.1.6 and 18.1.7). Since ι(Λ) is of finite index in Γ , the group of automorphisms of Xv (for v  S) determined by Γ  is relatively compact, therefore its fixed point set Fv is not empty; Fv is obviously contained in Fv and so in particular it is compact and (1) is proved. By Proposition 18.1.5, ϕ (Γ ) is contained in G(k) as a Zariski-dense subgroup. Therefore its normalizer in G(K)S is contained in G(k) and so it coincides with the normalizer N(ϕ (Γ )) of ϕ (Γ ) in G(k). Obviously, Fv is stable under the natural action of N(ϕ (Γ )) on Xv . Hence, for all v  S, N(ϕ (Γ )) is a relatively compact subgroup of G(k v ). From this we conclude that N(ϕ (Γ )) is  a discrete subgroup of G S := v ∈S G(k v ), and as it contains ϕ (Γ ), which is a

564

The Volume Formula

discrete subgroup of G S of finite covolume, the index of ϕ (Γ ) in it is finite. 2 This implies in particular that the normalizer N(Γ ) of Γ  in G S is S-arithmetic, which proves (2). For v ∈ T − S, let Pv be the (finite) set of parahoric subgroups of G(k v )  which fix some facet contained in Fv . For P = vS Pv , where Pv ∈ Pv if v ∈ T − S, and Pv is the hyperspecial parahoric subgroup as above if v ∈ V − T, let Λ P = G(k) ∩ P, ΛP = ι(Λ P ) and N(ΛP ) be the normalizer of ΛP in G S . As (by (1)) any arithmetic subgroup containing Γ  has a fixed point in Fv , v  S, it is contained in the normalizer of ΛP for a suitable P. Since according to (2), N(ΛP ) itself is an S-arithmetic subgroup, it follows that Γ  = N(ΛP ) for some P if Γ  is maximal. This proves (4). Also, since there are only finitely many P and, for each P, [N(ΛP ) : Γ ] is finite, we conclude that the S-arithmetic  subgroups of G S containing Γ  are finite in number, which proves (3).  Remark 18.1.12 Let Λ = G(k) ∩ vS Pv be a principal S-arithmetic subgroup of G S associated with (Pv )vS . Let Λ = ι(Λ) and Γ  be the normalizer of Λ in G S . Then a good upper bound for the order of Γ /Λ is given by [BP89, Proposition2.9].

18.2 Notations, Conventions and Preliminaries 18.2.1 For a finite X, we will write #X for its cardinality. For a linear algebraic group H over a perfect field, Ru (H) will denote its unipotent radical; that is, its maximal connected normal unipotent subgroup. For v ∈ V, k v is assumed to carry the Haar measure with respect to which the measure of ov is 1 if v is non-archimedean, the measure of the unit interval [0, 1] is 1 if v is real, and the measure of any square in k v ( C), with sides of length 1, is 2 if v is complex. 18.2.2 We shall denote by G an absolutely almost simple, simply connected algebraic group defined and quasi-split over k. Let n = dim G and r be the absolute rank of G . For v ∈ V, for simplicity, we will denote Gkv by G . This should not cause any confusion. If G /k is not a triality form of type 6 D4 , let  be the smallest extension of k over which G splits; then [ : k]  3. If G /k is a triality form of type 6 D4 , let  be a fixed extension of k of degree 3 contained in the Galois extension of k, of degree 6, over which G splits; there are three such extensions, all isomorphic to each other over k. 2

For a different proof, see [Pra79, §1.5].

18.2 Notations, Conventions and Preliminaries

565

If k is a number field, let Dk (respectively, D ) be the absolute value of the discriminant of k/Q (respectively, /Q). Let d(/k) denote the relative discriminant of  over k; it is an ideal in the ring of integers of k. It is well known that  Nk/Q (d(/k)) · Dk[:k] = D . If k is the function field of a curve over a finite field, let qk (respectively, q ) be the cardinality of the finite field of the constant functions in k (respectively, 2g −2 ) and gk (resp. g ) be the genus of k (respectively, ). Let Dk = qk k , 2g −2 D = q . 18.2.3 Let v be a non-archimedean place of k such that v :=  ⊗k k v is a ramified field extension of k v of degree 2. Here, as well as in what follows, for y ∈ v , y¯ will denote its conjugate over k v ; let v0 = {y ∈ v | y + y¯ = 0}. Let δv = sup {v(y) | y ∈ v , y + y¯ = 1} . Then δv  0 and δv = 0 if and only if the characteristic of the residue field of k v is odd. For later use, we fix a λv ∈ v such that λv + λ¯v = 1, v(λv ) = δv . We assert that there exists a uniformizer πv of v such that λv πv + λ¯v π¯ v = 0 (cf. [Tit79, 3.11]). This is a simple consequence of the following observation. First as k v · v0 = v0 , we see that Z + v(v0 − {0}) = v(v0 − {0}). We also know that δv  v(v0 ) (see Lemma 2.8.1, note that δv was denoted in that lemma by μ). Hence, v(v0 −{0}) = ( 21 +δv )+Z. Since for any unit u of v , v(λv u) = v(λv ) = δv and δv  v(v0 − {0}), we see that λv u does not belong to v0 , i.e., λv u + λ¯v u¯  0 for any unit of v . Now since λv λ¯v−1  −u u¯−1 , for every unit u of v , we must have λv λ¯v−1 = −π¯ v πv−1 for a uniformizer πv of v . Let πv and λv be as in the previous paragraph.     |d(v /k v )| v = (πv − π v )2 v = πv2 (1 + λv λ¯v−1 )2 v 2  = πv /λ¯v v = qv2v(λv )−1 = qv2δv −1, where d(v /k v ) is the relative discriminant of v /k v . So |πv /λv |v2 = |d(v /k v )| v . 18.2.4 The integer s(G ). If G splits over k, let s(G ) = 0. Now assume G does not split over k. On the relative root system k Φ of G , with respect to a maximal k-split torus T , consider the ordering associated with a Borel k-subgroup containing T . The integer s(G ) is then defined as follows. If k Φ is reduced (which is the case if, and only if, G is not a k-form of type 2 Ar with r even), then s(G ) is equal to the sum of the number of short roots and of short simple roots. If G is a k-form of type 2 Ar with r even, then k Φ is the non-reduced root system BCr/2 and s(G ) = 12 r(r + 3), which is equal to the number of all roots in k Φ plus the number of simple roots. Note that if G is a k-form of type 2 Ar (r odd), 2 Dr (r arbitrary), or 2 E6 ,

566

The Volume Formula

then the root system k Φ is the reduced root system of type C(r+1)/2 , Br−1 , F4 respectively and s(G ) is 12 (r − 1)(r + 2), 2r − 1, 26 respectively. If G is a triality form of type 3 D4 or 6 D4 , then k Φ is of type G2 and s(G ) = 7.

18.3 Tamagawa Forms on Quasi-split Groups 18.3.1 Let G be as in 18.2.2. We fix a non-zero left-invariant exterior form ω on G of maximal degree that is defined over k; such a form is unique up to multiplication by an element of k × and is called a Tamagawa form on G /k. As G is a semi-simple group, ω is bi-invariant. 18.3.2 For each non-archimedean place v of k, we fix, once and for all, a maximal parahoric subgroup Pv of G (k v ) with the following properties. (1) If G splits over an unramified extension of k v (this is the case for all but finitely many v ∈ Vf , see Proposition 18.1.8), then Pv is a hyperspecial parahoric subgroup. (2) If G does not split over any unramified extension of k v (then Gkv is a residually split group) and it is not of outer type Ar with r even, then Pv is special. (3) In case G is an outer form of type Ar , with r even, we assume moreover that the derivative (i.e.the vector part) of the affine simple root corresponding to this special parahoric subgroup is a divisible root. The maximal reductive quotient of the special fiber of the Bruhat–Tits group scheme associated with this parahoric subgroup is SOr+1 .   (4) v ∈V∞ G (k v ) · v ∈Vf Pv is an open subgroup of the adèle group G (A). 18.3.3 Let v be a non-archimedean place of k and Gv be the Bruhat–Tits smooth affine ov -group scheme associated with the parahoric subgroup Pv . The generic fiber of the group scheme Gv is isomorphic to Gkv , and its group of ov -rational points is isomorphic to Pv . Let cv ∈ k v× be such that cv ω induces an invariant exterior form on the ov group scheme Gv that is defined over ov and whose reduction to the special fiber G v := Gv ×ov fv is not zero. It is obvious that such a cv exists and is unique up to multiplication by a unit. In particular, γv := |cv | v is a well-defined positive real number; it is equal to 1 for all but finitely many v. 18.3.4 If k is a number field, for an archimedean place v of k, let cv be the positive real number such that with respect to the Haar measure determined by the form cv ω, the volume of any maximal compact subgroup of Rkv /R (G )(C)

18.3 Tamagawa Forms on Quasi-split Groups

567

is 1, and let γv = |cv | v . We recall here that if v is real, then any maximal compact subgroup of Rkv /R (G )(C) is isomorphic to the unique (up to isomorphism) compact, almost simple, simply connected real-analytic Lie group of the same type as G and if v is complex, then any maximal compact subgroup of Rkv /R (G )(C) is the direct product of two copies of this group. 18.3.5 Let r be the absolute rank of G and let m1, . . . , mr (m1  · · ·  mr ) be the exponents of the almost simple, simply connected, compact real-analytic Lie group of the same type as G . Note that dim G = r + 2 ri=1 mi . We list below the exponents (see [Bou02]). Type Exponents 1, 2, . . . , r. Ar 1, 3, 5, . . . , 2r − 1. Br 1, 3, 5, . . . , 2r − 1. Cr 1, 3, 5, . . . , 2r − 5, 2r − 3, r − 1 Dr r − 1 has multiplicity 2 when r is even. 1, 4, 5, 7, 8, 11. E6 1, 5, 7, 9, 11, 13, 17. E7 1, 7, 11, 13, 17, 19, 23, 29. E8 1, 5, 7, 11. F4 1, 5. G2 Theorem 18.3.6 We have   . 12 s(G ) r    D mi !  γv =   .  (2π)mi +1  Dk[:k] v ∈V v ∈V∞ i=1 v Proof Let L be any (not necessarily finite) Galois extension of k containing , where  is as in 18.2.2. Then G splits over L. Let L(G ) be the Lie algebra of left-invariant vector fields on G /k, and g = L ⊗k L(G ). Let T be a maximal k-split torus of G and Z be its centralizer. Then Z is defined over k and it is a torus since G is quasi-split over k. Moreover, it splits over L since G does. Let Φ be the root system of G with respect to Z , and let Π (⊂ Φ) be the set of simple roots with respect to the ordering on Φ obtained by fixing a Borel k-subgroup of G containing Z . Let {Ha | a ∈ Π} ∪ {Xb | b ∈ Φ} be a Chevalley basis of g, where the Ha constitute a basis of the Lie algebra L ⊗k L(Z ) of Z /L and for each b ∈ Φ, Xb is an element of the root space gb . We fix an enumeration of this Chevalley basis, and for 1  i  n (= dim G ), let Xi be its ith element. Let X i be the dual basis of the dual g∗ and let ωCh = X 1 ∧ · · · ∧ X n ; ωCh is a G -invariant exterior form on G of maximal degree. The form ωCh is defined over L and any other choice of Chevalley basis or its enumeration gives only ωCh or −ωCh .

568

The Volume Formula

Since the space of G -invariant exterior forms on G of maximal degree is 1-dimensional, there is an α ∈ L× such that ω = α−1 ωCh . As ω is defined over k, for every σ ∈ Gal(L/k), σ(ω) = ω. Now since σ(ωCh ) = ±ωCh , we conclude that σ(α)2 = α2 for every σ ∈ Gal(L/k) and hence α2 ∈ k × . If k is a number field, det( Xi , X j ), where Xi , X j = Tr(ad Xi ad X j ) is the inner product of Xi with X j with respect to the Killing form on g, is an integer. Let m be its absolute value. Then m is uniquely determined by the absolute root system of G ; it does not depend on the choice of the Chevalley basis of g. We fix a k-basis X1, . . . , Xn of the Lie algebra L(G ) so that if X 1, . . . , X n is the dual basis, ω = X 1 ∧ · · · ∧ X n . If k is a number field, for every archimedean place v of k, we fix a basis Y1v , . . . , Ynv of k v ⊗k L(G ) such that with respect to the Killing form , v on k v ⊗k L(G ), Yiv is orthogonal to Yjv for 1  i  j  n,   and moreover, if v is real, then  Yiv ,Yiv v v = 1, whereas, if v is complex, then Yiv ,Yiv v = 1 for all i  n. Now let Yv1, . . . ,Yvn be the dual basis and ωvK = Yv1 ∧ · · · ∧ Yvn . Then ωvK is an invariant exterior form on G ×k k v , of maximal degree, defined over k v ; it determines a Haar measure on G (k v ) as well as on every maximal compact subgroup of Rkv /R (G )(C). The volume of each of the latter subgroups with respect to this measure is equal to m1/2

r  (2π)mi +1 i=1

and

m

1/2

r  (2π)mi +1 i=1

mi !

.2

mi !

if v is real,

  r    1/2  (2π)mi +1  = m   mi !  i=1

if v is complex; v

(this was first proved by Harish-Chandra; for a published proof see, for example, [Ono66a] or [Mac80]). Let d = det( Xi , X j ), where Xi , X j = Tr(ad Xi ad X j ) is the inner product of Xi with X j under the Killing form on L(G ). Then it is obvious that if v is a complex place, ωvK ⊗ ωvK equals dω ⊗ ω, and if v is real, then ωvK ⊗ ωvK equals either dω ⊗ ω or −dω ⊗ ω. Now as the volume of any maximal compact subgroup of Rkv /R (G )(C) with respect to the Haar measure determined by ωvK is   r    1/2  (2π)mi +1  m  ,  mi !  v

i=1

we conclude that, for all archimedean v,   r   mi !  1/2  − 12 γv = |d| v m  .  (2π)mi +1  i=1

v

18.3 Tamagawa Forms on Quasi-split Groups

569   Ch 4 2 2  −1   since ω = αω, we find that α m = d , which implies that dm v =  2But α  , and hence for all archimedean v, v   r  2 1/2  mi !    γv = α v   .  (2π)mi +1  v

i=1

Therefore,  v ∈V

γv2 =

 v ∈V f

γv2 ·

 v ∈V∞

γv2

 2 r   4   m !  7 i 2 = · 5α v   8 mi +1   (2π) v ∈V f v ∈V∞ 6 i=1 v9  2 r    −1    m !  i α 2  · γv2 · =   v m +1 i   (2π) v ∈V v ∈V v ∈V i=1 

γv2

f

v

∞

f

(by the product formula (18.3.3); recall that α2 ∈ k × )  2 r    −1    m !  i α 2  γ 2 · =  .  v v  (2π)mi +1  v ∈V f

v ∈V∞ i=1

v

Next we will prove that   −1  α 2  γ 2 = v v

v ∈V f

-

D Dk[:k]

. s(G ) .

This will establish the theorem. Let v be a non-archimedean place of k. Let Gv be the smooth affine ov -group scheme associated with the parahoric subgroup Pv of G (k v ) and cv , γv = |cv |v be as in 18.3.3. Let L(Gv ) be its ov -Lie algebra. Since the generic fiber Gv ×ov k v of Gv is Gkv , it follows that k v ⊗ov L(Gv )  k v ⊗k L(G ). We use this to identify L(Gv ) with an ov -Lie subalgebra of the k v -Lie algebra k v ⊗k L(G ). We consider first the non-archimedean places v such that G splits over a finite unramified extension Lv of k v . (Then  ⊗k k v is a direct product of certain unramified extensions of k v ; we note here, for future use, that for any finite unramified extension Lv of k v , |d(Lv /k v )| v = 1.) Then Pv is a hyperspecial parahoric subgroup of G (k v ) and Gv is a reductive ov -group scheme. Let Ov be the ring of integers of Lv . Then Gv ×ov Ov is a Chevalley group scheme as the generic fiber of Gv splits over Lv . Hence, there is an Ov -basis Z1v , . . . , Zvn of Ov ⊗ov L(Gv ) which is a Chevalley basis of the Lie algebra Lv ⊗k L(G). Let bv ∈ Lv× be such that X1 ∧ · · · ∧ Xn = bv Z1v ∧ · · · ∧ Znv . As, up to sign, ωCh is independent of the choice of the Chevalley basis and its enumeration, bv ω =

570

The Volume Formula

±ωCh = ±αω. Therefore, αω is an exterior form on Gv ×ov Ov whose reduction to the special fiber is non-zero. So αcv−1 is a unit. Hence, |α2 |v = |cv | 2 = γv2 for every non-archimedean place v such that G splits over an unramified extension of k v . Now let R be the finite set of non-archimedean places v of k such that G does not split over an unramified extension of k v , or, equivalently, a factor of  ⊗k k v is a non-trivial ramified field extension of k v . For v ∈ R, there are the following two possibilities. (1)  ⊗k k v is a field, denoted by v ; it is a ramified extension of k v and [v : k v ] = [ : k]. (2) G is a form of type 6 D4 ,  ⊗k k v is a direct product of k v and a ramified field extension v of k v of degree 2. In this case the k-root system of G is of type G2 and s(G ) = 6 + 1 = 7, and Gkv is a form of type 2 D4 of rank 3. The k v -root system of Gkv is of type B3 that has 6 short roots and 1 short simple root.  −1 ) We now claim that that for v ∈ R, again α2  γ 2 = |d(v /k v )| −s(G . Assumv ing this claim we conclude that

v

v

  −1   ) α 2  γ 2 = |d(v /k v )| −s(G = v v v

v ∈V f

v ∈R

-

D Dk[:k]

. s(G ) ;

see the “discriminant quotient formula” at the end of this chapter. (Recall that for v ∈ Vf − R,  ⊗k k v is a direct sum of certain unramified extensions of k v , and for any unramified extension K of k v , |d(K/k v )| v = 1.) This proves the theorem. We will now prove our claim. So consider a v ∈ R. We fix a maximal k v -split torus Tv of Gkv and let Zv denote its centralizer. As G is quasi-split, Zv is a k v -torus. We fix a Borel subgroup of Gkv that contains Zv , this gives compatible orderings on the root systems Φ(Tv ) and Φ(Zv ) of Gkv with respect to Tv and Zv respectively. Let Φ(Tv )+ (respectively,Φ(Zv )+ ) be the set of positive roots in Φ(Tv ) (respectively, Φ(Zv )) and Π(Tv ) (resp.Π(Zv )) be the set of simple roots. We fix a minimal Galois extension Lv of k v containing v ; G (and so also Zv ) splits over Lv . The Galois group Γ of Lv /k v operates on the character group of Zv ; under this action Φ(Zv ), Φ(Zv )+ and Π(Zv ) are stable. In case Gkv is of outer type Ar , with r even, we will say that every root of Φ(Tv ) that is not divisible is short, and by definition a root is long if and only if it is divisible. The restriction of a root in Φ(Zv ) to Tv gives a bijective correspondence between the set of Γ-orbits in Φ(Zv ) and the set Φ(Tv ); under this correspondence the Γ-orbits in Π(Zv ) correspond to the roots in Π(Tv ). Also it is easily seen that

18.3 Tamagawa Forms on Quasi-split Groups

571

the restriction b of a root b in Φ(Zv ) to Tv is a long root if and only if b is fixed under Γ. For b ∈ Φ(Zv ), we will denote its isotropy subgroup in Γ by Γb and denote by vb the subfield of Lv that is fixed pointwise by Γb . For every b ∈ Φ(Tv ) we fix a root b in Φ(Zv ) such that (1) the restriction of b to Tv is b, (2) if b is short, vb = v , and (3) the root associated with −b is the negative of the root associated with b. Let πv be a uniformizer in v . In the case Gkv is of outer type Ar , with r even, let δv , λv be as in 18.2.3 and we assume πv chosen so that λv πv + λ¯v π¯ v = 0 (see 18.2.3). The ring of integers ov of v is the direct sum of πvi ov for 0  i < [v : k v ]. We fix a Chevalley–Steinberg basis {Xb | b ∈ Φ(Zv )} ∪ {Ha | a ∈ Π((Zv )} in the Lie algebra Lv ⊗k L(G ). It has, in particular, the following properties. (1) σ(Xb ) = Xσ(b) for all σ ∈ Γ and b ∈ Φ(Zv ) whose restriction to Tv is a nondivisible root in Φ(Tv ). (2) σ(Xb ) = −Xb for σ ∈ Γ, σ  1, and any b whose restriction to Tv is a divisible root. Now the above Chevalley–Steinberg basis gives a valuation ϕ = (ϕb )b ∈Φ(Tv ) of the root datum of Gkv with respect to Tv . This valuation determines a point o of the apartment corresponding to Tv in the building of G (k v ). Whenever Gkv is not of outer type Ar , with r even, we take Pv to be the parahoric subgroup of G (k v ) determined by o, and Gv will denote the Bruhat–Tits group scheme associated to Pv in these cases. The union of the following sets is a ov -basis of the Lie algebra of this group scheme : {Ha | a (∈ Π(Tv )) long} and {Xb | b (∈ Φ(Tv )) long}, ⎫ ⎪ ⎬ ⎪ σ(πvi )Hσ(a) | a (∈ Π(Tv )) short, 0  i < [v : k v ] , ⎪ ⎪ ⎪ ⎪ σ ∈Γ/Γa ⎭ ⎩ ⎧ ⎪ ⎨  ⎪

⎧ ⎪ ⎨  ⎪ ⎪ ⎪ σ ∈Γ/Γb ⎩

σ(πvi ) Xσ(b)

⎫ ⎪ ⎬ ⎪

| b (∈ Φ(Tv )) short, 0  i < [v : k v ] . ⎪ ⎪ ⎭

The exterior product of elements of the dual of this basis gives an invariant exterior form on Gv of top degree that is defined over ov and has a non-zero reduction to the special fiber. Since the basis above is written in terms of elements of a Chevalley–Steinberg basis, this invariant exterior form can be written as an explicit multiple of ωCh . Since ωCh = α ω, we easily see that |α2 |v−1 γv2 = |d(v /k v )|v−s(G ) .

572

The Volume Formula

We will now consider the case where Gkv is of outer type Ar with r (= 2n) even. We use the above Chevalley–Steinberg basis that determines the point denoted o and the valuation ϕ = (ϕb )b ∈Φ(Tv ) as above. We identify the apartment A corresponding to Tv with R ⊗Z X∗ (Tv ) by choosing o to be the origin and then the real valued affine functions on A with derivative b ∈ Φ(Tv ) are pairs (b, t); by definition, (b, t) takes the value b(x) + t at x ∈ A. For b ∈ Φ(Tv ), the affine roots of G /k v with derivative b are (b, t) for t ∈ 12 Z if b is neither multipliable nor divisible; for t ∈ 12 Z + 12 δv if b is multipliable; and for t ∈ Z + δv + 12 if b is divisible. We index the simple roots (∈ Π(Tv )) as a1, a2, . . . , an so that an is the unique multipliable root and ai , ai+1  0 for all i < n. Let v ∈ A be the unique point such that ai (v) = 0 for all i < n and an (v) = − 12 δv . We take Pv to be the parahoric subgroup of G (k v ) determined by the point v, and Gv will now denote the Bruhat–Tits group scheme corresponding to Pv . An affine root (b, t) takes the value b(v)+t at v. Note that b(v) = 0 or ±δv (∈ 12 Z) if b is neither multipliable nor divisible; if b is multipliable, then b(v) = − 12 δv or 12 δv according as b is positive or negative; and finally if b is divisible, then b(v) = −δv or δv according as b is positive or negative. We list below the smallest affine root ψb with a given derivative b ∈ Φ(Tv ) that takes non-negative value at v. (1) ψb = (b, −b(v)) if b is neither multipliable nor divisible; (2) if b is multipliable, then ψb = (b, 12 δv ) if b is positive, and ψb = (b, − 12 δv ) if b is negative; (3) if b is divisible, then ψb = (b, δv + 12 ) if b is positive, and ψb = (b, −δv + 12 ) if b is negative. Observe that if b is not divisible, then ψb (v) = 0. This shows that v is a vertex and the root system of the reductive quotient of the special fiber of the group scheme Gv consists of nondivisible roots in Φ(Tv ); this is the root system of type Bn . For b ∈ Φ(Tv ) the corresponding root group of the group scheme Gv is Uψb , with ψb as above. Now considering the structure of Uψb and its Lie algebra, we will display an ov -basis of the Lie algebra of Gv . For this purpose, we define a real valued additive function f on the root lattice as follows: f (ai ) = 0 for all i < n and f (an ) = − 12 . (Then for b ∈ Φ(Tv ), b(v) = f (b)δv .) Also, for z ∈ v , z¯ will denote its conjugate over k v and for an element b of X∗ (Zv ), b¯ will denote its conjugate. The union of the following sets is an ov -basis of the Lie algebra of Gv : (1)

'

( πvi Ha + π¯ vi Ha¯ | a ∈ Π(Tv ), i = 0 or1 ,

18.4 Volumes of Parahoric Subgroups 573 ; − f (b) i − f (b) i πv Xb + λ¯v π¯ v Xb¯ , for b ∈ Φ(Tv ), b neither multipliable nor λv divisible, i = 0 or 1, ( ' i πv Xb + π¯ vi Xb¯ | b ∈ Φ(Tv ) multipliable and positive, i = 0 or 1 , {λv−1 πvi Xb + λ¯v−1 π¯ vi Xb¯ | b ∈ Φ(Tv ) multipliable and negative, i = 0 or 1}, {λ ' v−1πv Xb | b ∈ Φ(Tv ) divisible and positive} ,( λv π¯ v Xb | b ∈ Φ(Tv ) divisible and negative . :

(2) (3) (4) (5) (6)

Now using this basis which is given in terms of a Chevalley–Steinberg basis, we easily see by a direct computation, and using |πv /λv |v2 = |d(v /k v )| v (see 18.2.3) that |α2 |v−1 γv2 = |d(v /k v )|v−s(G ) , where s(G ) is equal to the number of all roots plus the number of simple roots in the root system Φ(Tv )(= k Φ) (see 18.2.4). 

18.4 Volumes of Parahoric Subgroups We begin with the following general lemma. Lemma 18.4.1 Let F be an arbitrary field, G and G  be connected semisimple F-groups. Assume that G is an inner F-form of G  and G  is quasi-split over F. Let F  be a separable extension of F such that G is quasi-split over F . Then G and G  are isomorphic over F . Moreover, if F  is a Galois extension of F, we can find an F -isomorphism ϕ : G → G  such that for all σ in the Galois group of F /F, ϕ−1 · σ ϕ ∈ Int(G). Proof By assumption, there exists an isomorphism f : G → G  defined over a (fixed) separable closure Fs of F  such that for all σ in the Galois group Γ(Fs/F) of Fs/F, we have aσ := f −1 · σ f ∈ Int(G). Choose a Borel subgroup B of G (resp. B  of G ) and a maximal torus T (resp. T ) of B (resp. B ), all defined over F . Then we can arrange that f maps B and T onto B  and T  respectively. Then so does σ f for all σ in the Galois group Γ(Fs/F ) (⊂ Γ(Fs/F)) of Fs/F . Hence for σ ∈ Γ(Fs/F ), aσ preserves B,T and so it is of the form Inttσ (tσ ∈ T). Assume now that G is adjoint. Then tσ is uniquely determined and it follows that σ → tσ is a 1-cocycle on Γ(Fs/F ) with values in T. The Galois group Γ(Fs/F ) acts on X∗ (T) by permuting the simple roots. These form a basis of X∗ (T) since G is adjoint. Therefore Γ(Fs/F ) acts as a permutation representation and this implies that T is a direct product of certain tori of the form RL/F  (Gm ). Therefore it is cohomologically trivial over F  and so (aσ ) is a coboundary: there exists a t ∈ T such that tσ = t · σ t −1 . Then for σ ∈ Γ(Fs/F ), ( f · Intt)−1 · σ ( f · Intt) = Intt −1 · aσ · Intσ t = Id,

574

The Volume Formula

hence the isomorphism ϕ := f · Intt is defined over F . If G is not adjoint, let G → AdG be the canonical central isogeny. If  t belongs to the inverse image of the previous t, then again ϕ := f · Int t is defined over F . Moreover, since f −1 · σ f ∈ Int(G) for all σ ∈ Γ(Fs/F), it is obvious that if F  is a Galois extension of F, then for all σ in the Galois group of F /F,  ϕ−1 · σ ϕ ∈ Int(G). 18.4.2 Let G be as in 18.2.2. Let G be an absolutely almost simple, simply connected algebraic k-group which is an inner form of G . We know from Proposition 18.1.8 that G is quasi-split over k v for all but finitely many places v of k, so, according to the previous lemma, it is isomorphic to G over k v . We shall use the notation introduced in the previous sections. Thus, in particular, ω is the G -invariant exterior form on G of maximal degree, defined over k, fixed in 18.3.1. In the following, for any non-archimedean place v of k, Kv will denote its maximal unramified extension, and Ov will denote the ring of integers in Kv . Let ϕ : G → G be an isomorphism defined over a (not necessarily finite) Galois extension K of k such that for every σ in the Galois group Γ(K/k) of K/k, ϕ−1 · σ ϕ is an inner automorphism of G. Then ω∗ := ϕ∗ (ω) is an invariant exterior form on G of maximal degree defined over k. To see this let σ be an element of the Galois group of K/k. Then σω∗ = σ ϕ∗ (ω) = (σ ϕ∗ · ϕ∗ −1 )(ϕ∗ (ω)) = (σ ϕ∗ · ϕ∗ −1 )(ω∗ ) = ω∗, since ϕ−1σ ϕ is an inner automorphism of G. Thus ω∗ is defined over k. If ψ : G → G is some other isomorphism defined over an extension of k, then as any commutative quotient of Aut(G )/Int(G ) is of order  3, it is clear that ψ ∗ (ω) = u(ψ)ω∗ where u(ψ) is a root of unity of order  3. For each v ∈ V, ω (respectively, ω∗ ), together with the normalized absolute value | | v on k v (see 18.1.1), determines a Haar measure on G (k v ) (resp.G(k v )) which we shall denote by ωv (respectively, ωv∗ ). The Haar measure ωv∗ on G(k v ) is independent of the choice of the isomorphism ϕ : G → G . 18.4.3 Let v be a non-archimedean place of k and Pv be a parahoric subgroup of G(k v ) and G v be the Bruhat–Tits smooth affine ov -group scheme associated with Pv . Its generic fiber (= G v ×ov k v ) is isomorphic to G kv and its group of integral points is isomorphic to Pv . Let the parahoric subgroups Pv and the smooth affine ov -group scheme Gv associated with Pv be as in 18.3.2 and 18.3.3 respectively. We shall denote by G v (respectively,G v ) the special fiber Gv ×ov fv (respectively, G v ×ov fv ) of Gv (respectively, G v ). Since ov is Henselian, the “reduction mod pv ” homomorphisms Pv = Gv (ov ) → G v (fv ) and Pv = G v (ov ) → G v (fv ) are surjective. Let

18.4 Volumes of Parahoric Subgroups

575

M v = G v /Ru (G v ) and M v = G v /Ru (G v ) be the maximal reductive quotients of Gv and G v respectively. As fv is a finite field, both M v and M v are quasi-split over fv . We fix a Borel fv -subgroup B v of M v , Bv of M v and a maximal fv -torus T v of B v , Tv of Bv . Let U v (resp.Uv ) be the unipotent radical of B v (resp. Bv ). Let Iv and Iv be the inverse images in Pv and Pv of B v (fv ) and Bv (fv ) respectively under the natural homomorphisms Pv → B v (fv ) and Pv → Bv (fv ) respectively. Then Iv (respectively, Iv ) is an Iwahori subgroup of G (k v ) (respectively, G(k v )). Obviously, [Pv : Iv ] = [M v (fv ) : B v (fv )] and [Pv : Iv ] = [M v (fv ) : Bv (fv )]. Proposition 18.4.4 For v ∈ Vf , let Iv ⊂ Pv and Iv ⊂ Pv be as in 18.4.3. We will use the other notation introduced in 18.4.3. ωv∗ (Iv ) =

#Tv (fv ) #T v (fv )

× ωv (Iv )

and ωv∗ (Pv ) =

[M v (fv ) : Bv (fv )] [M v (fv ) : B v (fv )]

×

#Tv (fv ) #T v (fv )

× ωv (Pv ).

Proof According to the Bruhat–Tits theory, the Iwahori subgroup Iv (respectively, Iv ) determines a smooth affine ov -group scheme GIv (respectively, G Iv ) whose generic fiber is Gkv (respectively, G kv ) and GIv (ov )  Iv (respectively, G Iv (ov )  Iv ). Now since G is a quasi-split inner k-form of G, and G is quasi-split over Kv , we conclude that G is isomorphic to G over Kv and there is an isomorphism ϕv : G Kv → GKv such that ϕv−1 · γ ϕv is an inner automorphism of G for all γ ∈ Gal(Kv /k v ); see Lemma 18.4.1. The exterior form ϕv∗ (ω) is then defined over k v and the Haar measure on G(k v ) determined by it (and the absolute value | | v on k v ) is ωv∗ (18.4.2). Now let Ov be the ring of integers of Kv . Then G Iv (Ov ) (respectively ,GIv (Ov )) is an Iwahori subgroup of G(Kv ) (resp.G (Kv )) and in view of the conjugacy of Iwahori subgroups, we may (and we will) assume that ϕv (G Iv (Ov )) = GIv (Ov ). Then the isomorphism ϕv is obtained by base change Ov → Kv (see Corollary 2.10.10) from a unique isomorphism G Iv ×ov Ov → GIv ×ov Ov defined over Ov , which we denote again by ϕv . It is obvious that there is an av (∈ k v× ) such that the exterior form av ω induces an invariant exterior form on the group scheme GIv , which is defined over ov and whose reduction to the special fiber GIv := GIv ×ov fv is non-zero. Then as

576

The Volume Formula

ϕv is an isomorphism defined over Ov , the exterior form ϕv∗ (av ω) = av ϕv∗ (ω) on the ov -group scheme G Iv is defined over ov and its reduction to G Iv ×ov f v , and hence also to G Iv := G Iv ×ov fv , is non-zero. The inclusion Iv ⊂ Pv (respectively, Iv ⊂ Pv ) induces a homomorphism G Iv → M v (resp.GIv → M v ), where M v and M v are as in 18.4.3. Also, any maximal fv -torus of G Iv (respectively, GIv ) is mapped isomorphically onto Tv (respectively, T v ) under this homomorphism. So, G Iv /Ru (G Iv ) = Tv and GIv /Ru (GIv ) = T v . Now, as the Haar measure on G (k v ) given by av ω is |av | v ωv , we conclude ([Oes84, I,2.5]) that |av | v ωv (Iv ) = #GIv (fv ) · qv− dim G = #T v (fv ) · qv− dim T v . Similarly, as the Haar measure on G(k v ) given by ϕv∗ (av ω) = av ϕv∗ (ω) is |av | v ωv∗ , we have |av | v ωv∗ (Iv ) = #Tv (fv ) · qv− dim Tv . Now note that dim T v = Kv −rank G = Kv −rank G = dim Tv , and so we deduce from the above that ωv∗ (Iv ) =

#Tv (fv ) #T v (fv )

· ωv (Iv ).

Then ωv∗ (Pv ) = [Pv : Iv ] · ωv∗ (Iv ) ω∗ (Iv ) [Pv : Iv ] = × v × ωv (Pv ) [Pv : Iv ] ωv (Iv ) =

[M v (fv ) : Bv (fv )] [M v (fv ) : B v (fv )]

This proves the proposition.

×

#Tv (fv ) #T v (fv )

× ωv (Pv ). 

18.4.5 Let Δv be the basis of the absolute affine root system of G at v (i.e. the affine root system of G over the maximal unramified extension Kv of k v ) determined by the Iwahori subgroup Iv . Let Θv be the subset of Δv corresponding to the parahoric subgroup Pv . The Galois group of Kv /k v operates on Δv , leaving Θv stable. The Tits index [Tit66] of the reductive group M v /fv is obtained from the Dynkin diagram of Δv together with the action of the Galois group of Kv /k v (i.e. the “local

18.4 Volumes of Parahoric Subgroups

577

index” of G kv ) by deleting the vertices corresponding to the roots in Θv and all the edges containing such vertices (cf. §9.10.6). Note that there is a canonical identification of the Galois group of Kv /k v with the Galois group of f v /fv , where f v is the residue field of Kv : it is an algebraic closure of fv . 18.4.6 Fixing a Kv -isomorphism of G with G , we identify the root system as well as the affine root system of GKv with those of G Kv . Let dv ∈ Δv be the affine simple root corresponding to the parahoric subgroup Pv . Then Δv − {dv } can be identified with a basis of the absolute root system Ψv of the reductive group M v ; see Remark 4.1.23. Since Pv is a hyperspecial parahoric subgroup if G splits over an unramified extension of k v otherwise (G is residually split over k v and) Pv is a special parahoric subgroup, dv is special (cf. Definition 1.3.39). Ψv is then a reduced and irreducible root system of rank rv := Kv -rankG = Kv -rankG. Hence the reductive group M v is in fact absolutely almost simple and its absolute rank is rv . 18.4.7

Let γv be as in 18.3.3. Then ([Oes84, I,2.5]) γv ωv (Pv ) = #G v (fv ) · qv− dim G = #M v (fv ) · qv− dim M v

where G v and M v are as in 18.4.3. Proposition 18.4.8 For v ∈ Vf , #Tv (fv )

(1) γv ωv∗ (Iv ) =

qv(rv +dim M v )/2

(2) γv ωv∗ (Pv ) = Proof

.

#M v (fv )

.

qv(dim M v +dim M v )/2

(1) According to Proposition 18.4.4, ωv∗ (Iv ) =

#Tv (fv ) #T v (fv )

ωv (Iv )

and hence γv ωv∗ (Iv ) = = =

#Tv (fv ) #T v (fv ) #Tv (fv ) #T v (fv ) #Tv (fv ) #T v (fv )

× × ×

γv ωv (Pv ) [Pv : Iv ] #M v (fv ) × qv− dim M v [M v (fv ) : B v (fv )] #B v (fv ) qvdim M v

.

(see 18.4.3 and 18.4.7)

578

The Volume Formula

Now recall that U v (resp. Uv ) is the unipotent radical of B v (resp. Bv ). So B v (fv ) = T v (fv ) · U v (fv ),

Bv (fv ) = Tv (fv ) · Uv (fv ),

#U v (fv ) = qvdim U v ,

#Uv (fv ) = qvdim Uv ,

and dim M v = dim T v + 2 dim U v ;

dim M v = dim Tv + 2 dim Uv .

Moreover, as we have observed above, dim T v = rv = dim Tv . So γv ωv∗ (Iv ) = =

#Tv (fv ) #T v (fv )

·

#Tv (fv ) qvrv +dim U v

#B v (fv ) qvdim M v =

#Tv (fv ) qv(rv +dim M v )/2

.

(2) γv ωv∗ (Pv ) = = = = =

[M v (fv ) : Bv (fv )] [M v (fv ) : B v (fv )] [M v (fv ) : Bv (fv )] [M v (fv ) : B v (fv )] #M v (fv ) qvdim M v #M v (fv ) qvdim M v

× ×

× ×

#B v (fv ) #Bv (fv )

#Tv (fv ) #T v (fv ) #Tv (fv ) #T v (fv )

×

× γv ωv (Pv ) (by 18.4.4) ×

#M v (fv ) qvdim M v

(by 18.4.7)

#Tv (fv ) #T v (fv )

#U v (fv ) #Uv (fv )

#M v (fv ) qv(dim M v +dim M v )/2

. 

18.5 Covolumes of Principal S-Arithmetic Subgroups As in 18.4.2, G is an inner k-form of G and ϕ : G → G is an isomorphism defined over some Galois extension K of k such that for every σ in the Galois group of K/k, ϕ−1 · σ ϕ is an inner automorphism of G.

18.5 Covolumes of Principal S-Arithmetic Subgroups

579

Let ω be an invariant exterior form on G defined over k and of maximal degree. Then ω∗ = ϕ∗ (ω) is an invariant exterior form on G of maximal degree; it is defined over k (18.4.2). We shall use the notation introduced in the preceding sections. Thus, G S =  v ∈S G(k v ), and for v ∈ Vf , Pv is a parahoric subgroup of G(k v ). We assume  in what follows that G S × vS Pv is an open subgroup of G(A). Then Λ :=  G(k) ∩ G S × vS Pv diahonally embedded in G S is a principal S-arithmetic subgroup. We shall determine its covolume in this section. 18.5.1 ω∗ determines a Haar measure ω∗A on G(A), which coincides with the

    product measure v ∈V∞ ωv∗ · v ∈Vf ωv∗ | Pv on the open subgroup v ∈V∞ G(k v )·   ∗ v ∈V f Pv ; note that since G is semi-simple, the product v ∈V f ωv (Pv ) is abso   lutely convergent and hence the product measure v ∈V∞ ωv∗ · v ∈Vf ωv∗ | Pv is   a Haar measure on v ∈V∞ G(k v ) · v ∈Vf Pv (see [Ono66b] where this is proved over number fields; a similar proof applies in the case of global function fields). In what follows, we will let ω∗A also denote the finite invariant measure on G(A)/G(k) induced by the Haar measure ω∗A on G(A). 18.5.2 Let Dk be as in 18.2.2. The Tamagawa number τk (G) of G/k is by definition the positive real number Dk−(1/2) dim G ω∗A(G(A)/G(k)); in view of the product formula (see 18.1.1) it depends only on G/k and not on the choice of the invariant exterior k-form ω. It was conjectured by André Weil that τk (G) = 1. He and T. Tamagawa proved this for all inner forms of type A, and in case k is of characteristic different from two, for all k-forms of type B, C and all forms of type D except the triality forms of type D4 ; M. Demazure verified the conjecture for the forms of type G2 . J.G.M. Mars then proved the conjecture for outer forms of type A, all forms of type F4 and certain inner forms of type E6 over number fields. For split groups over number fields, the conjecture was proved by R.P. Langlands [Lan66]. Using some of his ideas, G. Harder [Har74] proved the conjecture for all split groups over global function fields, and K.F. Lai [Lai80] proved it for quasi-split groups over number fields . R. Kottwitz [Kot88], following a proposal of Jacquet–Langlands [JL70] that simply connected semi-simple groups over a given number field that are inner forms of each other have same Tamagawa number, proved the conjecture for groups over number fields, without any case-by-case considerations, modulo the Hasse principle for the Galois cohomology of simply connected semi-simple groups. The Hasse principle was known to hold for all groups of type other than E8 . V.I. Chernousov more recently proved it also for the groups of type E8 . Hence the work of Kottwitz [Kot88] implies that τk (G) = 1 if k is a number field.

580

The Volume Formula

D. Gaitsgory and J. Lurie have recently announced that they have proved Weil’s conjecture that τk (G) = 1 for any simply connected semi-simple group G over a global function field k; see [GL19]. 18.5.3 Let S be a finite non-empty set of places of k, containing all the archimedean places, such that for some v ∈ S, G(k v ) is non-compact, or,  equivalently, G is isotropic over k v . As before, let G S = v ∈S G(k v ). Then the strong approximation property ([Pra77] or [Mar91, Chapter II, §6]) implies that    Pv · G(k) = G(A). GS × vS

Let Λ be the principal S-arithmetic subgroup of G S associated with the coherent collection (Pv )vS of parahoric subgroups Pv of G(k v ); thus, Λ is simply the

  subgroup G(k) ∩ G S × vS Pv embedded diagonally in G S and it is a lattice in G S , i.e. it is a discrete subgroup of G S of finite covolume. We can also view 

 Λ as the image of G(k) ∩ G S vS Pv into G S , under the natural projection  map G S × vS Pv → G S . The object of this section is to compute the volume μS (G S /Λ) with respect to a natural measure μS (see 18.5.5 below.) 

Let ωS∗ denote the measure on G S /Λ that is induced by the product measure  ∗ v ∈S ωv on G S , which we recall is v ∈S G(k v ). As  

G(A) = G S × Pv · G(k), vS

  G(A)/G(k) has a natural identification with G S × vS Pv /Λ, and so there is  a (principal) fibration G(A)/G(k) → G S /Λ with fiber vS Pv . Hence,  1 dim G = ω∗A(G(A)/G(k)) = ωS∗ (G S /Λ) × ωv∗ (Pv ). Dk2 vS

Therefore, 1

ωS∗ (G S /Λ) = Dk2

dim G

×

 vS

ωv∗ (Pv )

 −1

.

18.5.4 Let v be an archimedean place of k. Let cv (∈ R× ) be as in 18.3.4 and γv = |cv | v . We recall that cv is such that under the Haar measure induced by the invariant exterior form cv ω, any maximal compact subgroup of Rkv /R (G )(C) has volume 1. We assert that the volume of any maximal compact subgroup of Rkv /R (G)(C) in the Haar measure induced by the invariant form cv ω∗ is also 1. For this purpose, we assume (as we may) that the isomorphism ϕ of Lemma 18.4.1 is defined over C. Then this isomorphism identifies Rkv /R (G )(C) with

18.5 Covolumes of Principal S-Arithmetic Subgroups

581

Rkv /R (G)(C) as real Lie groups. As cv ω∗ = cv ϕ∗ (ω), our assertion obviously holds. 18.5.5 For any archimedean place v of k, let μv be the Haar measure on G(k v ) determined by the invariant exterior form cv ω∗ (cv as in 18.3.4), and for v non-archimedean, let μv be the Tits measure on G(k v ); that is the Haar measure with respect to which every Iwahori subgroup of G(k v ) has volume 1. Of course, μv = ωv∗ (Iv )−1 ωv∗ for all v ∈ Vf , where Iv is an Iwahori subgroup of   G(k v ). Let μS = v ∈S μv be the product measure on G S (= v ∈S G(k v )); we shall denote the G S -invariant induced measure on G S /Λ also by μS . Let , D and s(G ) be as in 18.2.2 and 18.2.4 respectively. Theorem 18.5.6

We have the following 1 2

μS (G S /Λ) = Dk

dim G



   21 s(G )      r mi !   E  [:k] (2π)mi +1 

D Dk

v ∈V∞

v

i=1

where E =

 q(rv +dim M v )/2 v #Tv (fv )

v ∈S f

×

 q(dim M v +dim M v )/2 v #M v (fv )

vS

and S f = S ∩ Vf . Proof

Clearly,

μS (G S /Λ) =



|cv |v

v ∈V∞

(cf.18.5.3) =



 v ∈S f

γv



v ∈V∞ 1 2

= Dk

1

= Dk2

ωv∗ (Iv )

dim G

v ∈S f



ωv∗ (Iv )

γv ×

dim G



×

 −1



v ∈V

v ∈S f

D Dk[:k]

 −1

ωS∗ (G S /Λ) 1 2

Dk

dim G

γv ωv∗ (Iv )

 21 s(G )

×

 vS

×

 vS

ωv∗ (Pv )

γv ωv∗ (Pv )

 −1

 −1

      r mi !   × E  (2π)mi +1  v ∈V∞

i=1

(by Theorem 18.3.6)

v

582

The Volume Formula

where E =

 v ∈S f

=

γv ωv∗ (Iv ) ×

 vS

 q(rv +dim M v )/2 v v ∈S f

#Tv (fv )

γv ωv∗ (Pv )

×

 −1

 q(dim M v +dim M v )/2 v #M v (fv )

vS

(by Proposition 18.4.8). 

This proves the theorem. Remark 18.5.7 If V∞  ∅, i.e. if k is a number field, then     [k:Q] r r   mi !  mi ! .   =  (2π)mi +1  (2π)mi +1 v ∈V i=1 i=1 v

∞

Remark 18.5.8 The reductive groups M v , M v , and the tori Tv (⊂ M v ) can be described in terms of the local index of G /k v , of G/k v , and the subset Θv of 18.4.5 (§9.10.6). Thus, in principle, μS (G S /Λ) can be computed using the above formula. 18.5.9 Let S be the set of places of k containing S and the places v  S such that either (1) G is not quasi-split over k v , or (2) it does not split over an unramified extension of k v , or (3) Pv is not hyperspecial. Then S is finite and for v  S, G is quasi-split over k v , splits over an unramified extension and Pv is a hyperspecial parahoric subgroup of G(k v ). So for all v  S, G is isomorphic to G over k v , and there is a k v -isomorphism between these groups that carries Pv isomorphically onto Pv . Hence, for all v  S, M v is isomorphic to M v . Let E =

 q(rv +dim M v )/2 v v ∈S f

#T v (fv )

×

 q(dim M v +dim M v )/2 v vS

#M v (fv )

be as in the preceding theorem. We will denote the following product  qdim M v v v ∈V f

#M v (fv )

by Z . We can now rewrite E as follows: . (r +dim M v )/2  #M v (fv ) q(dim M v +dim M v )/2 4  #M v (fv ) qv v 7 v , · · Z ×5 8× dim M v dim M v #T v (fv ) #M v (fv ) q q v ∈S v ∈S−S f v v 6 9 which in turn equals .  #M v (fv ) 4  #M v (fv ) (rv −dim M v )/2 7 (dim M v −dim M v )/2 · qv · qv Z ×5 . 8× #Tv (fv ) #M v (fv ) v ∈S v ∈S−S f 6 9

18.5 Covolumes of Principal S-Arithmetic Subgroups

583

We state this value of E in the following proposition for convenient reference. Proposition 18.5.10 E equals

.  #M v (fv ) 4  #M v (fv ) (rv −dim M v )/2 7 (dim M v −dim M v )/2 · qv · qv Z ×5 , 8× #T v (fv ) v ∈S−S #M v (fv ) 6v ∈S f 9 where  qdim M v v . Z = #M v (fv ) v ∈V f 18.5.11 Now we note that since for all v ∈ Vf , Pv is either a special or a hyperspecial maximal parahoric subgroup of G (k v ), M v is actually a connected absolutely almost simple group over the finite field fv . Now using the orders of finite simple groups of Lie types, given, for example, in [Table 16, p. 251][Asc93], and the Euler product expression for the values of Dedekind zeta-function ζk of k and Dirichlet L-function L/k of /k at positive integers  2, we easily see that Z in the above proposition is a product of values of ζk and/or L/k at mi + 1, where the mi are the exponents of the absolute root system of G listed in 18.3.5 (recall that L/k (s) =: ζ (s)/ζk (s)). We give below the values of Z in all cases. (1) If G is k-split (i.e., it is a Chevalley group), equivalently, G is of “inner  type,” then Z = i ζk (mi + 1). (2)  is a quadratic Galois extension of k in the following cases. (i) If G is of type 2 An , with n even, then Z =

n/2 

{ζk (2 j) L/k (2 j + 1)}.

j=1

(ii) If G is of type 2 An , with n odd, then Z = ζk (n + 1)

(n−1)/2 

{ζk (2 j) L/k (2 j + 1)}.

j=1

(iii) If G is of type 2 Dn , then Z = L/k (n)

n−1 

ζk (2 j).

j=1

(iv) If G is of type 2 E6 , then Z = ζk (2) ζk (6) ζk (8) ζk (10) L/k (5) L/k (9).

584

The Volume Formula

(3)  is a separable cubic extension of k, that is G is a triality form of type either 3 D4 or 6 D4 , then Z = ζk (2) ζk (6) L/k (4). Remark 18.5.12 Let S be just the set of archimedean places of k. We denote  G S = v ∈V∞ G(k v ), by G∞ and μS by μ∞ . Let Λ∞ be a principal S-arithmetic subgroup of G∞ . Assume that the absolute rank of G∞ equals that of any maximal compact subgroup, then k is totally real and we get a very “attractive” formula for the volume μ∞ (G∞ /Λ∞ ) in terms of Dedekind zeta and Dirichlet Lfunction values at certain negative integers using Theorem 18.5.6, Proposition 18.5.10, and the value of Z given in 18.5.11 rewritten using the functional equations for the zeta and L-functions.

18.6 Euler–Poincaré characteristic of S-arithmetic subgroups. We assume here that S-arithmetic subgroups under consideration have a torsionfree subgroup of finite index. This is the case if and only if either k is a number field, or G is k-anisotropic when k is a function field ([Ser71, Theorem 4]). Then there exists a G S -invariant measure μEP S on G S called the Euler–Poincaré may be zero) such that for any S-arithmetic subgroup Γ measure (note that μEP S of G S , | χ(Γ)| = μEP S (G S /Γ), where χ(Γ) is the Euler–Poincaré characteristic of Γ in the sense of C.T.C.Wall ([Ser71, §§1.8, 3]). It follows from [Ser71, Proposition 25] that, up to sign, μEP S is the product of the Euler–Poincaré measures on the groups G(k v ), v ∈ S, introduced in [Ser71, §3], and to be denoted here by μEP v . It is known EP [Ser71, §3] that for every non-archimedean v, μv is non-zero and for an archimedean v, μEP v is non-zero if and only if G(k v ) contains a compact Cartan subgroup (equivalently, G contains a k v -anisotropic maximal torus) [Ser71, Proposition 23]. Thus if k is a global function field, then μEP S is non-zero, and EP if k is a number field and μS  0, then k is totally real. For archimedean v, the Hirzebruch proportionality principle [Ser71, §3.2] implies at once that, up to sign, μEP v equals χv μv , where μv is the Haar measure on G(k v ) defined in 18.5.5, and χv is the Euler–Poincaré characteristic of the compact dual of the symmetric space associated with G(k v ). Hence, if S is  the set of archimedean places of k and, as above, Λ = G(k) ∩ vS Pv , is a

18.7 Bounds for the Class Number of Simply Connected Groups  principal arithmetic subgroup of G S = v ∈S G(k v ), then .  χv × μS (G S /Λ). | χ(Λ)| =

585

v ∈S

So we can compute the Euler–Poincaré characteristic of Λ using the volume formula given in Theorem 18.5.6, together with the value of E provided by Proposition 18.5.10, and the value of Z given in 18.5.11. Computation of Euler–Poincaré characteristic of principal arithmetic subgroups was crucial for the determination of fake projective planes (in [PY07]) and their higherdimensional analogs.

18.7 Bounds for the Class Number of Simply Connected Groups As before in this chapter, G will denote an absolutely almost simple simply connected group defined over a global field k; A will denote the k-algebra of adèles. Let P := (Pv )v ∈Vf be a coherent collection of parahoric subgroups. It is known, due to Borel in the case k is a number field, and due to Harder if k is a global function field and G is k-anisotropic, that the set  

 G(k v ) × Pv \G(A)/G(k) v ∈V∞

v ∈V f

of double cosets is finite. The cardinality of this set is called the class number of G relative to P; it will be denoted by c(G, P) here.  18.7.1 If G is either k-isotropic, or k is a number field and v ∈V∞ G(k v ) is non-compact, then the strong approximation property implies that for every coherent collection of parahoric subgroups P = (Pv ) ,    G(k v ) × Pv · G(k) = G(A), v ∈V∞

v ∈V f

hence the class number in these cases is 1. So we will assume in the rest of this section that G is k-anisotropic, and, moreover, if k is a number field then  v ∈V∞ G(k v ) is compact.

   We will denote the compact open subgroup v ∈V∞ G(k v ) × v ∈V f Pv of G(A) by C, and continue to use the notation introduced earlier in this chapter. In particular, ω∗A is the Haar measure on G(A) defined in 18.5.1. 18.7.2

Fix representatives gi ∈ G(A) of the double cosets in C\G(A)/G(k).

586

The Volume Formula

Then G(A) = 1 2 dimG

Dk

c(G, P)

C gi G(k). Recall from 18.5.2 that ω∗A(G(A)/G(k)) =

i=1

. So, 1 2 dimG

Dk



 −1 ω∗A(C)

=

c(G, P) i=1

1 ; #Fi

(18.7.1)

where Fi := gi−1 Cgi ∩G(k) is a finite subgroup of G(k) since gi−1 Cgi is compact and G(k) is a discrete subgroup of G(A). 18.7.3 Using the results of §§18.4–18.5, and the fact that as G(k v ) is compact for every v ∈ V∞ , γv ωv∗ (G(k v )) = 1 for all v ∈ V∞ , we see that   . . 12 s(G ) r  −1   D mi !    ∗ −1 ∗ γ ω (P ) . (ω A(C)) =   v v v  (2π)mi +1  D[:k] v ∈V∞ i=1

k

Setting ξ(P) :=



v ∈V f

(ω∗A(C))−1

=

γv ωv∗ (Pv )

D

 −1

. 12 s(G ) -

Dk[:k]

v ∈V f

v

, rewrite the preceding equation as:

  . r   mi !    ξ(P).  (2π)mi +1  v ∈V i=1

(18.7.2)

v

∞

Applying Proposition 18.4.8(2), we infer that ξ(P) =

 q(dim M v +dim M v )/2 v #M v (fv )

v ∈V f

.

(18.7.3)

Using (18.7.1), we now see that   . . 12 s(G ) c(G, r    P) 1 1 D m ! dimG   i = Dk2   ξ(P). (18.7.4) mi +1  [:k]  #F (2π) i D v ∈V i=1 i=1 v

∞

k

18.7.4 We note that ξ(P) is equal to E as in 18.5.9 for S = V∞ . So from Proposition 18.5.10 we see that .  #M v (fv ) (dim M v −dim M v )/2 · qv ξ(P) = Z × , (18.7.5) v ∈S−S #M v (fv ) where Z =

 qdim M v v v ∈V f

#M v (fv )

;

(18.7.6)

and S is as in 18.5.9. The value of Z in terms of ζ- and L-functions is given in 18.5.11.

18.8 The Discriminant Quotient Formula for Global Fields

587

18.7.5 Let f be the smallest (finite) upper bound for the orders of finite subgroups of G(k). We conclude from (18.7.1) that 1

c(G, P)  Dk2

dimG

ω∗A(C)

 −1

 f −1 c(G, P).

These bounds imply the following: Proposition 18.7.6 1

f Dk2

dimG

1

(ω∗A(C))−1  c(G, P)  Dk2

dimG

(ω∗A(C))−1 .

The value of (ω∗A(C))−1 is given by the equations (18.7.3.1), (18.7.4.1) and (18.7.4.2).

18.8 The Discriminant Quotient Formula for Global Fields We shall use the notation introduced at the beginning of this chapter, however, in the following  will be an arbitrary finite separable extension of k. If k is a number field, we will now let A denote its ring of integers and B that of . For a place v of k (respectively, w of ), k v (respectively, w ) will denote the completion of k (respectively, ) at v (respectively, w). If v is non-archimedean and k is a number field, then Av (respectively, Bw ) will denote the closure of A (respectively, B) in k v (respectively, w ); Av is the same as the ring denoted by ov earlier. | | ∞ will denote the usual absolute value on Q, and for each rational prime p, | | p denotes the p-adic absolute value. For v ∈ Vf , the absolute value | | v extends to the fractional ideals of k if k is a number field and to the divisors of k if k is a function field. 18.8.1 In case k is a number field, let d(A/Z), d(B/Z) be the discriminants of A/Z, B/Z respectively, and Dk = |d(A/Z)| ∞ , D = |d(B/Z)| ∞ . The relative discriminant d(/k) of /k is by definition the discriminant d(B/A) of B/A, it is an ideal in A. 18.8.2 The group of divisors of function fields will be written multiplicatively.  Let K be a global function field. If a = av is the prime factorization of a divisor a of K, then its degree, to be denoted degK (a), is defined by  deg (a) |av | −1 (18.8.1) qK K = v , v

where qK is the cardinality of the field of constants of K. The discriminant DK 2g −2 of K is by definition equal to qK K , where gK is the genus of K.

588

The Volume Formula

If L is a finite separable extension of K, then D(L/K) will denote the different of L/K. The relative discriminant d(L/K) is by definition the divisor NL/K (D(L/K)) of K. 18.8.3 For a place w of  lying over a non-archimedean place v of k, let d(w /k v ) be the relative discriminant of w /k v . Then w /k v is unramified if and only if d(w /k v ) is trivial. The v-component of the discriminant d(/k) is   w |v d(w /k v ) and |d(/k)| v = w |v |d(w /k v )| v . Theorem 18.8.4 Let  be a finite separable extension of k. Then  |d(w /k v )| −1 D /Dk[:k] = v .

(18.8.2)

v ∈V f w |v

This theorem was proved by Moshe Jarden and Gopal Prasad in 1987. Proof Number fields and function fields will be treated separately. (1) k is a number field. We use the following relation for the relative discriminants of the ring of integers d(B/Z)/d(A/Z)[:k] = Nk/Q (d(B/A)). Taking the absolute value of both sides of the above, we obtain   D /Dk[:k] =  Nk/Q (d(/k))∞    Nk/Q (d(/k))−1 by the product formula in 18.1.1 = p p

=

 p

=

v |p

|d(/k)|v−1

 v ∈V f w |v

by [Cas67, Theorem in §11]

|d(w /k v )| −1 v

(cf. 18.8.3).

(2) k is a function field 3 . Let k be the field of constants of k, and l that of ; qk (respectively, ql ) is the cardinality of k (respectively, l). Let k  = k l. Then k  and  have the same field of constants, the genus gk  of k  equals the genus gk of k ([Deu73, Theorem 2 on p. 132]) and the different D(k /k) is trivial. Theorem 8 of [Che51] implies then that D(/k ) = D(/k). The Riemann–Hurwitz formula for /k  ([Che51]) gives 2g − 2 = [ : k ](2gk  − 2) + deg (D(/k )).

(18.8.3)

By a result in [Deu73, p. 110], we have [l : k] deg (D(/k )) = degk (d(/k)), 3

We are indebted to W.-D. Geyer for a simplification of an earlier version of the proof in this case.

18.8 The Discriminant Quotient Formula for Global Fields

589

since d(/k) = N/k (D(/k)). Now multiplying (18.8.3) by [l : k] we obtain [l : k](2g − 2) = [ : k](2gk − 2) + degk (d(/k)). As q = qk[l:k] , this leads to 2g −2

q

(2gk −2)[:k] deg k (d(/k) qk .

= qk

deg (d(/k))

By (18.8.1) and the last result of 18.8.3, qk k formula (18.8.2) follows therefore from (18.8.4).

=

(18.8.4)   v

w |v

|d(w /k v )| −1 v , 

PART FIVE APPENDICES

A Operations on Integral Models

Let k be a field given with a non-trivial discrete valuation ω. We denote the ring of integers of k by o and the maximal ideal of the latter by m. We assume that k is Henselian, cf. §2.1.

A.1 Base Change Let X be an affine scheme of finite type over k and let X be an integral model. If /k is a finite separable extension of k given with a valuation extending ω, and o is its ring of integers, then X ×o o is an integral model of X .

A.2 Schematic Closure Let X be an affine scheme of finite type over k and let X be an integral model. Given a closed subscheme Y ⊂ X the schematic closure of Y in X is the smallest closed subscheme Y of X through which the composition of the closed immersion Y → X and the natural inclusion X → X factors. If I ⊂ k[X] is the ideal defining Y then the ideal defining Y is I ∩ o[X ]. Therefore the coordinate ring of Y is the image of o[X ] under the natural map k[X] → k[Y ]. The process of schematic closure gives a map from the set of closed subschemes of X to the set of closed subschemes of X . This map is injective: the generic fiber of the schematic closure of Y is equal to Y . But this map is not surjective: any non-empty closed subscheme of the special fiber of X is also a closed subscheme of X , but not the schematic closure of a closed subscheme of X. However, if Y is a closed subscheme of X that is moreover flat over o, then Y is the schematic closure of its generic fiber. We record this as follows. 593

594

Operations on Integral Models

Lemma A.2.1 (1) If Y ⊂ X is a closed subscheme and X is an integral model of X, then the schematic closure Y of Y in X is an integral model of Y . (2) For any flat o-algebra R we have Y (R) = X (R) ∩ Y (R ⊗o k), the intersection being taken in X(R ⊗o k). (3) Y → Y := Yk is a bijection between the set of closed flat subschemes of X and the closed subschemes of X, with inverse given by the schematic closure in X . Proof The schematic closure Y of Y is affine and of finite type, since it is a closed subscheme of X . It is flat because its coordinate ring is a subring of k[Y ] and hence is m-torsion-free. The set X (R) ∩ Y (R ⊗o k) consists of those o-algebra homomorphisms ϕ : o[X ] → R whose k-linear extension kills the defining ideal I of Y in k[X], and this is equivalent to ϕ(I ∩ o[X ]) = 0, since R is m-torsion-free. If Y is a closed flat o-subscheme of X described by the ideal I ⊂ o[X ], and we let Y be its generic fiber, then we have the exact sequences of o-modules 0

/I

/ o[X ]

/ o[Y ]

/0

0

 / I ⊗o k

 / k[X]

 / k[Y ]

/0

The second is obtained by applying ⊗o k to the first. If Y is flat over o, then the  rightmost vertical map is injective and this implies I = (I ⊗o k) ∩ o[X ]. Lemma A.2.2 If /k is a finite extension, the schematic closure of Y ×k  in X ×o o is naturally identified with Y ×o o . Proof It is a basic fact that schematic closure commutes with flat base change. In the case at hand we can see this as follows. For two submodules N1, N2 of an R-module M and a flat R-algebra R  one has (N1 ⊗R R ) ∩ (N2 ⊗R R ) = (N1 ∩ N2 ) ⊗R R  inside M ⊗R R , as one sees by considering the map M → M/N1 ⊕ M/N2 with kernel N1 ∩ N2 . Using this we compute the coordinate ring of Y ×o o as (o[X ]/(I ∩o[X ]))⊗o o = o [X ]/(I ⊗o o ∩o [X ]) = o [X ]/(I ⊗k ∩o [X ]), which coincides with the coordinate ring of the schematic closure of Y ×k  in  X ×o o . Lemma A.2.3 Let G be a flat o-group scheme with generic fiber G and H be

A.3 Weil Restriction of Scalars

595

a subgroup of G. Then the schematic closure H of H in G is a flat o-subgroup scheme of G . Moreover, Lie(H ) = Lie(H) ∩ Lie(G ) inside Lie(G). Proof The product H × H is an o-flat closed subscheme of the o-flat scheme G × G and its generic fiber is H × H inside (G × G )k = G × G. The group law m : G × G → G is a flat map (being obtained from pr1 via the o-automorphism (x, y) → (xy, y) of the o-scheme G × G . So m−1 (H ) is an o-flat closed subscheme of G ×G . Since the bijection described in Lemma A.2.1(3) is inclusionpreserving in both directions, we have H × H ⊂ m−1 (H ) inside G × G if and only if the analogue holds over k inside G × G. In other words, m carries H ×H inside H if and only if mk carries H × H inside H. The same argument applies to the analogous assertions with inversion i : G → G and the identity section e : Spec(o) → G in place of m, so H is a closed o-subgroup scheme of G if and only if H is a k-subgroup of G. To prove the assertion about the Lie algebra of H , following the notation 1 1 ⊗ o[H ] → Ωo[H leads to of Remark 2.10.8, the surjection Ωo[G ]/o o[G ] ]/o a surjection ωG ,e → ωH , e and it is immediate that Homo (ωH , e,o) is the intersection of Homo (ωH , e, k) and Homo (ωG , e,o) in Homo (ωG ,e, k). For a different argument, use Lemma A.2.1(2) with the flat o-algebra o[]/ 2 and the identification Lie(H ) = ker(H (o[]/ 2 ) → H (o)). 

A.3 Weil Restriction of Scalars We review here the concept of Weil restriction of scalars in the special case that we need – the Weil restriction of an affine scheme of finite type along an extension of rings where the larger ring is a finitely generated free module over the smaller ring. For a more general discussion we refer to [BLR90, §7.6] and [CGP15, §A.5]. Let R  be a finite ring extension of R such that the R-module R  is free. The two cases that will be relevant for us arise when /k is a finite separable extension and either R = k and R  = , or R = o and R  = o . In the latter case the freeness of R /R is due to the fact that R is a principal ideal domain. Write S = Spec(R), S  = Spec(R ). Definition A.3.1 Let X be an S -scheme. A Weil restriction of scalars is a scheme over S, denoted by RS /S X , equipped with a bijection HomS (Z ×S S , X ) → HomS (Z , RS /S X ), functorial in schemes Z over S.

596

Operations on Integral Models

In other words, RS /S is right adjoint to the base change functor ×S S . Since this determines the functor of points, Yoneda’s lemma implies that if RS /S X exists, then it is unique up to unique isomorphism. Fact A.3.2 The Weil restriction functor is transitive; that is, given ring extensions R /R /R we have RS /S (RS /S X ) = RS /S X , provided these exist. 

Proof This follows from the transitivity of base change.

Fact A.3.3 If X is a group scheme, then so is RS /S X in a natural way, provided it exists. Proof This follows from the fact that RS /S is a functor that respects fiber products, the latter being immediate from the definition.  Example A.3.4 Consider X = A1S . If we choose a basis e1, . . . , en for the R-module R , then we can take RS /S X = ASn , where for any R-algebra A we identify the point (x1, . . . , xn ) ∈ Rn = ASn (A) with the point x1 e1 + · · · + xn en ∈ A ⊗R R  = A1S (A ⊗R R ). Let us give a construction in the special case that X is affine and of finite type. It will give an explicit description of RS /S X , which in particular shows that it is also affine and of finite type. This construction is an elaboration on Example A.3.4 and its compatibility with closed immersions. Construction A.3.5 Assume that X is affine and of finite type. Thus there exists a closed immersion X → AdR , whose image is described by an ideal I  ⊂ R [T1, . . . ,Td ]. Let e1, . . . , en be a basis of the R-module R . Consider the R -algebra homomorphism  α : R [Ti ]i=1,...,d → R [Ti, j ]i=1,...,d , Ti → e j Ti, j . j=1,...,n

j

The map R[Ti, j ] ⊗R R  → R [Ti, j ],

f ⊗ λ → λ · f

is an isomorphism of R -algebras, where the R-algebra structure on the left is the usual algebra structure on the tensor product, and the R -module structure is via the right factor. For each h the basis vector eh gives the R-linear form eh∗ : R  → R and hence the R-linear form eh∗ : R [Ti, j ] → R[Ti, j ]. Given f  ∈ R [Ti ] we can consider fh = eh∗ α( f ) ∈ R[Ti, j ]. The defining relation of fh is explicitly given by .     f e j T1, j , . . . , e j Td, j = e j · f j (Ti, j ). j

j

j

A.3 Weil Restriction of Scalars

597

Let I ⊂ R[Ti, j ] be the ideal generated by {e∗j α( f )| f  ∈ I , j = 1, . . . , n}. Let RS /S X be the closed subscheme of Adn R defined by I. For any R-algebra A we have the map HomR (R[Ti, j ], A) → HomR (R [Ti, j ], A ⊗R R ) → HomR (R [Ti ], A ⊗R R ) sending σ ∈ HomR (R[Ti, j ], A) to σ  := (σ ⊗ idR ) ◦ α. In other words σ (Ti ) =  j σ(Ti, j ) ⊗ e j . It is immediate that the map σ → σ is bijective, functorial in A, and σ ( f ) = 0 if and only if σ( f j ) = 0 for j = 1, . . . , n. Note that when X is finitely presented, then I is finitely generated, so RS /S X is also finitely presented. Remark A.3.6 From the above construction one sees that in the special case where R is a Henselian discretely valued field k or its ring of integers o, the identification RS /S (X )(S) = X (S ) is compatible with the analytic topologies (cf. Lemma 2.2.5) as well as with the notion of boundedness (cf. Definition 2.2.1). Indeed, it is enough to check this when X is an affine space, in which case it is immediate. Lemma A.3.7 Let X → Y be a closed immersion of affine schemes of finite type over S . Then RS /S X → RS /S Y is a closed immersion of schemes over S. Proof

This is readily visible from Construction A.3.5.



Remark A.3.8 Present the coordinate ring of Y as R [T1, . . . ,Td ]/I  as in Construction A.3.5. Let I  ⊂ J  ⊂ R [T1, . . . ,Td ] be the ideal cutting out X . Let e1, . . . , en be a basis of the R-module R . Then the coordinate ring of RS /S Y is R[Ti, j ]/I with I generated by { f j | f  ∈ I , j = 1, . . . , n} and the coordinate ring of RS /S X is R[Ti, j ]/J for the ideal I ⊂ J generated by {g j | g  ∈ J , j = 1, . . . , n}. Lemma A.3.9

Assume that X is affine and smooth. Then so is RS /S X .

Proof The infinitesimal lifting criterion for smoothness and the definition of the functor of points of RS /S X imply the formal smoothness of the latter. Since RS /S X is of finite presentation by construction, we see that it is smooth [BLR90, §2.2, Proposition 6].  Lemma A.3.10 Assume that X is affine, smooth, and with geometrically connected fibers. Then so is RS /S X . Proof This is [CGP15, Proposition A.5.9]. We note that, unless S  → S is étale, the smoothness assumption of X is essential, as shown in the example following [CGP15, Proposition A.5.9]. 

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Remark A.3.11 Weil restriction can exhibit counterintuitive behavior for non-smooth schemes. Consider for example an algebraically closed field F and the ring A = F[t]/t 2 of dual numbers. Let X be the affine A-scheme with coordinate ring A[u]/(u2 − t). We claim that R A/F X is empty. Since F is algebraically closed, it is enough to check that the set of F-points of that scheme is empty, that is, that X(A) is empty. The latter is clear since A does not contain a square root of t. A similar example can be made with the ring extension O L /O, where L/K is a quadratic (necessarily ramified) extension. Consider now the situation R  = o and R = o. Instead of writing RS /S we will now write Ro /o , and instead of X ×S S  we will write X ×o o . The same abuse of notation will be applied to R/k and X ×k . We would like to show that Ro /o respects the operation of passing to the generic fiber. For this we note that we have a second pair of adjoint functors. Namely, if Z  → Z is a morphism of schemes and X is a Z -scheme, then composing the structure map X → Z  with Z  → Z we obtain from X a Zscheme, which we denote by X | Z . We can call the functor X → X | Z the naive restriction. In the affine setting this is simply the forgetful functor that inputs a ring that is an O Z  -algebra and outputs the same ring, but now only considered as an O Z -algebra. The functor | Z is left-adjoint to × Z Z . Lemma A.3.12 Let X be an affine o -scheme of finite type. For an o-scheme Z write Z  = Z ×o o . Ro /o (X ) ×o Z = R Z  /Z (X ×o Z ). In particular, if X is an affine scheme of finite type over k and X is a smooth model of X, then Ro /o X is a smooth model of R/k X. Proof The second claim follows from the first, Lemma A.3.9, and the equality  = k ⊗o o . The first claim follows immediately from the formal properties of the adjoint pairs (base change, Weil restriction) and (naive restriction, base change) and the transitivity of fiber products. Spelled out, the transitivity of fiber products implies (Z × Z Z )|o = (Z |o ) ×o o for any Z-scheme Z , and hence Hom Z (Z , R Z  /Z (X ×o Z )) = Hom Z  (Z × Z Z , X ×o Z ) = Homo ((Z × Z Z )|o , X ) = Homo ((Z |o ) ×o o , X ) = Homo (Z |o, Ro /o X ) = Hom Z (Z , (Ro /o X ) ×o Z).

A.3 Weil Restriction of Scalars

599 

Corollary A.3.13 Let X be an affine scheme of finite type and smooth over a finite extension L of K and let Y = R L/K X. Let U ⊂ X(L) be an open, closed, and bounded subset. If U is schematic as a subset of X(L), then U is schematic as a subset of Y (K), under the identification X(L) = Y (K) as topological spaces. Proof Let X be the model of X with X (O L ) = U. Let Y = RO L /O X . Lemma A.3.12 gives the isomorphism Y ×O K → Y . We obtain the commutative diagram  / X (L)  / X(L) X (O L ) 

   Y (O)



 / Y (K)





 / Y (K)

The left and middle vertical isomorphisms come from the functorial isomorphism HomO (T, Y ) → HomO L (T ×O O L , X ) that is part of the data of the Weil restriction Y = RO L /O X , and the commutativity of the left square expresses the functoriality of that isomorphism. The commutativity of the right square is a consequence of Lemma A.3.12.  For a general S /S and S -schemes X and Y , there is no easy formula for computing RS /S (X * Y ). An exception to this is the situation where S  and S are infinitesimal, which we will explore in the setting of the Greenberg functor in §A.4, and the simple situation where S  = S * S, which we record as follows. Fact A.3.14 Let S be an affine base scheme and let X1, . . . , Xn be schemes over S. Consider X1 * · · · * Xn as a scheme over S * · · · * S. Then R(S*···*S)/S (X1 * · · · * Xn ) = X1 × · · · × Xn . Proof

Immediate from the functor of points description.



Remark A.3.15 Let /k be a finite Galois extension and let e1, . . . , en be a basis for the k-vector space . Enumerate Gal(/k) = {σ1, . . . , σn }. The n × n-matrix (σi (e j ))i, j with values in  has non-zero determinant. The square of this determinant is equal to the discriminant of /k. In particular, if /k is unramified and we choose e1, . . . , en ∈ o to be a basis of o over o, then the determinant of (σi (e j )) lies in o× . This implies that the map  , x ⊗ y → xσ(y)  ⊗k  → σ ∈Gal(/k)

is an isomorphism of -algebras, where the -vector space structure on the

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source is carried by the left tensor factor. In the same way, when /k is unramified, the map  o , x ⊗ y → xσ(y) o ⊗o o → σ ∈Gal(/k)

is an isomorphism of o -algebras. Slightly more generally, if /k is only assumed finite separable and  /k is finite Galois with  ⊂  , then the map    ⊗k  →  , x ⊗ y → xσ(y) σ

is an isomorphism of into   over k.

 -algebras,

where σ now runs over the embeddings of 

Lemma A.3.16 Consider one of the following two situations. (1) /k is a finite Galois extension, R = k, R  = . (2) /k is a finite unramified Galois extension, R = o, R  = o . Let X be an affine scheme of finite type over S . We have the isomorphism  RS /S (X ) ×S S  → X σ, σ ∈Gal(/k) σ

where X σ is the fiber product of S  −→ S  ←− X . Proof

Set Z = S  and apply Lemma A.3.12 to obtain the isomorphism RS /S (X ) ×S S  → RS ×S S /S (X ×S (S  ×S S )).

Note that in the subscript of R the map S  ×S S  → S  is the first projection, while in the argument of R the map S  ×S S  → S  is the second projection. According to Remark A.3.15 we have in the first case the isomorphism of < S -schemes S  ×S S  → σ ∈Gal(/k) S , while in the second case we have the < isomorphism of S -schemes S  ×S S  → σ ∈Gal(/k) (S )σ . The claim follows from this and Fact A.3.14.  Remark A.3.17 Lemma A.3.16 can be made more explicit using Construction A.3.5 as follows. Fix a basis e1, . . . , en of R /R and enumerate Gal(/k) = {σ1, . . . , σn }. If R [T1, . . . ,Td ]/( f1, . . . , fm ) is a presentation of the coordinate ring of X , then the coordinate ring of RS /S (X ) ×S S  is presented by R [Ti, j ]/( fh, j ), where the defining relation of fh, j is .    e j fh, j = fh e j T1, j , . . . , e j Td, j . j

j

j

The invertibility of the matrix (σi (e j )) implies that the R -span of ( fh, j )nj=1 in

A.3 Weil Restriction of Scalars 601 n . Since f R [Ti, j ] is equal to the R -span of ( j σi (e j ) fh, j )i=1 h, j has coefficients in R we have for any 1  r  n .. . -     σr (e j ) fh, j = σr e j fh, j = σr fh e j T1, j , . . . , e j Td, j j

j

j

j

= σr ( fh )(V1,r , . . . ,Vd,r ), where Vi,r = σr ( j e j Ti, j ) = j σr (e j )Ti, j . Using again the invertibility of the matrix (σi (e j )) we see that the elements Vi, j have the same R -linear span as the elements Ti, j . Thus the coordinate ring of RS /S (X ) ×S S  has the presentation  = R [V1,r , . . . ,Vd,r ]/(g1,r , . . . , gm,r ) , r

and gi,r (V1,r , . . . ,Vd,r ) = σr ( fi )(V1,r , . . . ,Vd,r ). We will write vi,r for the image of Vi,r in this quotient. Remark A.3.18 In the abstract situation of a pair L : C → D and R : D → C of adjoint functors, with L being left-adjoint to R, there are the two adjunction maps q : LR → idD and j : idC → RL, which we now recall. In terms of the natural bijection HomD (L X,Y ) = HomC (X, RY ) these arise by setting Y = L X and taking the identity map in HomD (L X, L X), or by setting X = RY and taking the identity map in HomC (RY, RY ). Note that the map L → LRL → L given by the composition Lq ◦ j L is the identity transformation of L, and the map R → RLR → R given by qR ◦ R j is the identity transformation of R. Construction A.3.19 In the situation at hand we have the adjoint pair of functors − ×S S  : SchS → SchS and RS /S : SchS → SchS . We obtain the adjunction map qX : (RS /S X ) ×S S  → X of schemes over S , functorial in X , and the adjunction map jY : Y → RS /S (Y ×S S ) of schemes over S, functorial in Y . Note that qX is not the norm map, which we will discuss below in Construction A.3.22. For X = Y ×S S  the composition X

jY × S S

/ RS /S (X ) ×S S 

qX

/X

is the identity of X . For Y = RS /S X the composition Y

jY

/ RS /S (Y ×S S )

R S  /S (qX )

/Y

is the identity of Y . In terms of the of functor of points, these maps were described explicitly

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in [CGP15, Proposition A.5.7]. Here we describe them in terms of coordinate rings. Starting with an affine scheme X of finite type over S , we present its coordinate ring as R [T1, . . . ,Td ]/I . The adjunction (RS /S X ) ×S S  → X is given on coordinate rings by the R -algebra morphism R [T1, . . . ,Td ]/I  → (R[Ti, j ]/I) ⊗R R  induced by α(Ti ) = j Ti, j ⊗ e j . Starting with an affine scheme Y of finite type over S, we present its coordinate ring as the quotient R[T1, . . . ,Td ]/J. Then the coordinate ring of Y ×S S  is R [T1, . . . ,Td ]/I , where I  ⊂ R [T1, . . . ,Td ] is the ideal generated by J, i.e. the image of J ⊗R R  under the multiplication map. Let I ⊂ R[Ti, j ] be the ideal constructed from I  as above. The adjunction Y → RS /S (Y ×S S ) is given on coordinate rings by the R-algebra morphism R[Ti, j ]/I → R[Ti ]/J sending Ti, j to e∗j (1) ·Ti , where if 1 = r1 e1 + · · · + rn en is the expression of 1 ∈ R  in terms of the basis e1, . . . , en of R /R, then e∗j (1) = r j . If the first basis element equals 1, then e∗j (1) = δ j,1 . We see in particular that this map of rings is surjective, which implies the following. Fact A.3.20 Let Y be an affine S-scheme of finite type. The adjunction jY : Y → RS /S (Y ×S S ) is a closed immersion. Proof This is stated in [CGP15, Proposition A.5.7], and is also visible from the description of this map on coordinate rings given in Construction A.3.19. Alternatively, one can fix a closed immersion of Y into affine space and use Lemma A.3.7 to reduce the problem to Y being affine space, in which case the claim is obvious.  The following lemma will be used in the explicit description of the coordinate ring of the standard integral model of an algebraic torus, see Lemma B.4.5. Lemma A.3.21 Let R = o and R  = o . Consider S = R/k Gm, and let v ∈ X∗ (S) be the adjunction q : S ×k  → Gm, . Then v ∈ [S]. Let e1, . . . , en be a basis of the free o-module o . Then e1, . . . , en is also a basis of the coordinate ring [S] as a module over k[S]. Write   e i ti , v −1 = ei ui v= i

i

for ti , ui ∈ k[S]. The coordinate ring of the integral model Ro /o Gm,o of S is the o-subalgebra of k[S] generated by ti , ui . Proof We present the coordinate ring of R/k Gm, as in Construction A.3.5, namely as k[T1, . . . ,Tn, U1, . . . , Un ]/( f1, . . . , fn ). Then the coordinate ring of Ro /o Gm,o is o[T1, . . . ,Tn, U1, . . . , Un ]/( f1, . . . , fn ). Again we use lower case letters ti and ui for the images of Ti and Ui in the quotient ring. We claim that in fact ti = ti and ui = ui . Indeed, the description of the

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603

adjunction map (R/k Gm, ) ×k  → Gm, in Construction A.3.19 shows that the corresponding map between coordinate rings [T, U]/(TU − 1) → [T1, . . . ,Tn, U1, . . . , Un ]/( f1, . . . , fn ) sends T to i ei Ti and U to i ei Ui. In other words, i ei ti ∈ [S] belongs to X∗ (S) and equals the adjunction map v. This shows ti = ti and in the same way  one sees ui = ui . Construction A.3.22 Let /k be a finite Galois extension. Consider either R = o and R  = o , or R = k and R  = . Let X be an affine scheme of finite type defined over R and let Y = RS /S (X ×S S ). For any R-algebra A we have Y (A) = XR (A ⊗R R ) = HomR (R [X ], A ⊗R R ) = HomR (R[X ], A ⊗R R ). Each σ ∈ Gal(/k) acts on this set by acting via id A ⊗ σ on A ⊗R R . This gives a functorial action of Gal(/k) on the functor of points of Y , hence by Yoneda’s lemma an action of Gal(/k) on the scheme Y over S. In terms of the description of the coordinate ring of Y given in Construction A.3.5 this action has the following explicit description. If σ(e j ) = k λ j,k ek then the action of σ on R[ti, j ] is by σ(ti, j ) = k λk, j ti,k . Note that the adjunction map jX identifies X with Y Gal(/k) . Assuming that X is a commutative group scheme we obtain a norm map N/k : Y → X  defined by N/k (y) = σ ∈Gal(/k) σ(y), which being fixed by Gal(/k) lies in X , which is identified with its image under the adjunction map jX . The norm map constructed above is frequently used to construct norm-1 tori. That is, one considers the kernel of N/k : R/k Gm → Gm . This kernel is an anisotropic torus, which is a good testing ground for various constructions.

A.4 The Greenberg Functor We assume in this section until §A.4(e) that char(f) = p > 0. Let X be an affine scheme of finite type over o/m n for some n > 0. If char(k) = p then there is a canonical embedding f → o/m n whose composition with the projection o/m n → f is the identity, cf. [Ser79, Chapter II, §4, Proposition 8]. We can form Y = R(o/m n )/f X , which is an affine scheme of finite type over f. This procedure can be used to reduce problems over the ring o/m n to problems over the field f. For example, if X is a smooth affine group scheme of finite type, sometimes

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we can solve problems involving X by accessing the theory of smooth affine groups over the field f. When the characteristic of k is zero and n > e, where e is the absolute ramification degree of k, then o/m n is not an F p -algebra, hence also not an f-algebra, so we cannot perform the same kind of Weil restriction. We would nonetheless like to have a “mixed characteristic” version of Weil restriction from o/m n to f. This was introduced by Greenberg in [Gr61, §4] (even beyond the affine setting) and is now called the Greenberg functor. This functor naturally associates to any affine scheme X of finite type over o/m n , or more generally over any artinian local ring R with residue field f, an affine scheme Gr(X) of finite type over f. When p = 0 in R (e.g., o/m n above with n ≤ e), so R is canonically an f-algebra, then Gr(X) will coincide with the Weil restriction RR/f (X) (see Example A.4.9). A motivating property will be to have, just as in the case of Weil restriction, the equality X(R) = Gr(X)(f). The construction of the Greenberg functor is based on the ring of Witt vectors, which we shall first review briefly. To carry out the construction in a coordinate-free manner, it will be convenient (as in the original approach of Greenberg in [Gr61]) to have a short discussion about “module schemes,” and then “ring schemes.” Such tools provide a convenient language for later use. We will need to go beyond affine X , but the real work is in the affine case and so we primarily focus on affine X , deducing results for non-affine X from the affine case afterwards. A thorough foundational treatment of the Greenberg functor in much broader generality than we need (e.g., omitting the hypothesis in the affine case that X is finite type over R) is given in [BG18] (especially §2 and §6–§9).

(a) Review of Witt Vectors There exists a complete discrete valuation ring with fraction field of characteristic 0, uniformizer p, and perfect residue field f, and such a discrete valuation ring is unique up to unique isomorphism: see [Ser79, Chapter II, §5, Theorem 3]. When f has finite degree f over F p then the corresponding discrete valuation ring is the valuation ring of the degree- f unramified extension of Q p . In general, this discrete valuation ring admits a unique multiplicative section a → [a] to its reduction map onto f [Ser79, Chapter II, §4, Proposition 8] and its ring structure can be described explicitly in terms of the resulting p-adic expansions i i0 [ci ]p with ci ∈ f. This explicit description is given by formulas  −i  −i  −i p p [ai ]pi + [bi ]pi = [Si (a, b) p ]pi ,

A.4 The Greenberg Functor and



p −i

[ai ]pi ·

605

 −i  −i p [bi ]pi = [Pi (a, b) p ]pi

for some universal polynomials Si , Pi ∈ Z[X0, . . . , Xi ,Y0, . . . ,Yi ] independent of f (but depending on p); see [Ser79, Chapter II, §4,§5] for a systematic treatment. Remark A.4.1 All monomials occurring in each Pi involve both X’s and Y ’s because (by design) each Pi (X, 0) and Pi (0,Y ) vanishes. Using these Si ’s and Pi ’s, we define the functor Wm of “length-m” Witt vectors (for m ≥ 1) on the category of F p -algebras by Wm (A) = Am equipped with addition and multiplication laws (a0, . . . , am−1 ) + (b0, . . . , bm−1 ) = (S0 (a, b), . . . , Sm−1 (a, b)), (a0, . . . , am−1 )(b0, . . . , bm−1 ) = (P0 (a, b), . . . , Pm−1 (a, b)); having additive identity given by (0, . . . , 0) and multiplicative identity given by (1, 0, . . . , 0). When m = 0 we set Wm (A) = {0}. If we define [a] = (a, 0, . . . , 0) m−1 p p2 p m−1 [ai ]pi = (a0, a1 , a2 , . . . , am−1 ). In then a → [a] is multiplicative and i=0 particular, if A is perfect (i.e., a → a p is bijective) then every w ∈ Wm (A) can be uniquely written in the form 0i e := deg(E), so n = me + r with 0  r  e − 1 and m  1. Then the uniformizer π satisfies π n+(e−r) = π (m+1)e ∈ pm+1 R× and π n ∈ pmT r R× . Hence, R = (o/(pm+1 ))/(π n ) equals Wm+1 (f) ⊕ · · · ⊕ Wm+1 (f)T r−1 ⊕ Wm (f)T r ⊕ · · · ⊕ Wm (f)T e−1 with algebra structure given by the vanishing of E(T) = T e + pwe−1T e−1 + · · · + pw1T + pu0 for some w1, . . . , we−1 ∈ W(f) and u0 ∈ W(f)× . Then for f-algebras A, the Wn (A)-algebra R(A) is functorially given by R(A) = Wm+1 (A) ⊕ · · · ⊕ Wm+1 (A)T r−1 ⊕ Wm (A)T r ⊕ · · · ⊕ Wm (A)T e−1 with algebra structure determined by the vanishing of E(T). If instead n ≤ e then R = f[T]/(T n ) and R(A) = A[T]/(T n ) naturally in f-algebras A. If f  is a perfect extension field of f and R  := W(f ) ⊗W (f) R is the associated artinian local ring with residue field f , then naturally in R we have R   Rf as Wn -algebra schemes over f . This is immediate from Example A.4.5 and the method of construction of R. Remark A.4.7 As a ring-theoretic analogue of Remark A.4.4 using Wn instead of W, beware that the natural map Wn (A) ⊗Wn (f) R = Wn (A) ⊗Wn (f) R(f) → R(A) is generally not an isomorphism when A is not perfect (so the definitions of the Greenberg functor in [BLR90, p.276] and [Sta12, §2] respectively resting on W(A) ⊗W (f) (·) and Wn (A) ⊗Wn (f) (·) are incorrect; neither reference gives a proof of representability based on such a definition). Likewise, the natural map Wn (A)[T]/(E,T n ) → R(A) is generally not an isomorphism when A is not perfect. For example, in the special case E = T 2 − p and odd n = 2m + 1 > 1 it is the map Wn (A)[T]/(T 2 − p,T n ) = (Wn (A) ⊕ Wn (A)T)/(pmT, pm+1 ) → Wm+1 (A) ⊕ Wm (A)T, which fails to be an isomorphism for non-perfect A because Wn (A)/(pr ) → Wr (A) is not an isomorphism for non-perfect A and 1  r < n. It is precisely

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for this reason that we take the roundabout approach through ring-schemes as in Example A.4.6 when defining R. In the same spirit, beware (as is noted in [BG18, Ex.2.5]) that M is generally not left-exact in M. For example, for the natural quotient map q : Wn+1 (f) → Wn (f) = Wn+1 (f)/(pn ), the kernel of q : Wn+1 → Wn is W1 = Ga via a → (0, 0, . . . , a) but the natural map W1 = ker(q) → ker(q) is not an isomorphism: it is instead identified with the n-fold relative Frobenius morphism ker(q) → ker(q)(n) of ker(q) = W1 = Ga (a quotient by the infinitesimal group scheme n αp n ) since pn [a] = (0, 0, . . . , a p ) in Wn+1 (A) for f-algebras A. This failure of left-exactness will not be directly relevant to us, but it will be lurking in the shadows in one important result later on (Theorem A.4.18).

(c) Definition and Properties of the Greenberg Functor Let R be an artinian local ring with perfect residue field f of characteristic p > 0. This is uniquely a W(f)-algebra and hence a Wn (f)-algebra for large n (with pn = 0 in R). We have seen above how to define the ring scheme R over f that is equipped with a Wn -algebra structure. This ring scheme will enable us to define a variant of Weil restriction, where the generally non-existent operation of base change from f to R is replaced with the functor of points of R on f-algebras. Definition A.4.8 For an affine scheme X of finite type over R, the Greenberg transform Gr(X) = Gr R/f (X) of X is the functor on f-algebras defined by Gr(X)(A) = X(R(A)) = HomR (Spec(R(A)), X); this is the set of scheme morphisms over R = R(f). We call X  Gr(X) the Greenberg functor on the category of such schemes X. In the special case R = o/m n for a complete discrete valuation ring o with residue field f (we allow o to be equicharacteristic p), for an affine o-scheme X of finite type we may write Grn (X) to denote Gr R/f (XR ). For most applications we will use this with varying n. Example A.4.9 In the special case that p = 0 in R, so R is an f-algebra, the functor R is A  R ⊗f A. This is ultimately due to Example A.4.3. Hence, from the functorial definition, in such cases Gr(X) is the Weil restriction of X through f → R by another name. The purpose of the Greenberg functor is to provide a suitable replacement for Weil restriction when p may not vanish in R. But whereas Weil restriction

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611

is defined rather generally for finite flat maps of noetherian base rings, the Greenberg functor is only defined here on schemes over an artinian local ring with perfect residue field of characteristic p > 0. The analogy with Weil restriction motivates the techniques to be used to prove the basic properties of the Greenberg functor, starting with the representability of Gr(X). Before we discuss general properties of the Greenberg functor, we record a crucial special case. Example A.4.10 Let X = AnR . In this case we claim that Gr(X) is represented by Afn·(R) , where (R) denotes the length of the artinian ring R. (In the special case n = 0 this says that the functor Gr(Spec(R)) on f-algebras is represented by Spec(f), which is easily checked directly, using the finality of Spec(R) in the category of R-schemes). By definition Gr naturally commutes with direct products, so we may assume X = A1R (this is done just to simplify notation in the calculation to be given). In this case we want to show that Gr(X) is . For f-algebras A we have represented by A(R) f Gr(X)(A) = X(R(A)) = R(A), so Gr(X) is represented by the ring scheme R (so Gr(AnR ) is represented by Rn ).  By construction, R = Wmi as Wn -module schemes over f, where R =  W (f) as a W(f)-module. Thus, R is an affine space over f with dimension mi mi , so we just need to check that mi coincides with the length of R. Since R is artinian local with residue field f, its length as a ring coincides with its length as a W(f)-module. But the latter length is mi since W (f) (Wm (f)) = m. Theorem A.4.11 For any affine R-scheme X of finite type, Gr(X) is represented by an affine f-scheme of finite type and its formation commutes with fiber products as well as perfect extension of the ground field: if f /f is a perfect extension field then Gr(X)f is represented by Gr R /f (XR ) for R  = W(f ) ⊗W (f) R. The following additional properties hold for affine R-schemes of finite type. (1) If f : X → Y is a smooth R-map then Gr( f ) is smooth. The same holds for the property of being étale. (2) If f : X → Y is a closed immersion over R then Gr( f ) is a closed immersion. The same holds for the property of being an open immersion. $ (3) If X = Xi is a finite disjoint union of affine R-schemes of finite type $ then Gr(Xi ) → Gr(X) is an isomorphism. (4) If {Ui } is an affine Zariski open cover of X then {Gr(Ui )} is an affine Zariski open cover of Gr(X). (5) If f : X → Y is smooth with its non-empty fibers of pure dimension d ≥ 0

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then the smooth map Gr( f ) has its non-empty fibers of pure dimension d · (R), and if f is a smooth surjection then so is Gr( f ). (6) If X → Y is a torsor for a smooth affine R-group G then Gr(X) → Gr(Y ) is a torsor for the smooth affine f-group Gr(G). In particular, if X → Y is a torsor for a finite constant group Γ then so is Gr(X) → Gr(Y ). The reader may readily check that the proof of this theorem is very similar to the proof of the analogous properties of Weil restriction given in [CGP15, A.5]. Proof To prove representability, we choose a closed immersion X → AnR over R, thus presenting the coordinate ring of X as R[t1, . . . , tn ]/( f1, . . . , fm ). We are going to show that the resulting map Gr(X) → Gr(AnR ) = Rn = Afn·(R) is represented by a closed subscheme of the target. By definition, for f-algebras A we have Gr(X)(A) = X(R(A)) = HomR (R[t1, . . . , tn ]/( f1, . . . , fm ), R(A)) = {(x1, . . . , xn ) ∈ R(A)n | f j (x) = 0} where the f j ’s have coefficients in R = R(f) → R(A). Since we are aiming to represent Gr(X) by a closed subscheme of Rn , our task is reduced to the representability by a closed immersion for the condition f = 0 on Rn with any f ∈ R[t1, . . . , tn ] (so then we may conclude by intersecting such closed immersions scheme-theoretically for f1, . . . , fm ). The ring scheme structure on R is really an R-algebra scheme structure via the ring homomorphism R = R(f) → R(A) for f-algebras A. Thus, any f ∈ R[t1, . . . , tn ] defines a natural transformation f : Rn → R via the natural R-algebra structure on each R(A). The condition f = 0 is represented by the pull-back f −1 (0) that is a closed subscheme of Rn . By the functorial definition, it is clear that Gr(X) commutes with fiber products in X. To analyze the behavior of Gr(X) under perfect ground field extension f → f , consider an f -algebra A. We have Gr(XR )(A) = XR (R (A)) = X(R (A)) where in the final term on the right we are forming points over R = R(f) relative to the R-algebra structure on the R -algebra R (A). Thus, it suffices to check that R  as a Wn -algebra scheme over f  is identified with the scalar extension of

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R as a Wn -algebra scheme over f. But this was treated at the end of Example A.4.6. We prove the assertions (1) through (6) in order. For (1), since Gr( f ) is a map between affine f-schemes of finite type, for smoothness it suffices to check the infinitesimal criterion. That is, if A → A0 is a surjection with square-zero kernel then we want to show that the natural map Gr(X)(A) → Gr(X)(A0 ) ×Gr(Y)(A0 ) Gr(Y )(A) is surjective. By definition, this map is identified with the natural map X(R(A)) → X(R(A0 )) ×Y(R(A0 )) Y (R(A)),

(A.4.2)

where all points are formed over R = R(f). Since X → Y is smooth and hence satisfies the infinitesimal criterion, it suffices to check that the R-algebra map R(A) → R(A0 ) is surjective with kernel consisting of nilpotent elements (as smooth morphisms satisfy the infinitesimal lifting criterion relative to such surjections of rings over the base). For some n  1 we have pn = 0 in R, so since R is artinian local with residue field f we can use an f-basis of R/pR consisting of 1 and nilpotent elements to build a Wn (f)-module presentation R  Wm1 (f)1 ⊕ Wm2 (f)r2 ⊕ · · · ⊕ Wmd (f)rd for nilpotent r2, . . . , rd ∈ R. Then for any f-algebra A we have naturally as a Wn (A)-module R(A) = Wm1 (A) · 1 ⊕ Wm2 (A) · r2 ⊕ · · · ⊕ Wmd (A) · rd

(A.4.3)

relative to the ring map R = R(f) → R(A). This description of R(A) has all summands apart from the first one consisting entirely of nilpotents, and Wm1 (A) · 1 is identified with the ring Wm1 (A). Thus, the nilpotence task reduces to the special case R = Wm (f). That is, it suffices to show that for any surjection of f-algebras A → A0 with square-zero kernel, the kernel J of Wm (A) → Wm (A0 ) satisfies J 2 = 0. By Remark A.4.1, for any w, w  ∈ Wm (A) whose Witt coordinates lie in a common ideal of A, the Witt coordinates of ww  lie in the square of that ideal of A. This completes the proof of preservation of smoothness under the Greenberg functor. The preservation of étaleness under the Greenberg functor goes in exactly the same way, except that we analyze (A.4.2) being bijective rather than surjective. Next, we prove (2). The proof of representability shows that if X → Afn is a closed immersion then the map Gr(X) → Gr(Afn ) = Rn = Afn·(R)

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is represented by a closed immersion. Hence, if X → Y is a closed immersion over R then upon choosing a closed immersion Y → AnR over R we get a composite closed immersion X → Y → AnR . Thus, in the resulting diagram of maps Gr(X) → Gr(Y ) → Gr(AnR ) the second map and the composite map are closed immersions. This forces the first map to be a closed immersion (since, in terms of rings, if A → A → A is a pair of ring maps for which the first and the composition are surjective then the second is surjective). Hence, the first part of (2) is settled. Since the Greenberg functor clearly takes monomorphisms to monomorphisms, and open immersions are precisely the same thing as étale monomorphisms [EGAIV4 , 17.9.1], the preservation of being an open immersion is established. For (3), we first recall that to check whether or not a map between f-algebras is an isomorphism it is sufficient to do so after a ground field extension. Hence, we may and do assume f is algebraically closed. To show that a map g : Y → Z between schemes of finite type over the algebraically closed field f is an isomorphism, we claim it suffices to show that Y (A) → Z(A) is bijective for all finite local f-algebras A. Grant such bijectivity. Using A = f, Y (f) → Z(f) is bijective. Varying through all A then yields that for any y ∈ Y (f) and the corresponding z ∈ Z(f) the induced local map O Z,z → OY ,y between local noetherian rings induces an isomorphism between completions (as the category of complete local noetherian f-algebras with residue field f is fully faithfully embedded into the category of functors on such A). The isomorphism condition on completions implies that g : Y → Z is étale at all f-points [EGAIV4 , 17.6.3], so it is étale (since the locus of points at which a map of finite type between noetherian schemes is étale is always open). The relative diagonal map Δg : Y → Y × Z Y is therefore étale (being a map between étale Z-schemes), yet is also a monomorphism and hence is an open immersion [EGAIV4 , 17.9.1]. But the open immersion Δg is clearly bijective on f-points and so it is an isomorphism (as f is algebraically closed). The diagonal Δg being an isomorphism says exactly that the original map g is a monomorphism, but g is étale, so it is an open immersion as well. Bijectivity of g on f-points thereby implies that g is an isomorphism, as claimed. To summarize, for the proof of (3) we have arranged that f is algebraically closed and we just need to check bijectivity on the set of points valued in each finite local f-algebra A. Since A is local, the map on A-valued points is the natural map >  > (A.4.4) Xi (A). Gr(Xi )(A) → Gr

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(The locality of A is crucial, since for the left side we are using that any map from Spec(A) into a disjoint union of schemes must factor through one of the constituents of the disjoint union; that would fail if Spec(A) is disconnected, which is ruled out by locality of A.) The map (A.4.4) is identified with $ $ (Xi (R(A))) → ( Xi )(R(A)). Provided that R(A) is local, this latter map is bijective by the same reasoning we just gave in connection with the left side of (A.4.4). The quotient map A → A/m A = f is surjective with kernel consisting of nilpotent elements, so (similarly to what we saw with square-zero kernel in the proof of (1)) the same holds for R(A) → R(f) = R. By locality of R, it follows that R(A) is local, as desired. This finishes the proof of (3). Turning to (4), by (2) each Gr(Ui ) is Zariski open in Gr(X), so the task is $ surjectivity of Gr(Ui ) → Gr(X). We may apply ground field extension so that f is algebraically closed, and then it suffices to check that Gr(X)(f) is covered by the subsets Gr(Ui )(f). This is the same as checking if X(R(f)) = X(R) is covered by the subsets Ui (R), and that in turn is clear since Spec(R) consists of a single point and the Ui are an open cover of X. For (5), consider a smooth map f : X → Y whose fibers are each empty or of pure dimension d ≥ 0. We want to show that Gr( f ) has all non-empty fibers with pure dimension d · (R), and that if instead f is a smooth surjection then so is Gr( f ). By compatibility with perfect ground field extension, we can assume f is algebraically closed. Hence, to analyze the fibers of the smooth Gr( f ) it suffices to check over f-points of Gr(Y ). Since Gr(Y )(f) = Y (R(f)) = Y (R), we can use the compatibility of the Greenberg functor with respect to the fiber product X ×Y Spec(R) for a section y ∈ Y (R) to reduce to showing that if X is R-smooth and non-empty then Gr(X) is f-smooth and non-empty, and if moreover X has pure dimension d then Gr(X) has pure dimension d · (R). By (1) we know Gr(X) is smooth over Gr(Spec(R)) = Spec(f), and to check it is non-empty or of pure dimension d · (R) we may work Zariski locally on X by (4). Thus, we can assume that there exists an étale map h : X → AdR , so Gr(h) expresses Gr(X) as étale over Gr(AdR ) = Afd ·(R) . This shows that the f-smooth Gr(X) has pure dimension d · (R), and we just need to show it is non-empty when X is non-empty. But Gr(X)(f) = X(R(f)) = X(R), and since f is algebraically closed we know that the smooth non-empty special fiber X0 = X mod mR has an f-point. The infinitesimal lifting property for the R-smooth X ensures that X(R) → X(f) = X0 (f) is surjective, so (5) is settled. To prove (6), we first note that a G-torsor f : X → Y is necessarily a smooth surjection, so Gr( f ) is a smooth surjection by (5). Hence, to show that the Gr(G)-action on Gr(X) over Gr(Y ) is a torsor structure it is equivalent to show

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that the natural map Gr(G) ×Spec(f) Gr(X) → Gr(X) ×Gr(Y) Gr(X) is an isomorphism. The compatibility of the Greenberg functor with respect to fiber products identifies this map with the output of applying the Greenberg functor to G ×Spec(R) X → X ×Y X, and this latter map is an isomorphism since X is a G-torsor over Y . In the special case that X is a Γ-torsor over Y for a finite group Γ, the same holds for Gr(X) → Gr(Y ) since naturally Gr(ΓR ) = Γf (as holds with any finite set in place of Γ because Gr(Spec(R)) = Spec(f)).  There is one further property that we need to prove: the preservation under the Greenberg functor for a smooth morphism to have geometrically connected (non-empty) fibers. The analogous result in the case of Weil restriction given in [CGP15, Proposition A.5.9] rests on a deformation retract argument involving the closed immersion X → R A /A(X A ) for a finite flat map A → A of noetherian rings and an affine A-scheme X of finite type. For the Greenberg functor there is no analogous closed immersion since the Greenberg functor does not rest on base change. Hence, the proof below involves an entirely different idea that is specific to residue characteristic p > 0. (This yields another proof of [CGP15, Proposition A.5.9] for Weil restriction: via spreading-out arguments over Z-schemes of finite type it is enough to treat the case of positive characteristic, where geometric fibers of Weil restrictions are an instance of the Greenberg functor.) Theorem A.4.12 If f : X → Y is a smooth surjection between affine Rschemes of finite type such that all fibers are geometrically connected, then the same holds for Gr( f ). In particular, if X is a smooth affine R-group then Gr(X 0 ) = Gr(X)0 . The notation X 0 was introduced in Definition 4.1.18. This result was originally proved by Greenberg in [Gr63, p.264, Corollary 2] in the essential case Y = Spec(R) with algebraically closed residue field f by expressing the adjunction morphism Spec(R(k[Gr(X)])) → X Zariski locally on X as a composition of torsors by vector groups. An alternative argument specific to the setting of group schemes (e.g., using short exact sequences) is given in [Sta12, Lemma 4.10] that is similar to one given for Weil restriction of smooth group schemes in [Oes84, Proposition A.3.7].

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Proof By the same base change argument in the treatment of smooth surjections in the proof of Theorem A.4.11(5), we can focus on the case of R-smooth X with f algebraically closed and Y = Spec(R). In this case we can write {Xi } for the set of connected components of X, and it suffices to show that {Gr(Xi )} is the set of connected components of Gr(X). By Theorem A.4.11(3) we may reduce to the case where the R-smooth X is connected, and we want to show that the f-smooth Gr(X) is connected. Since the connected X has the same underlying space as its special fiber X0 = X mod mR that is f-smooth, this space is irreducible. In particular, any non-empty affine open subschemes U,V ⊂ X have non-empty overlap, and Gr(U) ∩ Gr(V) = Gr(U) ×Gr(X) Gr(V) = Gr(U ×X V) = Gr(U ∩ V) as open subschemes of Gr(X). This is non-empty by Theorem A.4.11(5) since U ∩ V → Spec(f) is a smooth surjection. Thus, if {Ui } is an affine open cover of X by non-empty schemes (necessarily irreducible and smooth, hence connected), it suffices to prove the connectedness of each Gr(Ui ). In other words, our task is Zariski local on X. We can cover the R-smooth X by non-empty affine open subschemes that are étale over affine spaces over R, so we may and do assume there exists an étale R-map h : X → AdR for some d  0. Now comes the crucial step which is very specific to working in residue characteristic p > 0: we claim that there exists a finite étale R-map X → AdR . Beginning with the étale h over R, consider the associated map h0 : X0 → Afd after reduction modulo mR . This is an étale f-map from an affine f-scheme to an affine space over f. Since char(f) > 0, by [Ach17, Proposition 5.2.1] there exists a finite étale map h0 : X0 → Afd . Smoothness of AdR over R (or bare hands) and the affineness of X allows us to lift h0 to an R-map h  : X → AdR . Since h  is an R-map between smooth R-schemes and its reduction modulo mR is the map h0 that is étale by design, h  is étale [EGAIV4 , 17.8.2]. The map h  is also finite since finiteness for a map between noetherian schemes can be checked modulo nilpotents (so we can bootstrap from the finiteness of h0 ). Thus, by renaming h  as h we may and do arrange that h is finite étale. In particular, the image of h is both open and closed, so it is surjective since AdR is connected. That is, h : X → AdR is a finite étale cover. By Grothendieck’s theory of the étale fundamental group, every connected finite étale cover of a connected noetherian scheme is dominated by one which is a torsor for a finite group. Hence, there exists a finite group Γ and a Γ-torsor Y → AdR for which there is a (necessarily finite étale surjective) map g : Y → X over AdR . Since the étale Gr(g) is necessarily surjective (Theorem A.4.11(4)), the connectivity of Gr(X) reduces to that of Gr(Y ). In this way we can replace X with

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Y to reduce to the case that f : X → AdR is a Γ-torsor for a finite group Γ. Hence, by Theorem A.4.11(6), Gr( f ) : Gr(X) → Gr(AdR ) = Afd ·(R) is a Γ-torsor with smooth connected target. All connected components of Gr(X) are finite étale and hence surjective onto the smooth connected target, so every connected component of Gr(X) meets every fiber of Gr( f ). Thus, the connectedness of Gr(X) holds if Gr( f ) has some fiber that consists of a single topological point (this sufficiency does not require the Γ-torsor property, but we will verify this criterion for Gr( f ) by using the torsor property). Consider the Γ-torsor f0 : X0 → Afd over R/mR = f. The function field F of the integral scheme X0 is a Galois extension of  := f(t1, . . . , td ) with Galois group Γ (it will not be important to distinguish between Γ and its opposite group in what follows). Since the R-algebra R() is an infinitesimal thickening of Ga () =  (with R → Ga corresponding to R  f), we can choose a point y ∈ AdR (R()) lifting the generic point in Afd () (either by hand or by the R-smoothness of AdR ). Viewing y as an R(F)-valued point over R via the natural ring map R() → R(F) over R(f) = R, by étaleness of X → AdR a lift x ∈ X(R(F)) of y ∈ AdR (R(F)) is uniquely specified by a lift x0 ∈ X(F) = X0 (F) of y0 ∈ AdR (F) = Afd (F). But y0 as an F-valued point is exactly the composition Spec(F) → Spec() → Afd where the first map is the extension of function fields for the connected Galois cover X0 → Afd and the second map is the generic point. Thus, we can take x0 to be the generic point of X0 to specify a choice of x ∈ X(R(F)). The point x we have built belongs to X(R(F)) = Gr(X)(F), and by design it lies over a point in AdR (R(F)) = Gr(AdR )(F) arising from an -valued point (over f). The Γ-orbit of x is topologically the entire fiber through x over Gr(AdR ) (due to the Γ-torsor property), so if the Γ-action on Gr(X) leaves the topological image point for x fixed in place then the connectedness of Gr(X) will follow (as we have explained already). Hence, it suffices to prove that two ways of making Γ act on x coincide: through the Γ-action on Gr(X) (namely, the Greenberg functor applied to the Γ-action on X) and through the Γ-action on F (in its guise as a Galois extension of  with Galois group Γ). The reason this would do the job is that pre-composition of any f-map Spec(F) → Gr(X) with the Γ-action on Spec(F) keeps the topological image point in Gr(X) fixed in place! So our task reduces to checking equalities of various pairs in Gr(X)(F) = X(R(F)) (namely, applying to x the effect of each γ ∈ Γ either on X or on F). By the infinitesimal lifting property for étaleness applied to X → AdR , it is the same as equalities of the corresponding pairs in AdR (R(F)) and equalities of the corresponding pairs in X0 (F). But the Γ-action on X covers the trivial

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action on AdR , and the Γ-action on F is defined in terms of the Γ-action on X0 = X mod mR . Since x lies over the point y ∈ AdR (R(F)) belonging to the subset AdR (R()) whose elements are unaffected by the Γ-action on R(F) arising from the Γ-action on F (as  is unaffected by that action, due to Γ being a group  of automorphisms of X0 over Afd ), we are done.

(d) Beyond the Affine Case The definition of Gr(X) for affine X makes sense without affineness (and as such is useful for work with lft-Néron models; see Definition B.8.9): Definition A.4.13 For an R-scheme X locally of finite type, the functor Gr(X) on f-algebras is defined by Gr(X)(A) = X(R(A)). For any monomorphism X  → X between R-schemes locally of finite type, clearly Gr(X ) → Gr(X) is a monomorphism of functors on f-algebras. If U,V ⊂ X are open subschemes of X then the subfunctors Gr(U), Gr(V) ⊂ Gr(X) satisfy Gr(U) ∩ Gr(V) = Gr(U ∩ V) (as is readily checked by evaluating on an f-algebra). When X is separated and U and V are affine then U ∩V is also affine. We will use gluing from the affine case to establish representability of Gr(X) when X is separated. It is convenient to first verify a sheaf property relative to the Zariski topology on the category of f-algebras. Lemma A.4.14 The functor Spec(A)  Spec(R(A)) on the category of affine f-schemes carries Zariski open immersions to Zariski open immersions, and likewise for Zariski open covers compatibly with overlaps of Zariski open immersions. Proof It suffices to show that if {Spec(Ai )} is a Zariski open cover of Spec(A) for an f-algebra A then {Spec(R(Ai ))} is a Zariski open cover of Spec(R(A)) with each natural map R(Ai ) ⊗R(A) R(Ai ) → R(Ai ⊗ A Ai ) an isomorphism. Each affine open subscheme Spec(Ai ) ⊂ Spec(A) is covered by finitely many basic affine open subschemes that are also basic affine open in Spec(A). In this way, our task reduces to the case where each Spec(Ai ) is a basic affine open in Spec(A). For n large enough that pn = 0 in R, so R is a Wn (f)-algebra, by design R is not only a Wn -module scheme but it is even a Wn -algebra scheme. Thus, for any a ∈ A we have a canonical element [a] ∈ R(A) arising from [a] = (a, 0, . . . , 0) ∈ Wn (A) via the ring scheme map Wn → R. The natural map

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R(A) → R(Aa ) carries [a] to a unit because Wn (A) → Wn (Aa ) carries [a] to a unit. In this way we get a natural ring map R(A)[a] → R(Aa )

(A.4.5)

and it is enough to show that this map is always an isomorphism and that if {ai } generates 1 in A then {[ai ]} generates 1 in Wn (A) (hence also in the Wn (A)-algebra R(A)). To verify the assertion about generating 1 it is enough to show that the natural surjective ring map Wn (A) → A has kernel consisting of nilpotent elements. Via the quotient map A → Ared modulo nilpotents we may assume A is reduced, so then A injects into its perfection (the direct limit of copies of A under the p-power endomorphism). This reduces our task to the case that A is perfect, so then the kernel of Wn (A) → A is pWn (A), which consists of nilpotents since pn = 0 in Wn (A). The desired isomorphism property for (A.4.5) is a special case of the more general claim for any Wn (f)-module M and the associated Wn (A)-module M(A) that the natural map M(A)[a] → M(Aa ) of Wn (A)[a] -modules is bijective. Since M is a direct sum of Wn (f)-modules of the form Wmi (f), by naturality and additivity in M we may assume M = Wm (f) for 1 ≤ m ≤ n. This reduces our problem to showing that the natural map Wm (A)[a] → Wm (Aa ) is an isomorphism for any f-algebra (or even F p -algebra) A and a ∈ A. By definition of multiplication on Wm , we have [a] · (b0, . . . , bm−1 ) = (ab0, a p b1, . . . , a p

m−1

bm ).

Any element w ∈ Wm (Aa ) can be written in the form (c0 /a n, c1 /a np , . . . , cm−1 /a np

m−1

)

for c j ∈ A and sufficiently large n, so [a n ]w is in the image of Wm (A). Since [a n ] = [a]n , this shows that Wm (A)[a] → Wm (Aa ) is surjective. To prove Wm (A)[a] → Wm (Aa ) is injective we just have to check that any w = (b0, . . . , bm−1 ) ∈ Wm (A) with vanishing image in Wm (Aa ) is killed by [a]n for sufficiently large n. The vanishing condition on w means that each b j has vanishing image in Aa , so there is a large n for which a n b j = 0 in A for all 0  j  m − 1. Then [a]n w = [a n ](b0, . . . , bm−1 ) = (a n b0, a np b1, . . . , a np

m−1

bm−1 ) = (0, . . . , 0). 

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Theorem A.4.15 Let X be a separated R-scheme locally of finite type. The functor Gr(X) is represented by a separated f-scheme locally of finite type. If U → X is an open immersion then Gr(U) → Gr(X) is an open immersion, and if {Ui } is an open cover of X then the open subschemes Gr(Ui ) ⊂ Gr(X) constitute an open cover. The separatedness hypothesis can be removed, but we only need the separated case. Proof Let us first check that the functor Gr(X) is a sheaf for the Zariski topology on the category of f-algebras. If {Spec(Ai )} is an affine open cover of an affine f-scheme Spec(A) then for the coordinate ring Aii = Ai ⊗ A Ai of each Spec(Ai ) ∩ Spec(Ai ) the sheaf axiom amounts to the left-exactness of the diagram of sets   X(R(Ai )) ⇒ X(R(Aii )). X(R(A)) → (i,i )

Lemma A.4.14 gives that {Spec(R(Ai ))} is a Zariski open cover of Spec(R(A)) with ith and i th members having overlap Spec(R(Aii )), so the desired leftexactness is a consequence of the sheaf property for the functor of points of X relative to the Zariski topology on the category of R-algebras. This completes the verification that Gr(X) is a Zariski sheaf. Let {X j } be an affine open cover of X, so the affine schemes Gr(X j ) are subfunctors of Gr(X) that satisfy Gr(X j ) ∩ Gr(X j  ) = Gr(X j ∩ X j  ) with X j ∩ X j  also affine (since X is separated). Since X j ∩ X j  ⇒ X j , X j  are open immersions of affine R-schemes of finite type, it follows from Theorem A.4.11(2) that the maps Gr(X j ∩ X j  ) ⇒ Gr(X j ), Gr(X j  ) are open immersions of affine f-schemes of finite type. The gluing axiom is easily verified (since Gr(X j ) ∩ Gr(X j  ) ∩ Gr(X j  ) = Gr(X j ∩ X j  ∩ X j  )), so we can define an f-scheme T locally of finite type by gluing the Gr(X j ) along these “overlaps.” To see T is separated, which is to say that the diagonal ΔT /f is a closed immersion, it is enough to show that each map of affine schemes Gr(X j ∩ X j  ) → Gr(X j ) ×Spec(f) Gr(X j  ) = Gr(X j ×Spec(R) X j  ) (X j × X j  ) → is a closed immersion. But this holds because X j ∩ X j  = Δ−1 X/R X j × X j  is a closed immersion (by separatedness of X) and the Greenberg functor on affine R-schemes of finite type takes closed immersions to closed immersions (Theorem A.4.11(2)).

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Since Gr(X) is a Zariski sheaf, we can uniquely “glue” the natural transformations Gr(Xi ) → Gr(X) of functors on f-algebras to define a natural transformation Homf (·,T) → Gr(X) of functors on f-algebras. We claim that this is an isomorphism, so the desired representability result will be proved. Due to the way T was constructed from subfunctors of the Zariski sheaf Gr(X), it is enough to show that for any f-algebra A and ξ ∈ Gr(X)(A) there is an affine open Zariski cover {Uα } of Spec(A) such that each ξ |Uα factors through one of the subfunctors Gr(Xi ) (where i may depend on α). By definition, ξ corresponds to an f-morphism h : Spec(R(A)) → X, and we seek a covering of Spec(A) by affine open subschemes Uα = Spec(Aα ) such that the restriction of h to each Spec(R(Aα )) factors through some Xi . The preimages h−1 (Xi ) constitute a Zariski open cover of Spec(R(A)), so it is enough to show for any f-algebra A that a cofinal system of open covers of Spec(R(A)) consists of those arising from applying the functor R to Zariski open affine covers of Spec(A). The quotient map R  f gives rise to a natural transformation R → W1 = Ga that is surjective on A-valued points (as with any surjection M  M  of finitely generated Wn (f)-modules in place of R  f), so we get a closed immersion Spec(A) → Spec(R(A)) naturally in A. We saw in the proof of Lemma A.4.14 that R(Aa ) = R(A)[a] with {[ai ]} generating 1 in R(A) when {ai } generates 1 in A, so it is enough to show that the ideal ker(R(A)  A) consists of nilpotents. By (A.4.3) this reduces to the case R = Wm with 1  m  n. The nilpotence of all elements of ker(Wm (A) → A) was shown in the proof of Lemma A.4.14. This completes the proof of representability of Gr(X) by a separated f-scheme locally of finite type. Since we constructed a representing scheme for Gr(X) by gluing Gr(X j ) along Gr(X j ∩ X j  ) for any desired choice of affine open cover {X j } of X, by using such a cover extending one for an open subscheme U ⊂ X it is clear that Gr(U) → Gr(X) is an open immersion, and also that Gr transforms open covers to open covers.  Corollary A.4.16 If f : X → Y is a smooth map between separated Rschemes locally of finite type then Gr( f ) is smooth. In particular, if X is a smooth separated R-scheme locally of finite type then the separated f-scheme Gr(X) locally of finite type is f-smooth. Proof This can be done in two ways: copy the proof in the affine case (Theorem A.4.11(1)) verbatim, or reduce to the affine case. For the reader who prefers not to look back at the earlier proof, here is the second option. By definition, the Greenberg functor is compatible with fiber products of locally finite type R-schemes. Hence, if {Yj } is an open cover of Y by affine schemes then for the affine open subschemes Gr(Yj ) that cover Gr(Y ) we have

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Gr( f )−1 (Gr(Yj )) = Gr(X j ) for the (typically non-affine) open subschemes X j = f −1 (Yj ) that cover X. Since smoothness of Gr( f ) is Zariski local over Gr(Y ), we may replace f : X → Y with each X j → Yj to reduce to the case that Y is affine. Now pick an affine open cover {Xi } of X, so {Gr(Xi )} is an open cover of Gr(X). It is therefore sufficient to treat each Xi → Y separately, so that reduces us to the settled case when X and Y are both affine.  The following result provides a technique for studying Greenberg transforms via induction on the length of R, analogous to a result for Weil restriction [CGP15, Proposition A.5.12] (whose proof is similar to that given below). Theorem A.4.17 Let R → R  be a surjection between artinian local rings with residue field f, and let X be a smooth separated R-scheme locally of finite type. For X  := X ⊗R R , the natural f-map f : Gr(X) → Gr(X ) defined functorially by Gr(X)(A) = X(R(A)) → X(R (A)) = Gr(X )(A) for f-algebras A is a smooth surjection. Proof Once it is proved that f is smooth, for surjectivity it will be sufficient to check on f-points. But f on f-points is X(R(f)) → X(R (f)), for which the surjectivity is an instance of the infinitesimal lifting property for the R-smooth X since R(f) → R (f) is a surjection between artinian local R-algebras (namely, W(f) ⊗W (f) (·) applied to R  R ). Since f is an f-morphism between f-schemes locally of finite type, it is locally of finite presentation and it is smooth if and only if its scalar extension ff is smooth. The formation of Gr(X) commutes with extension of the perfect ground field for the same reason as in the affine case, so we may and do assume f is algebraically closed. Hence, to prove f is smooth it suffices to check the infinitesimal criterion using finite local f-algebras [EGAIV4 , 17.14.2]. Thus, for a surjection A → A0 with square-zero kernel between finite local f-algebras we want to show that Gr(X)(A) → Gr(X )(A) ×Gr(X  )(A0 ) Gr(X)(A0 ) is surjective. This map is identified with the natural map X(R(A)) → X(R (A)) ×X(R (A0 )) X(R(A0 )).

(A.4.6)

The natural map R(A) → W1 (A) = A is surjective with kernel consisting of nilpotents (use (A.4.3)), and likewise for R(A0 ), R (A), and R (A0 ). Thus, the rings R(A), R(A0 ), R (A), and R (A0 ) are infinitesimal thickenings of the

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common residue field f. Hence, anything in the target of (A.4.6) corresponds to a compatible triple of infinitesimal thickenings of a common f-point of X over R, so they all factor through a common affine open U ⊂ X (namely, any U containing that f-point). By reduction to the affine case with U, we thereby conclude that the natural map X(R (A) ×R (A0 ) R(A0 )) → X(R (A)) ×X(R (A0 )) X(R(A0 )) is bijective, where we use the ring-theoretic fiber product on the left side. This identifies (A.4.6) with the application of X to the map of R-algebras R(A) → R (A) ×R (A0 ) R(A0 )

(A.4.7)

whose kernel consists of nilpotent elements (because that is the case for the kernel of R(A) → R(f) = R → f). By direct limit considerations, the infinitesimal criterion for an R-smooth scheme is satisfied using surjective ring maps with kernel consisting of nilpotent elements. Thus, by R-smoothness of X it suffices to prove that (A.4.7) is surjective. More generally, for any surjection M  M  between finitely generated Wn (f)-modules we claim that the natural map M(A) → M (A) × M  (A0 ) M(A0 ) between Wn (A)-modules is surjective. Let us first reduce to the case where M is Wn (f)-free. Let {m1, . . . , mr } ⊂ M be a subset lifting an f-basis of M/pM, and let π : N = Wn (f)r  M be a surjection from a free Wn (f)-module carrying the standard basis to the mi ’s. The composite map q : W(f)r → N → M is surjective, and by the structure of finitely generated W(f)-modules we can precompose q with an automorphism α of W(f)r (and likewise precompose π with α mod pn ) so that ker q (and hence ker π) is a direct sum compatibly with the ambient one on W(f)r . In this manner, we arrive at a choice of π so that compatibly M  ⊕Wni (f) for 0  n1, . . . ,  nr  n. Hence, N(A0 ) → M(A0 ) is surjective (because Wn (A0 ) → Wn (A0 ) is surjective for any 0  n   n), so it is enough to treat N  M  M  rather than M  M . In this way, we have reduced to the case where M is Wn (f)-free. Since M is now Wn (f)-free, we can repeat the use of the structure theorem for finitely generated W(f)-modules with M  in place of M to arrive at a Wn (f)basis of M so that ker(M  M ) is a compatible direct sum. Thus, M  M  is a direct sum of natural maps Wn (f) → Wni (f) for 0  ni  n, so it suffices to treat the natural maps Wn (f) → Wn (f) for 0  n   n. That is, we want to show that the natural map Wn (A) → Wn (A) ×Wn (A0 ) Wn (A0 )

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is surjective for 0  n   n. The case n  = 0 is trivial, and the case 1  n   n is easily checked directly (by handling the first n  Witt coordinates separately  from the last n − n  such coordinates). Now let X be a smooth separated R-group scheme locally of finite type, so the smooth surjection in Theorem A.4.17 is visibly a homomorphism and its kernel is a smooth f-group. The kernel turns out to always be a connected unipotent group (in particular, affine). To see this, note first that the kernel J of f : R  R  is exhausted by a sequence of ideals Ji = miR J for i  0 (with mR the maximal ideal of R). More specifically, for N so large that JN = 0, we can express f as the composition of quotient maps R = R/JN → R/JN −1 → R/JN −2 → · · · → R/J0 = R/J = R , so if we define Xi = X ⊗R (R/Ji ) then Gr( f ) : Gr(X) → Gr(X ) is the composition of smooth surjective f-homomorphisms Gr(X) = Gr(X N ) → Gr(X N −1 ) → Gr(X N −2 ) → · · · → Gr(X ). Thus, ker(Gr( f )) is connected unipotent provided that the kernels of the smooth surjections Gr(Xi ) → Gr(Xi−1 ) are each connected unipotent (note that an fgroup scheme extension E of an affine f-group H of finite type by an affine f-group H  of finite type is necessarily also affine of finite type, since E → H is an H -torsor and hence represented by an affine morphism of finite type). By renaming Ri as R and Ri−1 = Ri /Ji−1 as R , this reduces us to the case where mR J = 0. In that special case there is the following more precise result. Theorem A.4.18 Let X be a smooth separated R-group scheme locally of finite type. If J ⊂ R is an ideal satisfying mR J = 0 then for R  = R/J and X  = X ⊗R R  the kernel of Gr(X) → Gr(X ) is a vector group that is a canonically a quotient of Lie(Xf ) ⊗f J modulo an infinitesimal subgroup. In the special case R = W2 (f), J = pR, and affine X, this is studied at length in [CGP15, A.6], where it is shown that the kernel is the Frobenius twist Lie(Xf )(p) . In general it is difficult to give such a precise description (keeping track of Frobenius twists) because the kernel is sensitive to the cyclic W(f)module decomposition of R and how J sits in that. For example, if R = o/m n for a complete discrete valuation ring (o, m) with mixed characteristic (0, p) then the absolute ramification of o is relevant to such twisting due to its role in defining m in Example A.4.6. Proof For an f-algebra A, we saw in the proof of Theorem A.4.15 that the natural map R(A) → f(A) = A is a surjection with kernel consisting of nilpotent elements, so R(A) → R (A) is a surjection between compatible infinitesimal

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thickenings of A. In particular, for the kernel functor J = ker(R → R ) the ideal J (A) consists of nilpotent elements. Hence, the kernel ker(Gr(X)(A) → Gr(X )(A)) = ker(X(R(A)) → X(R (A))) consists of certain R-morphisms Spec(R(A)) → X that are infinitesimal deforef

mations of the trivial map Spec(A) → Spec(f) −→ Xf to the identity section. Letting O = OX,e denote the local ring of X at its infinitesimal identity section over the artinian local ring R, we have O = R ⊕ I for the augmentation ideal I. Thus, ker(Gr(X)(A) → Gr(X )(A)) is identified with the set of R-module maps δ : I → J (A) that are multiplicative. Although the functor M is not leftexact in M (see Remark A.4.7), we claim that if 0 → M1 → M2 → M3 → 0 is a short exact sequence of Wn (f)-modules then M 1 → ker(M 2 → M 3 ) is an fppf surjection. To prove this claim, pick a finite free Wn (f)-module N2 with a quotient map N2  M2 . Using the structure theorem for finitely generated W(f)-modules (as near the end of the proof of Theorem A.4.17), this quotient map is a direct sum of natural quotient maps Wn (f) → Wni (f) for 0  ni  n. Hence, N 2 (A) → M 2 (A) is surjective for all f-algebras A, so by replacing M2 → M3 with the composition N2 → M2 → M3 and replacing M1 → M2 with the pull-back M1 × M2 N2 → N2 we reduce to the case that M2 is Wn (f)-free. Again applying the same considerations with the structure theorem for finitely generated W(f)-modules, now to the new quotient map M2  M3 with Wn (f)free M2 , we can find a possibly new basis of M2 so that M2 → M3 is a direct sum of natural quotient maps Wn (f) → Wnj (f) for 0  n j  n. But then the kernel M1 = ker(M2 → M3 ) also compatibly decomposes as a direct sum, so we are reduced to the special case of the short exact sequences 

0 → Wn−n (f) = pn Wn (f) → Wn (f) → Wn (f) → 0 for 0  n   n. The cases n  = 0 and n  = n are trivial, so we may assume 1  n   n − 1. In this case the induced diagram of Wn -module schemes is Wn−n → Wn → Wn where the second map is (a0, . . . , an−1 ) → (a0, . . . , an −1 ) and the first map is 

pn



pn



(b0, . . . , bn−n −1 ) → pn (b0, . . . , bn−n −1, 0, . . . , 0) = (0, . . . , b0 , . . . , bn−n −1 ). This latter map is visibly an fppf surjection onto ker(Wn → Wn ), so the claim involving the fppf topology is proved.

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In our situation, applying the preceding to 0 → J → R → R  → 0 yields that J → J is an fppf surjection. Hence, J is identified with the fppf-sheaf image of the map of commutative connected unipotent smooth f-groups J → R, so J is represented by such an f-group too. Also, J is a vector group since it is a quotient of the vector group J associated to the f-vector space J (see Example A.4.3). The A-valued points of ker(Gr(X) → Gr(X )) therefore consist of the multiplicative R-linear maps δ : I → J (A) to an R(A)-module that is in fact a module over f(A) = A, so such δ factors through I/mR I = If . Moreover, the ideal J (A) = ker(R(A) → R (A)) has vanishing square since J is the fppf image of J on which the Wn -bilinear composition mult

J × J → R × R −→ R vanishes (as J × J → R × R → R vanishes, due to J being a square-zero ideal because mR J vanishes by hypothesis). Thus, by multiplicativity, δ factors through an f-linear map If /If2 → J (A). Since If /If2 = Lie(Xf )∗ , the A-valued points of ker(Gr(X) → Gr(X )) are now identified with Homf (Lie(Xf )∗, J (A)) = Lie(Xf ) ⊗f J (A).

(A.4.8)

The group structure of X induces the additive group structure on (A.4.8) because the group law on any smooth f-group (such as Xf ) induces addition on the tangent space at the identity. In this manner we see that as group functors on f-algebras, ker(Gr(X) → Gr(X )) = Lie(Xf ) ⊗f J for the p-torsion commutative f-group image J of the vector group J in the Wn -module scheme R. In particular, this kernel is naturally the quotient of the vector group associated to Lie(Xf ) ⊗f J modulo an f-subgroup scheme that is infinitesimal because J → R is injective on f-points (see Example A.4.5). 

(e) Applications We continue to assume that f is perfect, but we now drop the condition that f has positive characteristic. The following three cases are possible. (1) char(k) = char(f) = 0. (2) char(k) = char(f) = p > 0. (3) char(k) = 0, char(f) = p > 0.

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Let X be a smooth separated o-scheme, locally of finite type (a generality that is of interest for work with lft-Néron models; see Definition B.8.9). Its base change X ×o (o/m n ) is again smooth, separated, and locally of finite type. In the cases where char(f) = p > 0 we have the Greenberg functor and hence the f-scheme Grn (X ) = Gr(o/m n )/f (X ×o (o/m n )). In fact, as discussed in Example A.4.9, in case (2) we have Grn (X ) = R(o/m n )/f (X ×o (o/m n )), where Weil restriction is taken with respect to the canonical f-algebra structure of o/m n given by [Ser79, Chapter II, §4, Proposition 8]. In case (1), o/m n also has the structure of an f-algebra, cf. [Ser79, Chapter II, §4, Proposition 6], but this structure is not canonical. We can choose such a structure uniformly for all n by applying the same reference to the completion of o and then composing o/ m n . We choose such an algebra structure it with the isomorphism o/m n →  arbitrarily and again set Grn (X ) = R(o/m n )/f (X ×o (o/m n )), bearing in mind that this is functorial in X but depends on a non-canonical choice of f-algebra structure for  o. Theorem A.4.19 (1) Grn (X ) is smooth, locally of finite type, and separated. (2) If X is affine, so is Grn (X ). (3) If X is a group scheme, then Grn+1 (X ) → Grn (X ) is a smooth surjective group homomorphism whose kernel is a vector group. (4) If X is a group scheme, then Grn (X 0 ) = Grn (X )0 . Proof In the cases when char(f) = p > 0, statements (1), (2), and (3), were proved in the preceding subsections, namely Theorems A.4.15, A.4.11, A.4.17, and A.4.18, and Corollary A.4.16. Statement (4) was proved in A.4.12 under the assumption that X is affine. This assumption can be removed as follows. By Theorem A.4.15, Grn (X 0 ) is an open subgroup scheme of Grn (X ), so it is enough to show that it is connected. The latter follows by induction on n: the base case is immediate since Gr1 (X 0 ) is the special fiber of X 0 , which is connected by definition, and the inductive step follows from part (3). In the case when char(f) = 0 these properties of the Weil restriction functor were proved in [CGP15, Appendix A.5]. In that reference it is assumed that X is quasi-projective, but this assumption can be replaced by the assumption that X is separated and locally of finite type. Indeed, the representability of the Weil restriction functor in that case is proved in the same way as the representability of the Greenberg functor, namely by the argument given in the proof of Theorem A.4.15. This takes care of (1), except for smoothness, which follows from the fact that a smooth X locally of finite type is also locally of finite presentation, which carries over to its Weil restriction, and moreover the formal smoothness of Weil restriction follows from that of X .

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The validity of (2) was discussed in Construction A.3.5; (3) is discussed in [CGP15, Proposition A.5.12], whose proof remains the same; (4) is [CGP15, Proposition A.5.2(4), Proposition A.5.11(3)].  As a first application, we obtain a generalization of the well-known concept of topological Jordan decomposition, which is usually defined under the assumption that f is finite. Definition A.4.20 x ∈ X (o).

Assume that X is a smooth affine group scheme. Pick

(1) The element x is called topologically semi-simple, respectively unipotent, if for all n > 0 its image in X (o/m n ) = Grn (X )(f) is semi-simple, respectively unipotent. (2) A topological Jordan decomposition of x is a pair of elements xts, xtu ∈ X (o) that commute with each other such that xts is topologically semisimple, xtu is topologically unipotent, and x = xts · xtu . Remark A.4.21 As we have noted above, the functor Grn is not canonical when char(f) = 0, as it depends on the choice of f-algebra structure of  o. We will show momentarily that the concept of “topologically unipotent” does not depend on any choices. The concept of “topologically semi-simple” on the other hand does seem to depend on the choice of an f-algebra structure of o, which is not unique when char(f) = 0. Thus, in the case char(f) = 0, the topological Jordan decomposition also depends on that choice, and the uniqueness statement in Proposition A.4.23(3) below holds after this algebra structure has been chosen. Remark A.4.22 In the case that f is finite and o is complete, the notions of topologically semi-simple and unipotent coincide with well-known notions from the literature, cf. [Spi08]. Indeed, it is well-known that when G is a smooth affine f-group, an element x ∈ G(f) is semi-simple if and only if its (finite) order is prime to p, and is unipotent if and only if its order is a power of p. Therefore, an element x ∈ X (o) is topologically unipotent according to Definition A.4.20 if and only if its order in every X (o/m n ) is a power of p, hence its order in the profinite group X (o) = lim X (o/m n ) is pro-p. One the other hand, an ←−− element x ∈ X (o) is topologically semisimple according to Definition A.4.20 if and only if its image in every X (o/m n ) has order prime to p. According to Theorem A.4.19, the kernel of Grn+1 (X ) → Grn (X ) is a unipotent group, and hence a p-group on f-points, so the orders of the images of x in X (o/m n+1 ) and in X (o/m n ) are equal. Thus, the order of x is equal to the order of its image in any X (o/m n ), and is thus finite and prime to p.

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Proposition A.4.23 Assume that X is a smooth affine o-group scheme. Pick x ∈ X (o). (1) The element x is topologically unipotent if and only if its image in X (f) is unipotent. (2) If o is complete then there exists a topological Jordan decomposition of x. (3) The topological Jordan decomposition is unique if it exists, and is respected by homomorphisms of o-group schemes. Proof Let xn ∈ X (o/m n ) = Grn (X )(f) be the image of x. Then x1 ∈ X (f) = Gr1 (X )(f) is the image of xn under the map Grn (X ) → Gr1 (X ). This map is a homomorphism of smooth affine f-group schemes. If x is topologically unipotent then xn is unipotent by definition, and x1 is then also unipotent. Conversely assume that x1 is unipotent. Let xn = xn,ss · xn,u be the Jordan decomposition of xn . Since f is perfect, xn,ss, xn,u ∈ Grn (X )(f). By assumption xn,ss lies in the kernel of the map Grn (X ) → Gr1 (X ). But this kernel is a unipotent group by Theorem A.4.19. Therefore xn,ss = 1 and we see that xn is unipotent. Since Grn (X ) = Grn (X ×o  o), for the proofs of (2) and (3) we may and do assume o is complete. The existence and uniqueness of the topological Jordan decomposition follows from the equality X (o) = lim Grn (X )(f), which ←−− holds by the completeness of o, and the existence and uniqueness of the usual Jordan decomposition in each Grn (X )(f). The fact that homomorphisms of group schemes respect the topological Jordan decomposition is immediate from the functoriality of Grn and the corresponding property of the usual Jordan decomposition.  Remark A.4.24 If X is a smooth affine commutative o-group then the topological Jordan decomposition gives a direct product decomposition X (o) = X (o)ts × X (o)tu . Note that the projection map X (o)ts → X (f)ss is an isomorphism. In the case when f is algebraically closed and the special fiber of X is connected, this can be seen as a lift into X (o) of the maximal torus in that special fiber. As a second application we now present two lemmas that deal with relative identity components. In their proofs we will use Theorem A.4.19 without comment. Lemma A.4.25 Assume that o is complete and f is algebraically closed. Let X → Y be a morphism of smooth separated commutative group schemes.

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Assume that the abstract group X (o)/X 0 (o) is finitely generated and that X (o) → Y (o) is surjective. Then X 0 (o) → Y 0 (o) is also surjective. Proof This is [BLR90, §9.6, Lemma 2], whose proof we reproduce here for the convenience of the reader. The completeness of o implies X 0 (o) = lim X 0 (o/m n ), as well as the analogous statement for Y 0 . To show the sur←−− jectivity of X 0 (o) → Y 0 (o) it is enough to show that (1) for each n the map X 0 (o/m n ) → Y 0 (o/m n ) is surjective and (2) the sequence of kernels of these surjective maps satisfies the Mittag-Leffler condition. For (2), the sequence of kernels is {K (o/m n )}, where K is the scheme-theoretic kernel of X 0 → Y 0 . But K is an o-scheme of finite type (since X 0 and Y 0 are of finite type, as their fibers over Spec(o) are quasi-compact, due to being connected group schemes locally of finite type over a field and hence of finite type). Thus, Grn (K ) is of finite type. For each m > n the morphism Grm (K ) → Grn (K ) is a group scheme homomorphism. Its image is therefore a closed subgroup. Since the topological space of Grn (K ) is noetherian, the Mittag-Leffler condition follows. It is therefore enough to show (1). We fix n and consider the homomorphism X 0 (o/m n ) → Y 0 (o/m n ), which translates to Grn (X 0 )(f) → Grn (Y 0 )(f), which in turn translates to Grn (X )0 (f) → Grn (Y )0 (f). The homomorphism Y (o) → Y (o/m n ) is surjective by Lemma 8.1.3. Therefore Grn (X )(f) → Grn (Y )(f) is surjective. The group Grn (X )(f)/Grn (X )0 (f) is a quotient of X (o)/X 0 (o) and is therefore finitely generated. This reduces the proof to showing that if X → Y is a homomorphism between smooth separated commutative f-group schemes that is surjective on f-points, and if X/X0 is finitely generated, then X0 (f) → Y0 (f) is also surjective. Then X0 → Y0 is a homomorphism of finite type f-groups, so its image Y  ⊂ Y0 is a closed connected subgroup. Since f is algebraically closed, (Y0 /Y )(f) = Y0 (f)/Y (f) is a subquotient of X(f)/X0 (f) and hence finitely generated. A smooth connected commutative f-group scheme G with a finitely generated group of f-points is trivial. Indeed, for an integer n > 1 not divisible by char(f), the multiplication homomorphism n : G → G is étale and hence surjective, so the finitely generated G(f) is n-divisible and thus finite. This forces G = 1 since G is smooth and connected over the field f that is algebraically  closed. Applying this to G = Y0 /Y , we are done. Lemma A.4.26 Assume that f is algebraically closed. Let X , Y1, . . . , Yn be group schemes over o of finite type and let Yi → X be morphisms of group schemes. Assume that X is smooth, that all the Yi ’s are smooth with connected fibers, and that the images of Yi (o) generate an open subgroup of X (o) of finite index. Then this subgroup equals X 0 (o).

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Proof Let U ⊂ X (o) be the subgroup generated by the Yi (o)’s. Since each Yi has connected fibers, the morphism Yi → X factors through X 0 and hence U ⊂ X 0 (o). By assumption U is open and of finite index in X 0 (o), hence also closed, so it is enough to show that it is also dense; that is, it surjects onto X 0 (o/m n ) for all n > 0. The image Un of U in X 0 (o/m n ) is generated by Yi (o/m n )’s since each Yi is o-smooth. Thus Un is the subgroup of Grn (X 0 )(f) generated by Grn (Yi )(f). By [Bor91, Corollary 1.4(a)], the image of each Grn (Yi ) → Grn (X 0 ) is Zariski closed and connected (smoothness of each Yi allows us to apply Theorem A.4.19(4)), so by [Bor91, Proposition 2.2] the group Un generated by these images is a closed connected (for the Zariski topology) subgroup of Grn (X 0 )(f). It is moreover of finite index, hence equals Grn (X )0 (f), which in turn equals  Grn (X 0 )(f), and the required density has been shown.

A.5 Dilatation We review here the concept of dilatation in the special case that we need – for an affine scheme, flat and of finite type, over a Henselian discrete valuation ring o. For a more general discussion we refer to [BLR90, §3.2] and [MRR]. Let X be an affine scheme over k and let X be an integral model. Given a closed subscheme Y of the special fiber of X consider the functor XY : {flat o−algebras} → {sets},

R → {x ∈ X (R)| xf ∈ Y(R ⊗o f)},

where xf ∈ X (R ⊗o f) is the image of x under the reduction map. We will prove below that the functor XY is represented by an integral model of X. That is, there is an integral model whose functor of points, when restricted to the category of flat o-algebras, is the functor XY . The restriction to flat o-algebras is motivated by Remark 2.10.2. Note that, since XY will itself be flat, it is uniquely determined by the restriction of its functor of points to the category of flat o-algebras, since Yoneda’s lemma can be applied within that category. Definition A.5.1 X.

The integral model XY is called the dilatation of Y within

Construction A.5.2 Before we prove that the functor XY is represented by an integral model of X we will give an explicit construction of the coordinate ring of that integral model. For this, let A ⊂ k[X] be the o-subalgebra that is the coordinate ring of X as in Fact 2.10.4. Let I ⊂ A be the ideal defining Y. Since Y is contained in the special fiber of X , I contains the maximal ideal of

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o. Choose a uniformizer π ∈ o and let AY ⊂ k[X] be the o-subalgebra generated by A and the set {π −1 f , f ∈ I}. It is called an affine blow-up algebra and denoted by A[ mI ], cf. Remark A.5.9. Proposition A.5.3 The scheme Spec(AY ) is an integral model of X and is equipped with a natural morphism iY : Spec(AY ) → X via which it represents the functor XY . Proof By construction AY is an o-subalgebra of k[X], hence flat over o. It contains A, so AY [π −1 ] = k[X]. It is enough to check that AY is of finite type. Since A is of finite type it is generated by finitely many elements a0, . . . , an ∈ A. Since A is noetherian the ideal I is generated by finitely many elements b0, . . . , bm ∈ A. Then AY is generated by a0, . . . , an, π −1 b0, . . . , π −1 bm . We have thus proved that Spec(AY ) is an integral model of X. The inclusion A ⊂ AY gives a morphism Spec(AY ) → X . Let R be a flat o-algebra. The π-torsion-freeness of R implies that Spec(AY )(R) → X (R) is injective since Spec(AY )(R) and X (R) are subsets of X (R ⊗o k). Let x ∈ X (R) = Homo (A, R). Then x extends to AY if and only if its unique extension to an element of Homk (k[X], R ⊗o k) sends AY into R, equivalently  x(I) ⊂ mR. This is equivalent to xf ∈ Y(R ⊗o f). Remark A.5.4 One might be tempted to expect that the special fiber of XY is isomorphic to Y, but this is rarely true. We will show in Lemma A.5.10 that if both X and Y are smooth, then so is XY . So the generic and special fibers of XY have the same dimension. This is also true for X , and since the generic fibers of X and XY are the same, we see that the special fibers of X and XY have the same dimension, which of course need not be the dimension of Y. We will provide some information about the special fiber of XY in Proposition A.5.22 when X and Y are smooth affine group schemes. Fact A.5.5 If Z is a flat scheme over o and f : Z → X is a morphism whose special fiber factors through the closed immersion Y → Xf , there exists a unique morphism fY : Z → XY such that f = iY ◦ fY . Proof The problem is local on Z , so we may assume that Z is affine, after which the claim follows from the definition of the functor of points of XY .  Lemma A.5.6 Dilatations commute with products. More precisely, the dilatation of Y1 ×f Y2 within X1 ×o X2 is (X1 )Y1 ×o (X2 )Y2 . Proof

This is immediate from the definition of the functor of points of XY .

Lemma A.5.7



If X is a group scheme and Y is a subgroup scheme of Xf ,

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then XY is a group scheme in a natural way and iY : XY → X is a morphism of group schemes. Proof The composition XY × XY → X × X → X of the multiplication map for X and (iY, iY ) has special fiber factoring through Y, so by Fact A.5.5 it lifts uniquely to a morphism XY × XY → XY . In the same way we obtain the inversion map XY → XY and the unit section Spec(o) → XY . The verification of the group scheme axioms and the fact that iY is a morphism of group schemes follow from the unicity statement in Fact A.5.5 and is left to the reader.  Example A.5.8 Consider X = Ga, k with X = Ga,o . Thus k[X] = k[t] and A = o[t]. The special fiber of X is Ga,f . Let Y be the closed subscheme of Ga,f = A1f given by the origin. Its corresponding ideal I ⊂ o[t] is the ideal I = (t, π) and thus AY = o[π −1 t]. The natural map XY → X identifies XY (o) with m ⊂ o = X (o) ⊂ X(k) = k. Let us write X1 for XY . At the same time, note that X1 is itself isomorphic to affine space A1 . Indeed, t] with  t = π −1 t. The map X1 → X becomes the coordinate ring of X1 is o[ 1 1 the map A → A given by multiplication by π. The special fiber of X1 is again isomorphic to the affine line. We can now continue performing dilatations. Taking the origin in the special fiber of X1 and dilating it inside X1 , we obtain the dilatation X2 . Inductively we have the dilatation Xn of Xn−1 . It comes equipped with a natural map Xn → X obtained by composing all intermediate maps Xn → Xn−1 . The image of Xn (o) in X (o) = o is m n . The coordinate ring of Xn is An = o[π −n t]. We will use this construction in Appendix C. Remark A.5.9 We will need to understand the smoothness properties of dilatations. Before we do so, it is useful to note the close relationship between dilatation and blow-up. As in Construction A.5.2, let A ⊂ k[X] be the coordinate ring of X and let I ⊂ A be the ideal defining Y. Recall [EGAII, §3.1, Definition 8.1.3] that the blow-up BlY (X ) of X along Y is the homogeneous ∞ n spectrum Proj(A) of the graded A-algebra A = n=0 I . The A-algebra structure on A induces the structure map h : BlY (X ) → X . The closed subscheme h−1 (Y) is called the exceptional divisor of the blow-up, and h is an isomorphism over X − Y. This can be given a coordinate description as follows. Since A is noetherian, we may generate I by finitely many elements g0, g1, . . . , gn ∈ I. Then BlY (X ) can be identified with the closed subscheme of PnA described by the homogeneous ideal I  that is the kernel of the homogeneous morphism A[T0, . . . ,Tn ] → A,

Ti → gi ,

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where gi is the element of A = A ⊕ I ⊕ I 2 · · · in degree 1. The open subscheme of PnA where Ti does not vanish is isomorphic to AnA and is thus an affine scheme. Its intersection with BlY (X ) is therefore also affine, with ring of global sections given by the maximal gi -torsion-free quotient " # g0 gn A ,. . ., of A[T0, . . . ,Ti−1,Ti+1, . . . ,Tn ]/(g j − gi Tj ) ji . gi gi Since Y is contained in the special fiber of X , I contains a uniformizer π of o and we may choose the generators g0, . . . , gn so that g0 = π. For i = 0 we then obtain the affine scheme with ring of global sections being the πtorsion-free A-algebra A[π −1 g1, . . . , π −1 gn ]. This is the same as the ring AY of Construction A.5.2. In other words, XY is the affine open subscheme of BlY (X ) consisting of those points at which the stalk of the invertible ideal sheaf of the exceptional divisor is generated by π. The restriction of the structure map h : BlY (X ) → X to this affine open subscheme recovers the dilatation map XY → X . By construction, the special fiber of this map factors through Y, so the special fiber of XY is contained in h−1 (Y), and in fact equals h−1 (Y) ∩ XY . Lemma A.5.10 Let X be a smooth affine o-scheme and let Y ⊂ Xf be a closed f-smooth subscheme. Then XY is o-smooth and (XY )f → Y is an open subbundle of a projective space bundle. If the fibers of X are connected and Y is connected, and if XY (o)  ∅, then the fibers of XY are connected. Proof As explained in Remark A.5.9, the dilatation XY is an affine open subscheme of the blow-up BlY (X ) and (XY )f ⊂ h−1 (Y). Since XY → X is an isomorphism on the generic fiber, the generic fiber of XY is smooth. Since XY is flat and of finite type, the smoothness of XY is equivalent to the smoothness of its special fiber. Our assumption that XY (o)  ∅ implies further that the connectedness of XY would follow from the connectedness of its special fiber. The assumption XY (o)  ∅ implies that the special fiber of XY is a non-empty open subscheme of h−1 (Y), so it is enough to prove the following: h−1 (Y) is smooth, and if Y is connected, then so is h−1 (Y). We claim that the closed immersion Y → X is a regular embedding in the sense that the ideal defining Y is locally generated by a regular sequence. Indeed, X is regular, being o-smooth, and Y is regular, being f-smooth. Using [FL85, Chapter IV, Proposition 3.1(i)(iv)] we reduce to the claim that the regularity of a quotient of a regular local ring by an ideal forces that ideal to be generated by a regular sequence, which is a well-known property of regular local rings, e.g., [Mat89, Theorem 21.2(ii)]. For the ideal I of Y in X , it follows that the conormal sheaf C = I/I 2 is

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locally free as an f[Y]-module [FL85, Chapter IV, Proposition 3.2(a)]. As for any blow-up along a regular embedding, the exceptional divisor is a projective space bundle over the blow-up locus. That is, h−1 (Y)  P(C ) as Y-schemes [FL85, Chapter IV, Proposition 4.3(a)]. More specifically, if d  1 denotes the fiber-rank of the locally free sheaf C over a connected component of Y, then Zariski locally on that connected component the scheme P(C ) is a projective (d − 1)-space. In particular, P(C ) → Y is a smooth surjection with connected fibers, so the f-smoothness of Y implies the f-smoothness of P(C ) and the  connectedness of Y implies the connectedness of P(C ). Lemma A.5.11 If U ⊂ X is an open subscheme and U is its special fiber, the dilatation of Y ∩ U within U is an open subscheme of XY . Proof We recall (in the setting of affine schemes) the standard fact that blowups respect flat morphisms. The blow-up of an affine scheme is constructed as Proj of the graded algebra given by the direct sum of non-negative powers of an ideal. But base change of Proj is identified with Proj of the base change of the graded algebra by [EGAII, 2.8.10]. Since flat base change carries ideals to ideals due to the preservation of injectivity of maps under flat base change, we are done. We use again Remark A.5.9. Since open immersions are flat and blow-ups respect flat morphisms we see that BlY∩U (U ) is an open subscheme of BlY (X ) and the claim follows.  We can generalize Example A.5.8 as follows. Definition A.5.12 and let n ∈ Z0 .

Let G be an affine flat group scheme of finite type over o

(1) Let G (m n ) = ker(G (o) → G (o/m n )). (2) Let G0 = G and for n  1 Gn be the dilatation within Gn−1 of the trivial subgroup of the special fiber of Gn−1 . We call Gn the nth congruence group scheme of G . Lemma A.5.13

The following identities hold.

(1) n G (m n ) = {1}. (2) o[Gn ] = o[G ][π −n I] as subrings of k[G ], where I ⊂ o[G ] is the ideal of the trivial subgroup and π ∈ o is a uniformizer. m+n (3) Gn (Mm R ) = G (M R ) inside G (R ⊗o k) for any flat o-algebra R, where we have set MR = mR. (4) Lie(Gn ) = Lie(G ) ⊗ m n as lattices inside of Lie(G).

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Proof (1) is immediately seen by choosing a closed immersion of G into an affine space. (2) let I be the ideal of the trivial subgroup of G . Note that I = Ik ∩ o[G ], where Ik ⊂ k[G ] is the ideal of the trivial subgroup of the generic fiber of G . Let Y be the trivial subgroup of the special fiber of G . The ideal of Y as a closed subscheme of G is then I, π and Construction A.5.2 shows o[G1 ] = o[G ][π −1 I]. The ideal of the trivial subgroup of G1 is Ik ∩ o[G ][π −1 I] = π −1 I. (2) now follows by induction, and (3) is immediate from (2). (4) Consider a flat o-algebra R. The ring of dual numbers R[]/ 2 is then also flat over o. We have Lie(Gn )(R) = ker(Gn (R[]/ 2 ) → Gn (R)), which according to (3) equals ker(Lie(G )(R) → Lie(G )(R/MnR )) = ker(Lie(G ) ⊗o R → Lie(G ) ⊗o R/MnR ). This functor is represented by the lattice Lie(G ) ⊗ m n in Lie(G), where G is the generic fiber of G .  Remark A.5.14 As observed in [MRR, Proposition 2.12], one can define the congruence group scheme Gn without iteration, namely as a single dilatation of the trivial subgroup of the closed subscheme G ⊗o o/m n of G The functor of points is given by R → ker(G (R) → G (R/MnR )), where R is any flat o-algebra and MR = m · R. The argument of Proposition A.5.3 shows that this functor is represented by the scheme with coordinate ring o[G ][π −n I] as in Lemma A.5.13. The formation of Gn is compatible with restriction of scalars in the following sense. Lemma A.5.15 Let o  be a discrete valuation ring finite over o with ramifi ) = cation degree e. If G  is a smooth affine o -group scheme, then Ro /o (Gen  Ro /o (G )n for n  0. Proof Let m  be the maximal ideal of o . For any flat o-algebra R one has  )(R) = G  (R ⊗ o  ) = ker(G  (R ⊗ o  ) → G  (R ⊗ o  /m en )) = Ro /o (Gen o o o en  ) ker(Ro /o G (R) → Ro /o G (R/m n R)) = Ro /o (G )n (R). Since both Ro /o (Gen and Ro /o (G )n are smooth (Lemmas A.3.9 and A.5.10), and a smooth affine o-scheme is determined by its functor of points restricted to flat o-algebras, the proof is complete.  Remark A.5.16 We continue with the setting of Lemma A.5.15 and make the coordinate rings explicit. Since o  is flat over o, it is a free o-module, and we can fix a basis e1, . . . , e N for it. We present o [G ] = o [T1, . . . ,Td ]/I  and let Ie ⊂ o [T1, . . . ,Td ] be the preimage of the ideal of the unit section (so I  ⊂ Ie ).

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 ), and a uniformizer Choose generators I  = ( f1, . . . , f) and Ie = (g1 , . . . , gm   π ∈ o. Then o [Gen ] is the maximal π-torsion-free quotient of  ,m o [T1, . . . ,Td ,Y1, . . . ,Ym ]/( fr, gs − π nYs )r=1 s=1 .

Applying Construction A.3.5 to this presentation we see that the coordinate  ) is the maximal π-torsion-free quotient of ring of Ro /o (Gen  m N o[T1, j , . . . ,Td, j ,Y1, j , . . . ,Ym, j ; 1  j  N]/( fr , j + (gs, j − π nYs, j ))r=1, s=1, j=1,

where fr , j for r = 1, . . . , l, and j = 1, . . . , N is determined by - N . N N     e j T1, j , . . . , e j Td, j = e j fr , j fr j=1

j=1

j=1

and gs, j for s = 1, . . . , m and j = 1, . . . , N is determined by .     gs e j T1, j , . . . , e j Td, j = e j gs, j . j

j

j

On the other hand, applying Construction A.3.5 to the ring o [T1, . . . ,Td ] and  ) we obtain the coordinate ring the ideals I  = ( f1, . . . , f) ⊂ Ie = (g1 , . . . , gm o[T1, j , . . . ,Td, j ; 1  j  N] and the ideals I = ( fr , j ) ⊂ Ie = (gs, j ). The coordinate ring of Ro /o G  is o[T1, j , . . . ,Td, j ; 1  j  N]/I, the ideal of the unit section is Ie /I, and o[T1, j , . . . ,Td, j ,Y1, j , . . . ,Ym, j ; 1  j  N]/( fr , j + (gs, j − π nYs, j ))  ). is again seen to be the coordinate ring of Ro /o (Gen

Remark A.5.17 Let R be a commutative ring with identity. Consider a finite R-algebra B and a finitely generated R-submodule V ⊂ B. The inclusion V → B extends canonically to an algebra homomorphism SymR (V) → B. This homomorphism is surjective if and only if V generates B. To discuss injectivity, consider generators ( f1, . . . , fd ) of V. We obtain a surjective R-module homomorphism ϕ : R d → V and hence a surjective R-algebra homomorphism Sym(ϕ) : R[T1, . . . ,Td ] = SymR (R d ) → SymR (V). If we denote by M the kernel of ϕ and by I the kernel of Sym(ϕ), then I is the ideal generated by M, and M is the submodule of homogeneous elements of degree 1 of the graded ideal I. The homomorphism SymR (V) → B is injective if and only if the kernel J of the composed homomorphism R[T1, . . . ,Td ] → B is generated by the homogeneous elements of degree 1 that belong to J. The following lemma deals with one situation where SymR (V) → B is an isomorphism, and will be used in the proof of the next proposition.

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Lemma A.5.18 Let G be a smooth affine o-group scheme. Let A be the coordinate ring of G , Ie ⊂ A the ideal of the identity section e : A → o and let M A = m · A. For a positive integer n the multiplication map embeds Ie /(Ie2 + Ie · MnA) ⊗o m−n as an o/m n -submodule of o[Gn ] ⊗o o/m n and induces an isomorphism Symo/m n (Ie /(Ie2 + Ie · MnA) ⊗o m−n ) → o[Gn ] ⊗o o/m n of Hopf algebras, where on the left we take the natural Hopf algebra structure on the symmetric algebra, cf. 2.10.20. Note that, even if the generic fiber of G is connected but the special fiber is possibly disconnected, the special fiber of Gn is connected, cf.Lemma A.5.10. Proof Write Ie,n = Ie + MnA. The coordinate ring of Gn is the affine blow-up algebra " # Ie,n A = A . MnA In this algebra we have Ie,n A = m n A, so o[Gn ] ⊗o o/m n = A/m n A =  , where I  = I  A/Ie,n e,n A . Let π ∈ o be a uniformizer. The natural A-algebra e,n homomorphism * h Ie,n ⊗o m−hn → A h0



that sends h bh ⊗ π −hn to h bh /π hn is surjective due to the construction of A. We claim that its kernel is generated by the element π n ⊗ π −n − 1. By induction it is enough to take an element b0 + b1 ⊗ π −n + · · · + bh ⊗ π −hn of this kernel with h  1 and show that we can decrease h by modifying this element by a multiple of π n ⊗ π −n − 1. Since b0 + b1 /π n + · · · + bh /π hn = 0 we have h−1 . The element bh = −π n (π (h−1)n b0 + · · · + bh−1 ), so bh := bh /π n ∈ Ie,n b0 + b1 ⊗ π −n + · · · + (bh−1 + bh ) ⊗ π −n(h−1) also lies in the kernel and differs from the previous element by (bh ⊗ π −n(h−1) ) · (π n ⊗ π −n − 1). The above homomorphism induces after applying ⊗ A A/Ie,n a surjective homomorphism of algebras over A/Ie,n = o/m n * h h+1  Ie,n /Ie,n ⊗o m−hn → A/Ie,n = A/m n A h0

whose kernel is still generated by π n ⊗ π −n − 1. Since G is smooth the ideal Ie ⊂ A is generated by a regular sequence. Adding π n to this regular sequence

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we obtain a regular sequence generating Ie,n . Therefore [FL85, IV, §2, Corollary 2.4] shows that the natural homomorphism of graded o/m n -algebras * 2 h h+1 )→ Ie,n /Ie,n Symo/m n (Ie,n /Ie,n h0

is an isomorphism. For every h we have the isomorphisms h h+1 2 /Ie,n ⊗o m−hn → SymhA/Ie, n (Ie,n /Ie,n ) ⊗o m−hn Ie,n 2 → SymhA/Ie, n (Ie,n /Ie,n ⊗o m−n ).

Since M A is generated by π ∈ o, we have Ie ∩ MnA = Ie · MnA. The equality Ie,n = Ie + MnA thus induces the direct sum decomposition 2 n = Ie /(Ie2 + Ie · MnA) ⊕ MnA/(M2n Ie,n /Ie,n A + Ie · M A). 2 ⊗ m −n ) decomposes as Accordingly, the algebra Symo/m n (Ie,n /Ie,n o n −n Symo/m n (Ie /(Ie2 + Ie · MnA) ⊗o m−n ) ⊗ Symo/m n (MnA/(M2n A + Ie · M A) ⊗o m )

and the element π n ⊗ π −n − 1 is translated under this decomposition to 1 ⊗ (π n ⊗ π −n − 1). The multiplication map in A induces an A-linear map n −n = A/(MnA + Ie ) = A/Ie,n = o/m n, MnA/(M2n A + Ie · M A) ⊗o m

and hence a surjective algebra homomorphism n −n n Symo/m n (MnA/(M2n A + Ie · M A) ⊗o m ) → o/m

whose kernel is generated by π n ⊗ π −n −1. Putting all homomorphisms together we obtain the isomorphism of A-algebras Symo/m n (Ie /(Ie2 + Ie · MnA) ⊗o m−n ) → A/m n A . We have thus proved that the multiplication map Ie ⊗o m−n → A induces an  of o/m n -modules injection Ie /(Ie2 + Ie · MnA) ⊗o m−n → A/m n A = A/Ie,n and that the resulting algebra homomorphism from the symmetric algebra is an isomorphism. To show that this is an isomorphism of Hopf algebras, it is enough to show that it preserves comultiplication (preservation of the antipode and counit are  is the symmetric algebra then automatic). Since we have proved that A/Ie,n n 2 n −n of the o/m -module Ie /(Ie + Ie · M A) ⊗o m , it is enough to check that  is sent via the comultiplication map x ∈ Ie /(Ie2 + Ie · MnA) ⊗o m−n ⊂ A/Ie,n       Δ : A /Ie,n → A /Ie,n ⊗ A /Ie,n to x ⊗ 1 + 1 ⊗ x. Thus, given b ∈ Ie we want to show that Δ(π −n b) − π −n b ⊗ 1 − 1 ⊗ π −n b ∈ A ⊗ A maps trivially   . That element equals π −n (Δ(b) − b ⊗ 1 − 1 ⊗ b). The ⊗ A/Ie,n to A/Ie,n

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comultiplication map Δ : A → A ⊗ A maps Ie to A ⊗ Ie + Ie ⊗ A and we claim that the induced map f : Ie /Ie2 → (A ⊗ Ie + Ie ⊗ A)/(A ⊗ Ie + Ie ⊗ A)2 is given by the map b → b ⊗ 1 + 1 ⊗ b. Indeed, the o-linear map ϕ : Ie /Ie2 ⊕ Ie /Ie2 → (A⊗ Ie + Ie ⊗ A)/(A⊗ Ie + Ie ⊗ A)2,

(i1, i2 ) → i1 ⊗ 1+1 ⊗i2

is an isomorphism, and composing f with the inverse of ϕ yields the diagonal Ie /Ie2 → (Ie /Ie2 ) ⊕ (Ie /Ie2 ), which is the dual of addition Te G ⊕ Te G → Te G that is the derivative of the multiplication map G × G → G . This claim implies that Δ(b) − b ⊗ 1 − 1 ⊗ b lies in (A ⊗ Ie + Ie ⊗ A)2 , which  + I  ⊗ A )2 = A ⊗ I 2 + I 2  is contained in (A ⊗ Ie,n e,n ⊗ Ie,n + Ie,n ⊗ A . But e,n e,n  n  2n   recall that Ie,n = m · A . Therefore the latter ideal is simply m · A ⊗ A . This implies that π −n (Δ(b) − b ⊗ 1 − 1 ⊗ b) ∈ m n · A ⊗ A, and hence this element  ⊗ A /I  .  has trivial image in A/m n ⊗ A/m n = A/Ie,n e,n The following Proposition is an adaptation to our special case of [MRR, Theorem 3.5]. Proposition A.5.19 Let G be a smooth affine o-group scheme. (1) For any integer n > 0 there is a functorial isomorphism of group schemes Gn |o/m n → Lie(Gn )|o/m n . In particular, the special fiber of Gn is a vector group, that is, isomorphic to Gra for some r  0, cf. 2.10.20. (2) The above isomorphism is compatible with restriction of scalars in the following sense. If k /k is a finite separable extension of ramification degree e, o  is the ring of integers of k , and H is a smooth affine o -group scheme, then the following diagram commutes R(o /me n )/(o/m n ) (Hen |(o /men ) )

/ R(o /men )/(o/m n ) (Lie(Hen )|(o /me n ) )

Ro /o (Hen )|(o/m n )

Ro /o (Lie(Hen )|(o/m n ) )

Ro /o (H )n |(o/m n )

/ Lie(Ro /o (H )n |(o/m n ) )

where the top horizontal arrow is given by R(o /me n )/(o/m n ) applied to the isomorphism for H , while the bottom horizontal arrow is the isomorphism for G = Ro /o (H ).

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(3) For two integers 2n  m  n > 0 and a flat o-algebra R, and MR = mR, there is an injective group homomorphism n m G (MnR )/G (Mm R ) → (Lie(G ) ⊗ M R )/(Lie(G ) ⊗ M R )

(A.5.1)

that is also surjective when (R, MR ) is local Henselian. It is functorial in G and R. Proof The functorial isomorphism of group schemes Gn |o/m n → Lie(Gn )|o/m n follows directly from Lemma A.5.18. Indeed, let A be the coordinate ring of G , Ie ⊂ A the ideal of the identity section, and M A = m · A the ideal of the special fiber. By Lemma A.5.13 (4) we have Lie(Gn ) = Lie(G ) ⊗o m n = (Ie /Ie2 )∨ ⊗o m n and the coordinate ring of the scheme underlying this o-module is Symo ((Ie /Ie2 ) ⊗o m−n ). Its base change to o/m n equals (using A/Ie = o, A/Ie,n = o/m n ) Sym A/Ie, n ((Ie /(Ie2 + Ie · MnA)) ⊗o m−n ) and Lemma A.5.18 establishes an isomorphism of Hopf algebras between this algebra and o[Gn ] ⊗o o/m n . The fact that the special fiber of Gn is a vector group is then immediate, and (1) is proved. For (2) we use the fact that the isomorphism Hen |o /me n → Lie(Hen )|o /me n is uniquely characterized as the functorial isomorphism fitting into the commutative diagram / Lie(Hen )(R )

Hen (R ) , r 2 +I en ) ⊗  m −en, R  ) Homo (IHe /(IH · M o H e H e

for any o /m en -algebra R , where both diagonal arrows are the restriction maps with respect to the o -algebra homomorphisms IHe ⊗o m −en → o [Hen ] and IHe ⊗o m −en → o [Lie(Hen )] = o [Lie(H )] ⊗o m −en (recall o [V] = Symo (V ∗ )). If we consider R  = R ⊗o/m n o /m en for an o/m n -algebra R, by Remark A.5.16 the preceding diagram becomes isomorphic to the diagram / Lie(Gn )(R)

Gn (R) + s Homo (IGe /(IG2e + IGe · MGn ) ⊗o m−n, R)

and (2) follows. To prove (3) it is enough to treat the case m = 2n: for the general case we use the functoriality of the isomorphism in G applied to the map Gm−n → G .

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Assuming m = 2n, Lemma A.5.13(3) gives G (MnR ) = Gn (R) and G (M2n R ) = n n Gn (MR ). The homomorphism Gn (R) → Gn (R/MR ) provides an injective homomorphism of abelian groups between Gn (R)/Gn (MnR ) and Gn (R/MnR ), which is also surjective if (R, MR ) is local Henselian by Lemma 8.1.3. From (1) we obtain the isomorphism Gn (R/MnR ) → Lie(Gn )(R/MnR ), whose target equals Lie(Gn )/Lie(Gn ) ⊗R MnR = Lie(G ) ⊗R MnR /Lie(G ) ⊗R M2n R by Lemma A.5.13(4).



Remark A.5.20 The homomorphism (A.5.1) can be described explicitly as follows. As in the proof we may inject G (MnR )/G (Mm R ) into m n Gn (R/Mm R ) = ker(G (R/M R ) → G (R/M R )).

An element ψ of that kernel can be written uniquely as e + δ, where e : R[G ] → n m R/Mm R is the identity section, and δ : R[G ] → M R /M R is an R-linear map satisfying δ(x y) = e(x)δ(y) + e(y)δ(x). In other words, δ is an MnR /Mm R -valued R-linear derivation of R[G ]. Since e vanishes on Ie , the restrictions of ψ and δ to Ie agree, and this restriction induces a linear form Ie /(Ie2 +Ie ·MnA) → MnR /Mm R, and hence an algebra homomorphism Symo/m n (Ie /(Ie2 + Ie · MnA) ⊗o m−n ) → R/Mm−n R . This homomorphism is the image of ψ under the isomorphism Gn |o/m n → Lie(Gn )|o/m n . An equivalent way to think about this image is as the element δ of n m Homo (Ie /(Ie2 + Ie · MnA), MnR /Mm R ) = Lie(G ) ⊗ M R /M R .

Example A.5.21 The group Gm,k has coordinate ring k[t, t −1 ] with comultiplication t → t ⊗ t, coinversion t → t −1 . The obvious o-subalgebra o[t, t −1 ], which is in fact a Hopf subalgebra, is the coordinate ring of the canonical smooth integral model Gm,o . The associated subgroup of Gm (K) = K × is O× . If we let G = Gm,o , then the first congruence group scheme G1 of Definition A.5.12 has the coordinate ring o[π −1 (t −1), t −1 ]. By construction G1 (o) = 1+m. More generally, for any n  0 the coordinate ring of Gn is o[π −n (t − 1), t −1 ] and Gn (o) = 1 + m n . If we introduce the variable u = π −n (t − 1) then the coordinate ring becomes o[u, (π n u + 1)−1 ] = o[u, v]/((π n u + 1)v − 1). Now assume n  1. Upon applying ⊗o f this becomes f[u, v]/(v − 1) = f[u]. Thus Gn is a smooth group scheme with generic fiber Gm,k and special fiber Ga,f . Let R be a flat o-algebra. Evaluating x ∈ G (R) = Homk−alg (k[t, t −1 ], R) at 1 gives an isomorphism G (R) = R× that identifies Gn (R) with 1 + MnR . The Lie algebra of Gm,k is the k-vector space Homk ((t − 1)/(t − 1)2, k), which is

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identified with k by sending the element t − 1 to 1. The Lie algebra of Gm,o is the lattice o ⊂ k under the same identification. Following Remark A.5.20 we see that in this case (A.5.1) can be explicated as (A.5.1)

(1 +

MnR )/(1 x(t) o

+

Mm R)

o



Gn (R/Mm R) 

x



/ Lie(Gn )(R/Mm ) R / x|(t−1) 



/ Mn /Mm R R / x(t − 1)

and thus becomes identified with the isomorphism (1 + MnR )/(1 + Mm R) → sending x to x − 1. MnR /Mm R Proposition A.5.22 Let G be a smooth group scheme over o and H a closed smooth subgroup scheme of the special fiber of G . Then under the map GH → G the special fiber of GH surjects onto H with kernel a vector group of dimension dim G − dim H. Proof By Lemma 8.1.3 the reduction map G (O) → G (f) is surjective. Lemma A.5.10 shows that the special fiber of GH maps smoothly onto H. Hence, the special fiber V of the kernel is a smooth f-group with dimension dim(G) − dim(H). It remains to show that V is a vector group. Let G1 be the first congruence subgroup of Definition A.5.12. The homomorphism G1 → G factors through the homomorphism GH → G , as one sees considering the functors of points. We have the subgroups G1 (O) ⊂ GH (O) ⊂ G (O). An element of V(f) lifts to an element x ∈ GH (O) by Lemma 8.1.3 and this element actually lies in G1 (O). Thus the image of (G1 )f → (GH )f is V and the claim follows from Proposition A.5.19(1).  We will now reinterpret the relative identity component G 0 as a dilatation when G is affine. Proposition A.5.23 Let G be a smooth affine group scheme over o with connected generic fiber. (1) Then G 0 is the dilatation of Gf0 within G . In particular, G 0 is also affine. (2) Any intermediate group G (M) ⊂ U ⊂ G (O), whose image in G (fs ) is Zariski closed, is schematic. The integral model is the dilatation within GO of the Zariski closure in Gf of the image of U in G (fs ). (3) Any intermediate group G 0 (O) ⊂ U ⊂ G (O) is schematic and the relative identity component of its integral model coincides with GO0 .

A.6 Smoothening

645

Proof (1) Let G  be the dilatation of (G ×o f)0 within G . Both G 0 and G  are flat over o, so it is enough to compare their functors of points in the category of flat o-algebras. That these functors agree is immediate from the definitions, see [SGA3, Exp. VIB , 3.1] and Definition A.5.1. (2) Let U ⊂ G (fs ) be the image of U under G (O) → G (fs ) and let G  be the dilatation of the corresponding closed subscheme of Gfs within GO . Then G (O) is the preimage of U ⊂ G (fs ) in G (O), which equals U due to the assumption G (M) ⊂ U. (3) Since G is o-smooth and the residue field of O is separably closed, we have G (O)/G 0 (O)  π0 (Gfs ). Hence, U/G 0 (O) is identified with a subgroup of the geometric component group of the special fiber. If {g j } is a set of G 0 (O)coset representatives in U then the translates g j GO0 constitute a collection of open subschemes of GO with full generic fiber GK . The union G  = ∪ j g j GO0 is an open O-subgroup of GO satisfying G (O) = U, and by design G 0 = GO0 . 

A.6 Smoothening Let X be an affine scheme of finite type over o with smooth generic fiber. There exists a smooth affine scheme X ∞ of finite type over o equipped with a morphism X ∞ → X which induces an isomorphism on generic fibers and induces a bijective map X ∞ (O) → X (O). This is shown in [BLR90, Theorem 3.1.3, Corollary 3.1.4]. We will briefly review it here in the special case of an affine flat group scheme of finite type over o. In that case X ∞ is a smooth affine group scheme. Note that it is uniquely determined by X by Corollary 2.10.11. Construction A.6.1 Let X be an affine flat group scheme defined over o with smooth generic fiber X. We construct inductively a sequence of integral models X (i) equipped with maps X (i+1) → X (i) as follows. Let X (0) = X . For every i, let Y(i) be the closed subscheme of the special fiber of X (i) obtained by taking the Zariski closure of the image of the reduction map X (i) (O) → X (i) (f). Note that Y(i) is a smooth group scheme inside Xf (i) . Let X (i+1) be the dilatation of Y(i) in X (i) as in Definition A.5.1. It is also an integral model of X and a group scheme, by Proposition A.5.3 and Lemma A.5.7, and X (i+1) (O) → X (i) (O) is bijective. Theorem A.6.2 There exists i such that X (i) is o-smooth. Sketch of Proof We first recall the Néron measure of the failure of smoothness. Since X (i) is an affine flat group scheme of finite type, it is smooth if and only if its special fiber is smooth at the identity element e ∈ X (i) (o). According

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to the Jacobi criterion this is the case if and only if its Jacobian matrix at e has maximal rank. That is, once we present the coordinate ring o[X (i) ] = o[t1, . . . , tn ]/( f1, . . . , fm ), we consider the matrix (∂ fi /∂t j )(e) with entries in O. If d = dim(X) then X (i) is smooth if and only if after reducing to f, this matrix has rank n − d. This is the case if and only if some n − d minor of this matrix does not vanish. If we consider this matrix before reducing to f, the requirement is that some n − d minor of this matrix does not lie in the maximal ideal of o. This allows to define the following defect of smoothness: δi = min(ω(Δ)), where Δ runs over the n − d-minors of the Jacobi matrix. Then X (i) is smooth if and only if δi = 0. The main observation now is that if δi > 0 then δi+1 < δi . This is the technical heart of the proof, which is unfortunately somewhat complicated. For the details we refer to [BLR90, §3.3, Proposition 5].  Note that once X (i) is smooth, the image of the reduction map X (i) (O) → X (i) (f) becomes surjective by Lemma 8.1.3, so X (i+1) = X (i) . Therefore the sequence X (n) stabilizes. Definition A.6.3 Let X be a flat affine group scheme of finite type over o. Its smoothening is X (n) for n . 0. We denote it by X ∞ . Lemma A.6.4 The natural morphism i ∞ : X ∞ → X satisfies the following universal property. If Z is a smooth group scheme and f : Z → X is a homomorphism, there exists a unique homomorphism f ∞ : Z → X ∞ such that f = i ∞ ◦ f ∞ . If f is a closed immersion, then so is f ∞ . Proof Lemma 8.1.3 states the surjectivity of the reduction map Z (O) → Z (f), which implies that f (Z (f)) lies in the Zariski closure in X (f) of the image of X (O). Fact A.5.5 implies the existence of a unique lift f 1 : Z → X (1) . The existence of f ∞ follows by induction. If i ∞ ◦ f ∞ is a closed immersion,  then so is f ∞ . For an example of the smoothening process, see Example B.7.5. Corollary A.6.5 Let X be a smooth affine k-group. A bounded open subgroup U ⊂ X(K) is schematic if and only if there exists an integral model X , not necessarily smooth, so that X (O) = U. If U is stable under Gal(K/k) then X uniquely descends to an integral model of X. Proof The smoothening X ∞ of X still satisfies X ∞ (O) = U.



A.7 Schematic Subgroups

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Corollary A.6.6 Let X be a smooth affine k-group. If U1, U2 ⊂ X(K) are schematic, then so is U1 ∩ U2 . Proof Let X1 and X2 be the smooth integral models with Xi (O) = Ui . Let X3 be the schematic closure of the diagonal of X ×k X in X1 ×O X2 . Then X3 is an integral model of X with X3 (O) = U1 ∩ U2 . Furthermore X3 is a subgroup scheme of X1 × X2 , because by flatness this can be checked on the generic fiber, where it is obvious. Apply Corollary A.6.5. 

A.7 Schematic Subgroups We reproduce here a proposition due to Bruhat–Tits and Deligne, cf. [BT84a, Lemma in 4.6.18]. It is used in §8.3 for the construction of integral models of the groups in the inclusion chain (7.7.1). Indeed, once an integral model for any of these groups has been constructed, this proposition will provide the integral models for all larger groups. We also use it in Appendix B to obtain the lft-Néron model of a k-torus (see Definition B.8.9) from its ft-Néron model (see Section B.7). Proposition A.7.1 Let H be a smooth affine K-group and let U ⊂ V ⊂ H(K) be two subgroups such that U is normal in V and bounded open in H(K). Assume that U is schematic with smooth affine integral model U . (1) There exists a (not necessarily affine) smooth separated group scheme V with generic fiber H such that V (O) = V. (2) It comes equipped with a morphism U → V , unique since V is separated, that extends the inclusion U → V → H(K). (3) The morphism U → V is an open immersion and induces an isomorphism between the relative identity components. (4) If [V : U] < ∞ then V is affine and thus a smooth integral model of H. (5) If R is any local ring extension of O, then V (R) = U (R) · V. (6) If H comes via base change from a smooth affine k-group and V is Γ-invariant, V descends to an o-group scheme. Recall that O is strictly Henselian. In particular, its residue field is infinite. For the notion of a relative identity component, see Definition 4.1.18. Proof We first describe a set-theoretic model of the construction. Let H = H(K). Let {va }a ∈U\V be a set of representatives for U \ V. Thus we have $ V = a ∈U\V Uva in H. Consider for each a the bijection Rva : H → H $ sending x to xva . Consider the disjoint union a ∈U\V H and impose on it the

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equivalence relation that xa ∼ xb if and only if Rva xa = Rvb xb , where xa lies in the ath copy of H, and xb lies in the bth copy. The quotient of the disjoint $ union a ∈U\V H by this equivalence relation is in natural bijection with H, by embedding H into the copy of the disjoint union indexed by the trivial element $ of U \ V. Under this bijection the image of U = a ∈U\V U in the quotient is identified with V. Under this identification, the group structure on V is identified with the following map U ×U → U: given (ua, ub ) ∈ U ×U, where ua lies in the ath component and ub lies in the bth component, let c ∈ U \ V be the unique index such that va vb vc−1 ∈ U. Then the pair (ua, ub ) is mapped to the element uc of the cth component of U defined by uc = ua · (va ub va−1 ) · (va vb vc−1 ). We have used here that U is normal in V. We now implement this model with schemes. Consider the disjoint union $ $ i ∈U\V U . This is a scheme whose generic fiber is i ∈U\V H. Consider for a pair of indices i, j ∈ U \ V the isomorphism fi, j = Rvi −1 v j : H → H sending x to xvi v j−1 . Evidently f j,k ◦ fi, j = fi,k . Let V be the scheme obtained from $ i ∈U\V U by gluing the generic fibers of all copies of U in this disjoint union via the system of isomorphisms fi, j . Each copy of U in the disjoint union is identified with an open subscheme of V , which we shall call Ui . These open subschemes cover V . Since they are smooth (in particular of finite type), V itself is locally of finite type and smooth. Moreover V is of finite type if [V : U] < ∞. The generic fiber of V is identified with the generic fiber of U , hence with H. By construction the special fiber of U is an open subscheme of the special fiber of V (in fact, the special fiber of V is a disjoint union of copies of the special fiber of U), whence the morphism U 0 → V 0 is an isomorphism. To simplify matters we now assume v1 = 1. Identify U with its image U1 in V . We now endow V with a group scheme structure as follows. The identity section of V will be given by the identity section of U . For a pair a, b ∈ U \ V of indices the morphism ca,b : H × H → H,

(x, y) → z = x · (va yva−1 ) · (va vb vc−1 )

maps the subgroup U × U to the subgroup U, since U is normal in V, and therefore extends uniquely, by Corollary 2.10.10, to a morphism ca,b : U × U → U . These morphisms respect the gluing of generic fibers and descend to a morphism V × V → V , which will be the multiplication morphism. The inversion morphism is defined analogously, by taking for an index a ∈ U \ V the morphism ia : H → H,

x → (va−1 x −1 va ) · (va−1 vc−1 ),

where c ∈ U \ V is the unique element such that vc va ∈ U. The axioms of a group scheme are easily verified.

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We now prove that V is separated. We have the cartesian diagram V ×O V Δ

V

f

/V O e

/ Spec(O),

where Δ is the diagonal morphism, e is the identity section, and f (x, x ) = x · x −1 . The property of being a closed immersion is stable under base change, so it is enough to show that the identity section is a closed immersion. For this $ we will use the affine open covering V = a Ua and will show that e induces a closed immersion e−1 (Ua ) → Ua , cf. [EGAI, Chapter I, Corollary 4.2.4]. Since e : Spec(O) → U1 is the identity section for U1 and the latter is separated, it is a closed immersion. Consider now a  1. The special fiber of Ua is disjoint from that of U1 , so e−1 (Ua ) = Spec(K). The morphism e : Spec(K) → Ua factors as the closed immersion eK : Spec(K) → H and the inclusion of H into Ua as its generic fiber. This composition is identified with the composition of the closed immersion va−1 : Spec(K) → H with the open immersion H → U as its generic fiber. This latter composition is a closed immersion if and only if it is topologically closed. Assume that it is not topologically closed, and let Z be its schematic closure. Since the image of va−1 is not closed by hypothesis but meets the open H in a closed set, Z meets the special fiber of U . Moreover Z is affine and flat, so its coordinate ring A injects into A ⊗O K. At the same time A ⊗O K  K, since the generic fiber of Z has the single point va−1 . Under the isomorphism A ⊗O K → K the image of A is an O-subalgebra of K, hence it is either O or K. It cannot be K, for then Z would not meet the special fiber of U . Thus A = O, which means that va−1 extends to an O-point of U , but that is a contradiction to the assumption va−1 ∈ H(K) − U. If [V : U] < ∞, then [PY06, Proposition 3.1] implies that V is affine. Let R be a local ring extension of O. A morphism x : Spec(R) → V sends the closed point of Spec(R) into one of the open subschemes Ui of V . Lemma 8.1.1 implies that x factors through a point x : Spec(R) → Ui . Under the group law of V , right multiplication by vi is an isomorphism U → Ui of O-schemes.  Therefore, xvi−1 is an R-point of U .

A.8 Reductive Models It is a consequence of Bruhat–Tits theory that a quasi-split connected reductive k-group G has a reductive model if and only if it is K-split, and that furthermore

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any such model has to have a connected special fiber, cf. Propositions 8.3.16 and 8.4.14. Due to the usefulness of this statement, we give here an alternative proof that does not involve Bruhat–Tits theory. Proposition A.8.1 Given a quasi-split connected reductive k-group G that splits over the maximal unramified extension K of k, there is a reductive o-group scheme G whose generic fiber is G. Proof Let G0 be the split form of the given quasi-split group G over k. Fix a Borel k-subgroup B0 of G0 and a (split) maximal k-torus T0 in B0 . Using this choice, we get a root datum R(G0,T0 ) equipped with a positive system of roots (via B0 ). For each simple positive root c ∈ Φ(G0,T0 ), choose a non-zero Xc ∈ Lie(Uc ). According to a precise form of the “isomorphism theorem” for split connected reductive groups over fields (see [Con14, Theorem 6.1.17] for a formulation over rings, which we will soon need), the group A = Aut(R(G0,T0 )) is naturally identified with the group of k-automorphisms of G0 that preserve (B0,T0 ) and permute the Xc (in their effect on the Lie algebras of the root groups for simple positive roots). The resulting map H1 (k, A) → H1 (k, Aut((G0 )ks )) is a bijection onto the subset of quasi-split forms of G0 over k; see the proof of [Con14, Proposition 7.2.11] (which is formulated over rings, but whose proof applies as written when the base rings are limited to just fields). In the same manner, the map H1 (K/k, A) → H1 (K/k, Aut((G0 )K ))

(A.8.1)

is a bijection on the set of quasi-split forms of G0 over k that split over K. By [Con14, Definitions 5.1.1, 6.1.1 and Theorems 6.1.16, 6.1.17], there is a reductive o-group scheme G0 with a split maximal o-torus T0 having generic fiber (G0,T0 ) such that A is naturally identified with a subgroup of Aut((G0 )O ) (recovering the inclusion of A into Aut((G0 )K ) used above). In this way we get a natural map H1 (O/o, A) → H1 (O/o, Aut((G0 )O ))

(A.8.2)

(using the profinite group Aut(O/o) = Gal(K/k)) compatible with (A.8.1). For a finite unramified Galois extension k /k and the corresponding finite étale extension of valuation rings o → o , Galois descent from affine o -schemes to affine o-schemes works just as it does over fields in terms of compatible isomorphisms among twists by Aut(o /o) [BLR90, 6.2, Ex.B]. Hence, passing to the direct limit over such extensions, the target of (A.8.2) is the set of isomorphism classes of o-forms G of G0 that become isomorphic to G0 over O; any such G is clearly a reductive o-group scheme. It follows from the compatibility of (A.8.1) and (A.8.2) that everything in the image of (A.8.1)

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651

is the generic fiber of a reductive o-group scheme. Since the image of (A.8.1) consists of the quasi-split forms of G0 which split over K, and G is such a form, we conclude that G is the generic fiber of a reductive o-group, as desired.  Proposition A.8.2 Let G be a smooth affine o-group scheme whose generic fiber G is a connected reductive k-group and whose special fiber G has reductive identity component G 0 . (1) The o-group G contains closed o-tori S ⊂ T such that S is o-split and is a maximal closed o-split torus of G , T splits over O and T is a maximal closed o-torus of G , the generic fiber T of T splits over K, the special fiber S of S is a maximal f-split torus, and the special fiber T of T is a maximal f-torus, of G 0 . (2) T is a maximal torus of G and since T splits over K, G K is split. (3) The special fiber G is connected. So G is a reductive group scheme. (4) If the residue field f is of dimension  1, then G is quasi-split (over k). Proof Let S be a maximal f-split torus of the reductive f-group G 0 and T be a maximal f-torus of G 0 containing S . Then there are closed o-tori S ⊂ T in G whose special fibers are S ⊂ T (this is seen by applying Proposition 8.2.1(1) twice). Let S ⊂ T be the generic fibers of S ⊂ T respectively. The character groups of S , S and S are all isomorphic to each other as Γ-modules. Since S is f-split, Γ acts trivially on its character group, hence S is o-split and S is k-split. Similarly, the character groups of T , T and T are all isomorphic to each other as Γ-modules. Hence, as T splits over the separably closed field fs , T splits over O and T splits over K. The maximality assertions about S and T follow from the corresponding maximality assertions for S and T . This settles (1). To prove (2), consider anyS and T as in (1). Since G is affine, so is G 0 (by Proposition A.5.23(1)). Let ZG 0 (S ) and ZG 0 (T ) respectively be the centralizers of S and T in the neutral component G 0 of G . Both these subgroup schemes are smooth (Proposition 8.2.1(1)), and hence their generic and special fibers are of equal dimension. Since G 0 is reductive, the centralizer of the maximal torus T in G 0 is itself, so the special fiber of ZG 0 (T ) is T . By dimension and connectedness considerations, this implies that ZG 0 (T ) = T , so the centralizer of T in G equals T. Hence T is a maximal torus of G. Since this maximal torus splits over K, we conclude that G splits over K. This settles (2). We will now prove that G is in fact connected. We may (and will) replace o with O so that T is split (by (2)) and the residue field f is separably closed. The centralizer ZG (T ) of T in G is a smooth subgroup scheme (Proposition

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8.2.1(1)) containing T as a closed subgroup scheme, and its generic fiber is ZG (T) = T. We see that the closed immersion T → ZG (T ) between smooth (and hence flat) o-schemes is an equality on generic fibers and hence is an equality. Therefore, the inclusion NG (T )(o) → NG (T)(k) gives an embedding NG (T )(o)/T (o) → NG (T)(k)/T(k), and ZG (T ) = T . As T is a maximal f-torus of G , by the conjugacy of maximal f-tori in G 0 under G 0 (f) (f is separably closed so every f-torus is split!), we see that G (f) = NG (T )(f) · G 0 (f). Proposition 8.2.1(3) implies that NG (T )(o) → NG (T )(f) is surjective, and hence NG (T )(o)/T (o) → NG (T )(f)/T (f) is surjective too. So the order of NG (T )(f)/T (f) is less than or equal to that of NG (T )(o)/T (o) (→ NG (T)(k)/T(k)). On the other hand, NG (T)(k)/T(k) is the Weyl group of the root system of (T, G) and NG 0 (T )(f)/T (f) is the Weyl group of the root system of (G 0, T ), but these root systems are isomorphic([SGA3, Exp.XXII, Proposition 2.8]), hence their Weyl groups are isomorphic. We conclude from these observations that the inclusion NG 0 (T )(f)/T (f) → NG (T )(f)/T (f) is an isomorphism. So NG (T )(f) = NG 0 (T )(f), and therefore, G (f) = NG (T )(f) · G 0 (f) = G 0 (f). This implies that G is connected and hence G = G 0 . This settles (3). Finally let us assume that the residue field f of k is of dimension  1. Then the reductive f-group G 0 is quasi-split, i.e., it contains a Borel subgroup defined over f (Corollary 2.3.8), or, equivalently, the centralizer in G 0 of the maximal f-split torus S is a torus, and hence this centralizer is T . Thus the special fiber of the group scheme ZG 0 (S ) is T . Since ZG 0 (S ) contains T we conclude that ZG 0 (S ) = T , so the centralizer of the maximal k-split torus S of G in the  latter is the torus T. Therefore, G 0 and G are quasi-split, settling (4).

B Integral Models of Tori

Let k be a field given with a discrete valuation ω. We denote the ring of integers of k by o and the maximal ideal of o by m, and assume that o is Henselian. We fix a maximal unramified extension K of k and denote its ring of integers by O. For finite extensions  of k, the ring of integers is denoted by o . In case L is a finite extension of K, the ring of integers in L is denoted by O L .

B.1 Preliminaries Let T be a k-torus. In this section we are going to show that the open bounded subgroups T(K)1 and T(K)0 of T(K) from Proposition 2.5.8 and Definition 2.5.13, are schematic, and that moreover T(K)0 is connected, in the sense of Definition 2.10.15. We will furthermore define descending filtrations of T(K)0 that are also schematic. We will try to be as explicit as possible about the corresponding integral models. Fact B.1.1 If T is an integral model of T, then T (o) ⊂ T(k)1 . Proof

This is immediate from Remark 2.10.3.



Recall from [SGA3, Exp. IX, Definition 1.3] that an o-torus is an o-group scheme T which is, locally for the fpqc topology, isomorphic to a group of the form Grm for some r  0. In fact, due to the special nature of the base ring o, this definition is equivalent to requiring the existence of a finite unramified extension /k such that T ×o o is isomorphic to Grm for some r  0, cf [SGA3, Exp. X, Corollary 4.6]. We will say that T is split if it is isomorphic to Grm . Let X ∗ (T ) = Hom(T , Gm ) and X∗ (T ) = Hom(Gm, T ), the homomorphisms being taken in the category of O-group schemes. The above discussion has the following immediate corollary. 653

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Fact B.1.2 Let T and T be the generic and special fibers of T , respectively. The homomorphisms of abelian groups X ∗ (T ) → X ∗ (T), X∗ (T ) → X∗ (T), X ∗ (T ) → X ∗ (T), and X∗ (T ) → X∗ (T), obtained by taking the generic respectively special fibers of a homomorphism T → Gm or Gm → T , are isomorphisms of abelian groups. These isomorphisms are equivariant with respect to the action of Gal(K/k) = Gal(O/o) = Gal(f/f). In particular, T is split if and only if T is split, if and only if T is split. We warn the reader that the integral models of a given k-torus T that we will construct in this section will rarely be o-tori. They will be flat affine ogroup schemes of finite type, sometimes smooth, but their special fiber will be a commutative f-group scheme that is often disconnected, and its identity component will often have a non-trivial unipotent radical. In order for such an affine o-group scheme to be an o-torus, it is necessary and sufficient that its special fiber is a torus, cf. [SGA3, Exp. X, Corollary 4.9]. Note that any integral model of a k-torus T is a commutative group scheme.

B.2 Split Tori Recall from Example A.5.21 that Gm, k has the canonical integral model Gm,o whose group of R-points is equal to R× for any o-algebra R. For each integer n > 0 the nth congruence group scheme Gm,o, n is the smooth group scheme with generic fiber Gm, k and special fiber Ga,f whose group of R-points for a flat o-algebra R is equal to 1 + MnR , where MR = m · R. We can generalize d. this example to any split k-torus T. There exists an isomorphism T → Gm std Via this isomorphism we obtain integral models Tn of T for any n  0. We can formulate these intrinsically in terms of T without using an isomorphism d as follows. T → Gm Definition B.2.1 The standard model of the split k-torus T is the smooth group scheme T std whose coordinate ring is the group ring o[X∗ (T)] of the free abelian group X∗ (T) over the coefficient ring o. The n-standard filtration model Tnstd is the nth congruence group scheme of T std in the sense of Definition A.5.12. Note that T std is smooth with connected fibers by construction, and Tnstd is smooth with connected fibers by Lemma A.5.10. Recall that, when T is split, we have T(k)0 = T(k)1 = X∗ (T) ⊗Z o× . Lemma B.2.2

We have

T std (o) = X∗ (T) ⊗Z o× and Tnstd (o) = X∗ (T) ⊗Z (1 + m n ) when n > 0.

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655

Proof We have X∗ (T) = HomZ (X∗ (T), Z). The universal property of the group ring and the freeness of X∗ (T) imply T std (o) = Homo−alg (o[X∗ (T)],o) = HomZ (X∗ (T),o× ) = HomZ (X∗ (T), Z) ⊗Z o× = X∗ (T) ⊗Z o× . In the same way one obtains T std (o/m n ) = X∗ (T) ⊗Z (o/m n )× . Tracing through these identifications one sees that the natural map T std (o) → T std (o/m n ) is identified with id ⊗ pn : X∗ (T) ⊗Z o× → X∗ (T) ⊗Z (o/m n )×, where pn : o → o/m n is the natural projection.



Assuming ω is normalized so that ω(k × ) = Z, we define for any r ∈ R0 the integral model Trstd to be Tnstd , where n is the smallest integer greater than or equal to r. If ω is not normalized, then Trstd should be defined as Tnstd , where n is the smallest integer greater than or equal to er, where e ∈ R>0 is defined by ω(k × ) = e−1 Z. In all cases we have the formula Trstd (o) = X∗ (T) ⊗Z {x ∈ k × | ω(x − 1)  r }.

(B.2.1)

The following is [PY06, Lemma 4.1]. It will be used a few times in this book. Lemma B.2.3 Let S be a split K-torus and S an integral model of S. The following are equivalent. (1) S (O) is the maximal bounded subgroup of S(K). (2) S = S std . (3) S is a torus. Proof (2) ⇒ (3) follows from the construction of S std . (3) ⇒ (2) Since S is an o-torus whose generic fiber is split, it is a split o-torus, i.e. isomorphic to Grm . Via this isomorphism we see that S (O) is the maximal bounded subgroup of S(K). Since S is also smooth, Corollary 2.10.11 and Lemma B.2.2 imply that S = S std . (2) ⇒ (1) follows from the first assertion of Lemma B.2.2. (1) ⇒ (2) we want to argue again using Corollary 2.10.11 and Lemma B.2.2, but at the moment we do not know that S is smooth. This is what we will now prove. Let i ∞ : S ∞ → S be the smoothening of Definition A.6.3. Then S ∞ (O) = S (O) equals the maximal bounded subgroup of S(K) by assumption, so S ∞ = S std and we want to show that i ∞ is an isomorphism.

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The group scheme S ∞ acts on S by multiplication via i ∞ , and hence it acts on the affine ring O[S ] of S . By [SGA3, Exp.I,4.7.3], O[S ] ⊂ O[S ∞ ] decomposes into a direct sum of weight modules. Since O[S ∞ ] is the group algebra O[X], X = X∗ (S), there are non-zero elements ax ∈ O, x ∈ X, such  that the affine ring O[S ] of S equals x ∈X Oa x e x . The comultiplication ∞ map of the Hopf algebra O[S ] sends ex to ex ⊗ ex , and hence that of O[S ] sends ax ex to ax ex ⊗ ex . This latter is in O[S ] ⊗ O[S ] if and only if a2x divides ax in O, i.e., ax ∈ O× . Since this holds for all x ∈ X, we conclude that  O[S ] = O[S ∞ ] and so i ∞ is an isomorphism. Proposition B.2.4 Let G be a smooth affine o-group scheme. We assume that the generic fiber G of G contains a k-torus S such that SK is split and the maximal bounded subgroup of S(K) is contained in G (O). Then the schematic closure S of S in G is an o-torus whose generic fiber is S. Conversely, if S is an o-split torus in G with generic fiber S, then it is closed. In-fact, S is the schematic closure of S in G . Proof As S is the schematic closure of S in G , S (O) = G (O) ∩ S(K). Now since G (O) is bounded (Remark 2.10.3), S (O) = G (O) ∩ S(K) is bounded. Since by hypothesis G (O) contains the maximal bounded subgroup S(K)1 of S(K), we see that S (O) = S(K)1 . Now Lemma B.2.3 applied to SK and SO implies that S is an o-torus. To prove the converse, we observe that by Lemma B.2.3, S (O) (→ G (O)) is the maximal bounded subgroup S(K)1 of S(K). Now let T be the schematic closure of S in G . Then S is the generic fiber of T . As T (O) = G (O) ∩ S(K) is bounded and it contains the maximal bounded subgroup S(K)1 of S(K), we infer from Lemma B.2.3 that T is smooth. Now since, S (O) = S(K)1 = T (O), we conclude that T = S . 

B.3 Induced Tori Let T be an induced torus (cf. Definition 2.5.1). There are two immediate ways in which one can produce an integral model. (1) The coordinate ring of T is ([X∗ (T)])Θ , where /k is the splitting field of T and Θ is the Galois group of /k. We could consider the o-subalgebra o [X∗ (T)]Θ . We will see in Lemma B.4.9 that this is most useful when /k is unramified. (2) The torus R/k Gm, has the obvious integral model Ro /o (Gm,o ). Writing a general induced torus T as a product of tori of the form R/k Gm, we

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obtain an integral model of T. This is the standard model, and also the finite type Néron–Raynaud model. The first construction has the appeal of being simple and explicit, while the second construction has the obvious advantage of producing a smooth integral model by Lemma A.3.9. The second construction seems to depend on the choice of an isomorphism between T and a product of restriction of scalars. We will show that this is not the case. Later, in Lemmas B.4.5 and B.4.9, we will give an explicit description of the coordinate ring, and compare it to the first construction. Lemma B.3.1 Let /k be a finite separable extension. Write T = Ro /o Gm,o and T = R/k Gm, . (1) T is smooth and its special fiber is (geometrically) connected. (2) T (O) is the unique maximal bounded subgroup of T(K). (3) Any automorphism of the k-group scheme T extends to a (necessarily unique) automorphism of the o-group scheme T . Proof (1) follows from Lemmas A.3.9 and A.3.10. (2) Let k  be the maximal unramified extension of k contained in  and let d = [k  : k]. Enumerating Homk (k , K) = {σ1, . . . , σd }, the map a ⊗ b →  (aσ1 (b), . . . , aσd (b)) extends to an isomorphism K ⊗k k  = dj=1 K. Writing L for the maximal unramified extension of  in k s we have K ⊗k  = (K ⊗k k ) ⊗k   =

d 

L.

j=1

 In the same way, O ⊗o o = dj=1 O L . This implies T(K) = (K ⊗k )× = (L × ) ⊕ d and T (O) = (O×L ) ⊕ d . (3) follows from (2) together with Corollary 2.10.10 and Lemma A.3.9, since any continuous automorphism of T(K) with continuous inverse must preserve the unique maximal bounded subgroup.  Remark B.3.2 The preceding lemma implies that the integral model from the second construction above is up to unique isomorphism independent of the choice of presentation of the induced torus as a product of restriction of scalars. Indeed, this amounts to showing that any composite k-group isomorphism   Ri /k Gm,i  T  R j /k Gm, j i

j

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uniquely extends to an o-isomorphism   Ro j /o Gm,o j . Roi /o Gm,oi  i

j

By Corollary 2.10.10 and Lemma B.3.1(2), it suffices to show that for any finite separable extensions ,   over k, the effect on K-valued points for any k-homomorphism R/k Gm, → R  /k Gm,  carries the maximal bounded subgroup into the maximal bounded subgroup. But for any map X → Y between affine K-schemes of finite type, X(K) → Y (K) carries bounded subsets onto bounded subsets, so we are done (by Proposition 2.2.13 and the definition of boundedness in Definition 2.2.1). Corollary B.3.3 Let T be an induced torus. The maximal bounded subgroup T(K)1 of T(K) is schematic and connected. Recall here that T(K)1 = T(K)0 according to Lemma 2.5.18. Definition B.3.4 We shall write T std for the smooth integral model with T (O) = T(K)1 and call it the standard integral model. Fact B.3.5 Any morphism of induced tori extends (uniquely) to a morphism between their standard integral models. Proof The argument is the same as for how Lemma B.3.1(2) is used in Remark B.3.2.  In the case of a split torus T we defined the nth standard filtration model Trstd for all r ∈ R0 , with the property (B.2.1). We will now generalize this definition to induced tori. Proposition B.3.6 Let T be an induced k-torus. (1) For r ∈ R>0 , the subgroup ' ( t ∈ T(K)0 | ∀ χ ∈ X∗ (T), ω( χ(t) − 1)  r is schematic and connected. The set of jumps of this filtration is discrete. (2) For r ∈ R>0 , write Trstd for the corresponding smooth model. The special fiber of Trstd is a vector group. std (O). (3) Write T0std = T std . For all r ∈ R0 we have Trstd (M) = Tr+1 Proof All statements are made over K, so we base change to K and consider an induced K-torus, noting that by unramified descent (Fact 2.10.16) the models we’ll construct descent from O to o. Thus T is now an induced K-torus. Since all statements are compatible with

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products, we may assume without loss of generality that T = R L/K Gm for some finite separable extension L/K. (1) We have ' ( Ur := t ∈ T(K)0 | ∀χ ∈ X∗ (T), ω( χ(t) − 1)  r = {x ∈ L × | ω(x − 1)  r }. The set of jumps of this filtration is given by ω(L × ) and is thus discrete. Lemma B.2.2 provides a smooth O L -model Srstd of the L-group Gm with connected fibers such that Srstd (O L ) = Ur , cf. (B.2.1). We set Trstd = RO L /O Srstd . By Lemmas A.3.9 and A.3.10, Trstd is smooth with connected fibers. It satisfies Trstd (O) = Ur . std , where e is the ramification (2) Consider the shifted indexing Srstd,∗ = Sre std,∗ std = Sn , where n is the smallest integer greater than index of L/K. Then Sr or equal to r, and this is the nth congruence group scheme of S std of Definition A.5.12. We use the analogous notation Trstd,∗ , so that Trstd,∗ = RO L /O Srstd,∗ . Lemma A.3.12 implies that the special fiber of Tnstd,∗ equals R A/f ((Snstd ) A), where A = O L ⊗O f is an artinian local quotient of O L with residue field f. We already know that (Snstd ) A has special fiber (Snstd )f  Ga . The algebra A is artinian, so we apply [CGP15, Proposition A.5.12] repeatedly and see that R A/f ((Gnstd ) A) is an extension of vector groups, hence a vector group. (3) For any non-negative ingeger n set r = n/e. By definition of Trstd we have Trstd (M) = Snstd,∗ (MeL ), which according to Lemma A.5.13(3) equals std,∗ std S std (Me+n  L ) = Se+n (O), which in turn by definition equals Tr+1 (O).

B.4 The Standard Model Let T be an arbitrary k-torus. Let /k be any finite separable extension splitting T. Then T = T ×k  is a split torus, and therefore R/k (T ) is an induced torus. The adjunction map T → R/k (T ) is a closed immersion by Fact A.3.20. Definition B.4.1 Let T std be the schematic closure of T in the standard integral model of R/k (T ) of Definition B.3.4. We call T std the standard integral model of T. Lemma A.2.1 shows that T std is indeed an integral model of T in the sense of Definition 2.10.1, in particular that it is affine, flat, and of finite type over o. In [BT84a] this model is referred to as the “canonical scheme with generic fiber T.” Lemma B.4.5 below shows that this is the same as the “standard integral model of T” of [VKM02]. We’ll see in Corollary B.4.6 that the o-model T std of T is independent of the

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choice of ; once this is known, its formation is clearly compatible with direct products in T. First however we check the consistency property that Definition B.4.1 applied to an induced torus T yields the (intrinsic) standard integral model from Definition B.3.4 (so at least in the induced case T std respects products in T):  Example B.4.2 Let T be an induced k-torus, so T = i Ri /k (Gm,i ) for finite separable extensions i /k. Let /k be a finite separable extension that splits T, or equivalently for which the finite étale -algebra i ⊗k  is a product of copies of  for every i. Let’s check that the schematic closure of T in the standard integral model of the induced R/k (T ) (in the sense of Definition B.3.4) coincides with the standard integral model of the induced T in the sense of Definition B.3.4. This is all compatible with direct products in T, so we may assume T = R0 /k Gm,0 for a finite separable extension 0 /k such that 0 ⊗k    d as algebras (the natural index set for the direct product is the set of k-embeddings of 0 into ). Hence, T has a standard integral model T std = Ro0 /o (Gm,o0 ) in the sense of Definition B.3.4 and the induced k-torus d d 4 7  Gm, 8 = R/k (Gm,o ) T  := R/k (T ) = R/k 5 j=1 j=1 6 9  has a standard integral model T std = dj=1 Ro /o (Gm,o ) in the sense of Definition B.3.4. There is a natural adjunction α : T → R/k (T ) = T  that is a closed immersion by Fact A.3.20, and we just need to check that α extends to a closed immersion T std → T std . By the transitivity of Weil restriction, α is the effect of R0 /k applied to the  natural map ι : Gm,0 → σ R/0 (Gm, ) whose components are adjunctions (where σ varies through the k-embeddings of 0 into ). Hence, it suffices to show that ι extends to a closed immersion  Ro /o0 (Gm,o ). Gm,o0 → σ

Take the components to be adjunctions X → Ro /o0 (X ×o0 ,σ o ) for each σ with X = Gm,o0 , and note that such adjunctions are closed immersions by Fact A.3.20. Fact B.4.3 Proof

We have T std (O) = T(K)1 .

The equality follows from Lemmas A.2.1, B.3.1, and Proposition 2.5.9. 

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661

Beware that although T std (O) as a subset of T(K) is thereby independent of /k, it doesn’t follow right away that T std is independent of /k as a model of T because we cannot apply Corollary 2.10.10 to the models built from two such choices of  (due to the possibility that these models may not be o-smooth). Lemma B.4.4 Let /k be a finite Galois extension splitting T. Under the identification Lie(T) = (X∗ (T) ⊗Z )Gal(/k) of Construction 2.5.22 we have the Gal(/k)-equivariant identification Lie(T std ) = (X∗ (T) ⊗Z o )Gal(/k) inside Lie(T ) = X∗ (T) ⊗Z .  Proof First suppose T = Ri /k (Gm,i ) is induced with each finite separable i /k even Galois. To treat such cases, we can reduce to the case T = R0 /k Gm and then may rename 0 as  (since we know that in the induced case the choice of  doesn’t matter and the formation of T std respects products in T). We have T std = Ro /o Gm . The functor R → Lie(T std )(R) then sends R to (Ro /o A1 )(R) = R ⊗o o , from which we conclude that Lie(T std ) is the o-module o , sitting as a lattice inside of the k-vector space Lie(T) = . On the other hand, X∗ (T) = Z[Gal(/k)]. This completes the proof when T is induced. Consider now a general torus T and let S = R/k (T ). We have the canonical “diagonal” embedding X∗ (T) → X∗ (S). According to Lemma A.2.3, Lie(T std ) = Lie(T) ∩ Lie(S std ) = (X∗ (T) ⊗Z )Gal(/k) ∩ (X∗ (S) ⊗Z o )Gal(/k) = (X∗ (T) ⊗Z o )Gal(/k) .  Lemma B.4.5 The coordinate ring o[T std ] is given as follows. For /k as in Lemma B.4.4, fix any basis e1, . . . , en of the free o-module o , so it is also a basis of the k[T]-module [T] (coordinate rings of the tori T and T respectively). Fix a basis χ1, . . . , χd of the free Z-module X∗ (T) and write   χi = e j ti, j , χi−1 = e j ui, j j

j

with ti, j , ui, j ∈ k[T]. Then o[T std ] = o[ti, j , ui, j ] ⊂ k[T]. Proof Write Z = R/k (T ). The choice of ( χ1, . . . , χd ) gives an isomorphism T → (Gm, )d and hence an isomorphism Z → (R/k Gm, )d . Write Z  = R/k Gm, . Recall from Lemma A.3.21 that the coordinate ring of the standard

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Integral Models of Tori

integral model of Z  is given by o[t1, . . . , tn, u1, . . . , un ] ⊂ k[Z ], where ti , ui are determined by the basis expansions v = e j t j and v −1 = e j u j of the adjunction v : Z  → Gm, and its reciprocal. We have the commutative diagram / Z

T id



adj

  T

 (χi )

/ (Z  )d  

adj

/ (Gm, )d

The top horizontal maps are obtained via base change from T → Z → (Z )d . Write v1, . . . , vd for the components of the right vertical adjunction map. The commutative diagram above shows that the image of vi in [T] is precisely χi . Write ti, j and ui, j for the coefficients in the basis expansions of vi and vi−1 in terms of the basis e j . By definition of T std and Lemma A.3.21, o[T std ] is the o-subalgebra of k[T] generated by the images of ti, j and ui, j under k[(Z )d ] → k[Z] → k[T]. But sending the entire equation vi = j e j ti, j through the map k[(Z )d ] → k[Z] → k[T] gives a basis expansion of the image of vi in terms of the images of ti, j . Thus the images of ti, j are the elements of k[T] in the expansion of the image χi of vi in [T] in terms of the  basis e j . Corollary B.4.6 The model T std is independent of the choice of finite separable extension /k splitting T. In particular, the formation of T std respects direct products in T. Proof It is enough to show that replacing  by a larger finite separable extension m does not change the model. We obtain an o-basis of om by combining an o-basis of o and an o -basis of om via multiplication (so in particular an o-basis of o is part of an o-basis of om , since 1 is part of an o -basis of om ). Hence, applying Lemma B.4.5 does the job.  Lemma B.4.7 A morphism of tori f : T1 → T2 over k extends (uniquely) to a morphism T1std → T2std . In other words, the assignment T → T std is a functor. If f : T1 → T2 is a closed immerison then its extension T1std → T2std is a closed immersion. In other words, T1std is the schematic closure of T1 in T2std . Proof Let /k be a finite separable extension splitting both T1 and T2 . We

B.4 The Standard Model

663

obtain the commutative diagram R/k ((T1 ) ) O

/ R/k ((T2 ) ) O

T1

/ T2

in which the vertical arrows are closed immersions by Fact A.3.20. The top horizontal map extends uniquely to a morphism of integral models by Fact B.3.5. It respects the schematic closures of T1 and T2 and therefore induces a morphism T1std → T2std . Assume now that the bottom map is a closed immersion, so the top map is too. Since schematic closure is a transitive operation this reduces to showing that the schematic closure of R/k ((T1 ) ) in Ro /o (X2 ), where X2 is the (smooth) standard model of the split torus (T2 ) , equals Ro /o (X1 ), where X1 is the (smooth) standard model of the split torus (T1 ) . Note that both Weil restrictions Ro /o (X ) are o-smooth and hence o-flat. The closed immersion (T1 ) → (T2 ) extends to a closed immersion X1 → X2 , and Fact A.3.20 shows that Ro /o (X1 ) → Ro /o (X2 ) is also a closed immersion. In this way both Ro /o (X1 ) and the schematic closure of R/k ((T1 ) ) in Ro /o (X2 ) are closed subschemes of Ro /o (X2 ), flat over o, with the same generic fiber, so by Lemma A.2.1 must be equal.  Lemma B.4.8 Let /k be any finite separable extension, S a torus over , and T = R/k (S). Then T std is the maximal flat subscheme of Ro /o (S std ), cf. Remark 2.10.7. Proof Choose  / to be a finite separable extension Galois over k that splits T. Then   also splits S. Let X = S  and consider the adjunction S → R  / (X). Then S std is the schematic closure of S in Ro  /o (X std ). Applying Ro /o to the closed immersion S std → Ro  /o (X std ), we obtain according to Lemma A.3.7 a closed immersion Ro /o (S std ) → Ro  /o (X std ). Let T  be the maximal flat closed subscheme of Ro /o (S std ). The closed immersion T  → Ro /o (S std ) induces an isomorphism on generic fibers. According to Lemma A.3.12, the closed immersion T  → Ro  /o (X std ) induces on generic fibers the closed immersion T → R  /k (X). Lemma A.2.1 implies that the closed immersion T  → Ro  /o (X std ) is the schematic closure of T in Ro  /o (X std ). The latter being the standard model of the induced torus R  /k (X), we conclude the proof by applying Lemma B.4.7.  Lemma B.4.9

Let T be an induced k-torus split by a finite Galois extension

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Integral Models of Tori

 /k with Galois group Θ = Gal( /k). We have o  [X∗ (T)]Θ ⊂ o[T std ], with equality if and only if the finite Galois splitting field for T over k is unramified. We are not requiring  /k to be the splitting field; it is an arbitrary finite Galois extension that contains the splitting field. Proof It is elementary to check that the assertion to be proved is insensitive to replacing   with an extension finite Galois over k. Also, by the construction of the anti-equivalence between tori and Galois lattices,  [X∗ (T)]Θ is naturally identified with the coordinate ring k[T] of T. Consider the case that T1,T2 are induced tori and T = T1 × T2 , hence X∗ (T) = ∗ X (T1 ) ⊕ X∗ (T2 ). Considering Θ-invariant bases of X∗ (T1 ) and X∗ (T2 ), and the resulting Θ-invariant basis of X∗ (T) obtained as the disjoint union, we see that the inclusion o  [X∗ (T1 )]Θ ⊗o o  [X∗ (T2 )]Θ ⊂ o  [X∗ (T)]Θ between o-lattices in k[T1 ] ⊗k k[T2 ] = k[T] is an equality. Hence, the assertion for T is equivalent to the combined assertions for the Tj ’s separately (the splitting field for T is the compositum over k of the splitting fields for the Tj ’s). Now we may assume that the induced torus T is given by R/k Gm, for a finite separable extension field /k Let χ1 : T → Gm be the adjunction map. Its Θ-orbit { χ1, . . . , χn } is a basis for X ∗ (T). Fix a basis e1, . . . , en for /k and present the coordinate ring o[T std ] as in Lemma B.4.5. Since each χi and χi−1 is given as an o  -linear combination of ti, j and ui, j we have X∗ (T) ⊂ o  [ti, j , ui, j ] and therefore, o  [X∗ (T)]Θ ⊂ o  [ti, j , ui, j ]Θ = o[ti, j , ui, j ] = o[T std ]. The opposite inclusion holds if and only if ti, j , ui, j ∈ o  [X∗ (T)]. The matrix expressing the vector ( χ1, . . . , χn ) in terms of the vector (t1,1, . . . , t1,n ) is given by (σi (e j ))i, j where σi :  →   varies through the k-embeddings. This is an n × n matrix with entries in o  and the square of its determinant is the discriminant disco (o ), which is a unit if and only if /k is unramified. The matrix expressing the vector (t1,1, . . . , t1,n ) in terms of the vector ( χ1, . . . , χn ) will therefore have entries in o  if and only if the discriminant of o over o is a unit in o, which is equivalent to the Galois closure of  over k being unramified. This Galois closure is the splitting field for T over k. The argument for arbitrary  (ti,1, . . . , ti,n ) and (ui,1, . . . , ui,n ) is the same. Lemma B.4.10 Let T be a k-torus and k1 /k a (possibly non-algebraic) extension of Henselian discretely-valued fields that is “unramified” in the sense that

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665

the ramification degree e(k1 /k) is equal to 1 and the (possibly non-algebraic) extension of residue fields is separable. The standard model of Tk1 is naturally identified with T std ×o o1 , where o1 is the ring of integers of k1 . We recall here that an arbitary field extension E/F is called separable if L ⊗F E is reduced for any field extension L/F, cf. [BLR90, §3.6, Definition 1]. Proof We consider first the special case where k  is contained in a finite Galois extension /k that splits T. Let T1std be the standard model of the torus T1 = T ×k k1 . We will use Lemma B.4.5 to show that o1 [T1std ] = o[T std ] ⊗o o1 . The argument will be similar to that used in the proof of Lemma B.4.9. Let e1, . . . , en be an o1 -basis of o and let f1, . . . , fm be an o-basis of o1 . Then {ei f j } is an o-basis of o . Let χ1, . . . , χd be a Z-basis of X∗ (T). Write χi = j e j ti, j and χi−1 = j e j ui, j with ti, j , ui, j ∈ k1 [T]. According to Lemma B.4.5 we have o1 [T1std ] = o1 [ti, j , ui, j ] ⊂ k 1 [T]. Write further ti, j = s fs ti, j,s and ui, j = s fs ui, j,s . Then o[T std ] = o[ui, j,s , ti, j,s ] and we see o[T std ] ⊗o o1 becomes identified under the isomorphism k[T] ⊗k k 1 = k 1 [T] with the subgring o1 [ui, j,s , ti, j,s ] of k1 [T]. This subring contains the elements ti, j , ui, j and hence the subgring o1 [T1std ] generated by them. To obtain the converse inclusion we consider the variouss k-embeddings σ : k 1 → . For each such we have the identity  fsσ · ti, j,s . ti,σj = s

Keeping i, j fixed we obtain a system of linear equations indexed by σ, thus consisting of [k  : k]-many equations, each considered in the variables ti, j,s where i, j is fixed, and s runs over a set of cardinality [k  : k]. The matrix of this system is ( fsσ )s,σ , and the determinant of this matrix belongs to o1× since k1 /k is unramified. We conclude that ti, j,s is contained in o1 [ti, j , ui, j ]. The same holds for ui, j,s . This completes the proof under the assumption that k1 is contained in a finite Galois extension /k that splits T. Consider now a general, possibly non-algebraic, unramified extension k1 /k. Choose a separable clsoure k1,s and a k-embedding k s → k1,s . Let /k be a finite Galois extension, contained in k s , that splits T. Let 1 =  · k1 ⊂ k1,s . Let k  =  ∩ k1 , intersection taken inside of 1 . Then k /k is an unramified extension contained in the finite Galois extension /k, and we can apply the proved special case to it. This allows us to assume that k = k , and hence that  and k 1 are linearly disjoint over k. Then 1 =  ⊗k k1 and o1 = o ⊗o o1 .

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Integral Models of Tori

Let A = T , B = R/k (A), A std = Spec(o [X∗ (T)]), and B std = Ro /o (A std ). Then T std is the schematic closure of the image of T → B → B std . Let T1 = Tk1 , A1 = (T1 )1 , B1 = R1 /k1 (A), so that T1std is the schematic closure of the image of T1 → B1 → B1std . Now A1 = T1 and A1std = Spec(o1 [X∗ (T)]) = A std ×o o1 . Therefore we have B1std = Ro1 /o1 (A1std ) = B ×o o1 , where we have used Lemma A.3.12 for the extension o1 /o and the identity o1 = o ⊗o o1 . We conclude that T1std is the schematic closure of the image of T ×k k1 → B ×k k 1 → B ×o o1 . Since schematic closure commutes  with flat base change, we see that this coincides with T std ×o o1 . Example B.4.11 The standard model need not be a connected group scheme, and may even fail to be smooth. The most basic example is the norm-1 torus 1 Gm = ker(N/k : R/k Gm → Gm ) T = R/k

for a separable quadratic extension /k generated by a uniformizing element  such that π =  2 is a uniformizing element of k. Then {1, } is an o-basis of o . According to Construction A.3.5, the coordinate ring of R/k Gm is given by k[t, t , u, u ]/(t u + πt  u  − 1, t u  + t  u). The norm map N/k was discussed in Construction A.3.22. It is given on coordinate rings by ) t → t 2 − πt 2 k[t, u]/(t u −1) → k[t, t , u, u ]/(t u + πt  u  −1, t u  +t  u), u → u2 − πu 2 . since at the level of the functor of points valued in general k-algebras R we have N(R ⊗k )/R (t · 1 + t  · ) = (t · 1 + t  · )(t · 1 − t  · ) = t 2 − πt 2 and similarly for the norm of u · 1 + u  ·  = (t · 1 + t  · )−1 . Functorially setting the norm to be 1 (which forces the unit condition), we 1 G is see that the coordinate ring of R/k m k[t, t ]/(t 2 − πt 2 − 1) realized as a quotient of k[t, t , u, u ]/(t u + πt  u  − 1, t u  + t  u) via u → t and u  → −t  since (t · 1 + t  )−1 = t · 1 − t   for functorial points with norm equal to 1. 1 G is the schematic According to Lemma B.4.7, the standard model of R/k m 1 G in the standard model of R closure of R/k m /k Gm . Its coordinate ring is therefore the image of o[t, t , u, u ]/(t u + πt  u  − 1, t u  + t  u) in k[t, t , u, u ]/(t u + πt  u  − 1, t u  + t  u, t 2 − πt 2 − 1, u2 − πu 2 − 1).

B.5 The Standard Filtration

667

Identifying the latter with k[t, t ]/(t 2 − πt 2 − 1), this image is seen to equal o[t, t ]/(t 2 − πt 2 − 1). The special fiber has coordinate ring f[t, t ]/(t 2 − 1); this is f[t]/(t 2 − 1) ⊗f f[t ] and by tracing through the comultiplication maps one sees that upon replacing t  with t /t it becomes the group scheme μ2 × Ga (i.e., Ga has standard coordinate t /t). When the residual characteristic of k is not 2, this shows that the special fiber is disconnected. When the residual characteristic is 2, the special fiber is connected but not smooth.

B.5 The Standard Filtration Definition B.5.1

The standard filtration of T(k) is defined by 0 T(k)std 0 = T(k)

and for r > 0 ∗ T(k)rstd = {t ∈ T(k)std 0 | ∀ χ ∈ X (T), ω( χ(t) − 1)  r }

where ω is normalized on k. It is clear that T(k)rstd is a decreasing filtration of T(k)1 by bounded open

subgroups. It is separated: r T(k)rstd = {1}. In fact, if we apply the same definition to K in place of k we obtain a decreasing separated filtration of T(K)1 and we have T(k)rstd = T(K)rstd ∩ T(k) for all r  0 (since the unique extension of ω to K is normalized). We now define a few desirable properties of a decreasing separated filtration of T(K)1 . Definition B.5.2 schematic, if

A decreasing separated filtration T(K)r∗ of T(K)1 is called

(1) for every r  0 the subgroup T(K)r∗ is schematic in the sense of Definition 2.10.15, ∗ , where T ∗ denotes the smooth model determined by (2) Tr∗ (M) ⊂ T(K)r+1 r ∗ ∗ Tr (O) = T(K)r , (3) the set of jumps of this filtration is discrete. ∗  T(K)∗ , where Recall that we say that r is a jump of the filtration if T(K)r+ r  ∗ ∗ T(K)r+ = s>r T(K)s .

668 Definition B.5.3

Integral Models of Tori A schematic filtration {T(K)r∗ }r0 is called

(1) connected, if all T(K)r∗ are connected in the sense of Definition 2.10.15, ∗ . (2) congruent, if Tr∗ (M) = T(K)r+1 Recall also the definition of a functorial filtration: see Definition 7.2.2. According to Proposition B.3.6, when the torus T is induced, the standard filtration is schematic, connected, and congruent. We will see in Proposition B.6.4 below that these properties continue to hold under a weaker assumption. For a general torus, however, the standard filtration satisfies only weaker properties, cf. Proposition B.10.1 below. We will therefore introduce a better filtration in §B.10.

B.6 Weakly induced tori While the standard model has the advantage of being explicit, as shown in Lemma B.4.5, it has an essential drawback – it is not always smooth, as was shown in Example B.4.11. Example B.6.1 We will now present an example of a torus such that the special fiber of its standard model has the non-reduced unipotent group scheme )) as a quotient. Let  ∈ Q√2 be a fourth root of 2. Define α2 = Spec(f[t]/(t 2 √ 4   = Q2 () = Q2 ( 2) and  = Q2 ( 2 ) = Q2 ( 2). We will denote the ring of integers of k and  by o and o respectively; and f will denote the residue field of o. Consider the tori S := R1  / Gm and T := R/k S. The computation in Example B.4.11 shows that the standard model S std of S has the affine ring o [t, t ]/(t 2 −  2 t 2 − 1). We will now compute Ro /o S std . We will in particular see that it is flat, so Lemma B.4.8 will imply that it equals T std . Using Construction A.3.5 and the basis {1,  2 } of o over o we see that the coordinate ring of Ro /o S std is the quotient of o[t1, t2, t3, t4 ] by the ideal generated by t12 + 2t22 − 4t3 t4 − 1, 2t1 t2 − t32 − 2t42 . Reducing modulo 2, we obtain the following as the affine ring of the special fiber T of T std . f[t1, t2, t3, t4 ] f[t3 ] f[t1 ] = 2 ⊗ f[t2 ] ⊗ 2 ⊗ f[t4 ]. (t12 − 1, t32 ) (t1 − 1) (t3 ) Therefore, T is isomorphic to μ2 × Ga × α2 × Ga as a f-scheme.

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A simple computation shows that the product of the elements (t1, t2, t3, t4 ) and (u1, u2, u3, u4 ) of the special fiber T equals (t1 u1, t1 u2 + t2 u1 + t3 u3, t1 u3 + t3 u1, t1 u4 + t2 u3 + t3 u2 + t4 u1 ). Therefore, the closed subschemes {(t, 0, 0, 0) ∈ T }

and

{(t1, t2, t3, t4 ) ∈ T | t1 = 1, t3 = 0}

are subgroup schemes isomorphic to μ2 and G2a , respectively, while the map T → α2,

(t1, t2, t3, t4 ) → t1 t3

is a quotient map whose kernel is the product of these two closed subgroup schemes. In Examples B.4.11 and B.6.1, the splitting field of the torus is a wildly ramified extension (i.e., it is not tamely ramified). This gives the hope that if the splitting field is a tamely ramified extension, the non-smoothness pathology would not occur. This is true. In fact, there is a somewhat larger class of tori for which this does not occur, namely the class of tori that are “weakly induced” in the sense of the following definition. Definition B.6.2 We say that a k-torus T is weakly induced if there exists a tamely ramified extension /k such that T is an induced torus. Note in particular that any induced k-torus is weakly induced. Remark B.6.3 If we let kt be the maximal tamely ramified subextension of a separable closure k s /k then a k-torus T is weakly induced if and only if Tkt is induced. Hence, being weakly induced over k equivalent to the existence of a basis of X∗ (T), equivalently of X∗ (T), on which the wild inertia subgroup Gal(k s /kt ) ⊂ Gal(k s /k) acts through a permutation action. Proposition B.6.4 Let T be a weakly induced k-torus and let ω be the normalized discrete valuation on k, extended uniquely to all algebraic extensions of k. (1) The standard model T std is smooth. (2) The standard filtration is schematic, connected, and congruent. We denote by Trstd the corresponding smooth model. (3) Let r > 0 and let k 1 /k be a (possibly non-algebraic) unramified extension of Henselian discretely valued fields. Then Trstd (o1 ) = {t ∈ T(k1 ) | ∀ χ ∈ X∗ (T), ω( χ(t) − 1)  r }. (4) For r > 0 the special fiber of Trstd is a vector group.

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Proof By Lemma B.4.10 and the definition of the standard filtration, all statements depend only on TK , so we base change to K. Let L/K be a tame finite Galois extension such that TL is an induced L-torus. Then S = R L/K (TL ) is an induced K-torus. The group Gal(L/K) acts on S by algebraic automorphisms. The adjunction map T → S identifies T functorially with the group of Gal(L/K)-fixed points in S. (1) By Lemma B.4.7, T std is the schematic closure of T in S std . The model S std is smooth by the agreement of two ways to define the standard integral model in the induced case (Example B.4.2). The action of Gal(L/K) on S extends to an action on S std by Lemma B.3.1. The subscheme of Gal(L/K)fixed points in S std is smooth by Proposition 2.11.5 (here we use that L/K is tamely ramified, so that Gal(L/K) has size [L : K] ∈ O× ⊂ O×L ). Lemma A.2.1 implies that this fixed subscheme equals the schematic closure of its generic fiber. The generic fiber being T, we conclude that T std is identified with the subscheme of Gal(L/K)-fixed points in S std and is thus smooth. (2) We begin by constructing and studying an integral model that will later show has O-points equal to T(K)rstd . For r > 0 let Trstd be the fixed subscheme of Srstd for the action of Gal(L/K). The same argument shows that Trstd is a smooth integral model of T. Lemma A.3.12 shows that its special fiber is equal to ((Srstd )f )Gal(L/K) . According to Proposition B.3.6, (Srstd )f is smooth, connected, and unipotent. Lemma 8.1.5 implies that the Gal(L/K)-fixed subgroup scheme is also a smooth connected unipotent group. Since f is perfect, this implies that Trstd is a vector group. The congruence property of the models Srstd that is Proposition B.3.6(3) implies the corresponding property for Trstd . In order to complete the proof of (2), and also (4), it is now enough, by Corollary 2.10.11, to prove (3) and to also show that the group {t ∈ T(K) | ∀ χ ∈ X∗ (T), ω( χ(t) − 1)  r } automatically lies in T(K)0 and hence equals T(K)rstd . (3) Let K1 be the strict Henselization of k 1 and let O1 be its ring of integers. Since Trstd (o1 ) = Trstd (O1 )∩T(k 1 ) we reduce the proof to the case that k1 = K1 . Choose a separable closure K1,s of K1 and a K-embedding Ks → K1,s . Let L1 = L · K1 inside of K1,s . Note that L and K1 are linearly disjoint over K. In particular, Gal(L1 /K1 ) = Gal(L/K). We have Trstd (O1 ) = (S(K1 )rstd )Gal(L/K) = {t ∈ T(K1 ) | ∀ χ ∈ X(S), ω( χ(t) − 1)  r } = {t ∈ T(K1 ) | ∀ χ ∈ X(T), ω( χ(t) − 1)  r }. since X(S) → X(T) is surjective. We have used that the corresponding claim is obvious for the induced torus S, by the explicit construction of its standard filtration.

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671

This proves (3), but we still need to argue that Trstd (O) = T(K)rstd , for which it is enough to show Trstd (O) ⊂ T(K)0 . We note first that Trstd (O) ⊂ S(K)0 , because S(K)0 = S(K)1 is the maximal bounded subgroup of S(K) and Trstd (O) is bounded, cf. Lemma 2.5.18 and Remark 2.10.3. By definition, T(K)0 is the image of S(K)0 under the norm map for the extension L/K. The restriction of this map to T(K)0 equals to [L : K]-power map. It will therefore be enough to show that the group Trstd (O) is [L : K]-divisible. To see this, let z : T → T be the [L : K]-power map. It clearly preserves the subgroup Trstd (O), hence extends to an endomorphism of Trstd by Corollary 2.10.10. To show the surjectivity of this homomorphism on the level of O-points we  of T(K)  Since the filtration Trstd (O)  is may, by Lemma 8.1.2, replace O by O. separated, it is enough to show that z induces a surjective map on the quotient std (O) for each r > 0. According to the congruence property proved Trstd (O)/Tr+1 above, this quotient equals Trstd (f), which we know is a vector group. Since z is the power map for an integer prime to the characteristic of f, the desired surjectivity follows.  Corollary B.6.5 Let T be a weakly induced torus. The formation of the standard model and the standard filtration is compatible with passage to a (possibly non-algebrac) unramified extension k 1 /k of Henselian discretely valued fields. Proof For r = 0 this is Lemma B.4.10. For r > 0 we need to show that Trstd (K1 ) = (T1 )rstd (K1 ) by Corollary 2.10.11, where K1 is the strict Henselization of k1 and (T1 )rstd is the integral model of the rth standard filtration group of T1 = T ×k k1 . This follows from Proposition B.6.4(3) applied to Trstd and  (T1 )rstd . The above proposition shows that the standard filtration behaves very well when T is weakly induced. In fact, it turns out that for weakly induced tori the standard filtration is the only possible filtration that has certain nice properties. This will be discussed precisely in Proposition B.10.5 below, and rests on the following result. Proposition B.6.6 Assume that k = K and T is weakly induced. Let L/K be a tame Galois extension such that TL is induced. Let e = e(L/K) be the ramification index and Θ L/K be the Galois group. std = R(K)std For each r > 0, the norm map R = R L/K (TL ) → T carries T(L)re r std surjectively onto T(K)r . Proof We want to show that Rrstd (O) → Trstd (O) is surjective. By Lemma =O  L (as  since O L ⊗O O 8.1.2 it is enough to prove this after replacing O by O std K is henselian) and the formation of the integral models {Tr }r0 respects

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 for weakly induced tori T by Corollary B.6.5. Thus, we may and do (−) ⊗O O assume that O is complete.  We have RL = σ ∈Θ L/K TL and the norm map R(L) → T(L) sends (tσ ) to  tσ . Thus, for r  > 0 the norm map R(L)rstd → T(L)rstd based on the filtrations using the normalized valuation of L is surjective (this is of interest for r  = re). Since both R and T are weakly induced, the proof of Proposition B.6.4 shows std )Θ L/K and T(K)std = (T(L)std )Θ L/K . that R(K)rstd = (R(L)re r re Let R  be the kernel of the norm map R → T, so the kernel of the surjective std is R(L)std ∩ R  (L) and the surjectivity of R(K)std → map R(L)rstde → T(L)re re r std std ∩ R  (L)). Now R T(K)r would follow from the vanishing of H1 (Θ L/K , R(L)re std is a product of groups of the form 1 + Mn for is an induced torus, so R(L)re M some finite separable field extensions M/L and integers n > 0. These groups are complete for their natural separated filtrations. The graded pieces of this filtration are f-vector spaces, hence multiplication by the order of Θ L/K is an std ∩ R  (L). Since multiplication by the order of Θ isomorphism on R(L)re L/K 1 std kills H (Θ L/K , R(L)re ∩ R (L)), this group must be zero.  Lemma B.6.7 Assume that T is weakly induced. The Lie algebra of Trstd , as a lattice in the Lie algebra t(k) of T is given by {X ∈ t(k) | for all χ ∈ X∗ (T), ω((dχ)(X))  r }. Proof The Lie algebra of Trstd , as a lattice in t(k), equals the intersection with t(k) of the Lie algebra of Trstd ×o O as a lattice in t(K). Using Lemma B.4.10 we reduce to the case k = K. Let L/K be a finite tamely ramified Galois extension such that S = TL is induced. In the proof of Proposition B.6.4 we saw that Trstd is the schematic closure of T in RO L /O Srstd . Lemma A.2.3 reduces the proof to the case that T is induced. Since the statement is compatible with products, we can further reduce to the case that T = R L/K Gm . As discussed in B.3, Tr = RO L /O Ger for r ∈ R0 with re ∈ Z, where Gn is the nth congruence group scheme of the standard integral model of Gm and e is the ramification index of L/K. Weil restriction of scalars commutes with the formation of the Lie algebra, in the sense that, under the identification Lie(T)(K) = L the lattice Lie(RO L /O Ger ) becomes identified with Lie(Ger ) = Mer L , where the last equality follows from Lemma A.5.13(4). The same is true for the lattice in the statement of the currrent lemma.  Remark B.6.8

Equivalently, we have Lie(Trstd ) = (X∗ (T) ⊗Z mer )Gal(/k),

where /k is a finite Galois extension such that T is split, e is the ramification

B.6 Weakly induced tori

673

degree of /k, and r ∈ e−1 Z. This follows at once from Lemma B.6.7 and the identification t(k) = t()Gal(/k) = (X∗ (T) ⊗k )Gal(/k) , cf. Construction 2.5.22. Similarly, for r ∈ e−1 Z>0 we have T(k)rstd = (X∗ (T) ⊗ (1 + mer ))Gal(/k) . This follows from Proposition B.6.4(3), which gives the identification T(k)rstd = T()rstd ∩ T(k) and the identification T()rstd = X∗ (T) ⊗ (1 + mer ) of Lemma B.2.2. The following result is commonly referred to as the Moy–Prasad isomorphism. Proposition B.6.9

Let r ∈ R>0 and r < s  2r. The functors

T → T(k)rstd /T(k)std s

and

T → t(k)rstd /t(k)std s

on the category of weakly induced tori with values in the category of abelian groups are isomorphic, where by definition t(k)r = Lie(Trstd ) for r > 0. Proof Note that if we adjust the standard filtration by using a valuation on k that may not be normalized, the effect is to multiply r and s by a common scaling factor. Such a common scaling is harmless (note that it preserves the condition “r < s  2r”), so we have the flexibility to make the calculation of the filtration with possibly non-normalized valuations since we allow general r > 0 and the resulting permitted s. This flexibility will be invoked later in the proof. We build the isomorphism in stages. Consider first a split torus T. We use the normalized valuation on k, so it suffices to consider r, s ∈ Z>0 . Then the map 1 + x → x provides an isomorphism (1 + mr )/(1 + m s ) → mr /m s , which we tensor with X∗ (T) to obtain the desired isomorphism, noting that T(k)rstd = X∗ (T) ⊗Z (1 + mr ) by Lemma B.2.2, and t(k)rstd = X∗ (T) ⊗Z mr by Lemma A.5.13(4). Before we move on, we note that if /k is a finite separable extension of ramification degree e, then mr /m s embeds into mer /mes , and (1 + mr )/(1 + m s ) embeds into (1 + mer )/(1 + mes ), and in this way one std std sees that the isomorphism T(k)rstd /T(k)std s → t(k)r /t(k)s is the restriction std std std std of the isomorphism T()r /T()s → t()r /t()s , where we have used the extension to  of the normalized valuation on k. Consider next a torus of the form T = Res/k S for a finite separable extension /k and a split -torus S. Then T(k)rstd = S()rstd and t(k)rstd = s()rstd , where we are using on  the extension of the given valuation on k. The isomorphism for the split -torus S constructed in the above paragraph provides an isomorphism for the torus T. More generally, an induced torus T is a product of tori of the

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above form, and we obtain an isomorphism of the desired form for such a torus T. In order to see that this isomorphism is independent of the way T is expressed as a product of such tori, and also functorial in T, it is enough to check that this isomorphism is the restriction of the corresponding isomorphism for the split torus T ×k  , where  /k is any finite separable extension splitting T. This in turn can be checked in the special case T = Res/k Gm , because the isomorphism for split tori respects products. Then X∗ (T) = IndΓΓk Z and evaluation at 1 ∈ Γk provides a homomorphism X∗ (T) → Z that, via Remark B.6.8, leads to the   isomorphisms T(k)rstd = (X∗ (T) ⊗ (1 + me r ))Gal( /k) → (1 + mer ) and t(k)rstd =   (X∗ (T) ⊗ me r )Gal( /k) → mer and the claim is proved. Finally consider a weakly induced torus T and let /k be a tame finite Galois extension over which T becomes induced, and let Θ = Gal(/k). By the settled induced case, we have a functorial isomorphism T()rstd /T()std s  std std t()r /t()s , where we use the valuation on  extending the given valuation on k. By functoriality, this isomorphism is equivariant for the action of Θ. Thus, it is enough to show the natural maps std std Θ T(k)rstd /T(k)std s → (T()r /T()s ) ,

std std Θ t(k)rstd /t(k)std s → (t()r /t()s )

are isomorphisms, for then we can use the isomorphism over  to obtain an isomorphism over k. Note that the resulting isomorphism is independent of the choice of , because the isomorphism over  is compatible with enlarging  by the claim proved in the previous paragraph and the discussion of the case of split tori. The required isomorphism property would follow from the vanishing of H1 (Θ,T()rstd ) and H1 (Θ, t()rstd ) for general r > 0, as we will now establish. For this, we will reuse the letters r, s to mean any positive real number, no longer subject to r < s  2r. Let I ⊂ Θ be the inertia subgroup. We have the inflation-restriction sequence 1 → H1 (Θ/I, (T()rstd )I ) → H1 (Θ,T()rstd ) → H1 (I,T()rstd ). We will show the outer H1 ’s vanish, so the middle one does too, as desired. To show that H1 (I,T()rstd ) = 0 we’ll show multiplication by the size of I is an automorphism of the abelian group T()rstd . std be the smooth connected integral model of the induced -torus Let T,r std (o ). It is now convenient to use on  the normalized T , so T()rstd = T,r  valuation, thereby reindexing the filtrations. Proposition B.6.4 states that the std is a vector group. Therefore any element of T std (o ) killed special fiber of T,r ,r  std (m ) = T std (o ). Inductively, such by multiplication by #I must lie in T,r   ,r+1

an element must lie in s>0 T,stds (o ) = {1}. For surjectivity of multiplication

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675

by #I, we shall use Lemma 8.1.2 to reduce to the case where  is complete. std is o = o since k is henselian, and the formation of T,r First note that o ⊗o  o by Corollary B.6.5. Furthercompatible with scalar extension along o →  more, multiplication by #I is a surjective endomorphism of T whose kernel is a finite multiplicative subgroup of T of order not divisible by the characteristic of k (due to tameness), so that kernel is étale. Therefore that endomorphism is smooth, so Lemma 8.1.2 ensures that the desired surjectivity on o -valued points holds if we establish the analogous result with  =  ⊗k  k in place of . We can now assume that , hence o , is complete. By completeness, it is enough to prove that multiplication by #I induces a surjective endomorphism std std on T()std s /T()s+1 = T, s (f ) for all s > 0. This follows from the fact that (T,stds )f is a vector group and #I is invertible in f (not just in , as used above). Finally, we turn to the vanishing of H1 (Θ/I, (T()rstd )I ). By Proposition B.6.4(3) applied over the maximal unramified extensions L and K, we have (T()rstd )I = T(k )rstd, where k  is the fixed field of I (so k /k is finite Galois unramified, as /k is tame and thus has a separable residue field extension). Letting o  and f  be the valuation ring and residue field of k , it follows from Corollary B.6.5 that T(k )rstd = Trstd (o ) for all r > 0. The vanishing of H1 (Θ/I, Trstd (o )) is reduced by Lemma 8.1.4 to the vanishing of H1 (Θ/I, Trstd (f )). Since (Trstd )f is a smooth connected commutative unipotent group over the perfect field f, the latter vanishing follows from Lemma 2.4.2 applied with the finite Galois extension f /f.  The vanishing of H1 (Θ, t()rstd ) is proved analogously.

B.7 The ft-Néron Model As we saw in Examples B.4.11 and B.6.1 the standard model of an algebraic torus T may fail to be smooth. It is therefore necessary to replace it by a smooth model. Definition B.7.1 The ft-Néron model T ft of T is the smooth integral model obtained by applying the smoothening process of Definition A.6.3 to T std . Proposition B.7.2 Let T be an algebraic torus defined over k. (1) T ft (o) = T std (o) = T(k)1 . (2) T ft (O) = T std (O) = T(K)1 . (3) o[T ft ] = { f ∈ k[T] | f (T(K)1 ) ⊂ O}.

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Proof Part (2) follows from the fact that T (i+1) (O) → T (i) (O) is bijective by construction, together with Fact B.4.3 and Lemma B.4.10 (applied with  = K). Part (1) follows from (2) by taking invariants under Gal(K/k). Part (3) follows from (2) and Lemma 2.10.9.  Remark B.7.3 The above proposition gives an explicit description of T ft in terms of its coordinate ring. This description makes no reference to the standard model and the smoothening process. However, we are not aware of an argument that shows that this coordinate ring provides a smooth integral model of T without applying the smoothening process to the standard model. Corollary B.7.4 The subgroup T(K)1 is schematic. 1 G when k = Q is We consider the norm-1 torus T = R/k m 2 √ equipped with its normalized valuation and  = Q2 ( 2). Write π = 2 ∈ Q2 . In Example B.4.11 we showed that the coordinate ring of T is

Example B.7.5

k[x, y]/(x 2 − π y 2 − 1), and the coordinate ring of T (0) = T std is o[x, y]/(x 2 − π y 2 − 1). The latter is not smooth over o. The identity section e is determined by e(x) = 1, e(y) = 0. The Jacobian matrix is (2x, −2π y) and is sent by the identity section to (2, 0). The Néron defect of smoothness is therefore δ0 = 1. To compute the smoothening, note that the special fiber of T (0) is isomorphic to μ2 × Ga and therefore T (0) (f) = {1} × f. A simple direct computation shows that the image of the reduction map T (0) (O) → T (0) (f) is the singleton {(1, 0)}. The schematic closure of this point has ideal of definition (2, (x − 1), y) in o[x, y]/(x 2 − π y 2 − 1). The dilatation of this closed subscheme within T (0) produces the model T (1) whose coordinate ring is the o-subalgebra of k[x, y]/(x 2 − π y 2 − 1) generated by π −1 (x − 1) and π −1 y. Let us check that this ring is smooth over o. Writing z = π −1 (x − 1) and w = π −1 y we see that the coordinate ring of T becomes k[z, w]/(z 2 + z − π w 2 ) and the coordinate ring of T (1) is o[z, w]/(z2 + z − π w 2 ). Its reduction to f equals f[z]/z(z + 1) ⊗f f[w] and the corresponding scheme is (Z/2Z) × Ga , the first factor being the constant étale group scheme of order 2 over f. For a computation of the relative identity component, see Example B.10.6. The point (1, 0) of Z/2Z × Ga is the reduction of −1 ∈ L 1 = T(k) = T(k)1 , as one sees by setting x = −1 and y = 0 and computing z = 1 and w = 0. This is in complete parallel with the case of odd residual characteristic, where T std

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677

is smooth with special fiber of μ2 × Ga and μ2 is étale, and the reduction of −1 ∈  1 is the point (−1, 0) of μ2 × Ga . We can also compute the Lie algebras of the non-smooth model T (0) . The ideal of the identity section of T (0) is (x − 1, y)/(x 2 − π y 2 − 1) and setting z  = x − 1 we see that the cotangent space of T (0) is the o-module generated by z , y with the relation 2z  = 0. Thus it is the direct sum of the free o-submodule of rank 1 generated by y and the torsion submodule generated by z . The ideal of the identity section of T (1) is (z, w)/(z 2 + z − π w 2 ), so the cotangent module is generated by (z, w) subject to the relation z = 0, so it is the free o-module of rank 1 generated by w. The relationship between the two is via z  = π z and y = πw. We conclude that Lie(T (1) ) = π · Lie(T (0) ) ⊂ Lie(T). Proposition B.7.6 Let /k be a finite extension, S an -torus, and T = R/k S. Then T ft = Ro /o S ft . Proof Write T = Ro /o S ft . By Lemma A.3.12, T is a smooth model of T. By Corollary 2.10.11 it is enough to show T (O) = T(K)1 . For each kembedding σ : k  → K, the extensions Lσ = K ⊗k  ,σ  are each -isomorphic to L and we have the diagram / σ O L O ⊗o o σ  K ⊗k 

 / σ Lσ

with bijective horizontal maps, the products being taken over Homk (k , K), where k  ⊂  is the largest subextension unramified over k. This diagram trans  lates the inclusion T (O) ⊂ T(K) into the inclusion S ft (O Lσ ) ⊂ S(Lσ ),   whose image is S(Lσ )1 = T(K)1 . Lemma B.7.7 Let T be a k-torus and let k /k be a (possibly non-algebraic) unramified extension and let o  be the ring of integers of k . The ft-Néron model of Tk  is (T ft )o . Proof By construction, the ft-Néron model of Tk  is the smoothening of the standard model of Tk  , and moreover that smoothening is obtained by a sequence of dilatations. Lemma B.4.10 shows that the standard model of Tk  is given by (T std )o . Let f  be the residue field of o . Then f  = o  ⊗o f and this implies that dilatation commutes with the unramified base change o /o, which in turn implies that (T ft )o is the smoothening of the standard model of (T std )o , and  hence equals the ft-Néron model of Tk  . Remark B.7.8

The ft-Néron model is not compatible with ramified base

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change. Indeed, let /k be a tamely ramified quadratic extension and T = 1 G . We saw in Example B.4.11 that the standard model of T is smooth if R/k m the residual characteristic of k is not 2. Therefore it equals the ft-Néron model. Its special fiber is disconnected and has unipotent neutral component. On the other hand, the ft-Néron model of T = Gm, has special fiber that is connected and reductive. Consider the special fiber T of T ft . It is a smooth commutative affine group over f. Its component group is described in Corollary 11.2.1. Here we focus on the connected component T 0 . According to [Bor91, Theorem 10.6(3)] it is the product of its unipotent radical Ru (T 0 ) and its unique maximal torus. To describe that torus it is enough to describe the group X∗ (T 0 ) = Hom(Gm,f, T 0 ). Note that we may as well take T in place of T 0 , since Gm is f connected. Proposition B.7.9 There is a natural Gal(K/k)-invariant identification X∗ (T ) = X∗ (T)I . Proof By definition one has the identification X∗ (T)I = HomK (Gm, K ,TK ). Lemmas B.4.7 and B.7.7 imply that the latter equals Hom(Gm,O, TOstd ), and Lemma A.6.4 implies that this further equals Hom(Gm,O, TOft ). Let us write in the rest of the proof T in place of TOft . According to [SGA3, Exp. XI, Corollary 4.2] the functor sending an Oalgebra R to the set Hom(Gm, R , TR ) is representable by a smooth scheme over O. Since O is Henselian, Lemma 8.1.3 implies that the reduction map Hom(Gm,O, T ) → Hom(Gm,f, Tf ) is surjective. Everything is equivariant for the evident actions of Gal(K/k) = Gal(f/f), so to complete the proof it remains to show that this reduction map is in fact bijective. Since this is a map of commutative groups, it is enough to show that its kernel is zero. Assume therefore given a morphism λ : Gm,O → T whose base change to f is the trivial morphism. We claim that λ must be trivial itself. This is equivalent to λ(O× ) = 1 ∈ T(K). By assumption we have λ(O× ) ⊂ T (M), thus λ factors uniquely through the morphism T1 → T , where T1 is the first congruence group scheme of Definition A.5.12. The connected component of the special fiber of T1 is a smooth unipotent algebraic group by Lemma A.2.2. Therefore the special fiber of λ1 : Gm,O → T1 is again trivial. Inductively, we see that λ(O× ) is contained

in T (Mn ) for all n. But n T (Mn ) = {1} by Lemma A.5.13.  Corollary B.7.10

The relative identity component T 0 of T ft

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679

(1) is a torus over o if and only if T splits over K, (2) has unipotent special fiber if and only if T remains anisotropic over K. Proof Since the special fiber of T 0 is a smooth connected commutative fgroup, it is the product of a torus with a vector group. The dimension of the toral factor equals the rank of X∗ (T)I by Proposition B.7.9. Thus, it is zero if and only if T remains anisotropic over K, hence (2), and equals the dimension of T if and only if T splits over K. But T is a torus if and only if its generic and special fiber are both tori. Since the special fiber has the same dimension as that of T, it is a torus if and only if the toral factor has dimension equal to that of T, hence (1).  Lemma B.7.11 A morphism of tori f : T1 → T2 over k extends (uniquely) to a morphism T1ft → T2ft . In other words, the assignment T → T ft is a functor. If f : T1 → T2 is a closed immersion and in addition T1 is weakly induced, then its extension T1ft → T2ft is a closed immersion. In other words, T1ft is the schematic closure of T1 in T2ft . Proof The existence and uniqueness of extension of f to a morphism of group schemes T1ft → T2ft follows from Lemma A.6.4. Assume now that T1 is weakly induced. Proposition B.6.4 implies T1ft = T1std . The statement now follows from Lemma B.4.7 and Lemma A.6.4.  Corollary B.7.12 Let T be a k-torus, let S ⊂ T be the maximal unramified subtorus, and let A ⊂ S be the maximal k-split subtorus. The embeddings A → S → T extend to closed immersions A ft → S ft → T 0 , which identify S ft with the maximal subtorus of T 0 and A ft with the maximal o-split torus in T 0 . The special fibers A and S are identified with the maximal f-split torus, and the maximal torus, of the special fiber T 0 , respectively. In particular, T 0 = S × Ru (T 0 ). Proof By Proposition B.6.4 we have A ft = A std and S ft = S std . Thus A ft  (Gm,o )d A , where d is the rank of A, and (by Lemma B.4.10) (S ft )O  (Gm,O )dS , where dS is the rank of S. In particular, A and S are tori over o, with A split. Lemma B.7.11 gives closed immersions A → S → T ft . Since A and S have connected fibers, these embeddings factor through T 0 . Proposition B.7.9 implies X∗ (S ) = X∗ (S) = X∗ (T)I = X∗ (T ) and hence X∗ (A ) = X∗ (S)Gal(K/k) = X∗ (T )Gal(f/f) , which shows that S is the maximal torus of T and A is the maximal f-split torus in T . It follows that S is a maximal torus in T and A is a maximal o-split torus in T . 

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B.8 The Néron Mapping Properties and the lft-Néron Model Let T be a k-torus. Proposition B.7.2 implies that the maximal bounded subgroup T(K)1 of T(K) (cf. Proposition 2.5.8) is schematic. In this section we will show that the Iwahori subgroup T(K)0 (cf. Definition 2.5.13) is schematic, and also connected, cf. Corollary B.8.7. For this, we will consider other smooth models of T beyond T ft , and allow ourselves to drop the affineness assumption. Let T a smooth o-scheme (hence locally of finite type over o) equipped with an isomorphism Tk  T. We do not assume that T is quasi-compact (equivalently, of finite type) nor separated. In what follows we will use the standard fact that an o-group locally of finite type with connected fibers is of finite type. (This assertion is equivalent to quasi-compactness, so it is sufficient to check it for the fibers, and connected groups locally of finite type over a field are always quasi-compact [SGA3, Exp.VIA , Prop.2.4].) Definition B.8.1

We say that T satisfies the following.

(1) The Néron mapping property, if for every smooth o-scheme Y and every k-morphism fk : Yk → T there exists a unique extension fo : Y → T . (2) The ft-Néron mapping property, if T is of finite type and for every smooth o-scheme Y of finite type and every k-morphism fk : Yk → T satisfying fk (Y (O)) ⊂ T(K)1 there exists a unique extension fo : Y → T . (3) The connected Néron mapping property, if T has connected fibers and for every smooth o-scheme Y with connected fibers and every k-morphism fk : Yk → T satisfying fk (Y (O)) ⊂ T(K)0 there exists a unique extension fo : Y → T . Remark B.8.2 If T satisfies the Néron mapping property, then the natural inclusion T (o) → T(k) is an isomorphism. Indeed, one applies this property to Y = Spec(o). Lemma B.8.3 A scheme T satisfying any of the properties of Definition B.8.1 is uniquely an o-group extending the k-group structure on T. Proof The k-group structure follows by taking Y = Spec(o) and fk to be the identity section of T over k to get the identity section over o, taking Y = T ×T and fk to be the multiplication map T × T → T to get the multiplication on T , and taking Y = T and fk to be inversion on T to get inversion on T . Uniqueness of extensions ensures that these constructions satisfy the axioms for an o-group.  We will see in Proposition B.8.8 below that each of these properties specifies

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a unique scheme T . But first, we prove existence. It is known that a scheme satisfying the lft-Néron mapping property exists, cf. [BLR90, §10.1, Proposition 6]. We will now give a slightly different construction, whose generalization is applied in §8.3 to the more general case of connected reductive groups. The key arguments in the proofs below are “borrowed” from arguments in [BLR90, §§9,10]. Proposition B.8.4

Let T be a k-torus.

(1) The model T ft of Definition B.7.1 satisfies the ft-Néron mapping property. (2) The smooth o-group scheme T lft obtained by applying Proposition A.7.1 to T ft and the (possibly infinite index) inclusion T(K)1 ⊂ T(K) satisfies the lft-Néron mapping property. (3) The relative identity components of T lft and T ft coincide. The resulting model T 0 satisfies the connected Néron mapping property. Proof (1) follows at once from Corollary 2.10.13 and Fact B.4.3. (2) Write T lft for the smooth separated o-group scheme obtained by applying Proposition A.7.1 to T ft and the (possibly infinite index) inclusion T(K)1 ⊂ T(K). In particular, (T lft )O is the analogous model for TK due to Lemma B.7.7 applied to K/k. We claim that for a discrete valuation ring o  and a local homomorphism o → o  of ramification degree 1, T lft (o ) = T(k ). Let us admit this claim for now and use it to prove the Néron mapping property. Given a smooth o-scheme Y and a morphism f : Yk → T, consider a generic point ζ in the special fiber of Y . The local ring OY , ζ is a discrete valuation ring. Its ramification degree over o equals 1, since locally around any point of Y the morphism Y → Spec(o) factorizes as an étale map Y → Aon for some n, followed by the structure map of Aon . Applying T lft (o ) = T(k ) for o  = OY ,ζ , we obtain a morphism fζ : Uζ → T lft extending f , where Uζ is an open neighborhood of ζ in Y . We do this for all finitely many generic points ζ in the special fiber of Y , and choose the open neighborhoods Uζ so that they intersect non-trivially only in the generic fiber of Y . Then the morphisms fζ glue to a morphism from the union of all Uζ to T lft . This union is an open subscheme whose complement has codimension at least 2 in Y . Therefore, since T ft is separated, the Weil extension theorem [BLR90, §4.4, Theorem 1] applies and gives an extension to a morphism Y → T lft . Uniqueness is immediate since T lft is separated and Y is o-flat. The Néron lifting property for T lft has thus been proved, modulo the proof of the claim T lft (o ) = T(k ), to which we now turn. We will first assume k = K and write o = O and o  = O. Let K  be the field of fractions of O. According to Lemma B.7.7 we have T ft (O) = T(K )1 . As stated in Propo-

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sition A.7.1(5), we have T lft (O) = T ft (O) · T lft (O) = T(K )1 · T(K). For any finite Galois extension E/K and E  = EK  we have the identification Gal(E /K ) = Gal(E/K), since E/K and K /K are linearly disjoint. This identifies I = Gal(Ks /K) with a quotient of I  = Gal(Ks /K ). The action of I  on X∗ (T) factors through this quotient and we have the identification X∗ (T)I  = X∗ (T)I . At the same time, the ramification degree of O/O being 1 also implies that the construction of the Kottwitz homomorphism in §11.1 is compatible with base change from K to K  (see Corollary 11.1.6). Altogether we see that the natural map T(K)/T(K)0 → T(K )/T(K )0 is an isomorphism. This implies T(K )0 · T(K) = T(K ), which in turn implies T(K )1 · T(K) = T(K ), proving the claim that T lft (O) = T(K ). We now drop the assumption that k = K. Let O be the strict Henselization of o , so we can pick a local map O → O over o → o . Given x ∈ T(k ) we consider it as an element of T(K ), which according to the above paragraph (applied to K /K) lies in T lft (O). The map x : Spec(O) → T lft maps the closed point of Spec(O) into some open affine subscheme U of T lft , hence by Lemma 8.1.1 factors through it and is thus an element of U (O). If the o-algebra A is the coordinate ring of U , then the o-algebra map x ∗ : A → O composed with O → K  takes image in k  by assumption. But O ∩ k  = o , and we conclude that x lies in U (o ) ⊂ T lft (o ). The proof of (2) is complete. To prove (3), note that Proposition A.7.1 states that T ft is an open subgroup scheme of T lft and shares the same relative identity component T 0 . In particular, T 0 is affine, since T ft is. The same argument used to prove the ft-Néron mapping property for T ft now proves the connected Néron mapping property for T 0 , provided we know T 0 (O) = T(K)0 . This is proved in Corollary B.8.7 below, whose proof relies on part (2) of the current proposition, but not on part (3).  Remark B.8.5 If Y is a smooth affine o-group scheme and fk : Y → T is a homomorphism of k-groups between the generic fibers, then there exists a unique homomorphism of o-group schemes Y → T ft . Indeed, Y (O) is a subgroup of Y (K), and is bounded by Remark 2.10.3. Since T(K)1 is the maximal bounded subgroup of T(K) (cf. Proposition 2.5.8), fk (Y (O)) must be contained in T(K)1 . Therefore Proposition B.8.4 applies. Moreover, the extension f : Y → T ft is a group homomorphism due to its uniqueness. If in addition Y is connected, then the group homomorphism f : Y → T ft factors through the open immersion T 0 → T ft . Corollary B.8.6 Let 1 → A → B → C → 1 be an exact sequence of k-tori. Then B 0 (O) → C 0 (O) is surjective.

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683

  → B(K)  → C(K)  → 1 is Proof The sequence of K-points 1 → A(K) exact due to Lemma 2.5.4 and Example 2.3.2. This implies the surjectivity 0 (O)  → C lft (O)  by Proposition B.8.4(2). We claim that B lft (O)/B   of B lft (O) ft ft 0   is finitely generated. Since B is of finite type the quotient B (O)/B (O) ft (O)   is finitely generated. is finite, and it is enough to show that B lft (O)/B According to Proposition B.7.2, this quotient equals B(K)/B(K)1 . By Lemma 11.1.3, the latter is a subgroup of X∗ (B)I , which is visibly finitely generated.  → C 0 (O)  follows from Lemma A.4.25, we may Since surjectivity of B 0 (O) conclude by Lemma 8.1.2.  We are now ready to show that the Iwahori subgroup T(k)0 of Definition 2.5.13 is schematic and connected. Corollary B.8.7

For any k-torus T the equality T 0 (o) = T(k)0 holds.

Proof We have T(k)0 = T(K)0 ∩ T(k) by Definition 2.5.13, and T 0 (o) = T 0 (O) ∩ T(k) by taking fixed-points under Gal(K/k). This reduces to considering K-tori. If T is induced the claim follows from Lemmas 2.5.18 and B.3.1. For a general torus T let S = R L/K (TL ) for a finite Galois extension L/K splitting T and let S → T be the norm map. By definition T(K)0 is the image of S(K)0 . Since S is induced, S(K)0 = S 0 (O). Since the kernel of S → T is a  torus, Corollary B.8.6 shows that this image is T 0 (O). Proposition B.8.8 A scheme T satisfying any of the properties of Definition B.8.1 is determined up to unique isomorphism. In particular, the scheme T lft constructed in Proposition B.8.4 coincides with the lft-Néron model constructed in [BLR90, §10.1]. Proof For (1) we consider T1, T2 both satisfying the Néron mapping property, and apply this property to the isomorphism (T1 )k → T → (T2 )k and its inverse. The uniqueness statement in this property guarantees that the resulting morphisms T1 → T2 and T2 → T1 are inverse to each other. For (2), the argument is the same, but in addition we use that the isomorphism (T1 )k → T → (T2 )k maps T1 (O) into T2 (K)1 , since T1 (O) is bounded. Therefore, the ft-Néron mapping property guarantees that this isomorphism extends to a homomorphism T1 → T2 . For (3), the argument is again the same, but in addition we use that the isomorphism (T1 )k → T → (T2 )k maps T1 (O) into T2 (K)0 . Indeed, the isomorphism (T1 )k → T extends to a homomorphism T1 → T ft . Since T1 has connected fibers, this homomorphism factors through the open immersion T 0 → T ft . Therefore it maps T1 (O) into T(K)0 by Corollary B.8.7. The functoriality of the Iwahori subgroup (Lemma 2.5.19) implies that the isomorphism

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T → (T2 )k maps T(K)0 to T2 (K)0 , and we conclude that (T1 )k → T → (T2 )k maps T1 (O) to T2 (K)0 . The connected Néron mapping property guarantees that this isomorphism extends to a homomorphism T1 → T2 . we need to know that the natural map T (O) → T(K) lands in T(K)0 . We have from (2) that fk extends unique to fo : Y → T ft . Due to the connectedness of Y , this extension factors through the open immersion T 0 → T ft To show (3) characterizes such an o-model T , it has to be shown that We are granting knowledge that (2) is satisfied by T ft , so any T as in (3) admits a unique o-morphism T → (T ft )0 as models of T. Hence, we get an induced map on o-valued points of the form T (O) → (T ft )0 (O) = T(K)0 , and its composition with the inclusion T(K)0 → T(K) is clearly the natural map T (O) → T(K).  Definition B.8.9 The (unique by Proposition B.8.8) scheme T satisfying one of the properties in Definition B.8.1 is called respectively (1) an lft-Néron model, denoted by T lft , (2) an ft-Néron model, denoted by T ft , (3) a connected Néron model, denoted by T 0 . For a description of the component groups of T ft and T lft , see Corollary 11.2.1. Corollary B.8.10 Let k /k be a (possibly non-algebraic) extension of Henselian discretely valued fields with ramification degree 1, and let o  be the ring of integers of k . Then (T lft )o is the lft-Néron model of Tk  . Proof It is enough to show that (T lft )o satisfies the Néron mapping property. In the proof of Proposition B.8.4(2) it was shown that this property follows from the identity T lft (o ) = T(k ) for any discrete valuation ring o  and local homomorphism o  → o  of ramification degree 1, where k  is the fraction field of o . But then o → o  is a local homomorphism of ramification degree 1, and the identity T lft (o ) = T(k ) was established in the proof of that proposition. 

B.9 The pro-unipotent radical Let T be a k-torus and T 0 the relative identity component of T ft . Proposition B.9.1 The preimage of Ru (T 0 )(f) under the reduction map T(k)0 = T 0 (o) → T 0 (f) equals T(k)0 ∩ T(k)std 0+ .

B.9 The pro-unipotent radical

685

Remark B.9.2 When f is finite, the statement is elementary. Indeed, the maximal reductive quotient of T 0 is isomorphic to S and its group of f-points is the maximal quotient of T(k)0 whose order is prime to p. Therefore the subgroup of T(k)0 of elements whose image in T 0 (f) is unipotent is the prop-Sylow subgroup of T(k)0 , which is equals T(k)0 ∩ T(k)std 0+ by definition of std T(k)0+ . Proof of Proposition B.9.1 The statement is unaffected by passing from k to K. Consider first the case that T is induced, and assume without loss of generality that T = R L/K Gm . The reduction map T(K)0 → T 0 (f) → S (f) is × translated to the natural reduction map O×L → O×L /(1 + M L ) = f . On the other hand one sees right away that T(K)std 0+ = 1 + M L . Consider now a general K-torus T and the natural embedding T → T , where T  = R L/K (TL ) and L/K is a finite Galois extension splitting T. Let S ⊂ T be the maximal K-split subtorus. We have the embeddings S → T → T . They both extend to homomorphisms S → T 0 → T  between the relative identity components of the corresponding ft-Néron models by Lemma B.7.11, whose composition is a closed immersion by Corollary B.7.12, so S → T 0 is a closed immersion. On special fibers we obtain S → T 0 → T , and the first of these homomorphisms, as well as their composition, is a closed immersion. Since the image of S in T 0 is the maximal torus of the smooth connected commutative group scheme T 0 by Corollary B.7.12, we conclude that an element of T 0 (f) is unipotent if and only if its image in T (f) is unipotent. Therefore, the inverse image of Ru (T 0 )(f) in T(K)0 is T(K)0 ∩ T (K)std 0+ by the settled induced case. std . ∩ T(K) = T(K)  But by definition T (K)std 0+ 0+ The above proposition justifies calling T(K)0 ∩ T(K)std 0+ the pro-unipotent radical of T(K)0 . More generally, there is a topological Jordan decomposition, cf. Proposition A.4.23. Now we consider a special case that is useful in applications: Lemma B.9.3 Assume that k is complete with finite residue field f of characteristic p and that T is weakly induced. The group T(k)0+ is the pro-p-Sylow subgroup of the profinite group T(k)1 . Proof By assumption and Proposition B.6.4 we have T ft = T std . Let T 0 be the relative identity component of this model. According to that same proposition, for each n  1 the special fiber of each congruence group scheme Tn of T 0 is unipotent. Therefore Tn (f) = Tn (o)/Tn+1 (o) is a p-group, where we have used Lemmas 8.1.4 and 2.4.2. Since the filtration T(k)r is separated, we conclude that T(k)1 is a pro-p group. According to Proposition B.9.1, T(k)0+

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is the preimage of the Sylow-p-subgroup of T 0 (f), hence equal to the pro-pSylow subgroup of T(k)0 . It remains to show that T(k)1 /T(k)0 does not have elements of pro-p order. According to Corollary 11.1.6, this is a subgroup of X∗ (T)I . Since T is weakly induced, the wild inertia subgroup P ⊂ I preserves a basis of X∗ (T), and hence X∗ (T) P is a free abelian group. Since the pro-order  of I/P is prime to p, it follows that X∗ (T)I does not have p-torsion. Lemma B.9.4 Assume that k is complete with finite residue field f of characteristic p, and let 1 → A → B → C → 1 be an exact sequence of k-tori with A weakly induced. Then 1 → A(k)0+ → B(k)0+ → C(k)0+ → 1 is still exact. Proof According to Lemma 2.5.20, the natural map B(K)0 → C(K)0 is surjective. We have A(K)0 ⊂ A(K) ∩ B(K)0 ⊂ A(K)1 by Lemma 2.5.19 and Proposition 2.5.9. Lemma B.9.3 implies that the sequence 1 → A(K)0+ → B(K)0+ → C(K)0+ → 1 is exact. The claim follows from Lemma 8.1.4, which implies H1 (Gal(K/k), A(K)0+ ) = H1 (Gal(K/k), A0+ (O)) = {1}. 

B.10 The Minimal Congruent Filtration Let T be a k-torus. In §B.5 we introduced the standard filtration of T(K). In Proposition B.6.4 we showed that this filtration has very good properties when T is induced. For a general torus, we have the following weaker result. Proposition B.10.1 The standard filtration is functorial and schematic. Proof Functoriality is immediate from Lemma B.4.7. For the rest of the proof we extend scalars from k to K and consider a K-torus T. Corollary B.8.7 implies that T(K)std 0 is schematic. For r > 0, Corollary A.6.6 reduces the proof that T(K)rstd is schematic to showing that Ur := {t ∈ T(K) | for all χ ∈ X∗ (T), ω( χ(t) − 1)  r } is schematic. Let L/K be the splitting field of T and let S = R L/K (TL ). Then S is an induced torus and we have the adjunction T → S. Since X∗ (S) → X∗ (T) is surjective, Ur = T(K) ∩ Srstd (O). Let Trstd be the schematic closure of the image of

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687

the adjunction T → S in Srstd , where Srstd is the smooth integral model of Proposition B.3.6. Lemma A.2.1 implies that Trstd (O) = T(K) ∩ Srstd (O) = Ur , so Ur is schematic by Corollary A.6.5. The resulting closed immersion Trstd → Srstd (that was just constructed for r > 0, and is given by definition of T std when r = 0, see Lemma B.4.7), maps Trstd (M) into the intersection of std (O), the latter being equal to T std (O).  T(K) with Srstd (M) = Sr+1 r+1 For a moment we return to the case of weakly induced tori. We will now prove that the standard filtration is the unique filtration characterized by a simple set of desirable properties. Definition B.10.2 Consider a k-torus T varying through a full subcategory of the category of k-tori. A decreasing separated functorial filtration T(K)r∗ of T(K)1 is called admissible, if (1) it coincides with the standard filtration when T is induced; (2) T(K)∗0 is the Iwahori subgroup T(K)0 and T(K)∗0+ is the group of elements having unipotent image in T 0 (f), where T is the connected Néron model of T. Remark B.10.3 While not needed in this book, the following observation is often useful in applications. If T(K)r∗ is an admissible filtration and S ⊂ T is the maximal unramified subtorus, then the natural map S(K)∗0 /S(K)∗0+ → T(K)∗0 /T(K)∗0+ is an isomorphism. This follows at once from Corollary B.7.12. Lemma B.10.4

If T(K)r∗ is an admissible filtration, then T(K)r∗ ⊂ T(K)rstd .

Proof This follows by defining S = R L/K (T ×K L) for a finite Galois extension L/K splitting T and using functoriality for the adjunction map T → S.  Proposition B.10.5 On the category of weakly induced tori, the standard filtration is the unique admissible filtration. Proof Proposition B.10.1 shows that the standard filtration is functorial and schematic, in particular separated. By Proposition B.6.4 the standard model is smooth, hence equal to the ft-Néron model. Corollary B.8.7 implies T(K)std 0 = is the group of elements T(K)0 , while Proposition B.9.1 states that T(K)std 0+ 0 of T(K)std 0 having unipotent image in T (f). We conclude that the standard filtration is admissible. Conversely, let T(K)r∗ be an admissible filtration. Let L/K be a finite tamely ramified Galois extension such that TL is induced. Define S = R L/K (TL ). Then T(K)r∗ ⊂ T(K)rstd by Lemma B.10.4, while T(K)rstd ⊂ T(K)r∗ by using functoriality for the norm map S → T together with Proposition B.6.6 for r = 0, and Corollary B.8.6 for r > 0. 

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We saw in Examples B.4.11 and B.6.1 that when T is not weakly induced, the standard model can fail to be smooth and needs to be replaced by its smoothening, the ft-Néron model. We will now see an example in which the standard filtration fails to be connected or congruent. For this, we revisit the setting of Example B.7.5. 1 G Example B.10.6 Let k = Q2 and  = Q2 () where  2 = 2. Let T = R/k m and let S = R/k Gm , so that T ⊂ S. By definition of the standard filtration we have T(k)r = T(k) ∩ S(k)r . The jumps of the filtration S(k)r lie in 12 Z0 . For n ∈ Z0 let (Gm )n be the nth congruence group scheme of Gm . Then Sn/2 = Ro /o ((Gm )n ) has the property Sr (o) = S(k)r for r ∈ 12 Z. Moreover, if we let Tr be the schematic closure of T in Sr we obtain a model Tr of T with Tr (o) = T(k)r for r ∈ 12 Z. These identities also hold with K in place of k. Let us compute the coordinate ring of Tr . For this, we recall from Example A.5.21 that in the coordinate ring [ t,  u]/( t u − 1) of the -group Gm the coort − 1),  u]. It is dinate ring of the o -model (Gm )n is the o -subalgebra o [ −n ( t − 1). Then the coordinate ring convenient to make the substitution  xn =  −n ( xn,  u]/( n  xn  u+ u − 1) and the coordinate ring of of the -group Gm becomes [ xn,  u]/( n  xn  u+ u − 1). According the o -model (Gm )n is the o -subalgebra o [ to Construction A.3.5 we see that the coordinate ring of the k-group R/k Gm is k[xn, xn , u, u ]/( f1, f2 ), where the variables are related by xn + xn =  xn and u + u  =  u, and the polynomials f1, f2 are given as follows depending on whether n is even or odd. When n is even

f1 = 2n/2 xn un + 2(n/2)+1 xn u  + u − 1,

f2 = 2n/2 (xn u + xn u ) + u ,

and when n is odd f1 = 2(n+1)/2 (xn u + xn u ) + u − 1,

f2 = 2(n−1)/2 xn u + 2(n+1)/2 xn u  + u  .

We have computed in Example B.4.11 that the inclusion T → R/k Gm induces on coordinate rings the surjective k-algebra map k[t, t , u, u ]/(tu + 2t u  − 1, tu  + t u) → k[t, t ]/(t 2 − 2t 2 − 1) given by u → t and u  → −t . We have already expressed the source of the t = t +t , above map as k[xn, xn , u, u ]/( f1, f2 ). Here we have the relationships   n t=  xn + 1, which lead to  xn = xn + xn , and  t = 2n/2 xn + 1,

t  = 2n/2 xn

when n( 0) is even and t = 2(n+1)/2 xn + 1,

t  = 2(n−1)/2 xn

B.10 The Minimal Congruent Filtration

689

when n is odd. The target becomes k[xn, xn ]/ f3 , where f3 = 2(n/2)−1 xn2 + xn − 2n/2 xn2 when n( 2) is even, f3 = 2xn2 + 2xn − xn2 when n = 1, and f3 = 2(n−1)/2 xn2 + xn − 2(n−3)/2 xn2 when n > 1 is odd. The reason for working with the coordinates xn, xn in place of t, t  is that now the integral model Ro /o ((Gm )n ) has coordinate ring generated over o by xn, xn , u, u , and the image of this o-algebra in the coordinate ring of T, which equals the coordinate ring of Tn/2 , is directly seen to be o[xn, xn ]/ f3 . We can now examine the polynomial f3 and its reduction modulo m. When n = 2 this reduction is x22 + x2 = x2 (x2 + 1) and we see that T1 is smooth, but disconnected. When n > 2 is even this reduction is xn , and Tn/2 is smooth and connected. When n > 1 is odd the model Tn/2 is smooth and connected. When n = 1 the reduction of f3 is xn2 and we see that T1/2 is connected but not smooth. To compute its smoothening, we first check by a direct computation that the image of the group of O-points in the group of f-points consists of the two points x1 = 0, x1 = 0 and x1 = 0, x1 = 1. The ideal of this image is (2, x1, x1 (x1 − 1)) in f[x1, x1 ]/x12 , and the corresponding dilatation is O[x2, x1 ]/(x1 (x1 + 1) − 2x22 ), which is smooth and disconnected. Note that, according to our computation in Example B.7.5, T1 is the ftNéron model of T. We can now easily compute the relative identity component of this model. According to Proposition A.5.23, it is the dilatation of the identity component of the special fiber. The ideal of that identity component is (2, xn ), and the corresponding dilatation has coordinate ring generated over o by 2−1 x2 = x3 and x2 = x3 . Thus, this identity component is T3/2 . Let Trstd be the smooth integral model such that Trstd (O) = T(K)rstd . Thus Trstd = Tr for all r except r = 0 and r = 1/2, where one must take the  = 2−1 xn for all smoothening of Tr . The identities xn+2 = 2−1 xn and xn+2 std n  0 show that T(K)r is congruent for all r  0 except for r = 0 and r = 1/2. Remark B.10.7 One can consider the modification of the standard filtration obtained by taking the relative identity component of the corresponding smooth integral models. While this is an improvement, it does not yield a fully satisfactory filtration. For example, it does not restore the congruence property. We now present a different filtration, introduced by J.K. Yu in [Yu15, §5],

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called the minimal congruent filtration. It is connected and congruent, and satisfies the Moy–Prasad isomorphism, at least for r  1. We write T 0 for the relative identity component of the ft-Néron model. Definition B.10.8 For r  0 define the subgroup T(K)rmc of T(K)1 and the smooth model Trmc as follows. 0 (1) If r = 0, let T(K)mc 0 = T(K) be the Iwahori subgroup of Definition 2.5.13 mc 0 and T0 = T . 0 (2) If 0 < r < 1, let T(K)rmc be the subgroup of T(K)mc 0 = T(K) generated 0 std by T (M) and the images of R(K)r for all K-homomorphisms R → T, where R is an induced torus. We will show momentarily that T(K)rmc is schematic; call its smooth model Trmc . (3) If r  1, write r = r0 + n with n ∈ N and 0  r0 < 1. Let Trmc be the nth congruence subgroup (cf. Definition A.5.12) of Trmc and let 0 T(K)rmc = Trmc (O).

Lemma B.10.9 Let 0 < r < 1. Then T(K)rmc is schematic and connected. Proof The subgroup T(K)rmc /T 0 (M) ⊂ T 0 (f) is the subgroup generated by the images of the special fibers of Rr0 → T 0 . These images are Zariski closed by [Bor91, Corollary 1.4(a)] and connected, since the special fiber of Rr0 is connected. The group they generate is closed and connected by [Bor91, Proposition 2.2]. Therefore the dilatation of that subgroup within T 0 is a  smooth connected model of T whose O-points equal T(K)rmc . Proposition B.10.10 The filtration T(K)rmc is functorial, schematic, connected, and congruent. Proof Definition B.10.8, which rests on Lemma B.10.9, already 12345includes the fact that the filtration is schematic and congruent. Connectedness follows from Lemma A.5.10. To prove functoriality, let f : T1 → T2 be a morphism of tori. Lemma B.7.11 states that f extends to a morphism T10 → T20 , hence carries T10 (M) to T20 (M). It is then clear that f (T1 (K)rmc ) ⊂ T2 (K)rmc for 0  r < 1. For r  1 this statement follows from the fact that forming the nth congruence group scheme is a functorial operation.  0 std Lemma B.10.11 We have T(K)mc 0+ = T(K) ∩ T(K)0+ and this group is the 0 0 preimage under the reduction map T (O) → T (f) of the unipotent radical Ru (T 0 )(f).

Proof The second statement was already proved in Proposition B.9.1. To mc prove the equality T(K)0 ∩ T(K)std 0+ = T(K)0+ , we first consider the inclusion

B.10 The Minimal Congruent Filtration

691

std T(K)mc 0+ ⊂ T(K)0 ∩ T(K)0+ . For any homomorphism R → T with R an induced std std torus we have R(K)0+ → T(K)std 0+ by Proposition B.10.1, as well as R(K)0+ ⊂ std 0 0 0 R(K)0 = R(K) → T(K) from Fact B.4.3. On the other hand, T (M) consists of elements whose image in T 0 (f) is trivial, hence unipotent. For the opposite inclusion let T  = R L/K (TL ), for a finite Galois extension L/K splitting T (T  is induced!) and consider the norm map T  → T. Its kernel is a torus and therefore Proposition 2.3.5 implies that the homomorphism T (K) → T(K) is surjective. Let T  and T denote the lft-Néron models. Thus T (O) → T (O) is surjective and Corollary B.8.6 implies that T 0 (O) → T 0 (O) is surjective. According to the topological Jordan decomposition, Proposition A.4.23, the subgroup of T 0 (O) of elements with unipotent image in T 0 (f) maps surjectively onto the subgroup of T 0 (O) of elements with unipotent reduction image in T 0 (f). Together with Proposition B.9.1 this std means that the image of the norm map T (K)std 0+ → ∩T(K)0+ is contained in T(K)0 since the special fiber of T std is connected. So it is enough to check that mc  (under the norm map) T (K)std 0+ lands in T(K)0+ . But as T is induced, Definition B.10.8(2) does the job. 

Proposition B.10.12 For r > 0 the special fiber of Trmc is a vector group. The unipotent radical of the special fiber of T 0 = T0mc is equal to the image mc → T 0 . More precisely, the preimage in T 0 (O) of of the special fiber of T0+ Ru (T 0 )(f) equals T(K)mc 0+ . mc (O) was shown Proof That the preimage of Ru (T 0 )(f) in T0mc (O) equals T0+ in Lemma B.10.11. For r > 0, we already know that Trmc is smooth (by definition) and connected (by Proposition B.10.10). It is furthermore commutative, being an integral model of the torus T. Since f is assumed perfect, in order to show that the special fiber of Trmc is a vector group, it is enough to prove that it is unipotent. Let 0 < r < 1. By definition T(K)rmc is the group generated by the images of S(K)rstd for maps S → T with S induced, and T10 (O), where T1 is the first congruence group scheme (cf. Definition A.5.12) of T 0 . Therefore Trmc (f) is generated by the images of the special fibers of Srstd → Trmc and T10 → Trmc . By Proposition B.6.4 the special fiber of Srstd is unipotent (since induced tori are weakly induced!). By Lemma A.2.2 the special fiber of T10 is unipotent. Therefore Trmc (f) is the product of (finitely many commuting) unipotent groups, hence (Trmc )f is unipotent. mc , hence its For r  1, Trmc is the first congruence group scheme of Tr−1 special fiber is unipotent by Proposition A.5.19 and induction. 

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Corollary B.10.13 The minimal congruent filtration is admissible. In particular, if T is weakly induced then T(K)rmc = T(K)rstd . Proof If T is induced, then the standard filtration is functorial, connected, and congruent by Propositions B.10.1 and B.6.4 and moreover T(K)std 0 is the Iwahori subgroup by Lemma B.3.1. Therefore the standard filtration and the minimal congruent filtration coincide (by the proof of Proposition B.10.5). If 0 mc T is general, then T(K)mc 0 = T(K) by definition, while T(K)0+ is the group of elements of T(K)0 with unipotent image in T 0 (f) by Proposition B.10.12. The first point is thus proved, and the second follows from Proposition B.10.10 (for functoriality) and Proposition B.10.5.  Corollary B.10.14 Assume that k is complete. Let T be a k-torus and let n > 0 be an integer. For any r > 0 the group Hn (Γ,T(K)rmc ) vanishes. In addition, if dim(f)  1, then the group Hn (Γ,T(K)mc 0 ) vanishes. Proof Let k /k be a finite unramified extension. Let o  denote the ring of integers of k  and m  denote the maximal ideal of o  and f  denote the residue field of o . Consider first r > 0. Set T(k)rmc ∩T(k ). The group T(k )rmc is filtered  mc  mc by the subgroups T(k )mc s for s > r. The quotient T(k )r /T(k )r+1 is the group  of f -points of a connected commutative unipotent f-group U by Propositions B.10.10 (for congruent) and B.10.12 (for unipotence). Lemma 2.4.2 implies the vanishing of Hn (f /f, U(f )). So T(k )rmc = Trmc (o ) = lim(Trmc (o /m n )). ← −− n The Gal(k /k) action on this is on the second factor of Lie(Trmc ) ⊗o m n /m n+1 . As m n /m n+1 is a f -vector space with a semi-linear action of Gal(k /k), the Galois cohomology group H(f /f, Lie(Trmc ) ⊗o m n /m n+1 ) vanishes for all n. Now the completeness of k and [Ser79, Chapter XII, §3, Lemma 3] imply the vanishing of Hn (k /k,T(k )rmc ), which in turn implies the vanishing of Hn (Γ,T(K)rmc ) = lim  Hn (k /k,T(k )rmc ). −−→k /k mc Assume now dim(f)  1. The quotient T(K)mc 0 /T(K)0+ is the group of fpoints of the f-torus T that is the reductive quotient of the special fiber of the connected Néron model of T, cf. Proposition B.10.12. We have the exact sequence n mc n Hn (Γ,T(K)mc 0+ ) → H (Γ,T(K)0 ) → H (Γ, T(f)).

We have proved the vanishing of the first term. The vanishing of the third term follows from Lemma 2.5.4. 

B.10 The Minimal Congruent Filtration

693

Proposition B.10.15 Given 0  r < s, let t = max(s, r + 1). Then

ker(T smc → T rmc ) = im(T tmc → T smc )

and is in particular a smooth connected unipotent group.

Proof In this proof we drop the superscript “mc” to ease notation. We may also base change to K and hence assume k = K. We will use the statements recalled in §2.4(a) without further mention. We will first prove that im(Tt → Ts ) is equal to ker(Ts → Tr )red . For this it is enough to compare f-points. Thus, given x ∈ Ts (f) we want to show that its image in Tr (f) is trivial if and only if it lifts to Tt (f). By Lemma 8.1.3 we can choose a lift x ∈ Ts (O) of x. By the congruence property (Proposition B.10.10) of the T mc , the vanishing of the image of x ∈ Ts (O) ⊂ Tr (O) in Tr (f) holds if and only if x ∈ Tr+1 (O). Therefore, x dies in $osTt (f) if and only if x ∈ Tt (O). If these equivalent conditions hold, the reduction of x in T t (f) is a lift of x. Conversely, if x lifts to T t (f), then it has vanishing image in T r (f) since T t → T r vanishes (as t  r + 1). It remains to prove that ker(Ts → Tr ) is smooth. When s  r + 1 then the previous paragraph implies ker(Ts → Tr )red = Ts , which forces ker(Ts → Tr ) = Ts . This reduces the problem to the case 0  r < s < r + 1, which we now assume. Since T s and T r have the same dimension (namely, dim T, we have

dim(ker(Ts → Tr )) = dim(cok(Ts → Tr ))  dim(cok(Lie(Ts → Tr )))

and our goal is to prove that this inequality is an equality. Note that Lie(Ts ) = t(k)s ⊗o f and thus cok(Lie(Ts → Tr )) = cok(t(k)s ⊗o f → t(k)r ⊗o f) = t(k)r /t(k)s . The last quotient is a f-vector space since r < s < r + 1 and yt(k)r /tr+ = Lie(T r ) as a general feature of congruent filtration (Lemma A.5.13(4) applied to G = Tr ). We have the exact sequence

ker(Tr+1 → Ts ) → Tr+1 → ker(Ts → Tr )

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which leads to the chain of inequalities dimk (T) = dim(Tr+1 )  dim(ker(Tr+1 → Ts )) + dim(ker(Ts → Tr )) = dim(cok(Tr+1 → Ts )) + dim(cok(Ts → Tr ))  dim(cok(Lie(Tr+1 → Ts ))) + dim(cok(Lie(Ts → Tr ))) = dim(t(k)s /t(k)r+1 ) + dim(t(k)r /t(k)s ) = dim(t(k)r /t(k)r+1 ) = dim(T r ) = dimk (T). We conclude that all inequalities must be equalities, and the smoothness of  ker(Ts → Tr ) follows. We now formulate the Moy–Prasad isomorphism without the assumption that T is weakly induced. The statement in this generality is slightly weaker. While for weakly induced tori Proposition B.6.9 provides a functorial isomorphism mc mc T(k)rmc /T(k)mc s → t(k)r /t(k)s for all 0 < r  s  2r, for general tori one obtains such a functorial isomorphism only when 1  r, and with a slightly sharper condition on s. In the range 0 < r  1, one still obtains an isomorphism, but it may not be functorial. Proposition B.10.16 Let 0  r0 < 1, n a positive integer, r = r0 + n, and r0 + n  s  r0 + 2n. There exists a functorial isomorphism T(k)rmc /T(k)mc s → mc mc t(k)r /t(k)s (as abelian groups), which coincides with the isomorphism of Proposition B.6.9 when restricted to the category of weakly induced tori. and obProof We apply Proposition A.5.19(3) to the group scheme Trmc 0 mc mc /t(k)mc . Let /T(k) → t(k) tain a functorial isomorphism T(k)rmc r0 +n 0 +n r0 +2n r0 +2n s0 = s − n. By functoriality, this isomorphism identifies the image of the quomc mc mc tient T(k)mc s0 +n /T(k)s0 +2n with the image of the quotient t(k)s0 +n /t(k)s0 +2n , and mc mc therefore induces a functorial isomorphism T(k)rmc /T(k)mc s → t(k)r /t(k)s as desired. Consider now the case when T is weakly induced (so the mc-filtration is the std-filtration by Corollary B.10.13). Let /k be a tamely ramified extension over which T becomes induced. The isomorphism of Proposition B.6.9 is by construction the restriction of the analogous isomorphism over . The same is true for the isomorphism of Proposition A.5.19, by virtue of its functoriality in the coefficient ring. This reduces the claim to the case when T is induced. Both isomorphisms are compatible with restriction of scalars (see Proposition A.5.19(2) and the proof of Proposition B.6.9 in the induced case) and with

B.10 The Minimal Congruent Filtration

695

products, so the claim is further reduced to T = Gm , in which case it follows from Example A.5.21.  Proposition B.10.17 Let 0 < r < s  r + 1. There exists a possibly nonmc mc functorial isomorphism T(k)rmc /T(k)mc s → t(k)r /t(k)s (as abelian groups), compatible with unramified algebraic extensions of k. mc mc Proof We have T(k)rmc /T(K)mc s = Tr (O)/Ts (O) = cok(Ts → Tr )(f). By Proposition B.10.12, Tr is a vector group, and then so is cok(Ts → Tr ). For any vector f-group A we have a non-canonical Gal(f/f)-equivariant isomorphism A(f) → Lie(A)(f); it is obtained by choosing an isomorphism A  Gan of fgroups, where n is the dimension of A, and using the natural identifications Ga (f) = f and Lie(Ga )(f) = f. On the other hand, Proposition B.10.15 shows that Lie(cok(Ts → Tr )) = cok(Lie(Ts ) → Lie(Tr )) = cok(t(k)mc s ⊗o f → mc mc mc  t(k)r ⊗o f) = t(k)r /t(k)s , cf. §2.4(a).

C Integral Models of Root Groups

C.1 Introduction In the original construction of parahoric integral models of a connected reductive K-group G by F.Bruhat and J.Tits, in [BT84a], the integral models of root groups played an important role. We will give here an explicit description of these integral models. In this book, in the case of parahoric integral models of G, the integral models of root groups are constructed abstractly in §8.3. The construction in this appendix offers a different point of view that may be useful for computations. Let K be a field given with a non-trivial discrete valuation ω : K → R∪ {∞}. We denote by O its ring of integers and by f its residue field. We assume that f is algebraically closed. Let G be a connected reductive K-group. Recall that due to Corollary 2.3.8 it is automatically G is quasi-split. Let S be a maximal K-split torus of G and let A(S) be the corresponding apartment as in §6.1. Let a ∈ R(S, G) be a relative root and let Ua be the corresponding root subgroup. For any x ∈ A(S) and r ∈ R we have the filtration subgroup Ua, x,r ⊂ Ua (K) of Definition 6.1.2. We will construct explicitly an integral model Ua, x,r of Ua so that the subgroup Ua, x,r (O) ⊂ Ua (K) equals the filtration subgroup Ua, x,r ⊂ Ua (K). Since this specifies the model uniquely due to Corollary 2.10.11, this model is the same as the one constructed abstractly in Chapter 8. When 2a is also a root, we may consider the mixed filtration subgroup Ua, x,r , s := Ua, x,r · U2a, x, s ⊂ Ua (K) for r, s ∈ R and will construct explicitly an integral model Ua, x,r , s of Ua such that Ua, x,r , s (O) = Ua, x,r , s . Let us recall that for any y ∈ A(S) we have Ua, y,r = Ua, x,r+ a,x−y due to Lemma 6.1.6. Therefore we may choose the point x to be any convenient point of A(S), and use the above recipe to obtain an integral model for arbitrary x. 696

C.2 Integral Models for Filtration Subgroups of Ga

697

Such a convenient choice will be given by a Chevalley valuation, or an adjusted Chevalley valuation, in the sense of Definition 6.1.20. Recall also that U2a,x,2r = Ua,x,r ∩ U2a (K), and in particular Ua,x,r ,2r = Ua,x,r , when a is multipliable. The structure of the root group Ua was discussed in 2.6.8. Using Notation 2.6.11 and Construction 2.9.16, we see that fixing a Chevalley–Steinberg system leads to an isomorphism between Ua and (1) R L/K Ga , if a is of type R1, (2) R L/K UL /L , if a is of type R2, (3) R L/K UL0  /L , if a is of type R3, where L/K is a finite separable extension, and L /L is a quadratic separable extension. In fact, we can reduce the construction of the integral models Ua, x,r and Ua, x,r , s to the case L = K, as follows. The root group Ua and the apartment A(S) do not change if we replace G by Gsc and S by its inverse image in Gsc , which allows us to assume that G is semi-simple and simply connected. Then G is the product of K-simple factors, and Ua and A(S) are products of the corresponding objects for each factor. This reduces the problem to the case that G is K-simple. Then G = R L/K G  for an absolutely simple and simply connected L-group G , S = R L/K S  for a maximal L -split torus S  ⊂ G , and Ua = R L/K Ua , where we have a ∈ Φ(S, G) = Φ(S , G ) and Ua is the root subgroup of G  corresponding to the relative root a. We have the identification A(S) = A(S ) of 7.9.1, which we use to interpret x as a point of A(S ). With this interpretation, the subgroup Ua, x,r of Ua (K) corresponds under the   identification Ua (K) = Ua (L) to the subgroup Ua, x,r of Ua (L). Given an O L    model Ua, x,r of Ua with O L -points equal to Ua, x,r , Corollary A.3.13 shows that RO L /O Ua, x,r is an O-model of Ua whose O-points equal Ua, x,r . The analogous discussion applies to the subgroup Ua, x,r , s when a is multipliable. In §C.2–§C.4 below, we will assume L = K, and for simplicity will denote the quadratic extension L  by L. We will return to the general set-up L /L/K in §C.5.

C.2 Integral Models for Filtration Subgroups of Ga Consider first X = Ga . Then its coordinate ring K[X] is the polynomial algebra K[t] in one variable with comultiplication t → t ⊗ 1+1 ⊗ t, coinversion t → −t, and counit t → 0. We have the obvious sub-O-bialgebra A := O[t] ⊂ K[t]. It is smooth over O and the associated integral model X satisfies X (O) =

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O ⊂ K = X(K). More generally, choosing a uniformizer π, for any fixed n ∈ Z we have the sub-O-bialgebra An := O[π −n t] ⊂ K[t] and the associated smooth integral model Xn satisfies Xn (O) = Mn ⊂ K. Note that An , hence also Xn , is independent of the choice of π. If ω(K × ) = Z then Xn (O) = {x ∈ K |ω(x)  n} for all n ∈ Z. The case when ω is not normalized can be reduced to the case of a normalized ω by a simple rescaling. In this way we obtain for an arbitrary ω and for each real number r a smooth integral model Xr of X which is determined up to isomorphism by Xr (O) = {x ∈ K | ω(x)  r }. Fact C.2.1 Let s > r. The morphism Xs → Xr induced by Corollary 2.10.10 induces a morphism Xs → Xr between the special fibers which is an isomorphism when Xs (O) = Xr (O) and is trivial when Xs (O)  Xr (O). Proof We may reduce to the case ω(K × ) = Z, after which the claim is immediate from the description of the coordinate rings.  C.2.2 Lie algebra Assuming ω(K × ) = Z the set J of jumps of the filtration X(K)r equals Z. Given r ∈ R we let r  = inf{h ∈ J | h  r } and have  Lie(Xr ) = Mr inside Lie(X) = K.

C.3 Integral Models for Filtration Subgroups of R0L/K Ga Consider next X = R0L/K Ga . Recall that we denote by L 0 the kernel of the trace map L → K. Then a choice of η ∈ L 0 gives the isomorphism of algebraic groups Ga → R0L/K Ga,

x → η · x.

Let r0 = ω(η) and Y = Ga . For each r ∈ ω(K × ) we define the smooth model Xr by taking the smooth model Yr−r0 defined using Section C.2 (with Y here as X there) and composing the isomorphism (Yr−r0 )K → Y with the above isomorphism Y → X. Then Xr (O) = {x ∈ L 0 | ω(x)  r }. Fact C.3.1 Let s > r. The morphism Xs → Xr induced by Corollary 2.10.10 induces a morphism Xs → Xr between the special fibers which is an isomorphism when Xs (O) = Xr (O) and is trivial when Xs (O)  Xr (O). Proof

Immediate from Fact C.2.1.



C.3.2 Lie algebra. Let λ ∈ L tr=1,max , cf. (2.8.2), and let μ = ω(λ). Recall that we can take λ = 1/2 unless p = 2. According to Lemma 2.8.1 we can choose a uniformizer  ∈ L so that λ ∈ L 0 . Multiplication by λ induces an

C.4 Integral Models for Filtration Subgroups of UL/K

699

isomorphism Y → X that identifies Yt−ω(λ) = Yt−μ−1/2 with Xt . Assuming ω(K × ) = Z, the set J of jumps of the filtration X(K)r equals Z + μ + 12 . Given t  −μ−1/2 . t ∈ R let t  = inf{h ∈ J | h  t}. Then Lie(Xt ) = λ · m L

C.4 Integral Models for Filtration Subgroups of UL/K Consider finally X = UL/K . This case is more involved. We begin by recalling the relevant notation. Recall from (2.7.2) that for any K-algebra R we have UL/K (R) = {(u, v) ∈ (R ⊗K L)2 | uu = v + v} with group law (u, v)(u , v ) = (u +u , v + v  +uu ), where 0 u → u is the generator of Gal(L/K). Recall also the central subgroup UL/K 0 0 defined by UL/K (R) = {(0, v) ∈ (R ⊗K L)2 | v + v = 0}, so that UL/K = R0L/K Ga . 0 . We abbreviate Y = R L/K Ga and Z = R0L/K Ga and identify Z with UL/K Recall the filtrations of X(K) and Z(K) introduced in §3.2(b) during the discussion of SU3 . As in (3.2.3) we have the function ϕa : X(K) → R defined by ϕa (u, v) = 12 (ω(v) − μ), giving the filtration X(K)r = ϕ−1 a ([r, ∞]) for any r ∈ R. Here μ is defined in (2.8.1). The jumps of this filtration are the set Ja . As in (3.2.3) we also have the function ϕ2a : Z(K) → R defined by ϕ2a (0, v) = −1 ([s, ∞]) whose jumps are the set ω(v) − μ. It gives the filtration Z(K)s = ϕ2a J2a . Finally we have the set Ja/2a of jumps of the filtration of X(K)/Z(K) induced from the filtration of X(K). We recall from Corollary 3.2.4 (using that the separable quadratic L/K is ramified, as K is strictly Henselian) that Ja = 14 Z, Ja/2a = 12 Z, J2a = Z + 21 , assuming that ω(K × ) = Z, which will be our convention in this section. Our goal in this section will be to construct and study an integral model Xr , s whose O-points equal X(K)r · Z(K)t , for any pair of real numbers (r, s). The next lemma shows that we can assume without loss of generality that r ∈ Ja/2a , s ∈ J2a , and s  2r + 12 . Lemma C.4.1 Let r, s ∈ R. Let r  = inf{h ∈ Ja/2a | h  r } and s  = inf{h ∈ J2a | h  inf(2r, s)}. (1) r  ∈ Ja/2a , s  ∈ J2a . (2) s   2r  + 21 . (3) X(K)r · Z(K)s = X(K)r  · Z(K)s . Note that the product sets in (3) are subgroups since Z is central in X. Proof (1) follows from the fact that Ja/2a and J2a are discrete, while (2) follows from the relation 2Ja/2a + 12 ⊂ J2a .

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(3) From r   r and s   inf(2r, s) we see X(K)r  ⊂ X(K)r and Z(K)s ⊂ X(K)r ∪ Z(K)s ⊂ X(K)r · Z(K)s , therefore X(K)r  · Z(K)s ⊂ X(K)r · Z(K)s . To see the converse, note that Z(K)s ⊂ Z(K)s and Z(K)2r ⊂ Z(K)s by definition of s . Now pick x ∈ X(K)r . The definitions of Ja/2a and r  imply the existence of x  ∈ X(K)r  and x  ∈ Z(K) such that x = x  x . Then x  ∈ Z(K) ∩ X(K)r = Z(K)2r ⊂ Z(K)s .  Therefore X(K)r ⊂ X(K)r  · Z(K)s . The integral model Xr ,s will be constructed by using integral models of Y and Z. For this, we relate X to Y × Z via the following coordinate change. Choose λ ∈ L tr=1 , where we are using the notation (2.8.2). If (u, v) ∈ X(R) then x = v − uuλ ∈ R satisfies x + x = 0. This gives the isomorphism of pointed schemes X → Y × Z,

(u, v) → (u, v − uuλ).

(C.4.1)

If we transport the group structure from the source to the target of this isomorphism we obtain on Y × Z the group law (u, x) · (u , x ) = (u + u , x + x  − λuu  + λuu ).

(C.4.2)

We write X λ for the group scheme with underlying scheme Y × Z and this modified group law. The integral models of Y and Z that we will use are the smooth connected models Yr and Zs with the properties that Yr (O) = {u ∈ L | ω(u)  r } and Zs (O) = {x ∈ L 0 | ω(x)  s + μ}, where μ is as in (2.8.1). The integral model Yr is obtained by applying RO L /O to the integral model of §C.2 that was denoted by Xr there, applied to the unique extension to L of the normalized valuation on K. The integral model Zs is that obtained in §C.3, but shifted by μ, i.e. the model Xs+μ in the notation of §C.3. Let us abbreviate Yr = Yr (O) = {u ∈ L | ω(u)  r } and Zs = Zs (O) = {x ∈ L 0 | ω(x)  s + μ}. Lemma C.4.2 Let r ∈ Ja/2a and s ∈ J2a satisfy s  2r + 12 . The isomorphism (C.4.1) identifies X(K)r · Z(K)s with Yr × Zs . Proof We first show that under the isomorphism (C.4.1) we have the inclusions X(K)r ∪ Z(K)s ⊂ Yr × Zs and Yr ∪ Zs ⊂ X(K)r · Z(K)s . It is obvious that the isomorphism (C.4.1) identifies Z(K)s with Zs . Given ¯ Then ω(v) = 2r + μ, so (u, v) ∈ X(K)r . u ∈ Yr let v = uuλ. ¯ We have u ∈ Yr by Lemma 3.2.2, from Given (u, v) ∈ X(K)r let x = v − uuλ. which it follows that ω(x)  2r + μ. But x ∈ L 0 and ω(L 0 ) = Z+ μ+ 12 according

C.4 Integral Models for Filtration Subgroups of UL/K

701

to Lemma 2.8.1. Since 2r ∈ Z we conclude ω(x)  2r + μ + 12  s + μ, thus x ∈ Zs . We have now shown the desired inclusions. The proof will be complete if we show that X(K)r · Z(K)s is a subgroup of X(K) and Yr × Zs is a subgroup of X λ (K). Since X(K)r is a subgroup of X(K), Z(K)s is a subgroup of Z(K), and Z(K) is a central subgroup of X(K), it is clear that X(K)r · Z(K)s is a subgroup of X(K). To see that Yr × Zs is a subgroup of X λ (K) it suffices to ¯ . Now show that for u, u  ∈ Yr we have ω(z − z¯)  s + μ, where z = λuu ω(u)  r implies ω(z)  2r + μ, hence also ω(z − z¯)  2r + μ. We have 2r ∈ Z, z − z¯ ∈ L 0 , and ω(L 0 ) = Z + μ + 12 as we have already used above,  hence ω(z − z¯)  2r + μ + 21  s + μ, as required. Corollary C.4.3 For any r, s ∈ R the subgroup X(K)r · Z(K)s of X(K) is schematic and connected. Proof By Lemma C.4.1 we may assume that r ∈ Ja/2a , s ∈ J2a , and s  2r + 21 . Choose λ ∈ L tr=1,max and consider the isomorphism (C.4.1). According to Lemma C.4.2, this isomorphism identifies X(K)r · Z(K)s with Xrλ,s (O), where Xrλ,s = Yr × Zs . Composing the isomorphism Xrλ,s ×O K → X λ with the inverse of (C.4.1) we obtain an isomorphism Xrλ,s ×O K → X λ , hence an integral model Xr ,s of X with Xr ,s (O) = X(K)r · Z(K)s . By construction this model is smooth and connected. According to Corollary 2.10.11 it is a-priori independent of the choice of λ up to isomorphism.  Let Xr ,s denote the smooth model of X with Xr ,s (O) = X(K)r · Z(K)s . Lemma C.4.4 Let r2  r1 and s2  s1 . The morphism Xr2 , s2 → Xr1 , s1 induced by Corollary 2.10.10 induces a morphism Xr2 , s2 → Xr1 , s1 whose kernel is a smooth connected unipotent group. Proof The integral model Xr ,s is constructed as the product Yr × Zs of the integral models of Y and Z, using the isomorphism (C.4.1). Write Yr and Zs for the special fibers of the models Yr and Zs , respectively. It is enough to treat individually the case r2 = r1 and s2 > s1 and the case r2 > r1 and s2 = s1 . In the case r2 = r1 and s2 > s1 the kernel is identified with the kernel of Zs2 → Zs1 and the claim follows from Fact C.3.1. In the case r2 > r1 and s2 = s1 the kernel is identified with the kernel of Yr2 → Yr1 . Lemma A.3.12 converts this to the kernel of R A/f ((Xr2 ) A) → R A/f ((Xr1 ) A), where Xr is the integral model of Ga over o with Xr (O) = MrL , as discussed in the case of roots of type R1 near the start of §C.4, base changed to A = O L /mK O L = O L /m2L = f[]/( 2 ). Since Weil restriction commutes with fiber products ([CGP15, Proposition

702

Integral Models of Root Groups

A.5.2]), this kernel is identified with R A/f (ker((Xr2 ) A → (Xr1 ) A)). We may assume without loss of generality that r1 = n1 /2 and r2 = n2 /2 for some integers n1  n2 . The homomorphism (Xr2 ) A → (Xr1 ) A is given on coordinate rings by the inclusion A[ −n1 t] → A[ −n2 t]. When n = n2 − n1 > 1 this inclusion factors through the augmentation A[ −n1 t] → A and the unit A → A[ −n2 t], so the homomorphism (Xr2 ) A → (Xr1 ) A factors through the identity section, and its kernel is all of (Xr2 ) A. Applying R A/f to this gives back Yr2 . When n = 1, the coordinate ring of the kernel of (Xr2 ) A → (Xr1 ) A is equal to  A ⊗ A[ −n1 t] A[ −n2 t] = A[u]/u. Applying R A/f to it produces Ga . C.4.5 Lie algebra The unipotent group X = UL/K is the closed subvariety of (R L/K A1 )2 consisting of pairs (u, v) satisfying the equation uu = v + v. Its tangent space at the identity element (0, 0) is therefore the subspace of L ⊕ L consisting of pairs (U,V) satisfying the equation 0 = V +V; this is the subspace L ⊕ L 0 . It is also the Lie algebra of the group X λ , and the differential of the isomorphism X → X λ given by (C.4.1) is the identity on L ⊕ L 0 . Given r, s ∈ R we let r  ∈ Ja/2a and s  ∈ J2a be as in Lemma C.4.1. Then Lie(Xr ,s ) = Lie(Yr  ) × Lie(Zs ) by the proof of Corollary C.4.3. Let   be a uniformizing element of L. Then multiplication by  2r induces an isomorphism Y0 → Yr  , hence also Lie(Y0 ) → Lie(Yr  ), and we obtain the  equality Lie(Yr  ) = M2r L of O-modules inside Lie(Y ) = K. On the other hand, applying C.3.2 and paying attention to the shift by μ by which the filtration of Z(K) used here differs from the filtration used in §C.3, we obtain the identity  Lie(Zs ) = λ · m Ls −(1/2) of O-modules, where λ ∈ L tr=1,max and  is chosen as in §C.3.2 so that λ ∈ L 0 . Unlike in §C.2 and §C.3, where we worked directly with filtrations induced by the valuation ω, in this section we worked with a filtration shifted by μ. In preparation for combining the results here with the results in §C.2 and §C.3, we now undo this shift. Let  r = X(K)r−μ/2 = {(u, v) ∈ UL/K (K)|ω(v)/2  r }, X(K) and  s = Z(K)s−μ = {(0, v) ∈ U 0 (K)|ω(v)  s}. Z(K) L/K ,r ,s be the integral model associated to the subgroup X(K)  r · Z(K)  s . Then Let X 



,r ,s ) = M2r ⊕ λ · Ms −(1/2), Lie(X L L

(C.4.3)

with r  = inf{h ∈ Ja/2a | h  r−μ/2} and s  = inf{h ∈ J2a | h  min(2r, s)−μ}.

C.5 Summary

703

C.5 Summary 0 , and In §C.2-§C.4 we constructed integral models for the groups Ga , UL/K UL/K , which are the building blocks of the relative root groups of a quasisplit reductive group. We will now use these integral models to obtain integral models of such relative root groups. Thus, let G be a connected reductive K-group, S ⊂ G a maximal split K-torus, a ∈ Φ = Φ(S, G) a relative root, x ∈ A(S), and r, s ∈ R.

Proposition C.5.1 The root filtration subgroup Ua, x,r is schematic and connected, say with smooth connected integral model Ua, x,r . If a is of type R2, then the group Ua, x,r · U2a, x, s is likewise schematic and connected, say with smooth connected integral model Ua, x,r , s . Moreover, assuming ω(K × ) = Z, these models are compatible with the formation of congruence group schemes as follows: given an integer n  0, (Ua, x,r )n = Ua, x,r+n whenever a is not multipliable (i.e., is of type R1 or R3), and (Ua, x,r , s )n = Ua, x,r+n, s+n whenever a is multipliable (i.e., is of type R2) and s  2r. We warn the reader that, when a is multipliable, the relation (Ua, x,r )n = Ua, x,r+n does not hold since (Ua, x,r )n = (Ua, x,r ,2r )n = Ua, x,r+n,2r+n . Proof We revisit some of the discussion of §C.1. All statements are compatible with passage to Gsc , and with respect to taking products, so we may assume that G is simply connected and K-simple. Furthermore, the identity Ua, y,r = Ua, x,r+ a,x−y for x, y ∈ A(S) due to Lemma 6.1.6 shows that we may replace x by any convenient point in A(S). Choose a Chevalley–Steinberg system and a Borel K-subgroup of G containing S and let x ∈ A(S) be the corresponding adjusted Chevalley valuation as in Definition 6.1.20. We will assume that a is positive for the chosen Borel subgroup. As discussed in §C.1, we have Ua = R L/K Ua for a finite separable extension L/K, and the choice of Chevalley–Steinberg system identifies Ua with one of Ga , UL /L , or UL0  /L , where L /L is a separable quadratic extension.

On each of Ga (L) = L, UL0  /L (L) = (L )0 , and UL /L (L) = {(u, v) ∈ L × L | uu¯ = v + v¯ } we have the natural filtrations, indexed by r ∈ R, stemming from the valuation ω L of L normalized so that ω L (L × ) = Z. We recall that these filtrations are given by Ga (L)r 0 UL /L (L)r

= {x ∈ L | ω L (x)  r } = {x ∈ L 0 | ω L (x)  r + μ L }

UL /L (L)r = {(u, v) ∈ UL /L (L) | ω L (v)  2r + μ L }.

704

Integral Models of Root Groups

Here μ L is given by (2.8.1) relative to the quadratic extension L /L. According   to Definition 6.1.20, the root filtration subgroup Ua, x,r of Ua (L) is equal to the corresponding filtration subgroup above, provided we use the valuation ω L for the construction of A(S ). If instead we use the valuation ω normalized so that ω(K × ) = Z, then the above three filtration subgroups correspond to  , where e is the ramification index of L/K. The reason we want to use Ua, x,r/e the latter valuation is that then under the identification Ua (K) = Ua (L) we have  Ua, x,r = Ua, x,r . In §C.2-§C.4 we constructed O L -models of the three L-groups Ga , UL0  /L , and UL /L . The construction for UL0  /L did not account for the μ L -shift, so the indexing needs to be shifted accordingly. The construction for UL /L already had the μ L -shift built in. These models provide an O L -model Ua, x,r of Ua with  the property Ua, x,r (O L ) = Ua, x,r , provided we are using the valuation ω L to  construct A(S ). We reindex these models by divding r by e in order to use the valuation ω instead. Then we set Ua, x,r = RO L /O Ua, x,r and obtain a O-model of Ua such that Ua, x,r (O) = Ua, x,r . Analogously, when a is multipliable, we obtain an O-model Ua, x,r , s of Ua such that Ua, x,r , s (O) = Ua, x,r , s . By construction and Lemma A.3.10, the models Ua, x,r and Ua, x,r , s are affine, smooth, with connected fibers. We now come to the congruence property. Since it is invariant under changing the point x, it is sufficient to check it for the fixed adjusted Chevalley valuation. Lemma A.5.15 reduces to checking this property for the O L -group schemes Ua, x,r and Ua, x,r , s . We are working with the normalized valuation ω L (L × ) = Z, so we may assume L = K, drop the prime decoration, and write L in place of L , which was the notation used in §C.2-§C.4. The congruence property is 0 . Consider now the case of UL/K . Since the immediate when Ua is Ga or UL/K condition s  2r is invariant under replacing (r, s) by (r + 1, s + 1), we reduce the proof by induction to the case of n = 1. Let r , s  be obtained from r, s by Lemma C.4.1. Let (r + 1) and (s + 1) be the corresponding numbers obtained from r + 1, s + 1 by Lemma C.4.1. Then (r + 1) = r  + 1, and the assumption s  2r implies that (s + 1) = s  + 1. It is therefore enough to check that (Xr  , s )1 = Xr  +1, s +1 , where Xr  , s is the O-model constructed in §C.4. By construction we have Xr  , s = Yr  × Zs as schemes. While the group structure on the left is not translated to the direct product group structure on the right, but rather the twist of it given by (C.4.2), the relation ker(Xr  , s (O) → Xr  , s (O/M)) = ker(Yr  (O) → Yr  (O/M)) × ker(Zs (O) → Zs (O/M)) still holds, where on the right we are using the group structures on Yr  and Zs . Since these group structures satisfy the congruence  relation, we obtain Yr  +1 (O) × Zs +1 (O) = Xr  +1, s +1 (O), as desired.

C.5 Summary Corollary C.5.2

705

Let a ∈ Φ, Ω ⊂ A(T) a bounded subset, r ∈ R. The group Ua, x,r Ua,Ω,r = x ∈Ω

has a smooth connected integral model Ua,Ω,r . Proof Since Ω is bounded, it intersects a finite number of facets. The intersection can thus be taken over a finite number of points x ∈ A(T), one for each facet meeting Ω. The resulting filtration subgroups of Ua (K) are nested in each other, so the intersection equals the smallest of them.  Lemma C.5.3 For s  r the kernel of the natural map U a, x, s → U a, x,r between special fibers is a smooth connected unipotent group. Proof

This follows from Facts C.2.1 and C.3.1 and Lemma C.4.4.



C.5.4 Lie algebra. We continue with a Chevalley–Steinberg system {Xa}a ∈Φ  for Lie(G) as in Definition 2.9.15 and let x be the corresponding Chevalley valuation as in Definition 6.1.20. Pick a ∈ Φ(S)nd (i.e., not of type R3). Using the discussion of C.2.2 and C.4.5 we can give an explicit description of the Lie algebras of the integral models of the root groups for a and (if a is of type R2) for 2a, based at the point x. The description for general points is obtained at once from Lemma 6.1.6.  lying over a and let Oa be the ring of If a is of type R1, choose  a ∈ Φ integers of the field Ka and let Ma be the maximal ideal of Oa. Let e be the (ramification) degree of Ka/K and recall from Fact 6.4.5 that Ja = e−1 Z. Given r ∈ R, let r  = inf{h ∈ Ja | h  r }. Then Ua, x,r = Ua, x,r  is constructed in §C.2 by applying RO a /O to the (er )th congruence group scheme of the O L -group Ga , from which we conclude that   z → σ(z · Xa) Mer a  → Lie(U a, x,r ), σ ∈Gal(K s /K)/Gal(K s /K a )

(with target contained in Lie(Ua ) ⊗K Ks ) is an isomorphism of O-modules.  lying over If a is of type R2 (so 2a is of type R3) then we choose  a,  a ∈ Φ   Let ea and e = e2a be the (ramification) a such that  b :=  a+ a belongs to Φ. degrees of Ka/K and Kb/K; thus ea = 2e (recall the discussion in §2.6.8). Let σa be the generator of Gal(Ka/Kb). Let λ ∈ Ka be an element satisfying λ + σa (λ) = 1 and whose valuation is maximal among such elements. Let μ = ω(λ). According to Lemma 2.8.1(2) we have either p  2, in which case μ = 0, or p = 2, in which case μ < 0, and in both cases there exists a uniformizer  of Ka such that λ + σa (λ) = 0 (note however that the formula for μ given in Lemma 2.8.1 must be multiplied by e since ω(Ka× ) = eZ). Let Oa and

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Integral Models of Root Groups

Ob be the rings of integers of Ka and Kb, and let Ma and Mb be the maximal ideals of Oa and Ob, respectively. Given r, s ∈ R, define ' ( r  = inf h ∈ Ja/2a | h  r − 12 μ and s  = inf {h ∈ J2a | h  min(2r, s) − μ} . Note that these formulas are those of Lemma C.4.1, but shifted by μ because we are now using the unshifted filtration on UL/K , which in turn corresponds to x being a Chevalley valuation, rather than an adjusted Chevalley valuation. 1 Z and J2a = e1 (Z + 12 ) by Fact 6.4.5. Using the fact that Recall Ja/2a = 2e Ua, x,r , s is constructed by applying ROb /O to the Ob-integral model of the group UK a /Kb that was constructed in §C.4, and the computation of the Lie algebra of that latter model performed in §C.4.5, we see that the O-module Lie(Ua, x,r , s ) is the direct sum of the two O-submodules L1 and L2 given as the images of the maps of O-modules   → Lie(U ) ⊗ K , z →  σ(z · Xa) M2er a K s a  σ ∈Gal(K s /K)/Gal(K s /K a )

and 

(λ)(Mb)e(s −1/2) → Lie(Ua )⊗K Ks ,

z →

 σ ∈Gal(K s /K)/Gal(K s /K b )

σ(z·Xb).

Note that L2 = Lie(U2a, x, s ) ⊂ Lie(Ua, x,r , s ) (with 2a of type R3). Remark C.5.5 The constructions in this appendix can be phrased in an alternative way, which makes them applicable to a base field k that may not be strictly Henselian. We do not use this approach in this book, but we mention it for completeness. Recall from 2.10.20 the concept of a vector group. Given r ∈ R, the subset M := {x ∈ k | ω(x)  r } is an o-lattice in the k-vector space V = k. We can define G M to be the vector group over o associated to M. Then its generic fiber is GV , which is canonically isomorphic to Ga . Analogously, given a separable quadratic extension /k and r ∈ R we can consider the o-submodule M := {x ∈  0 | ω(x)  r } of the k-vector space V =  0 , where  0 = {x ∈  | x + x = 0}. Then V is a 1-dimensional k-vector 0 G . space, and G M is a smooth model of GV = R/k a Continuing with a separable quadratic extension /k we consider the group U/k . Lemma C.4.1 is valid over k with the same proof, but in part (2) we have to replace s   2r  + 12 by s   2r  + c where c = 1/2 when /k is ramified, and c = 0 when /k is unramified. This lemma again reduces the problem

C.5 Summary

707

of finding an integral model Xr ,s of U/k such that Xr ,s (o) = X(k)r · Z(k)s for given r, s ∈ R to the case that s  2r + c. We consider the isomorphism U/k → Y × Z of (C.4.1). We have the smooth models Yr and Zs of Y and Z obtained in the previous paragraphs (we again apply Ro /o to obtain from the smooth model for Ga a smooth model for R/k Ga ). The same argument shows that Y (k)r · Z(k)s is a subgroup of X λ (k). In particular, the bilinear mapping f : X(k) × X(k) → Z(k) sending (u, u ) to −λuu  + λuu  sends X(k)r × X(k)r to Z(k)s . As discussed in 2.10.20, this bilinear mapping leads to a smooth group scheme structure on Xr × Zs , which was denoted in 2.10.20 by Xr × f Zs . Via (C.4.1) this o-group scheme becomes an o-model of U/k whose o-points are equal to X(k)r · Z(k)s . This is the desired model Xr ,s .

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712 [Mat89]

[MRR] [Mil80] [MP94] [MP96] [NSW08]

[Oes84] [Ono65] [Ono66a] [Ono66b]

[Pop01] [PR84]

[PR85] [PR08]

[Pra77] [Pra79]

[Pra82] [Pra89]

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References [Pra20a]

[Pra20b] [PY02] [PY06] [PY07] [Ree10] [RY14] [Ric82] [Ric16a] [Ric16b] [Rou77]

[Sch85] [SS97]

[Sch86]

[Ser71]

[Ser79]

[Ser97] [Ser03] [SGA3]

713

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714 [Sha72]

[Spi08] [Spr79]

[Spr09] [Sta12]

[Ste65] [Ste68]

[Thi19] [Tit66]

[Tit74] [Tit79]

[VKM02] [Wei60] [Yu15]

[Zhu11]

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Index of Symbols

x , 142 x , 142 ≺, 26 κ, 421, 429 Θ χ , 96 Θ a , 96 μ, 111 Ξ, 44, 398, 402 π1 (G), 427 ρΩ, Ω , 302 ρΩ, Ω , 302 Φ, 15, 96  96 Φ, ΦΩ , 304 Φ∨ Ω , 303 Φ(S) f , 377 Φ f , 314  246 Φ, Φ(S, G), 93 Φ(T , G), 93 Φ(S) x , f , 377 Φ x , f , 314 ϕ + v, 200 Ψ, 14, 159, 218, 351 Ψ , 159, 218 Ψ , 351 ΨΦ , 25 ΨΩ , 304 Ψ(C)+ , Ψ(C)− , 21 Ψ(C)0 , 22 ΨF , 26 ΨK , 359 Ψk , 359 Ψnd , 16 Ψnm , 16 Ψx , ΨF , 26

Ψ(S) x , f , 377 Ψx , f , 314 Ψx , r , 468 ψau , 159, 345 ω, 157, 195 ωG , 102 ωX , e , 123 A, 326 A∗Ω , 304 Aψ , 345 A[ mI ], 632 A(S), 157, 204, 208  ), 178 A(T a∨ , 94 B, 326 B(G), 157, 263  B(G), 178  39 D,  k , 403 D  K , 402 D f, 68, 157 f + , 314 Fχ , 96 G, 286 G , 286 G 0 , 133, 162 GΩ0 , GΩb , GΩ1 , GΩ† , GΩ , 295 Gder , Gsc , Gad , 93 G(k)  , 104 G(k)0 , 104, 431 G(k)1 , 102, 430

G(k)Ω , 255 0 G(k)Ω , 255

b , G(k)1 , G(k)† , G(k) , 267 G(k)Ω Ω Ω Ω G(k) b , 106

715

716 G(k) b , 73

G(k) x , f , 247 G(k) x , r , 468 G (m), 292 Gr(X), 610 G x , f , 311 Hψ , 218, 345 Hu , 159, 212 Ja , Ja/2a , 224 K, 70, 157 k, 157, 195 k a , 96  k, 68 k s , 68, 195  tr=1 , 110  tr=1, max , 111 M, 157 m, 68, 157 m(u), 136 integral, 293 O, 89, 121, 157 o, 68, 157 PΩ , 255 P x , f , 247 P x , r , 468 R S  /S , 595 ru , 212 Ru s ,f , 286 s, 111 T (k) b , 88 T (k)0 , 90 T (k)0 , 245 T (k)1 , 88 T (k)r , 245 T (k)std r , 667 Trmc , 690 T ft , 675 T std , 659 Uψ , 159, 217, 345 Uψ+ , 217 Ua , 93, 95 Ua , 93, 96 Ua , ϕ , r , 198 Ua , Ω, 0 , 255 Ua , Ω, 0+ , 303 U[ a] , 96 Ua , F, 0 , 222 Ua (F)∗ , 95 Ua , x , f , 247 Ua , x , r , 198 U/k , 110

Index of Symbols va , 230 V (AG ), 178 V (S), 197, 205 V (S  ), 209 ,0 , W ,1 , W ,sc , W ,ad , 231 W W (Ψ), 14 W aff , W ext , 231 XY , 632 T (k), 142

General Index

additive norm, 514 dual, 526 self-dual, 527 splitting, 514 splitting basis, 514 admissible facet, 61 parabolic subgroup, 61 affine blow-up algebra, 632 affine root hyperplane, 16 affine root system, 14, 218 absolute, 359 relative, 351, 359 affine Weyl group, 143 extended, 44 of a quasi-split adjoint group, 231 of a reductive group, 231 of a simply connected group, 231, 362 of an affine root system, 15 algebraic fundamental group, 427 apartment, 145 standard, 142 associated subgroup, 563 barycenter, 5 bounded, 3 bounded subset, 70 Bruhat–Tits building, 145 Bruhat–Tits group scheme, 163 building Bruhat–Tits, 263, 329 enlarged, 178 of a Tits system, 54 restricted, 62 chamber Γ-chamber, 326

K-chamber, 326 k-chamber, 326 of affine root system, 19 of chamber complex, 54 chambers adjacent, 62 Chevalley involution, 113 Chevalley system, 114 weak, 114 Chevalley valuation, 202, 207 adjusted , 207 Chevalley–Steinberg system, 117 weak, 116 coherent collection of parahoric subgroups, 562 complex chamber complex, 54 thick, 54 thin, 54 polysimplicial, 54 simplicial, 54 concave function, 246 group scheme, 311 group, 247 congruence group scheme, 636 convex, 5 descent of group schemes, 327 diameter, 3 dilatation, 632 Dynkin diagram affine, 39 absolute, 403 folding, 403 relative, 403 finite absolute, 98 folding, 98

717

718

General Index relative, 98

equipollent, 201 facet Γ-facet, 326 K-facet, 326 k-facet, 326 of affine root system, 19 of chamber complex, 54 filtration admissible, 687 congruent, 667 connected, 667 minimal congruent, 690 schematic, 667 standard, 667 fixed point theorem, 6 gallery, 54, 62 Galois group action on building, 165, 325 geodesic, 4 Greenberg functor, 610 Henselian field, 68 induced ramification, 90, 426, 434 induced torus, 86 induced wild ramification, 669 integral model, 121 Iwahori decomposition, 468 Iwahori subgroup, 142, 158, 256, 329 of torus, 90 Iwahori–Weyl group, 231, 272 k-pinning, 112 lattice chain dual, 536 graded, 519 periodic, 519 self-dual, 536

parahoric subgroup, 158, 256, 329 hyperspecial, 566 special, 566 pinning, 112 principal S-arithmetic subgroup, 563 R1,R2,R3, 96 radius, 5 reduced affine root system, 16 relative identity component, 162, 440 residually quasi-split, 381 residually split, 384 retraction, 63 root datum 0 , 305, 357 GΩ G x , f , 317 G x , f , 378 absolute, 95 relative, 95 root group a-root group, 134 integral, 134 root space integral, 133 S–arithmetic subgroup, 561 S-arithmetic subgroup principal, 563 schematic closure, 593 schematic subgroup, 126 smoothening, 646 special, 29, 379 absolutely special, 278 extra special, 29 hyperspecial, 278, 380 superspecial, 278, 380, 391 special k-apartment, 326 special k-torus, 326 standard model, 654, 658, 659

maximally split, 93, 95 Moy–Prasad filtration Lie algebra, 470 parahoric subgroup, 468 torus, 466 Moy–Prasad filtration subgroup, 146

Tamagawa form, 566 Tits group, 115 Tits measure, 581 type, 48

Néron model connected, 680 ft, 675 lft, 680 open cell integral, 290 open subgroups of G(A S ), 562

valuation homomorphism, 102 valuation of root datum, 198, 364 vector group, 128 vertex Γ-vertex, 326 K-vertex, 326 k-vertex, 326

panel, 57, 62

Weil restriction of scalars, 273, 385, 595

unipotent radical k-split, 136