p-adic Banach Space Representations: With Applications to Principal Series 3031226836, 9783031226830

Table of contents :
Preface
Contents
1 Introduction
1.1 Admissible Banach Space Representations
1.2 Principal Series Representations
1.3 Some Questions and Further Reading
1.4 Prerequisites
1.5 Notation
1.6 Groups
Part I Banach Space Representations of p-adic Lie Groups
2 Iwasawa Algebras
2.1 Projective Limits
2.1.1 Universal Property of Projective Limits
2.1.2 Projective Limit Topology
Cofinal Subsystem
Morphisms of Inverse Systems
2.2 Projective Limits of Topological Groups and oK-Modules
2.2.1 Profinite Groups
Topology on Profinite Groups
2.3 Iwasawa Rings
2.3.1 Linear-Topological oK-Modules
Definition of Iwasawa Algebra
Fundamental System of Neighborhoods of Zero
Embedding oK[G0], G0, and oK into oK[[G0]]
2.3.2 Another Projective Limit Realization of oK[[G0]]
2.3.3 Some Properties of Iwasawa Algebras
Zero Divisors
Augmentation Map
Iwasawa Algebra of a Subgroup
3 Distributions
3.1 Locally Convex Vector Spaces
3.1.1 Banach Spaces
3.1.2 Continuous Linear Operators
3.1.3 Examples of Banach Spaces
Banach Space of Bounded Functions
Continuous Functions on G0
Mahler Expansion
3.1.4 Double Duals of a Banach Space
3.2 Distributions
3.2.1 The Weak Topology on Dc(G0,oK)
3.2.2 Distributions and Iwasawa Rings
3.2.3 The Canonical Pairing
3.3 The Bounded-Weak Topology
3.3.1 The Bounded-Weak Topology is Strictly Finer than the Weak Topology
The Weak Topology on V'
The Bounded-Weak Topology on V'
3.4 Locally Convex Topology on K[[G0]]
3.4.1 The Canonical Pairing
3.4.2 p-adic Haar Measure
3.4.3 The Ring Structure on Dc(G0,K)
A Big Projective Limit
4 Banach Space Representations
4.1 p-adic Lie Groups
4.2 Linear Operators on Banach Spaces
4.2.1 Spherically Complete Spaces
4.2.2 Some Fundamental Theorems in Functional Analysis
4.2.3 Banach Space Representations: Definition and Basic Properties
4.3 Schneider-Teitelbaum Duality
4.3.1 Schikhof's Duality
4.3.2 Duality for Banach Space Representations: Iwasawa Modules
K[[G0]]-module structure on V'
4.4 Admissible Banach Space Representations
4.4.1 Locally Analytic Vectors: Representations in Characteristic p
Locally Analytic Vectors
Unitary Representations and Reduction Modulo pK
4.4.2 Duality for p-adic Lie Groups
Part II Principal Series Representations of Reductive Groups
Notation in Part II
5 Reductive Groups
5.1 Linear Algebraic Groups
5.1.1 Basic Properties of Linear Algebraic Groups
More Examples of Linear Algebraic Groups
Unipotent Subgroups
Identity Component
Tori
5.1.2 Lie Algebra of an Algebraic Group
Lie Algebras
Lie Algebra of an Algebraic Group
5.2 Reductive Groups Over Algebraically Closed Fields
5.2.1 Rational Characters
5.2.2 Roots of a Reductive Group
Weyl Group
Abstract Root Systems
Simple Roots
5.2.3 Classification of Irreducible Root Systems
5.2.4 Classification of Reductive Groups
Cocharacters
Root Datum of a Reductive Group
Abstract Root Datum
5.2.5 Structure of Reductive Groups
Root Subgroups
Borel Subgroups and Parabolic Subgroups
5.3 F-Reductive Groups
5.4 Z-Groups
5.4.1 Algebraic R-Groups
5.4.2 Split Z-Groups
Root Subgroups
5.5 The Structure of G(L)
5.5.1 oL-Points of Algebraic Z-Groups
5.5.2 oL-Points of Split Z-Groups
5.5.3 Coset Representatives for G/P
5.6 General Linear Groups
6 Algebraic and Smooth Representations
6.1 Algebraic Representations
6.1.1 Definition and Basic Properties
6.1.2 Classification of Simple Modules of Reductive Groups
Abstract Weights
Weights of a Reductive Group
Dominant Bases of X(T)
Weights of a Module
Algebraic Induction
Simple Modules
6.2 Smooth Representations
6.2.1 Absolute Value
6.2.2 Smooth Representations and Characters
6.2.3 Basic Properties
Isomorphic Fields
Absolutely Irreducible Representations
Contragredient
Tensor Product of Representations
6.2.4 Admissible-Smooth Representations
6.2.5 Smooth Principal Series
Normalized Induction
Composition Factors of Principal Series
6.2.6 Smooth Principal Series of GL2(L) and SL2(L)
7 Continuous Principal Series
7.1 Continuous Principal Series Are Banach
7.1.1 Direct Sum Decomposition of IndP0G0(χ0-1)
7.1.2 Unitary Principal Series
7.1.3 Algebraic and Smooth Vectors
Algebraic Characters
Smooth Characters
7.1.4 Unitary Principal Series of GL2(Qp)
7.2 Duals of Principal Series
7.2.1 Module M0(χ)
7.3 Projective Limit Realization of M0(χ)
7.4 Direct Sum Decomposition of M(χ)
7.4.1 The Case G0=GL2(Zp)
7.4.2 General Case
8 Intertwining Operators
8.1 Invariant Distributions
8.1.1 Invariant Distributions on Vector Groups
8.1.2 ``Partially Invariant'' Distributions on Unipotent Groups
8.1.3 T0-Equivariant Distributions on Unipotent Groups
8.2 Intertwining Algebra
8.2.1 Ordinary Representations of GL2(Qp)
8.3 Finite Dimensional G0-Invariant Subspaces
8.3.1 Induction from the Trivial Character: Intertwiners
8.4 Reducibility of Principal Series
8.4.1 Locally Analytic Vectors
Reducibility Question for G(Qp)
Reducibility Question for G(L)
8.4.2 A Criterion for Irreducibility
A Nonarchimedean Fields and Spaces
A.1 Ultrametric Spaces
A.2 Nonarchimedean Local Fields
A.2.1 p-Adic Numbers
A.2.2 Finite Extensions of Qp
A.2.3 Algebraic Closure Qp
A.3 Normed Vector Spaces
B Affine and Projective Varieties
B.1 Affine Varieties
B.1.1 Zariski Topology on Affine Space
B.1.2 Morphisms and Products of Affine Varieties
B.2 Projective Varieties
References
Index

Citation preview

Lecture Notes in Mathematics  2325

Dubravka Ban

p-adic Banach Space Representations With Applications to Principal Series

Lecture Notes in Mathematics Volume 2325

Editors-in-Chief Jean-Michel Morel, CMLA, ENS, Cachan, France Bernard Teissier, IMJ-PRG, Paris, France Series Editors Karin Baur, University of Leeds, Leeds, UK Michel Brion, UGA, Grenoble, France Annette Huber, Albert Ludwig University, Freiburg, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK Angela Kunoth, University of Cologne, Cologne, Germany Ariane Mézard, IMJ-PRG, Paris, France Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg Mark Policott, Mathematics Institute, University of Warwick, Coventry, UK Sylvia Serfaty, NYU Courant, New York, NY, USA László Székelyhidi , Institute of Mathematics, Leipzig University, Leipzig, Germany Gabriele Vezzosi, UniFI, Florence, Italy Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany

This series reports on new developments in all areas of mathematics and their applications - quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. The type of material considered for publication includes: 1. Research monographs 2. Lectures on a new field or presentations of a new angle in a classical field 3. Summer schools and intensive courses on topics of current research. Texts which are out of print but still in demand may also be considered if they fall within these categories. The timeliness of a manuscript is sometimes more important than its form, which may be preliminary or tentative. Titles from this series are indexed by Scopus, Web of Science, Mathematical Reviews, and zbMATH.

Dubravka Ban

p-adic Banach Space Representations With Applications to Principal Series

Dubravka Ban School of Mathematical and Statistical Sciences Southern Illinois University Carbondale, IL, USA

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-031-22683-0 ISBN 978-3-031-22684-7 (eBook) https://doi.org/10.1007/978-3-031-22684-7 Mathematics Subject Classification: 22E50, 20G25, 11F70, 11F85, 46S10 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To the memory of my parents, Anka and Nebomir

Preface

This book grew out of a course taught in Spring 2021 at Southern Illinois University. Its purpose is to lay the foundations of the representation theory of p-adic groups on p-adic Banach spaces, explain the duality theory of Schneider and Teitelbaum, and demonstrate its applications to continuous principal series. This monograph is intended to serve both as a reference book and as an introductory text for students entering the area. In addition, it could be of interest to mathematicians who are working in the representation theory on complex vector spaces and would like to learn more about p-adic Banach space representations. The participants in the course were Devjani Basu, Jeremiah Roberts, Layla Sorkatti, Oneal Summers, An Tran, Manisha Varahagiri, and Menake Wijerathne. They prepared and presented lectures based on the first draft of the book. I would like to thank them for their patience in navigating through a half-finished book and for their corrections and comments. Following the suggestions of the three referees, this monograph includes many improvements, broadening the scope of exposition. I would like to thank the referees for their detailed reviews and invaluable comments. Finally, I would like to thank Brian Conrad, Matthias Strauch, and Marie-France Vignéras for their contributions to the final version of the book. Carterville, IL, USA September 2022

Dubravka Ban

vii

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Admissible Banach Space Representations . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Principal Series Representations .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Some Questions and Further Reading . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Notation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Part I

1 2 3 5 6 7 7

Banach Space Representations of p-adic Lie Groups

2 Iwasawa Algebras .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Projective Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.1 Universal Property of Projective Limits . .. . . . . . . . . . . . . . . . . . . . 2.1.2 Projective Limit Topology . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Projective Limits of Topological Groups and oK -Modules . . . . . . . . . . 2.2.1 Profinite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Iwasawa Rings .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Linear-Topological oK -Modules .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 Another Projective Limit Realization of oK [[G0 ]] . . . . . . . . . . 2.3.3 Some Properties of Iwasawa Algebras . . .. . . . . . . . . . . . . . . . . . . .

11 11 13 15 19 21 24 25 30 32

3 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Locally Convex Vector Spaces .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.1 Banach Spaces .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.2 Continuous Linear Operators . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.3 Examples of Banach Spaces . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.4 Double Duals of a Banach Space . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 The Weak Topology on D c (G0 , oK ) . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.2 Distributions and Iwasawa Rings . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.3 The Canonical Pairing .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

35 35 37 37 40 41 43 43 46 50

ix

x

Contents

3.3

The Bounded-Weak Topology . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.1 The Bounded-Weak Topology is Strictly Finer than the Weak Topology . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Locally Convex Topology on K[[G0 ]] . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.1 The Canonical Pairing . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.2 p-adic Haar Measure . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.3 The Ring Structure on D c (G0 , K) . . . . . . .. . . . . . . . . . . . . . . . . . . .

50

4 Banach Space Representations . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 p-adic Lie Groups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Linear Operators on Banach Spaces .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.1 Spherically Complete Spaces . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 Some Fundamental Theorems in Functional Analysis. . . . . . . 4.2.3 Banach Space Representations: Definition and Basic Properties.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Schneider-Teitelbaum Duality . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.1 Schikhof’s Duality.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.2 Duality for Banach Space Representations: Iwasawa Modules.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Admissible Banach Space Representations . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.1 Locally Analytic Vectors: Representations in Characteristic p . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.2 Duality for p-adic Lie Groups.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

63 63 64 64 65

3.4

Part II

53 55 56 57 59

68 73 73 78 81 84 85

Principal Series Representations of Reductive Groups

5 Reductive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Linear Algebraic Groups .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.1 Basic Properties of Linear Algebraic Groups . . . . . . . . . . . . . . . . 5.1.2 Lie Algebra of an Algebraic Group .. . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Reductive Groups Over Algebraically Closed Fields . . . . . . . . . . . . . . . . . 5.2.1 Rational Characters .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.2 Roots of a Reductive Group .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.3 Classification of Irreducible Root Systems .. . . . . . . . . . . . . . . . . . 5.2.4 Classification of Reductive Groups .. . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.5 Structure of Reductive Groups .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 F -Reductive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Z-Groups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.1 Algebraic R-Groups .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.2 Split Z-Groups .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 The Structure of G(L). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5.1 oL -Points of Algebraic Z-Groups .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5.2 oL -Points of Split Z-Groups . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5.3 Coset Representatives for G/P . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.6 General Linear Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

91 91 92 95 96 97 98 103 105 107 109 111 112 113 114 114 115 118 119

Contents

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6 Algebraic and Smooth Representations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Algebraic Representations . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.1 Definition and Basic Properties .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.2 Classification of Simple Modules of Reductive Groups .. . . . 6.2 Smooth Representations.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.1 Absolute Value.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.2 Smooth Representations and Characters .. . . . . . . . . . . . . . . . . . . . 6.2.3 Basic Properties.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.4 Admissible-Smooth Representations . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.5 Smooth Principal Series . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.6 Smooth Principal Series of GL2 (L) and SL2 (L) . . . . . . . . . . . .

123 123 124 124 132 133 134 135 137 138 143

7 Continuous Principal Series . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Continuous Principal Series Are Banach .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . G 7.1.1 Direct Sum Decomposition of IndP00 (χ0−1 ) . . . . . . . . . . . . . . . . . . 7.1.2 Unitary Principal Series . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.3 Algebraic and Smooth Vectors .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.4 Unitary Principal Series of GL2 (Qp ) . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Duals of Principal Series . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . (χ) 7.2.1 Module M0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . (χ) 7.3 Projective Limit Realization of M0 . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Direct Sum Decomposition of M (χ) . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.1 The Case G0 = GL2 (Zp ) . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.2 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

147 148 149 153 154 155 156 156 164 168 168 169

8 Intertwining Operators.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Invariant Distributions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.1 Invariant Distributions on Vector Groups . . . . . . . . . . . . . . . . . . . . 8.1.2 “Partially Invariant” Distributions on Unipotent Groups . . . . 8.1.3 T0 -Equivariant Distributions on Unipotent Groups . . . . . . . . . . 8.2 Intertwining Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.1 Ordinary Representations of GL2 (Qp ). . .. . . . . . . . . . . . . . . . . . . . 8.3 Finite Dimensional G0 -Invariant Subspaces . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.1 Induction from the Trivial Character: Intertwiners . . . . . . . . . . 8.4 Reducibility of Principal Series . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4.1 Locally Analytic Vectors .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4.2 A Criterion for Irreducibility . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

173 174 174 175 177 181 183 184 185 186 187 189

A Nonarchimedean Fields and Spaces . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.1 Ultrametric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2 Nonarchimedean Local Fields . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2.1 p-Adic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2.2 Finite Extensions of Qp . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2.3 Algebraic Closure Qp . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.3 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

193 193 195 195 197 199 199

xii

Contents

B Affine and Projective Varieties . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . B.1 Affine Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . B.1.1 Zariski Topology on Affine Space . . . . . . . .. . . . . . . . . . . . . . . . . . . . B.1.2 Morphisms and Products of Affine Varieties .. . . . . . . . . . . . . . . . B.2 Projective Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

201 201 202 203 204

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 207 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 211

Chapter 1

Introduction

We study the representation theory of p-adic groups on p-adic Banach spaces. This is an active field of research whose foundations were laid by Peter Schneider and Jeremy Teitelbaum in [64]. We start with a sequence of finite field extensions Qp ⊆ L ⊆ K and their rings of integers Zp ⊆ oL ⊆ oK . Our group G is the group of L-points of an algebraic group—a typical example is G = GLn (L). More specifically, G = G(L), where G is a split reductive Z-group. We also consider G0 = G(oL), which is a maximal compact subgroup of G. We study continuous representations of G and G0 on KBanach spaces. There are many types of representations of G. In general, a representation of G on a vector space V is a homomorphism π : G → Aut(V ). What we now call classical representations of p-adic groups are representations on complex vector spaces. The relevant category is the category of admissible-smooth representations, where smooth means locally constant and being admissible-smooth requires an additional finiteness condition (see Definition 6.12). The topology on V does not play a role in the definition and properties of smooth representations. Hence, we can replace complex numbers by other fields. For instance, if  is a prime number, we can consider -modular representations of G, which are smooth representations on vector spaces over the fields of characteristic . We just mention that the case  = p differs significantly from  = p. Enter K. When we look at the representations on K-vector spaces, it becomes immediately clear that there are many more interesting representations beyond smooth. For instance, we have algebraic representations, with the action of G described by polynomial functions. Algebraic representations of G are nice and natural, and most of them do not exist over C because there are no continuous polynomial functions L → C except the constant ones. Similarly, we have locallyanalytic representations of G, which again do not exist over C. The theory of representations of G on K-vector spaces branches into two lines of research: Banach space representations and locally analytic representations. The © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Ban, p-adic Banach Space Representations, Lecture Notes in Mathematics 2325, https://doi.org/10.1007/978-3-031-22684-7_1

1

2

1 Introduction

two lines are interconnected and interdependent, but the foundations are quite separated. In this book, we build the theory for Banach space representations and just occasionally mention locally analytic representations.

1.1 Admissible Banach Space Representations When stepping into a new area, it is a challenge to decide what are the right objects to study. The category of K-Banach space representations carries certain pathologies (see Remark 4.36), and to avoid them, Schneider and Teitelbaum introduce in [64] an additional finiteness condition called admissibility. To explain it, we first introduce the Iwasawa algebra of G0 , defined as the projective limit oK [[G0 ]] = lim oK [G0 /N], ← − N

where N runs over the set of open normal subgroups of G0 . In addition, we define K[[G0 ]] = K ⊗oK oK [[G0 ]] with the locally convex topology as in Definition 3.42. Then K[[G0 ]] can be identified with the convolution algebra D c (G0 , K) of continuous distributions on G0 (Theorem 3.44). Suppose that V is a K-Banach space representation of G, with continuous dual V  . Both V and V  carry the K[[G0]]-module structures induced by the G-actions. The K[[G0 ]]-module structure on V  , together with the Schikhof duality [58], is a basis for the Schneider-Teitelbaum duality (Theorem 4.35). The fundamental result by Lazard that oK [[G0 ]] and K[[G0 ]] are noetherian rings [44] leads to the notion of admissibility. The K-Banach space representation V of G is called admissible if V  is finitely generated as a K[[G0]]-module. We denote by Banadm G (K) the category of all admissible K-Banach space representations of G. This category is algebraic in nature. Namely, V → V 

(1.1)

defines an anti-equivalence between Banadm G0 (K) and the category of finitely generated K[[G0 ]]-Iwasawa modules (Theorem 4.43). Finitely generated modules over a noetherian ring form a category with nice properties. By duality (1.1), these properties then also hold in Banadm G0 (K).

1.2 Principal Series Representations

3

Behind the formulas and theorems, we recognize the beauty in the duality described above, justifying our choice to work with the category Banadm G (K). However, we have more to offer for justification. Most notably, admissible Banach space representations appear in Colmez’ p-adic Langlands correspondence for GL2 (Qp ) [20, 22]. For us, it will be important that Banadm G (K) contains all continuous principal series. Thus, the category Banadm (K) is rich in representations. G adm The size and diversity of the category BanG (K) is discussed in Remark 4.51.

1.2 Principal Series Representations Using the duality (1.1), we can study K-Banach space representations of G by considering the corresponding K[[G0 ]]-modules. We apply this approach in Part II to study principal series representations. Let P be a Borel subgroup of G, having unipotent radical U and split maximal torus T ⊂ P. Denote by U− the opposite subgroup of U. Let P = P(L) and P0 = P(oL ), and U0 = U(oL ). If χ : P → K × is a continuous character, we define −1 IndG P (χ ) = {f : G → K continuous | f (gp) = χ(p)f (g) for all p ∈ P , g ∈ G}

−1 with the action of G by left translations. We prove that IndG P (χ ) has a natural structure as a Banach space representation of G (Proposition 7.5). × In this introduction, we use the same letter χ for χ|P0 : P0 → oK . Similarly G0 −1 as above, we define IndP0 (χ ). This is a Banach space with respect to the sup −1 norm and the restriction map defines a topological isomorphism from IndG P (χ ) to G0 −1 IndP0 (χ ) (Proposition 7.3). −1 The space IndG P (χ ) can be described explicitly using a set of representatives of G/P . Let B be the standard Iwahori subgroup of G0 . We denote by W = W (G, T) the Weyl group of G relative to T. For each w ∈ W we select a representative w˙ ∈ G(Z). Let U − 1 = w˙ −1 B w˙ ∩ U0− . Then we have the disjoint union decomposition w, 2

G=



wU ˙ − 1P

w∈W

w, 2

as in Proposition 5.45, which gives us the direct sum decomposition −1 ∼ IndG P (χ ) =

 w∈W

C(U − 1 , K) w, 2

as in Proposition 7.3. Here, C(U − 1 , K) is the Banach space of continuous functions f :

U− 1 w, 2

w, 2

→ K equipped with the sup norm.

4

1 Introduction

The character χ extends to a character of K[[P0 ]]. We denote by K (χ) the K[[P0 ]]-module structure on K induced by this character. We prove that the dual of −1 IndG P (χ ) is isomorphic to M (χ) = K[[G0 ]] ⊗K[[P0 ]] K (χ) (see Theorem 7.12). As a K[[G0]]-module, M (χ) is generated by a single element −1 1 ⊗ 1, which implies that IndG P (χ ) is admissible (Corollary 7.13). −1 Next, we want to use the duality (1.1) to obtain results about IndG P (χ ). Our (χ) first step is to describe the structure of M . We give a projective limit realization (χ) (χ) of M0 = oK [[G0 ]] ⊗oK [[P0 ]] oK (Proposition 7.20) and prove a K[[B]]-module decomposition M (χ0 ) ∼ =

 w∈W

K[[B]] ⊗K[[P w,± ]] K (wχ) 1 2

where P 1w,± = B ∩ wP0 w−1 (Corollary 7.24). 2

In Chap. 8, we study intertwining operators on principal series representations. For any two continuous characters χ1 and χ2 of P , we want to compute the space G0 −1 −1 0 HG0 (χ1 , χ2 ) = HomcG0 (IndG P0 (χ1 ), IndP0 (χ2 )) G

G

of continuous intertwining operators between IndP00 (χ1−1 ) and IndP00 (χ2−1 ). We first compute the space of K[[G0 ]]-linear maps HomK[[G0 ]] (M (χ1 ) , M (χ2 ) ) (see Corollary 8.13). Then we can use duality to get HG0 (χ1 , χ2 ). It turns out that it is equal to HG (χ1 , χ2 ), and we have  HG (χ1 , χ2 ) = HG0 (χ1 , χ2 ) =

0

if χ1 = χ2 ,

K · id if χ1 = χ2

(1.2)

(see Proposition 8.15). The description of HG (χ1 , χ2 ) in (1.2) may come as a surprise to a mathematician working with smooth representations of G. Still, the length of a smooth principal series of G is at most the order of the Weyl group, and the corresponding space of intertwining operators is always finite-dimensional. A full-blown surprise is the equality HG (χ1 , χ2 ) = HG0 (χ1 , χ2 ) and the description of HG0 (χ1 , χ2 ). Here is why. Consider the case when χ1 = χ2 = χ is a smooth G character. Let V = IndP00 (χ −1 ) and let U be the smooth part of V . Then U is dense in V (see Lemma 7.7). As a G0 -representation, U decomposes as a countable direct sum of finite dimensional representations ρ with finite multiplicities m(ρ): U∼ =

 ρ

m(ρ)ρ.

1.3 Some Questions and Further Reading

5

Each ρ is closed in V , so V contains countably many closed subrepresentations. Clearly, U has numerous self-intertwining operators that are not scalar multiples of the identity. These operators, however, cannot be extended continuously to V . An example is worked out in Sect. 8.3.1.

1.3 Some Questions and Further Reading Equality (1.2) illustrates how the theory of p-adic representations differs from the classical theory of smooth representations. It also shows that some of the standard methods from the classical theory are not suitable in this new context. For instance, intertwining operators are of no use for studying principal series on p-adic Banach spaces. Also, a serious difficulty comes from the lack of a p-adic Haar measure on G (see Sect. 3.4.2), thus making many of the integration-based methods inapplicable. Still, in both theories we can ask similar questions, despite the fact that the answers are often completely different. The fundaments of the theory of smooth representations are well-established and can indicate interesting problems about p-adic Banach space representations. Cartier’s survey [16] is a good place to start reading about admissible smooth representations, together with Casselman’s lectures [17] and the first chapter of [14]. One of the concepts to investigate is parabolic induction. Namely, principal series representations are induced from a Borel subgroup, and we would like to know more about the induction from an arbitrary parabolic subgroup. For p-adic Banach space representations, this problem is mostly unexplored at the time of writing. Even for principal series, there is no comprehensive theory yet. Schneider’s conjecture on irreducibility of continuous principal series and some related results are discussed in Sect. 8.4. A true challenge will be to come up with replacements to integration-based methods. For instance, is there an analogue of the Harish-Chandra characters (distribution characters [36]) for p-adic Banach space representations? An irreducible representation that does not appear as a subquotient of a parabolically induced representation is called supercuspidal. Construction and classification of smooth supercuspidal representations is a deep problem, addressed by the theory of types. We cannot but wonder what will be discovered about supercuspidal representations on p-adic Banach spaces. Answers, of course, are not just around the corner—what is hard in the smooth case can only get harder over Banach spaces. We have a nice picture for unitary representations of GL2 (Qp ), where unitary means norm-preserving (see Sect. 4.4.1 for definitions). An admissible K-Banach space representation of GL2 (Qp ) is called ordinary if it is a subquotient of a continuous principal series induced from a unitary character. The classification of ordinary representation of GL2 (Qp ) is given in Sect. 8.2.1. Non-ordinary representations of GL2 (Qp ) are covered by the p-adic Langlands correspondence, which is a correspondence between Galois representations and Banach space representations (see Remark 4.51 for a precise statement). The idea

6

1 Introduction

of the p-adic Langlands correspondence was introduced by Breuil in [11]. Berger’s expository paper [6] is an excellent introduction to the p-adic Langlands correspondence for GL2 (Qp ). For other groups, see [15] and [12]. The correspondence for GL2 (Qp ) offers, among other things, an insight into the inner structure of the Banach space representations—particularly those containing smooth or locally algebraic vectors [7, 21]—thus indicating possible research directions for other groups. Locally analytic vectors make an important part of the picture. Many of the results on p-adic Banach space representations rely on considerations of locally analytic vectors. An overview of some of the results and connections between the two theories can be found in [61]. For foundations of the theory of locally analytic representation, see [32] and [65]. Locally analytic principal series are well understood, thanks to the works of Orlik and Strauch [50, 51]. Finally, the theory of mod p representations can be useful for studying representations over K. Namely, if V is a unitary K-representation of G, then we can reduce it mod pK (which is briefly explained in Sect. 4.4.1) to obtain a smooth representation V over the residue field κ = oK /pK . As before, we consider two types of representations: components of parabolically induced representations and supercuspidal (also called supersingular) representations. The classification of smooth Fp -representations of G in terms of supercuspidals is given in [1]. As the authors put it, “By contrast, supercuspidal mod p representations remain a complete mystery, apart from the case of GL2 (Qp ) [11] and groups closely related to it.” Still, as the theory of mod p representations develops, it will tell us more about unitary Banach space representations. How the two theories relate is beautifully illustrated by the compatibility of the p-adic Langlands correspondence for GL2 (Qp ) with reduction mod p [5].

1.4 Prerequisites The only prerequisites for Part I are basic topology and some elementary properties of nonarchimedean fields (as summarized in Appendix A). The theory we present is built on projective limits and nonarchimedean functional analysis. Everything we need from these two areas is covered in the book. For projective limits, this is done at the beginning, in Sects. 2.1 and 2.2. For nonarchimedean functional analysis, we take a different approach (so that the introductory sections do not take too long). We cover it gradually, piece by piece, intertwined with the theory of Iwasawa algebras, continuous distributions, and Banach space representations. Sections 3.1, 3.3, and 4.2 are on general functional analysis, while the rest of Chaps. 3 and 4 deals with more specific topics directed towards explaining the admissible Banach space representations and the Schneider-Teitelbaum duality. In Part II, we work with reductive Z-groups. The structure theory of reductive groups is extensive and we give an overview (with no proofs) in Chap. 5. For

1.6 Groups

7

the reader interested only in general linear groups, we summarize in Sect. 5.6 the structural components of GLn needed in Chaps. 7 and 8.

1.5 Notation In this book, p is a fixed prime number, and Qp is the field of p-adic numbers. Throughout the book, K and L are finite extensions of Qp . We denote by oK the ring of integers of K and by pK its unique maximal ideal. We define oL and pL similarly. The absolute value | | = | |K on K is given by −1 |K | = qK , where K is a uniformizer of K and qK is the cardinality of the residue field of K (see Appendix A.2.2). If X is a set, 1X denotes the characteristic function of X. We write N for the set of natural numbers, N = {1, 2, 3, . . . }. A more extensive notation is used in Part II, listed on page 89.

1.6 Groups The groups will evolve throughout the book, because the theory we develop will impose more and more conditions. G0 :

Throughout the book, G0 is a compact group. In addition, it is

• profinite: in Chaps. 2, 3, • a compact p-adic Lie group: in Chap. 4, • the group of oL -points of a split reductive Z-group: in Part II. G:

The group G is • a p-adic Lie group: in Chap. 4, • the group of L-points of a split reductive Z-group: in Part II.

Ultimately, we are interested in applying the theory on G0 = G(oL ) and G = G(L), where G is a split connected reductive Z-group. Then G(oL ) is a profinite group, and also a compact p-adic Lie group. Hence, any statement in the book that involves G0 holds for G0 = G(oL). Even more specifically, it holds for G0 = GLn (oL ). Similarly, G(L) is a p-adic Lie group, and the results of Chap. 4 for G hold for G = G(L) and also for G = GLn (L).

Part I

Banach Space Representations of p-adic Lie Groups

Chapter 2

Iwasawa Algebras

Throughout the book, K and L are finite extensions of Qp . In this chapter, G0 is a profinite group. We define the Iwasawa algebra of G0 (page 26) and study its properties. The Iwasawa algebra is defined using projective limits. Moreover, the group G0 itself is defined as a projective limit. Hence, our first step in understanding Iwasawa algebras is to understand projective limits, first for topological spaces, and then for topological groups and oK -modules.

2.1 Projective Limits In this section, we define projective limits and describe some of their basic properties. Our presentation follows Chapter 1 in [54]. We are in particular interested in the projective limits of compact Hausdorff topological spaces—the kind of projective limit we will encounter in the definition of an Iwasawa algebra. Definition 2.1 A directed partially ordered set or directed poset is a set I with a binary relation ≤ satisfying, for all i, j, k ∈ I , (i) (ii) (iii) (iv)

i ≤ i, if i ≤ j and j ≤ i, then i = j , and if i ≤ j and j ≤ k, then i ≤ k. for any i, j ∈ I there exists some k ∈ I such that i ≤ k and j ≤ k.

Definition 2.2 Let (I, ≤) be a directed poset. An inverse system or projective system of topological spaces over I (Xi , ϕij )I

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Ban, p-adic Banach Space Representations, Lecture Notes in Mathematics 2325, https://doi.org/10.1007/978-3-031-22684-7_2

11

12

2 Iwasawa Algebras

is a family of topological spaces {Xi | i ∈ I } together with continuous maps ϕij : Xi → Xj , for all i ≥ j, called the connecting maps, such that (i) ϕii = idXi , (ii) Compatibility condition: if i ≥ j ≥ k, then ϕj k ◦ ϕij = ϕik , that is, the following diagram commutes

We sometimes suppress ϕij from notation and write simply (Xi )I or (Xi ). Example 2.3 Take the set of natural numbers N with standard ≤. For n ∈ N, we consider the finite group Z/pn Z (with discrete topology). For n ≥ m, let ϕn,m : Z/pn Z → Z/pm Z be the reduction modulo pm . The maps ϕn,m clearly satisfy the compatibility condition, and hence (Z/pn Z, ϕn,m )N is an inverse system. Example 2.4 We equip N with the partial order  defined by mn

⇐⇒

m|n.

For n ∈ N, we consider the finite group Z/nZ (with discrete topology). If m|n, let ϕn,m : Z/nZ → Z/mZ be the reduction modulo m. Similarly as in Example 2.3, ϕn,m satisfy the compatibility condition, and (Z/nZ, ϕn,m )(N,) is an inverse system. Definition 2.5 Let (Xi , ϕij )I be an inverse system of topological spaces and let Y be a topological space. Suppose that for every i ∈ I we have a continuous map ψi : Y → Xi . We say that the maps ψi are compatible if for every i ≥ j ϕij ◦ ψi = ψj ,

2.1 Projective Limits

13

that is, the following diagram commutes

2.1.1 Universal Property of Projective Limits Definition 2.6 Let (Xi , ϕij )I be an inverse system of topological spaces. A topological space X together with compatible continuous maps ϕi : X → Xi is said to be a projective limit or an inverse limit of (Xi , ϕij )I if the following universal property is satisfied: • for any topological space Y equipped with compatible continuous maps ψi : Y → Xi there exists a unique continuous map ψ : Y → X such that ϕi ◦ ψ = ψi ,

for all i ∈ I.

In this case, the following diagram commutes for all i ≥ j

Theorem 2.7 Let (Xi , ϕij )I be an inverse system of topological spaces. Then its projective limit exists and it is unique in the following sense: if (X, ϕi )I and (X , ϕi )I are two projective limits of (Xi , ϕij )I , then there is a unique homeomorphism ϕ : X → X such that ϕi ◦ ϕ = ϕi for all i ∈ I . Proof For proving existence,  we will construct a projective limit (X, ϕi )I as a subspace of the direct product i∈I Xi . Set X = {(xi )i∈I ∈

 i∈I

Xi | ϕij (xi ) = xj if i ≥ j }

(2.1)

14

2 Iwasawa Algebras

and for j ∈ I , define ϕj ((xi )i∈I ) = xj .  We equip i∈I Xi with the product topology and X with the subspace topology. Then  ϕj is continuous, because it is the restriction to X of the canonical projection i∈I Xi → Xj . The requirement ϕij (xi ) = xj from the definition of X assures that the maps ϕj are compatible. Now, assume that  Y is a topological space with compatible maps ψi : Y → Xi . Define ψ : Y → i∈I Xi by ψ(y) = (ψi (y))i∈I . Then ψ is continuous, by general properties of the product topology. The condition ϕij ◦ ψi = ψj assures that the image of ψ is contained in X. Let us denote by the same letter ψ the corresponding map ψ : Y → X which is clearly continuous. The uniqueness of ψ follows from the condition ϕi ◦ ψ = ψi . The proof of uniqueness of a projective limit is standard, and it is left as an exercise.  If (Xi , ϕij )I is an inverse system, we denote its projective limit by lim Xi . ← − i∈I

The maps ϕi : lim Xi → Xi are called projections . They are not necessarily ← −i∈I surjective. However, as we see from the proof of Theorem 2.7, they come from the canonical projections i∈I Xi → Xj . Example 2.8 Let (Z/pn Z, ϕn,m )N be the inverse system defined in Example 2.3, where for n ≥ m the map ϕn,m : Z/pn Z → Z/pm Z is the reduction modulo pm . Then lim Z/pn Z = Zp , ← −

n∈N

the ring of p-adic integers. We can identify Z/pn Z with Zp /pn Zp , and the map ϕn : Zp → Z/pn Z = Zp /pn Zp , for n ∈ N, is the reduction modulo pn .

2.1 Projective Limits

15

Take Y = Z and let ψn : Z → Z/pn Z be the reduction modulo pn . Then the diagram

commutes for all n ≥ m, so the maps ψn are compatible. By the universal property of projective limits, there exists the corresponding map Z → Zp . This map is actually the embedding ι : Z → Zp .

2.1.2 Projective Limit Topology Definition 2.9 Let X be a topological space. A family B of open subsets of X is called a subbase of the topology on X if all finite intersections of elements of B form a base of the topology on X. Lemma 2.10 Let (Xi , ϕij )I be an inverse system of topological spaces, and X = lim Xi , with projection maps ϕi : X → Xi . The sets ← −i∈I ϕj−1 (Uj ), where j ∈ I and Uj is an open subset of Xj , form a subbase of the topology on X. Proof As in the proof  of Theorem 2.7, we realize X as a topological subspace of the direct product i∈I Xi . For j ∈ I , denote by prj the canonical projection prj :



Xi → Xj .

i∈I

If Uj is an open subset of Xj , then pr−1 j (Uj ) = {(xi )i∈I ∈



Xi | xj ∈ Uj } ∼ = Uj ×

i∈I



Xi

i=j

 is an open set in i∈I Xi . The sets pr−1 j (Uj ) form a subbase of the topology on  i∈I Xi . In this context, the projection map ϕj : X → Xj is just the restriction to X of prj . The sets ϕj−1 (Uj ) = pr−1 j (Uj ) ∩ X,

16

2 Iwasawa Algebras

then form a subbase of the topology on X.



Lemma 2.11 Let (Xi , ϕij )I be an inverse  system of Hausdorff topological spaces, and X = lim Xi . Then X is closed in i∈I Xi . ← −i∈I  Proof We  will prove that the complement of X is open in i∈I Xi . Take x = (xi )i∈I in i∈I Xi and assume x ∈ / X. Then there exist j, k ∈ I , j > k, such that ϕj k (xj ) = xk . In Xk , take the disjoint neighborhoods Uk of xk and Vk of ϕj k (xj ). By continuity of ϕj k , there exists an open neighborhood Uj of xj in Xj such that ϕj k (Uj ) ⊂ Vk . Define   Xi | yj ∈ Uj , yk ∈ Uk } ∼ Xi . U = {(yi )i∈I ∈ = Uj × Uk × i∈I

i=j,k

This is an open neighborhood of x in



i∈I

Xi disjoint from X.



Corollary 2.12 If (Xi , ϕij )I is an inverse system of compact Hausdorff topological spaces, then lim Xi is also a compact Hausdorff topological space. ← −i∈I  Proof As a direct product of compact spaces, i∈I Xi is compact. Then lim Xi ← −i∈I is compact because it is a closed subspace of a compact space.  Proposition 2.13 If (Xi , ϕij )I is an inverse system of compact Hausdorff nonempty topological spaces, then lim Xi is nonempty. In particular, the projective limit of ← −i∈I an inverse system of nonempty finite sets is nonempty. 

Proof This is Proposition 1.1.4 in [54].

Cofinal Subsystem Given an inverse system (Xi , ϕij )I , we sometimes want to reduce the number of spaces Xi in a way that we still obtain the same projective limit. Definition 2.14 Let (I, ≤) be a directed poset and J ⊂ I such that (J, ≤) is also a directed poset. We say that J is cofinal in I if for every i ∈ I there exists j ∈ J such that i ≤ j . The following is Lemma 1.1.9 from [54]. Lemma 2.15 Let (Xi , ϕij )I be an inverse system of compact topological spaces and let J be a cofinal subset of I . Then lim Xi ∼ = lim Xj . ← − ← − i∈I

j ∈J

2.1 Projective Limits

17

Morphisms of Inverse Systems Definition 2.16 Let (Xi , ϕij )I and (Yi , ψij )I be two inverse systems of topological spaces indexed by the same directed poset I . A morphism of inverse systems θ : (Xi , ϕij )I → (Yi , ψij )I is a family of continuous maps θi : Xi → Yi which are compatible, meaning that the following diagram commutes for all i ≥ j

Suppose that we have a morphism of inverse systems θ = (θi )I : (Xi , ϕij )I → (Yi , ψij )I . Let X = lim Xi ← − i∈I

and Y = lim Yi . ← − i∈I

By the definition of the projective limit, we have the compatible maps ϕi : X → Xi . Then θi ◦ ϕi : X → Yi are compatible continuous maps. By the universal property of projective limits, we obtain the corresponding map X → Y , denoted by lim θi . ← −i∈I Hence, lim θi : ← − i∈I

lim Xi ← −

→

i∈I

lim Yi . ← − i∈I

Lemma 2.17 Let θ : (Xi , ϕij )I → (Yi , ψij )I be a morphism of inverse systems of topological spaces. If each component θi : Xi → Yi is injective, then lim θi : ← − i∈I

lim Xi ← − i∈I

→

lim Yi ← − i∈I

is also injective. Proof Let X = lim Xi , Y = lim Yi , and θ = lim θi . As in Eq. (2.1), we ← −i∈I −i∈I ← −i∈I  ←  identify X with a subspace of i∈I Xi and Y with a subspace of i∈I Yi . Suppose that x, x are two different points in X. Write x = (xi ) and x = (xi ). There exists j ∈ I such that xj = xj . Then θj (xj ) = θj (xj ), because θj is injective. It follows θ (x) = θ (x ), thus proving injectivity of θ .  Example 2.18 The statement for surjective maps, analogous to Lemma 2.17, does not hold in general. Namely, let Z carry the usual discrete topology. Take the

18

2 Iwasawa Algebras

constant inverse system (Z, id)N and the inverse system (Z/pn Z, ϕnm )N discussed in Example 2.8. Let θn : Z → Z/pn Z be the reduction modulo pn . Then (θn )N is a morphism of inverse systems such that θn is surjective for all n ∈ N. As discussed earlier, lim θi : ← −

Z = lim Z ← −

n∈N

Zp = lim Z/pn Z ← −

→

n∈N

n∈N

is the embedding ι : Z → Zp . The group Zp is compact because it is equal to the projective limit of compact groups (Corollary 2.12). On the other hand, Z is not compact. It follows that ι : Z → Zp is not surjective. We will be working primarily with compact Hausdorff spaces, and for them, the following holds. Lemma 2.19 Let  : (Xi , ϕij )I → (Yi , ψij )I be a morphism of inverse systems of compact Hausdorff topological spaces. If each component θi : Xi → Yi is surjective, then lim θi : ← − i∈I

lim Xi ← −

→

i∈I

lim Yi ← − i∈I

is also surjective. Proof Let X = limi∈I Xi , Y = limi∈I Yi , and θ = limi∈I θi . ← − ← − ← − Take y = (yi ) ∈ Y . For each i ∈ I , let X˜ i = θi−1 (yi ). Observe that ϕij (X˜ i ) ⊆ X˜ j , so (X˜ i )I is an inverse system of topological spaces. Set X˜ = lim X˜ i . ← − i∈I

˜ then θ (x) ˜ Hence, Then X˜ ⊂ X and if x˜ ∈ X, ˜ = y, by construction of X. ˜ to prove surjectivity, it suffices to show that X is nonempty. For this, we first observe that each X˜ i is compact, as a closed subspace of the compact set Xi . Then Proposition 2.13 implies that X˜ = ∅.  Corollary 2.20 Let (Xi , ϕij )I be an inverse system of compact Hausdorff spaces and let Y be a compact Hausdorff space. Suppose that θi : Y → Xi are compatible continuous surjective maps. Then the corresponding map θ : Y → lim Xi is ← −i∈I surjective. Proof Apply Lemma 2.19 on the constant inverse system(Y, idY )I .



2.2 Projective Limits of Topological Groups and oK -Modules

19

Proposition 2.21 Let (Xi , ϕij )I be an inverse system of compact Hausdorff spaces. Let X = lim Xi , with the projections ϕi : X → Xi . If Y is a subspace of X, ← −i∈I then (i) Y is dense in lim ϕi (Y ). ← −i∈I (ii) If Y is a closed subspace of X, then Y = lim ϕi (Y ). ← −i∈I (iii) Suppose that ϕi (Y ) is closed in Xi , for all i. Then Y = lim ϕi (Y ), ← − i∈I

where Y is the closure of Y in X. Proof The statements for empty spaces clearly hold, so we will assume that both X and Y are nonempty. Set Yi = ϕi (Y ). Then Y ⊆ lim Yi . ← −i∈I (i) Take y0 ∈ lim Yi . As in the proof of Theorem 2.7, we realize X as a ← −i∈I   subspace of i∈I Xi . Then y0 = (yi )i∈I ∈ i∈I Yi . Take an arbitrary neigborhood of y0 in lim Yi . It contains an open neighborhood of y0 of the ← −i∈I form ⎛ ⎞   U =⎝ Uj × Yi ⎠ ∩ lim Yi ← − j ∈J

i∈I \J

i∈I

where J is a finite subset of I and Uj is an open neighborhood of yj in Yj . Select an index k ∈ I such that j ≤ k for all j ∈ J and choose y ∈ Y such that ϕk (y) = yk . Then y ∈ U , thus proving density. Assertion (ii) follows immediately from (i).   (iii) If Yi is closed in Xi , for all i, then i∈I Yi is closed in i∈I Xi . It follows that lim Yi is closed in X. By density (i), it follows that Y = lim ϕi (Y ). ← −i∈I ← −i∈I 

2.2 Projective Limits of Topological Groups and oK -Modules An inverse system of topological spaces can carry additional algebraic structures. Recall that a topological group is a group G together with a topology on G such that both the product G × G → G and the inverse map G → G, g → g −1 , are continuous. Here, G × G carries the product topology. Similarly, a topological ring is a ring R together with a topology on R such that both the addition and multiplication are continuous maps R × R → R, where R × R carries the product topology. We will work with the following two: • An inverse system of topological groups is an inverse system of topological spaces (Hi , ϕij )I such that each Hi is a topological group and that the connecting

20

2 Iwasawa Algebras

maps ϕij are group homomorphisms. The projective limit H = lim Hi ← − i∈I

has a natural group structure and, with the projective limit topology, it is a topological group (Exercise 2.22). The projection maps ϕi : H → Hi are group homomorphisms, and they are called projection homomorphisms . • We define similarly an inverse system of topological rings and inverse system of topological oK -modules . Exercise 2.22 Let (Hi , ϕij )I be an inverse system of topological spaces such that each Hi is a topological group and that the connecting maps ϕij are group homomorphisms. Let H = limi∈I Hi be the topological projective limit. Prove that ← − there is a unique group structure on H such that the projection maps ϕi : H → Hi are group homomorphisms for all i ∈ I . Exercise 2.23 Formulate the statement for topological rings and topological oK modules as in Exercise 2.22. Prove it. A sequence 0 → (Ai )I → (Bi )I → (Ci )I → 0 of inverse systems of topological groups (respectively, topological oK -modules) is said to be exact if the corresponding sequence of group homomorphisms (respectively, homomorphisms of topological oK -modules) 0 → Ai → Bi → Ci → 0 is exact for all i ∈ I . Proposition 2.24 Let 0 → (Ai )I → (Bi )I → (Ci )I → 0 be an exact sequence of inverse systems of topological groups (respectively, topological oK -modules). Then 0 → lim Ai → lim Bi → lim Ci ← − ← − ← − i∈I

i∈I

i∈I

is exact. Proof The proof is left as an exercise. Injectivity follows from Lemma 2.17.



Remark 2.25 We can consider the projective limit as a functor between appropriate categories. Proposition 2.24 tells us that lim is left exact on compact topological ← −i∈I groups and compact topological oK -modules. There is a condition, called the Mittag-Leffler condition, which assures that the functor lim is exact (see [43, ← −i∈I Proposition 10.3]). For us, it will be important that exactness holds for compact Hausdorff topological groups and compact Hausdorff topological oK -modules (see Proposition 2.26 below).

2.2 Projective Limits of Topological Groups and oK -Modules

21

Proposition 2.26 Let 0 → (Ai )I → (Bi )I → (Ci )I → 0 be an exact sequence of inverse systems of compact Hausdorff topological groups (respectively, compact Hausdorff topological oK -modules). Then 0 → lim Ai → lim Bi → lim Ci → 0 ← − ← − ← − i∈I

i∈I

i∈I

is exact. Proof Follows from Proposition 2.24 and Lemma 2.19.



We have already learned that the inverse systems built on the projections Z → Z/pn Z show all kind of things that can go wrong in the projective limit, among them the exact sequence in the exercise below. Exercise 2.27 Prove that the exact sequences 0 → pn Z → Z → Z/pn Z → 0, for n ∈ N, are compatible with respect to canonical maps. Hence, they define the exact sequence of inverse systems 0 → (pn Z)N → (Z)N → (Z/pn Z)N → 0,

(2.2)

where (Z)N is the constant inverse system (Z, idZ )N . Show that the corresponding sequence of projective limits is 0 → 0 → Z → Zp . As discussed in Example 2.18, the embedding Z → Zp is not surjective. Hence, applying the projective limit on the short exact sequence (2.2) does not result in a short exact sequence. It follows that the projective limit functor is not right exact.

2.2.1 Profinite Groups Definition 2.28 (i) A profinite group is a topological group that is isomorphic to the projective limit of an inverse system of discrete finite groups. (ii) A pro-p group is a topological group that is isomorphic to the projective limit of an inverse system of discrete finite p-groups. Example 2.29 (a) Consider the p-adic integers Zp as an abelian group. Since Zp = lim Z/pn Z, ← − n∈N

Zp is a profinite and pro-p group.

22

2 Iwasawa Algebras

(b) Similarly, let oL be the ring of integers in the p-adic field L and let pL be the unique maximal ideal in oL . Then [49, Proposition 4.5] oL = lim oL /pnL . ← − n∈N

It follows that oL is a pro-p group (see Exercise 2.31). × (c) Let us denote by oL the group of units in oL . For n ∈ N, define U (n) = 1 + pnL . × This is a subgroup of the multiplicative group oL . Then [49, Proposition 4.5] × × oL = lim oL /U (n) . ← − n∈N

× The group oL is profinite, but not pro-p (see Exercise 2.31).

Exercise 2.30 Using literature in algebraic number theory, find the orders of the following groups: × (a) oL /pnL , (b) oL /U (n) , (c) U (n) /U (n+1) . Exercise 2.31 (a) Let G be a profinite group. Suppose that G contains an element g of finite order such that p does not divide |g|. Prove that G is not a pro-p group. × (b) Prove that oL is not a pro-p group. (This could be done using (a).) Topology on Profinite Groups For our applications, it is convenient to denote the profinite group in the proposition below by G0 , and the groups introduced in assertion (ii) by Gi (see Examples 2.35 and 2.37). To avoid confusion, we assume either 0 ∈ / I or H0 = 1. Proposition 2.32 Let (Hi , ϕij )I be an inverse system of discrete finite groups and let G0 = lim Hi , ← − i∈I

with projection homomorphisms ϕi : G0 → Hi . Then (i) G0 is a compact Hausdorff group. (ii) For any i ∈ I , the group Gi = ker ϕi is a normal compact open subgroup of G0 . (iii) The groups Gi , i ∈ I , form a fundamental system of open neighborhoods of the identity in G0 .

2.2 Projective Limits of Topological Groups and oK -Modules

23

(iv) Let N(G0 ) be the set of all open normal subgroups of G0 . Then G0

∼ =

lim ← −

G0 /N

∼ =

N∈N(G0 )

lim G0 /Gi . ← − i∈I

Proof (i) Follows from Corollary 2.12, because Hi are compact Hausdorff groups. (ii) As the kernel of a continuous homomorphism into a Hausdorff group, Gi is a closed normal subgroup of G0 . Compactness of G0 implies that Gi is compact as well. It is open because it is precisely an open subset of G of the form Gi = ϕi−1 (1) as described in Lemma 2.10. Here, 1 is the trivial subgroup of Hi , and it is an open subgroup in the discrete group Hi . (iii) It follows from Lemma 2.10 that G0 has a fundamental system of open neighborhoods of the identity consisting of the subgroups of the form Gi1 ∩ · · · ∩ Gis where {i1 , . . . , is } is a finite subset of I . Take j ∈ I such that j ≥ ik for all ik ∈ {i1 , . . . , is }. Then Gj is an open subgroup of Gi1 ∩ · · · ∩ Gis . (iv) The projections G0 → G0 /Gi give us, using the universal property of projective limits, a continuous homomorphism θ : G0 → lim G0 /Gi . ← −i∈I Corollary2.20 tells us that θ is surjective. To find its kernel, we realize G0 as a subset of i Hi , as in Eq. (2.1). Then ker θ = {(hi )i∈I ∈ G0 | (hi )i∈I ∈ Gj ∀j ∈ I } = {(hi )i∈I ∈ G0 | hi = 1 ∀i ∈ I } = 1.

∼ lim G0 /Gi . To show that lim G0 /Gi ∼ It follows G0 = = limN∈N(G ) ← −i∈I ← −i∈I ← − 0 G0 /N, we observe that (Gi )I is cofinal in N(G0 ), and apply Lemma 2.15.  Exercise 2.33 Prove that the projective limit topology on oL coincides with the topology induced by the nonarchimedean absolute value on L. Prove that the ideals pnL form a neighborhood basis of 0 in oL . × Exercise 2.34 Prove that the projective limit topology on oL coincides with the topology induced by the nonarchimedean absolute value on L. Prove that the × subgroups U (n) = 1 + pnL form a neighborhood basis of 1 in oL .

Example 2.35 Let G = GLn (L) be the group of n × n matrices with coefficients in L. Let G0 = GLn (oL ) be the subgroup of GLn (L) consisting of all matrices × g = (gij ) such that the coefficients gij ∈ oL and det g ∈ oL . For n ∈ N, define Gn = {g ∈ G0 | g ≡ 1 mod pnL }.

24

2 Iwasawa Algebras

Then Gn is a normal subgroup of G0 , G0 /Gn is finite, and G0 = lim G0 /Gn . ← − n∈N

We consider G0 equipped with the projective limit topology (which is by Exercise 2.36 equal to the standard topology on G0 ). The groups Gn , n ∈ N, are compact and open, and they form a neighborhood basis of 1 in G. Exercise 2.36 Let Mn×n (L) be the space of n × n matrices with coefficients in L. Then Mn×n (L) is an n2 -dimensional L-vector space and it is equipped with the standard norm: if g = (gij ) ∈ Mn×n (L), then ||g|| = max |gij |. i,j

Prove that the topology on G0 induced by this norm is equal to the projective limit topology. The following example is central for the theory developed in Chaps. 7 and 8. Example 2.37 Similarly to Example 2.35 for GLn , we can consider a general split reductive Z-group G. Such groups will be explained in detail in Sect. 5.5. Here, we just mention their connection with projective limits. Let G0 = G(oL ) and let Gn be the kernel of G0 → G(oL /pnL ). Then Gn is a normal subgroup of G0 , G0 /Gn is finite, and G0 = lim G0 /Gn . ← − n∈N

We equip G0 with the projective limit topology. The groups Gn , n ∈ N, are compact and open, and they form a neighborhood basis of 1 in G (see Lemma 5.40). Exercise 2.38 Let G0 be a profinite group, and let N be an open subgroup of G0 . Prove that G0 /N is finite. Exercise 2.39 A nonempty topological space X is said to be totally disconnected if the connected components in X are the one-point sets. Let G0 be a profinite group. Prove that G0 is totally disconnected.

2.3 Iwasawa Rings We are now ready to define Iwasawa rings. An important property of Iwasawa rings is that their topology is algebraic in nature, given by ideals. This falls under the umbrella of linear-topological modules defined below.

2.3 Iwasawa Rings

25

2.3.1 Linear-Topological oK -Modules Definition 2.40 A topological oK -module is called linear-topological if the zero element has a fundamental system of open neighborhoods consisting of oK submodules. Example 2.41 The ring oK is a linear-topological oK -module because the ideals pnK form a fundamental system of open neighborhoods of zero. Lemma 2.42 Let H = {h1 , . . . , hn } be a finite group. The group ring oK [H ] = {a1 h1 + · · · + an hn | ai ∈ oK } has the natural topology as a free oK -module of rank n. With this topology, oK [H ] is a topological ring and a linear-topological oK -module. The ideals pm K [H ], m ∈ N, form a neighborhood basis of zero. n , and the topology on o [H ] is defined as Proof As an oK -module, oK [H ] ∼ = oK K n the product topology on oK . It is then easy to show that oK [H ] is a topological ring and a topological oK -modules. To show that the ideals pm K [H ], m ∈ N, form a neighborhood basis of zero, we m m n first observe that for every m, pm K is open in oK . Then pK × · · · × pK is open in oK m and hence pK [H ] is open in oK [H ]. Let U be an open neighborhood of zero in oK [H ]. Then there exist open neighborhoods of zero U1 , . . . , Un ⊂ oK such that U1 h1 + · · · + Un hn ⊂ U. For i m each i, there exists mi such that pm K ⊂ Ui . Let m = maxi {mi }. Then pK ⊂ Ui , for all i = 1, . . . , n, and m m pm K [H ] = pK h1 + · · · + pK hn ⊂ U1 h1 + · · · + Un hn ⊂ U.

This proves that the ideals pm K [H ] form a neighborhood basis of zero in oK [H ], thus also proving that oK [H ] is a linear-topological oK -module.  Definition of Iwasawa Algebra Let G0 be a profinite group. From Proposition 2.32, it follows that we can write G0 as G0 ∼ = lim G0 /Gn , ← − n∈N

where N is a directed poset and {Gn | n ∈ N} is a family of compact open subgroups of G0 such that Gn ≤ Gm for n ≥ m. For instance, we may take N to be the indexing set for all open normal subgroups of G0 . It is convenient to assume G0 ∈ {Gn | n ∈ N}, with the index 0 ∈ N.

26

2 Iwasawa Algebras

Fix n ∈ N. Then G0 /Gn is finite by Exercise 2.38, and the group ring oK [G0 /Gn ] is a topological ring and a linear-topological oK -module, as described in Lemma 2.42. If n, m ∈ N and n ≥ m, we denote by ϕn,m the natural projection ϕn,m : oK [G0 /Gn ] → oK [G0 /Gm ]. More specifically, if {g1 , . . . , gt } is a set of coset representatives of G0 /Gn , an element of oK [G0 /Gn ] can be written as a1 g1 Gn + · · · + at gt Gn . Then ϕn,m (a1 g1 Gn + · · · + at gt Gn ) = a1 g1 Gm + · · · + at gt Gm . The maps ϕn,m are clearly compatible. Hence, (oK [G0 /Gn ], ϕn,m )N is an inverse system of topological rings and oK -modules. Define oK [[G0 ]] = lim oK [G0 /Gn ]. ← − n∈N

We equip oK [[G0]] with the projective limit topology. Then oK [[G0 ]] is a topological ring and a topological oK -module. Clearly, it is torsion-free as an oK -module. As a projective limit of compact rings, oK [[G0]] is compact. The ring oK [[G0 ]] is called the completed group ring or the Iwasawa algebra of G0 over oK . It was introduced and studied by Lazard in [44].  As in Eq. (2.1), we can realize oK [[G0 ]] as a subset of n∈N oK [G0 /Gn ]. Then we can write μ ∈ oK [[G0 ]] as μ = (μn )n∈N ,

μn ∈ oK [G0 /Gn ].

Let m, n ∈ N such that n > m. We select a set of coset representstives {g1 , . . . , gs } of G0 /Gm , and a set of coset representstives {h1 , . . . , hr } of Gm /Gn . Then {gi hj | 1 ≤ i ≤ s, 1 ≤ j ≤ r} is a set of coset representatives of G0 /Gn . We can write μm =

s 

ai gi Gm

i=1

Then ϕn,m (

r

j =1

r s  

bij gi hj Gn .

i=1 j =1

j =1 bij gi hj Gn ) r 

and μn =

= ai gi Gm , and hence

bij = ai ,

for all i ∈ {1, . . . , s}.

2.3 Iwasawa Rings

27

The map ϕn,m can be represented by the following diagram: μn = · · · + bi1 gi h1 Gn + bi2 gi h2 Gn + · · · bir gi hr Gn + · · ·



↓ μm = · · · +

 r

j =1 bij



gi Gm

+···

Fundamental System of Neighborhoods of Zero Lemma 2.43 For n ∈ N, let ϕn be the projection homomorphism oK [[G0 ]] → oK [G0 /Gn ], and for m ∈ N, let prm denote the projection prm : oK → oK /pm K and also the induced map prm : oK [G0 /Gn ] → oK /pm [G /G ]. 0 n K (i) The two-sided ideals Jm,n (G0 ) = ker(prm ◦ ϕn ), for m ∈ N and n ∈ N, form a fundamental system of open neighborhoods of zero in oK [[G0]]. In particular, oK [[G0 ]] is a linear-topological oK -module. (ii) The ideal m(G0 ) = J1,0 (G0 ) is a maximal ideal in oK [[G0 ]]. Proof From Lemma 2.10, we know that the sets ϕn−1 (Un ), where n ∈ N and Un is an open set in oK [G0 /Gn ], form a subbase of the topology on oK [[G0 ]]. By Lemma 2.42, any neighborhood of zero in oK [G0 /Gn ] contains a neighborhood of the form pm K [G0 /Gn ]. Since ϕn−1 (pm K [G0 /Gn ]) = Jm,n (G0 ), it follows that finite intersections of the ideals Jm,n (G0 ), for m ∈ N and n ∈ N, form a fundamental system of open neighborhoods of zero in oK [[G0 ]]. Now, let J1 = Jm1 ,n1 (G0 ) and J2 = Jm2 ,n2 (G0 ). Set n ≥ {n1 , n2 } and m = max{m1 , m2 }. Then Jm,n (G0 ) ⊂ J1 ∩ J2 .

28

2 Iwasawa Algebras

This proves that the ideals Jm,n (G0 ), for m ∈ N and n ∈ N, actually form a fundamental system of open neighborhoods of zero in oK [[G0 ]]. (ii) Since oK [[G0 ]]/m(G0 ) ∼  = oK /pK , the ideal m(G0 ) is maximal. Embedding oK [G0 ], G0 , and oK into oK [[G0 ]] Lemma 2.44 The canonical projections θn : oK [G0 ] → oK [G0 /Gn ] are compatible and induce in the limit an injective ring homomorphism θ : oK [G0 ] → oK [[G0 ]] The image of θ is dense in oK [[G0]]. We use this homomorphism to identify oK [G0 ] with its image in oK [[G0 ]], endowed with the subspace topology of oK [[G0 ]]. Proof The projections θn are clearly compatible, so by the universal property of projective limits, there exists a continuous ring homomorphism θ : oK [G0 ] → oK [[G0 ]]. To show that θ is injective, take a nonzero element μ ∈ oK [G0 ], μ = a1 g1 + · · · + as gs , where ai ∈ oK \ {0} and gi ∈ G. There exists n ∈ N such that gi Gn ∩ gj Gn = ∅, for all i = j . Then θn (μ) = 0, and θ (μ) = 0. Finally, Proposition 2.21 (iii) tells us that the image of θ is dense in oK [[G0 ]], because the projections θn : oK [G0 ] → oK [G0 /Gn ] are surjective.  Since oK [G0 ] is dense in oK [[G0 ]], for every μ ∈ oK [[G0 ]] and every Jm,n (G0 ) there exists η ∈ oK [G0 ] such that μ − η ∈ Jm,n (G0 ). We can construct such η explicitly. Write μ = (μn )∞ n=1 ∈



oK [G0 /Gn ].

n∈N

We select a set of representatives {g1 , . . . , gs } of G0 /Gn . Then we can write μn = a1 g1 Gn + · · · + as gs Gn , where a1 , . . . , as ∈ oK . Define η = a1 g1 + · · · + as gs . Then μ − η ∈ Jm,n (G0 ). Moreover, μ − η ∈ J,n (G0 ), for all  ∈ N. Hence, we have proved.

2.3 Iwasawa Rings

29

Lemma 2.45 Let n ∈ N and let Jn be the kernel of the canonical projection oK [G0 ] → oK [G0 /Gn ]. Then  (i) Jn = m∈N Jm,n . (ii) For every μ ∈ oK [[G0 ]] there exists ν ∈ oK [G0 ] such that μ − ν ∈ Jn . Exercise 2.46 Given μ ∈ oK [[G0 ]], construct explicitly a sequence η(n) in oK [G0 ] such that limn→∞ η(n) = μ. (Let X be a topological space, and let xn , n ∈ N be a sequence in X. We say that xn converges to x ∈ X if for every neighborhood U of x there exists nU ∈ N such that xn ∈ U for every n ≥ nU .) Since G0 ⊂ oK [G0 ], the embedding oK [G0 ] → oK [[G0 ]] from Lemma 2.44 gives us the inclusion G0 → oK [[G0 ]]. Lemma 2.47 The inclusion G0 → oK [[G0 ]] is a homeomorphism onto its image. Proof The set {1 + Jm,n (G0 ) | m ∈ N, n ∈ N} is a neighborhood basis of 1 in oK [[G0 ]]. Since m ≥ 1, we have G0 ∩ (1 + Jm,n (G0 )) = Gn . The lemma then follows immediately, because {Gn | m ∈ N} is a neighborhood basis of 1 in G0 .  It is interesting to notice that the intersection G0 ∩ (1 + Jm,n (G0 )) = Gn is same for all m ≥ 1. It is customary to denote by 1 both the identity in oK and the identity in G0 , and we will mostly do the same. At few places, however, we will denote the identity in G0 by e. We have the standard embedding oK → oK [G0 ] given by a → a · 1. We write simply a for a · 1. In particular, 1 = 1 · 1 (or 1 = 1 · e), where 1



∈oK [[G0 ]]

=  1 ·  1 ∈oK

∈G0

Also, for g ∈ G0 , we write −g for (−1)g. This notation makes sense because (−1)g is the additive inverse of g in the ring oK [[G0 ]]. However, this should not be confused with matrix operations. Suppose that G0 is a group of matrices, such as GLn (oK ). For a matrix g = (aij ) ∈ G0 , the element −g = (−1)g ∈ oK [[G0 ]] is not equal to the matrix (−aij ). Similarly, for c ∈ oK , the element cg ∈ oK [[G0 ]] is not equal to the matrix (caij ). Notice that G0 ⊂ oK [G0 ]× ⊂ oK [[G0 ]]× where oK [G0 ]× and oK [[G0 ]]× are the groups of units in oK [G0 ] and oK [[G0 ]], respectively.

30

2 Iwasawa Algebras

Exercise 2.48 Prove that 1 + K oK [[G0]] ⊂ oK [[G0 ]]× . Example 2.49 (Iwasawa Rings for G0 = G(oL )) Foreshadowing what will be done in Sect. 5.5, let G be a Z-group and G0 = G(oL). As mentioned in Example 2.37, G0 = limn∈N G0 /Gn , where Gn is the kernel of G0 → G(oL /pnL ). Then ← − oK [[G0 ]] =

lim oK [G0 /Gn ] = lim oK [G0 /Gn ]. ← − ← − n∈N

n∈N∪{0}

2.3.2 Another Projective Limit Realization of oK [[G0 ]] Notice that oK [[G0 ]] = lim oK [G0 /Gn ] = lim ( lim oK /pm K )[G0 /Gn ]. ← − ← − ← − n∈N m∈N

n∈N

How we can combine the two projective limits on the right hand side into a single projective limit is described in the following proposition. Proposition 2.50 Suppose that m : N → N ∪ {0, ∞} is a function satisfying (i) m(n) ≤ m(n ) whenever n < n , and (ii) For any chain n1 < n2 < · · · < ni < · · · in N, we have limi→∞ m(ni ) = ∞. Then m(n) oK [[G0 ]] ∼ = lim oK /pK [G0 /Gn ]. ← − n∈N

Here, p0K = oK and p∞ K = 0. We allow m(n) = 0 because of our applications in Chap. 7 (see m(χ, n) defined on page 157). For the proof, we need the following lemma. m(n)

Lemma 2.51 For n ≥ n , define χn,n : pK

m(n )

[G0 /Gn ] → pK

[G0 /Gn ] by

χn,n (a1 g1 Gn + · · · + as gs Gn ) = a1 g1 Gn + · · · + as gs Gn . m(n)

Then (pK

[G0 /Gn ], χn,n )N is an inverse system of topological oK -modules, and lim pm(n) [G0 /Gn ] = 0. ← − K

n∈N

2.3 Iwasawa Rings

31 m(n )

Proof The maps χn,n , over all n ≥ n , are compatible, so (pK [G0 /Gn ], χn,n )N is an inverse system of topological rings.  Let Y = lim pm(n) [G0 /Gn ] and X = n pm(n) K K [G0 /Gn ]. Then Y ⊂ X. Take n∈N ← − a nonzero μ ∈ X. We claim that μ ∈ / Y . Assume, on the contrary, that μ ∈ Y . Write μ = (μn )n∈N . There exists n0 such that μn0 = 0. It follows that there exists  ∈ N such that μn0 ∈ / pK [G0 /Gn0 ]. By the properties of m(n), there exists n1 ∈ N such that m(n1 ) ≥ . Take n2 ≥ n0 , n1 . Then μn2 ∈ pK [G0 /Gn2 ]. It follows μn0 = χn2 ,n0 (μn2 ) ∈ pK [G0 /Gn0 ], a contradiction. Hence, μ ∈ / Y , thus proving Y = 0.



Proof of Proposition 2.50 For every n ∈ N, we have the following exact sequence m(n) 0 → pm(n) K [G0 /Gn ] → oK [G0 /Gn ] → oK /pK [G0 /Gn ] → 0.

We want to consider the corresponding inverse systems. For n ≥ n , we have the canonical projections ϕn,n : oK [G0 /Gn ] → oK [G0 /Gn ] and m(n)

ψn,n : oK /pK

m(n )

[G0 /Gn ] → oK /pK

[G0 /Gn ].

We also have the homomorphism m(n)

χn,n : pK

m(n )

[G0 /Gn ] → pK

[G0 /Gn ],

as in Lemma 2.51. Since all the maps are natural, it is easy to see that the following diagram is commutative, for all n ≥ n

m(n)

0 → pK

m(n)

[G0 /Gn ] → oK [G0 /Gn ] → oK /pK [G0 /Gn ] → 0 ↓ ↓ ↓

) m(n )



0 → pm(n [G /G ] → o [G /G ] → o /p [G0 /Gn ] → 0 0 K 0 K K n n K

Hence, we have an exact sequence of morphisms of inverse systems m(n)

0 → (pK

m(n)

[G0 /Gn ])N → (oK [G0 /Gn ])N → (oK /pK

[G0 /Gn ])N → 0.

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2 Iwasawa Algebras

In the projective limit, using Proposition 2.26 and Lemma 2.51, we get the exact sequence of topological rings m(n)

0 → 0 → oK [[G0 ]] → lim oK /pK ← −

[G0 /Gn ] → 0.

n∈N

m(n)

It follows oK [[G0 ]] ∼ = limn∈N oK /pK [G0 /Gn ]. ← − Corollary 2.52 If G0 ∼ = limn∈N G0 /Gn , then ← −



oK [[G0 ]] ∼ = lim oK /pnK [G0 /Gn ]. ← − n∈N

Proof Define m : N → N by m(n) = n and apply Proposition 2.50.



2.3.3 Some Properties of Iwasawa Algebras We conclude Chap. 2 with remarks on zero divisors, the augmentation map, and the Iwasawa algebra of a subgroup. Some additional important properties are listed in Proposition 5.42.

Zero Divisors Suppose g ∈ G0 is a torsion element, so g n = 1 for some n ∈ N. Then in oK [G0 ] ⊂ oK [[G0 ]] (1 − g)(1 + g + g 2 + · · · + g n−1 ) = 0, so 1 − g is a zero divisor.

  −1 0 . 0 −1 Notice that f = −e in oK [[G0 ]] (see the notation introduced on page 29). Then 1 + f = e + f = 0 ∈ oK [[G0 ]]. We have Example 2.53 Let e be the identity element in G0 = GL2 (oL ) and f =

(1 + f )(1 − f ) = 1 − f 2 = e − e = 0, so 1 + f and 1 − f are zero divisors in oK [[G0 ]]. For description of torsion elements in reductive groups, see [73]. If G0 is a compact Lie group, then Zp [[G0 ]] has no zero divisors if and only if G0 is torsion free [2, Theorem 4.3].

2.3 Iwasawa Rings

33

Augmentation Map For the projective limit of topological rings, the projection maps are continuous ring homomorphisms. This was supposedly stated and proved in the solution to Exercise 2.23. Then for oK [[G0 ]] = lim oK [G0 /Gn ] = ← − n∈N

lim ← −

oK [G0 /N],

N∈N(G0 )

the projection maps ϕn : oK [[G0 ]] → oK [G0 /Gn ] are continuous ring homomorphisms. In particular, for N = G0 , we obtain the augmentation map aug : oK [[G0 ]] → oK . The augmentation map can be computed easily for μ ∈ oK [G0 ]: if μ = a1 g1 + · · · + as gs , then aug(μ) = aug(a1 g1 + · · · + as gs ) = a1 + · · · + as . If m(G0 ) = J1,0 (G0 ) as in Lemma 2.43, then m(G0 ) = {μ ∈ oK [[G0 ]] | aug(μ) ∈ pK }. We know that m(G0 ) is a maximal ideal in oK [[G0 ]].

Iwasawa Algebra of a Subgroup Proposition 2.54 Let G0 be a profinite group and H a compact open subgroup of G0 . (i) The index of H in G0 is finite. (ii) oK [[H ]] is a closed subalgebra of oK [[G0 ]]. (iii) Let {g1 , . . . , gs } be a set of coset representatives of G0 /H. Then oK [[G0 ]] = g1 oK [[H ]] ⊕ · · · ⊕ gs oK [[H ]]. (iv) Let {g1 , . . . , gs } be a set of coset representatives of H \ G0 . Then oK [[G0 ]] = oK [[H ]]g1 ⊕ · · · ⊕ oK [[H ]]gs . Proof (i) The set of cosets gH is an open cover of G0 . By compactness, it has a finite subcover. Since the cosets are disjoint, it follows that H has a finite number of cosets in G, that is, |G0 : H | is finite.

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2 Iwasawa Algebras

(ii) Denote by N(H, G0 ) the set of open normal subgroups of G0 contained in H . From Proposition 2.32 and Lemma 2.15, we have oK [[H ]] = lim o [H /N] and ← −N∈N(H,G0 ) K oK [[G0 ]] =

lim ← −

N∈N(G0 )

oK [Go /N] =

lim ← −

oK [Go /N].

N∈N(H,G0 )

Then the compatible embeddings oK [H /N] → oK [G0 /N], for N ∈ N(H, G0 ), induce in the projective limit the continuous embedding oK [[H ]] → oK [[G0 ]]. Since oK [[H ]] is compact, it is closed in oK [[G0]]. (iii) The decompositions oK [Go /N] = g1 oK [H /N] ⊕ · · · ⊕ gs oK [H /N], for N ∈ N(H, G0 ), are compatible with respect to the natural projections. Passing to the projective limit gives us the statement. 

Chapter 3

Distributions

In this chapter, K is a finite extension of Qp and G0 is a profinite group G0 ∼ = lim G0 /Gn , ← − n∈N

where N is a directed poset and {Gn | n ∈ N} is a family of open normal subgroups of G0 such that Gn ≤ Gm for n ≥ m. The Iwasawa algebra oK [[G0 ]] can be identified with the unit ball in the space of continuous distributions on G0 . Before explaining this identification, we need some background material on locally convex K-vector spaces.

3.1 Locally Convex Vector Spaces In this section, we review basic definitions and properties of locally convex vector spaces. A detailed development of the theory of locally convex vector spaces can be found in Schneider’s book [60]. Definition 3.1 Let V be a K-vector space. (i) A subset A ⊂ V is called convex if either A is empty or is of the form A = v + A0 for some vector v and some oK -submodule A0 ⊂ V . (ii) A lattice in V is an oK -submodule L which satisfies the condition that for any v ∈ V there is a nonzero scalar a ∈ K × such that av ∈ L.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Ban, p-adic Banach Space Representations, Lecture Notes in Mathematics 2325, https://doi.org/10.1007/978-3-031-22684-7_3

35

36

3 Distributions

Let (V ,  ) be a normed K-vector space. For v ∈ V and a real number  > 0,1 we define the closed ball (or simply a ball) of radius  centered at v B (v) = {w ∈ V | w − v ≤ } and the open ball B− (v) = {w ∈ V | w − v < }. It is a property of ultrametric spaces that every closed ball B (v) is both open and closed (see Proposition A.3 in Appendix A). Similarly, every open ball B− (v) is both open and closed. Exercise 3.2 (i) Prove that B (0) and B− (0) are open lattices in V . (ii) Prove that B (v) and B− (v) are convex subsets. Definition 3.3 Let V be a K-vector space. Let (Lj )j ∈J be a nonempty family of lattices in V such that (lc1) (lc2)

for any j ∈ J and a ∈ K × , there exists a k ∈ J such that Lk ⊆ aLj , and for any two i, j ∈ J there exists a k ∈ J such that Lk ⊆ Li ∩ Lj .

Then the convex subsets of the form v + Lj for v ∈ V and j ∈ J form a basis of the topology on V called the locally convex topology on V defined by the family (Lj )j ∈J . Definition 3.4 A locally convex K-vector space is a K-vector space equipped with a locally convex topology. Exercise 3.5 Let V be a locally convex K-vector space. Prove that the addition V × V → V and scalar multiplication K × V → V are continuous maps. If (V ,  ) is a normed K-vector space, then the balls centered at 0 satisfy conditions (lc1) and (lc2). It follows that the convex subsets of the form v + B (0), for v ∈ V and  ∈ R+ , form a basis of the locally convex topology on V . Since v + B (0) = B (v), this topology is equal to the topology on V induced by the norm  . We will always consider the normed space V equipped with the locally convex topology described above. Hence, any normed vector space is locally convex. In general, if V is a locally convex K-vector space, then the topology on V can be defined by a family of seminorms [60, Proposition 4.4]. Definition 3.6 Let V be a locally convex K-vector space. A subset B ⊂ V is said to be bounded if for any open lattice L ⊆ V there is an a ∈ K such that B ⊆ aL.

1

Yes, real.

3.1 Locally Convex Vector Spaces

37

For normed vector spaces, this definition coincides with the classical definition of boundedness, as we can see from the following exercise. Exercise 3.7 Let (V ,  ) be a normed K-vector space. A subset B ⊂ V is bounded if and only if the set B = {x | x ∈ B} is bounded in R.

3.1.1 Banach Spaces The concept of completeness of a locally convex K-vector space V can be defined using Cauchy nets, as in [60, §7]. If V is metrizable, this is equivalent to the standard definition: V is complete if and only if every Cauchy sequence in V is convergent [60, Remark 7.2]. A Banach space is usually defined as a complete normed vector space. However, following the approach from [60], we do not consider the norm to be part of the structure. Definition 3.8 A K-Banach space is a complete locally convex vector space whose topology can be defined by a norm.

3.1.2 Continuous Linear Operators Suppose that V and W are locally convex K-vector spaces. We denote by L(V , W ) the space of all continuous linear maps f : V → W . In particular, if W = K, we denote the space L(V , W ) by V and call it the dual space or continuous dual of V. If V and W are normed K-vector spaces, for any K-linear map f ∈ HomK (V , W ), we define the operator norm f  = inf{c ∈ [0, ∞) | f (v) ≤ cv for all v ∈ V },

(3.1)

where the infimum of the empty set is ∞. If V = 0, then f  = sup {

f (v) | v ∈ V , v = 0}. v

Exercise 3.9 Suppose that V and W are normed K-vector spaces, and f ∈ HomK (V , W ). Prove that f is continuous if and only if f  < ∞.

38

3 Distributions

Notice that for every f ∈ L(V , W ) and v ∈ V , we have f (v) ≤ f  v. Exercise 3.10 Suppose that V and W are normed K-vector spaces. (a) Prove that Eq. (3.1) defines a nonarchimedean norm on L(V , W ). (b) If W is a Banach space, prove that L(V , W ) is also Banach. (See [60], Prop. 3.3.) Remark 3.11 The formula f 0 = sup {f (v) | v ∈ V , v ≤ 1} defines a norm on L(V , W ) which is equivalent to the operator norm. Contrary to the real or complex case, f 0 and f  may not be equal. For instance, take W = Q3 with the 3-adic absolute value as the norm and also V = Q3 , but with v = 2|v|3 . Let f = idQ3 : V → W . Then f  = 1/2, but f 0 = 1/3. Set V  = {v | v ∈ V } and |K| = {|a| | a ∈ K}. Lemma 3.12 If V and W are normed K-vector spaces such that V  = |K|, then the operator norm on L(V , W ) can be computed by the formula f  = sup {f (v) | v ∈ V , v ≤ 1} = sup {f (v) | v ∈ V , v = 1}. Proof Since V  = |K|, for every nonzero v ∈ V , there exists a ∈ K such that av = 1.   From Exercise 3.10 we see that, for normed vector spaces V and W , L(V , W ) carries the locally convex topology induced by the operator norm. As we will see below, this is just one among many. We review here a general method for defining different locally convex topologies on L(V , W ). Details can be found in [60, §6]. Suppose that V and W are locally convex K-vector spaces. For a bounded subset B ⊂ V and an open lattice M ⊂ W , we define L(B, M) = {f ∈ L(V , W ) | f (B) ⊆ M}. This is a lattice in L(V , W ). If W is a normed space, we write L(B, ) for L(B, B− (0)), i.e., L(B, ) = {f ∈ L(V , W ) | f (v) <  for all v ∈ B}. Let now B be a fixed family of bounded subsets of V which is closed under finite unions. Then {L(B, M) | B ∈ B, M an open lattice in W }

3.1 Locally Convex Vector Spaces

39

is a family of lattices satisfying conditions (lc1) and (lc2) from Definition 3.3 (Exercise 3.13). It therefore defines a locally convex topology on L(V , W ), called the B-topology on L(V , W ). We denote by LB (V , W ) the space L(V , W ) equipped with this topology. Exercise 3.13 With notation as above, (a) Prove that L(B, M) is a lattice in L(V , W ). (b) Prove that {L(B, M) | B ∈ B, M an open lattice in W } is a family of lattices satisfying conditions (lc1) and (lc2) from Definition 3.3 Exercise 3.14 Prove that for a bounded subset B ⊂ V , an open lattice M ⊂ W , and a nonzero a ∈ K, L(aB, M) = L(B, a −1 M). Important examples of B-topologies on L(V , W ) are the weak topology and the strong topology defined below. Example 3.15 Let B be the family of all finite subsets of V . We denote LB (V , W ) by Ls (V , W ). We call the corresponding B-topology the weak topology or the topology of pointwise convergence. In particular, we write Vs for the dual space V equipped with the weak topology. In literature, this topology on V is also called the weak* topology (pronounced “weak star topology”), with “star” referring to the dual space, sometimes denoted by V ∗ . Exercise 3.16 For v ∈ V , we define the evaluation map evv : L(V , W ) → W by evv : f → f (v),

f ∈ L(V , W ).

Prove that the weak topology is the weakest topology on L(V , W ) making all evaluations evv : L(V , W ) → W continuous. Example 3.17 Let B be the family of all bounded subsets of V . We write Lb (V , W ) for LB (V , W ). The corresponding B-topology is called the strong topology or the topology of bounded convergence. In particular, we write Vb for the dual space V equipped with the strong topology.

40

3 Distributions

Exercise 3.18 Suppose that V and W are normed vector spaces. Prove that the strong topology is same as the topology on L(V , W ) induced by the operator norm.

3.1.3 Examples of Banach Spaces For the basic example of K n , see Example A.10 in Appendix A.

Banach Space of Bounded Functions Let X be a nonempty set. Define ∞ (X) = {f : X → K | f bounded}. Then ∞ (X) is a K-vector space with pointwise addition and scalar multiplication. The formula f ∞ = sup |f (x)| x∈X

defines a norm on ∞ (X) which gives it the structure of a Banach space. Let c0 (X) be the subspace of ∞ (X) consisting of all f ∈ ∞ (X) such that for any  > 0 there exists only finitely many elements x ∈ X such that |f (x)| ≥ . Then c0 (X) is a closed subspace of ∞ (X) and hence c0 (X) is also a Banach space. From [60, §3], we have Lemma 3.19 Let X be a nonempty set. Then the dual of c0 (X) is isomorphic to ∞ (X). We write ∞ and c0 for ∞ (N) and c0 (N), respectively. Then ∞ is the space of bounded sequences and c0 = {(a1 , a2 , . . . ) ∈ ∞ | lim an = 0} n→∞

is the space of null sequences. Continuous Functions on G0 Let C(G0 , K) be the space of continuous K-valued functions on G0 . We equip C(G0 , K) with the Banach space topology induced by the sup norm f  = sup |f (g)|, g∈G0

3.1 Locally Convex Vector Spaces

41

which is by compactness of G0 equal to maxg∈G0 |f (g)|. Exercise 3.20 Denote by C ∞ (G0 , K) the subspace of C(G0 , K) consisting of smooth (i.e., locally constant) functions. Prove that C ∞ (G0 , K) is dense in C(G0 , K).

Mahler Expansion Let G0 = Zp and consider C(Zp , K). For x ∈ Zp and n ≥ 0, define   x x(x − 1) · · · (x − (n − 1)) . = n! n Theorem 3.21 (Mahler Expansion) (i) Let f ∈ C(Zp , K). Then there exists a unique null sequence (a1 , a2 , . . . ) in K such that f (x) =

∞  n=1

  x an , n

x ∈ Zp .

The series converges uniformly and f  = maxn |an |.  x  ∞ (ii) If (a1 , a2 , . . . ) is a null sequence in K, then x → n=1 an n defines a continuous function Zp → K.  

Proof Schikhof [59], Theorem 51.1.

Corollary 3.22 The dual of C(Zp , K) is isomorphic to the space of bounded sequences. Proof By Mahler expansion, C(Zp , K) C(Zp , K) ∼ = ∞ .

∼ =

c0 , and Lemma 3.19 implies  

3.1.4 Double Duals of a Banach Space In this section, we discuss the double duals of V . We assume that V is a K-Banach space—for simplicity of exposition, and also because our applications of the double duals will be in the duality theory for Banach spaces (Sect. 4.3). The double duals of V depend on the topology we put on V . Suppose that the topology on VB is finer than the topology on Vs . It is also coarser than the topology

42

3 Distributions

on Vb . Hence, for f ∈ HomK (V , K), we have f continuous on Vs

⇒

f continuous on VB ,

f continuous on VB

⇒

f continuous on Vb .

It follows (Vs ) ⊆ (VB ) ⊆ (Vb ) .

(3.2)

For v ∈ V , we have the evaluation map evv : V → K given by evv () = (v), for all  ∈ V . By Exercise 3.16, evv : Vs → K is continuous, so evv ∈ (Vs ) . Then Eq. (3.2) tells us that evv belongs to (VB ) . The linear map ε : V → (VB ) given by ε(v) = evv is called a duality map. We would like to select B and equip (VB ) with a locally convex topology so that ε : V → (VB ) is a topological isomorphism. The two obvious candidates, (Vs ) s and (Vb ) b , usually do not give topological isomorphisms. Definition 3.23 A K-Banach space V is called reflexive if the duality map ε : V → (Vb ) b is a topological isomorphism. Proposition 3.24 Let V be a K-Banach space. (i) The duality map ε : V → (Vs ) s given by ε(v) = evv is a continuous bijection. (ii) The duality map ε : V → (Vb ) b given by ε(v) = evv induces a topological isomorphism between V and im ε. (iii) V is reflexive if and only if it is finite dimensional. Proof All statements can be found in [60], where they are proved in a more general context. (i) follows from Proposition 9.7, (ii) from Lemma 9.9, and (iii) from Proposition 11.1.   Since ε : V → (Vs ) s is a continuous bijection, we can refine the topology on so that we obtain a homeomorphism ε : V → (Vs ) . It turns out that the strong topology on (Vs ) works well.

(Vs ) s

3.2 Distributions

43

Proposition 3.25 The duality map ε : V → (Vs ) b given by ε(v) = evv is a topological isomorphism. Proof A locally convex vector space V is called barrelled if every closed lattice in V is open. Since our space V is Banach, it is barrelled (see Example 2 on page 35 in [60]). Then the proposition follows from Corollary 13.8 in [60].  

3.2 Distributions As introduced earlier, C(G0 , K) is the Banach space of continuous K-valued functions on G0 . Let D c (G0 , K) be the continuous dual of C(G0 , K). We have the canonical pairing  ,  : D c (G0 , K) × C(G0 , K) → K given by µ, h = µ(h). From Exercise 3.10, we know that D c (G0 , K) is a Banach space with respect to the operator norm. This topology is equal to the strong topology (Exercise 3.18). Another important locally convex topology on D c (G0 , K) is of course the weak topology. In the theory of p-adic Banach space representations, another locally convex topology plays a prominent role. It is called the bounded weak topology and it is defined in Sect. 3.3. It coincides with the weak topology on the bounded sets. The weak topology on the unit ball in D c (G0 , K) is studied below.

3.2.1 The Weak Topology on D c (G0 , oK ) In this section, we restrict our attention to distributions which are oK -valued on the oK -valued functions. More specifically, with C(G0 , oK ) denoting the oK -module of all continuous functions f : G0 → oK , define D c (G0 , oK ) = {µ ∈ D c (G0 , K) | µ, f  ∈ oK for all f ∈ C(G0 , oK )}. Notice that D c (G0 , oK ) is the unit ball in D c (G0 , K), with respect to the operator norm. We equip D c (G0 , oK ) with the subspace topology coming from the weak topology on D c (G0 , K). We call this topology the weak topology on D c (G0 , oK ). To describe a fundamental system of open neighborhoods of zero in D c (G0 , oK ), we first recall that the ideals pm K , m ∈ N, form such a system in K. For a finite subset

44

3 Distributions

c F ⊂ C(G0 , K) and m ∈ N, we write L0 (F, m) for L(F, pm K )∩D (G0 , oK ). Hence,

L0 (F, m) = {µ ∈ D c (G0 , oK ) | µ, f  ∈ pm K for all f ∈ F }. The oK -modules L0 (F, m) form a fundamental system of open neighborhoods of zero in D c (G0 , oK ). Lemma 3.26 The weak topology on D c (G0 , oK ) has a fundamental system of open neighborhoods of zero consisting of the lattices L0 ({1U1 , . . . , 1Us }, m), where U1 , . . . , Us are disjoint compact open subset of G0 and m ∈ N. Proof Take an open lattice L0 ({f }, m), for some f ∈ C(G0 , oK ) and m ∈ N. By continuity of f , for every g ∈ G0 there exists a compact open neighborhood Ug such that f (u) − f (g) ∈ pm K,

∀u ∈ Ug .

By compactness of G0 , there is a finite subset {g1 , . . . , gs } of G0 such that the sets Ugi , for i = 1, . . . , s, cover G0 . We can further reduce this to a disjoint cover by compact open subsets Ui ⊂ Ugi . Set ai = f (gi ) ∈ oK . We claim that L0 ({a1 1U1 , . . . , as 1Us }, m) ⊆ L0 ({f }, m). To prove the claim, take µ ∈ L0 ({a1 1U1 , . . . , as 1Us }, m). We can write f = f1 + · · · + fs , where fi = f |Ui . Then µ, f  = µ, f1  + · · · + µ, fs . Fix i ∈ {1, . . . , s}. By continuity of µ, there exists δ > 0 such that fi − h < δ

⇒

µ, fi − h ∈ pm K,

for any h ∈ C(G0 , oK ). Repeating a similar process as above, we can find a disjoint cover of Ui by compact open subsets Uij and points gij ∈ Uij such that the function hi = bi1 1Ui1 + · · · + bir 1Uir satisfies bij = f (gij ) and fi − hi  < δ. Since the sets Uij are disjoint, we have 1Ui = 1Ui1 + · · · + 1Uir . Note that gij ∈ Uij ⊂ Ui and hence bij − ai = f (gij ) − f (gi ) ∈ pm K.

3.2 Distributions

45

Then µ, hi − ai 1Ui  = µ, (bi1 1Ui1 + · · · + bir 1Uir ) − ai (1Ui1 + · · · + 1Uir ) =

r  (bij − ai )µ, 1Uij  ∈ pm K, j =1

(3.3) because µ ∈ D c (G0 , oK ), so µ, 1Uij  ∈ oK . It follows µ, f  = µ, f −



a i 1 Ui  +

i

= µ, f − =

 i



hi +



i

µ, fi − hi  +

i



hi −

 i

µ, ai 1Ui 

i



a i 1 Ui  +

i

µ, hi − ai 1Ui  +



µ, ai 1Ui 

i



µ, ai 1Ui .

i

  Then i µ, fi − hi  ∈ pm i µ, hi − K because fi − hi  < δ, Eq. (3.3) gives m a 1  ∈ p , and the assumption µ ∈ L ({a 1 , . . . , a 1 }, m) implies that i U 0 1 U s U s i 1 K  m m i µ, ai 1Ui  ∈ pK . Hence, µ, f  ∈ pK , proving the claim. Since ai ∈ oK , for all i, we have L0 ({1U1 , . . . , 1Us }, m) ⊆ L0 ({a11U1 , . . . , as 1Us }, m) ⊆ L0 ({f }, m). Next, suppose that we have another element f ∈ C(G0 , oK ). Then there exist disjoint compact open subsets U1 , . . . , Ur of G0 such that L0 ({1U , . . . , 1Ur }, m) ⊆ 1 L0 ({f }, m). Define Vij = Ui ∩ Uj . Then L0 ({1Vij | i = 1, . . . , s, j = 1, . . . , r}, m) ⊂ L0 ({f }, m) ∩ L0 ({f }, m). It follows that for any finite set F ⊂ C(G0 , oK ) and any m ∈ N there exist disjoint compact open subsets U1 , . . . , Ur of G0 such that L0 ({1U1 , . . . , 1Us }, m) ⊆ L0 (F, m). The final result is obtained by scaling. Namely, if F is a finite subset of C(G0 , K), there exists a nonzero a ∈ oK such that aF ⊂ C(G0 , oK ). For m ∈ N, take  ∈ N such that pK ⊂ apm K . Then there exist disjoint compact open subsets U1 , . . . , Ur of G0 such that L0 ({1U1 , . . . , 1Us }, ) ⊆ L0 (aF, ) ⊆ L0 (F, m), finishing the proof.

 

Corollary 3.27 Write G0 ∼ = limn∈N G0 /Gn , as on page 35. The weak topology on ← − D c (G0 , oK ) has a fundamental system of open neighborhoods of zero consisting of

46

3 Distributions

the lattices L0 ({1gGn | gGn ∈ G0 /Gn }, m), where m ∈ N, n ∈ N. Proof Take a lattice L0 ({1U1 , . . . , 1Us }, m), where U1 , . . . , Us are disjoint compact open subset of G0 and m ∈ N. We can decompose each Ui as a disjoint union of cosets of Gn , for some n ∈ N (Exercise 3.28). Then L0 ({1gGn | gGn ∈ G0 /Gn }, m) ⊆ L0 ({1U1 , . . . , 1Us }, m) and the statement follows from Lemma 3.26.

 

Exercise 3.28 Suppose that U1 , . . . , Us are disjoint compact open subset of G0 . Prove that there exists n ∈ N such that each Ui can be written as a disjoint union of cosets of Gn .

3.2.2 Distributions and Iwasawa Rings For g ∈ G0 , we denote by δg the corresponding Dirac distribution. This is the distribution in D c (G0 , oK ) ⊂ D c (G0 , K) defined by δg (f ) = f (g), for all f ∈ C(G0 , K). Let D Dir (G0 , oK ) denote the oK -linear span in D c (G0 , oK ) of all Dirac distributions. This is an oK -submodule of D c (G0 , oK ). Any element of D Dir (G0 , oK ) is a finite oK -linear combination of Dirac distributions. Lemma 3.29 Let n ∈ N. Define Dn = {µ ∈ D c (G0 , oK ) | µ, 1gGn  = 0 for all gGn ∈ G0 /Gn }. Then

 (i) Dn = m∈N L0 ({1gGn | gGn ∈ G0 /Gn }, m). (ii) For every µ ∈ D c (G0 , oK ) there exists ν ∈ D Dir (G0 , oK ) such that µ − ν ∈ Dn . (iii) If  ≥ n, then D ⊆ Dn . Proof Assertions (i) and (iii) are obvious. For (ii), take µ ∈ D c (G0 , oK ). Let {g1 , . . . , gr } be a set of coset representatives of G0 /Gn . Define ν = µ, 1g1 Gn δg1 + · · · + µ, 1gr Gn δgr .

3.2 Distributions

47

This is an element of D Dir (G0 , oK ). For every j ∈ {1, . . . , r}, we have ν, 1gj Gn  = µ, 1g1 Gn δg1 (1gj Gn ) + · · · + µ, 1gr Gn δgr (1gj Gn ) = µ, 1gj Gn . It follows µ − ν ∈ Dn .

 

Corollary 3.30 D Dir (G0 , oK ) is dense in D c (G0 , oK ), where D c (G0 , oK ) is equipped with the weak topology. Proof Take an arbitrary neighborhood U of µ. By Corollary 3.27, U contains µ + L0 ({1gGn | gGn ∈ G0 /Gn }, m), for some m ∈ N, n ∈ N. By Lemma 3.29, there exists ν ∈ D Dir (G0 , oK ) such that µ − ν ∈ Dn ⊂ L0 ({1gGn | gGn ∈ G0 /Gn }, m).   Proposition 3.31 We consider oK [G0 ] equipped with the subspace topology coming from oK [[G0 ]] and D Dir (G0 , oK ) equipped with the subspace topology coming from the weak topology on D c (G0 , oK ). The oK -linear map oK [G0 ] → D Dir (G0 , oK ) given by g → δg ,

g ∈ G0 ,

is a topological isomorphism. Proof Define ϕ : oK [G0 ] → D Dir (G0 , oK ) by ϕ(a1 g1 + · · · + as gs ) = a1 δg1 + · · · + as δgs . This is an oK -linear map and clearly a bijection. We denote the inverse of ϕ by ψ. Then ψ(δg ) = g. To show that ϕ is a homeomorphism, we first recall that oK [[G0 ]] has a fundamental system of open neighborhoods of zero consisting of the two-sided ideals Jm,n (Lemma 2.43). We claim that ϕ(Jm,n ∩ oK [G0 ]) = L0 ({1gGn | gGn ∈ G0 /Gn }, m) ∩ D Dir (G0 , oK ).

(3.4)

To prove the claim,  take η ∈ Jm,n ∩ oK [G0 ]. As an element of oK [G0 ], η is a finite sum η = ti=1 ci hi for some hi ∈ G0 , ci ∈ oK . Let {g1 , . . . , gr } be a set of coset representatives of G0 /Gn . Putting together hi ’s belonging to the same coset of G0 /Gn , we can write η as η=

r 



j =1 hi ∈gj Gn

ci hi .

48

3 Distributions

Then the projection ηn of η to oK [G0 /Gn ] is r 

ηn =

⎛ ⎝

j =1

ϕ(η), 1gj Gn  = 

hi ∈gj Gn ci

r 

ci ⎠ gj Gn .

hi ∈gj Gn



The condition η ∈ Jm,n implies we have

⎞





∈ pm K for all j . For any j ∈ {1, . . . , r},

ci δhi , 1gj Gn  =

j =1 hi ∈gj Gn



ci ∈ p m K.

hi ∈gj Gn

Hence, ϕ(η) ∈ L0 ({1g1 Gn , . . . , 1gr Gn }, m) ∩ D Dir (G0 , oK ). The arguments for the converse inclusion are similar. Take η ∈ L0 ({1g1 Gn , . . . , 1gr Gn }, m) ∩ D Dir (G0 , oK ). Then η = write it as

t

i=1 ci δhi , for some hi

η=

∈ G0 and some coefficients ci ∈ oK , and we can r 



ci δhi .

j =1 hi ∈gj Gn

For each j ∈ {1, . . . , r}, we have 

η, 1gj Gn  =

ci ∈ p m K.

hi ∈gj Gn

This implies ψ(η) ∈ ϕ(Jm,n ∩ oK [G0 ]), proving the claim. It follows that both ϕ and ψ are continuous at zero. By linearity, both ϕ and ψ are continuous, thus proving that ϕ is a topological isomorphism.   Theorem 3.32 The map g → δg ,

g ∈ G0 ,

extends oK -linearly and by continuity to a topological isomorphism of oK -modules oK [[G0 ]] ∼ = D c (G0 , oK ), where D c (G0 , oK ) carries the weak topology. Proof Let ψ : D Dir (G0 , oK ) → oK [G0 ] be, as in the proof of Proposition 3.31, the topological isomorphism of oK -modules given by ψ(δg ) = g. Then for n ∈ N

3.2 Distributions

49

we have the corresponding continuous map ψn : D Dir (G0 , oK ) → oK [G0 /Gn ] obtained by composing ψ with the canonical projection oK [G0 ] → oK [G0 /Gn ]. Let Jn and Dn be as in Lemmas 2.45 and 3.29, respectively. Equation (3.4) implies that for all n ∈ N, ψ(Dn ∩ D Dir (G0 , oK )) = Jn ∩ oK [G0 ] and ψn (Dn ∩ D Dir (G0 , oK )) = 0. Then we have a well-defined oK -linear map

n : D c (G0 , oK ) → oK [G0 /Gn ] given by n (µ) = ψn (ν), where ν is any element of D Dir (G0 , oK ) such that µ − ν ∈ Dn . By Lemma 3.29, such ν exists. The maps n , n ∈ N, are continuous, compatible, and induce in the projective limit the continuous surjection

: D c (G0 , oK ) → lim oK [G0 /Gn ] = oK [[G0 ]] ← − n∈N

(see Corollary 2.20). Notice that |D Dir (G0 ,oK ) = ψ. It is easy to show that is injective, so is a continuous bijection We claim that (L0 ({1gGn | gGn ∈ G0 /Gn }, m)) = Jm,n (G0 ), for any m ∈ N, n ∈ N. Take µ ∈ L0 ({1gGn | gGn ∈ G0 /Gn }, m). There exists ν ∈ D Dir (G0 , oK ) ∩ L0 ({1gGn | gGn ∈ G0 /Gn }, m) such that µ − ν ∈ Dn . Then (ν) = ψ(ν) ∈ Jm,n (G0 ) and (µ − ν) ∈ Jn imply (µ) ∈ Jm,n (G0 ). This proves one containment. For the converse containment, take µ ∈ / L0 ({1gGn | gGn ∈ G0 /Gn }, m). Then for some coset gGn we have µ, 1gGn  ∈ / pm K . By continuity of , there exists a neighborhood U of µ such that (η) − (µ) ∈ Jm,n (G0 ) for all η ∈ U . Applying Lemma 3.29 once again, we can find ν ∈ D Dir (G0 , oK ) ∩ U such that µ − ν ∈ Dn . Then ν, 1gGn  = µ, 1gGn  ∈ / pm / Jm,n (G0 ) and hence K . It follows (ν) = ψ(ν) ∈

(µ) ∈ / Jm,n (G0 ). It follows that is open, and hence a homeomorphism.   Corollary 3.33 D c (G0 , oK ) is compact in D c (G0 , K), in the weak topology. Proof Since oK [[G0]] is compact, Theorem 3.32 implies that D c (G0 , oK ) is compact as well.  

50

3 Distributions

3.2.3 The Canonical Pairing It follows from Theorem 3.32 that we can identify oK [[G0 ]] with D c (G0 , oK ) by identifying g ∈ G0 with the Dirac distribution δg . Then from  ,  : D c (G0 , oK ) × C(G0 , oK ) → oK we obtain the canonical pairing  ,  : oK [[G0 ]] × C(G0 , oK ) → oK . We can describe the pairing explicitly (see §12 in [67]). Let µ ∈ oK [[G0 ]] and h ∈ C(G0 , oK ). Write µ = (µn )∞ n=1 , where µn ∈ oK [G0 /Gn ]. On the other hand, h can be uniformly approximated by a sequence {hn }∞ n=1 of smooth functions such that hn is right Gn -invariant. If g1 Gn = g2 Gn , then δg1 (hn ) = δg2 (hn ). It follows that we have a well-defined pairing µn , hn . More specifically, if {g1 , . . . , gs } is a set of representatives of G0 /Gn , we can write µn = a1 g1 Gn + · · · + as gs Gn

and hn = b1 1g1 Gn + · · · + bs 1gs Gn ,

where ai ∈ oK and bi ∈ oK for all i. Then µn , hn  = a1 b1 + · · · + as bs .

(3.5)

It can be shown that {µn , hn }∞ n=1 is a Cauchy sequence whose limit is independent of the choice of {hn }∞ (Exercise 3.34). Then n=1 µ, h = lim µn , hn . n→∞

Observe that hn ∈ C(G0 , oK ), so we can apply the above formula to evaluate µ, hn . It is easy to show that µ, hn  = µn , hn . Exercise 3.34 Let µn , hn  be as in Eq. (3.5). Prove that {µn , hn }∞ n=1 is a Cauchy sequence whose limit is independent of the choice of {hn }∞ . n=1

3.3 The Bounded-Weak Topology By a norm-bounded subset of V , we mean a subset B ⊂ V bounded with respect to the operator norm, which is equivalent to B being bounded in Vb . Definition 3.35 Let V be a normed K-vector space. The bounded-weak topology on V is the finest locally convex topology on V that coincides with the weak topology on norm-bounded subsets of V . We denote by Vbs

the space V equipped with the bounded-weak topology.

3.3 The Bounded-Weak Topology

51

In the Schneider-Teitelbaum duality theory described in Sects. 4.3 and 4.4, the dual V of a p-adic Banach space representation V carries the bounded weak topology. It is therefore important for us to understand this topology well, and we will devote the next two sections to its study, with Sect. 3.4 taking care of some properties of K[[G0 ]] needed in Chap. 4. More general properties of the boundedweak topology can be found in [57], where this topology is called the bounded weak star topology. The following lemma gives several criteria for determining if a lattice is open . The conditions are very similar, but it will be useful for us having them all in Vbs stated. Lemma 3.36 Let V be a normed K-vector space and M the unit ball in V . Then the following are equivalent for any lattice L ⊆ V : ; (i) L is open in Vbs (ii) for every norm-bounded set B ⊂ V , the set L ∩ B is open in B, where B carries the weak topology; (iii) for any 0 = c ∈ oK , the set cL ∩ M is open in M, where M carries the weak topology; (iv) for any k ∈ N, the set L ∩ K−k M is open in K−k M, where K−k M carries the weak topology.

Proof (i) ⇔ (ii) by the definition of the bounded-weak topology. (iii) ⇔ (iv) follows from the property that scalar multiplication K × V → V is continuous in any locally convex topology (Exercise 3.5), using the fact that a nonzero c ∈ oK can be written as c = Kk u, for some k ∈ N ∪ {0} and a unit × element u ∈ oK . To prove the equivalence (ii) ⇔ (iv), assume first (ii). For any k ∈ N, the set K−k M is norm-bounded. Then (ii) implies that L ∩ K−k M is open in K−k M. Conversely, assume (iv). Let B be a norm-bounded subset of V . Then B ⊂ −k K M, for some k ∈ N. By (iv), L ∩ K−k M is open in K−k M. Then L ∩ B is open in B by the definition of subspace topology.   Lemma 3.37 Let V be a normed K-vector space and M the unit ball in V . Then the bounded-weak topology on V is the finest locally convex topology such that the inclusion of M, with its weak topology, is continuous. Proof Let us denote by Vlc (just in this proof) the space V equipped with the finest locally convex topology such that the inclusion of M, with its weak topology, is continuous. Hence, Vlc contains as many open lattices as possible. The only obstruction comes from the requirement that ι : M → V is continuous. If L is an open lattice in Vlc , then necessarily L ∩ M is open in M. This condition, however, is not sufficient. Recall that from the condition (lc1) in Definition 3.3, if L is an open lattice in Vlc , then aL is also open for any a ∈ K × .

52

3 Distributions

Hence, aL∩M must be open in M for any a ∈ K × . This reduces to the requirement that cL ∩ M is open in M for any 0 = c ∈ oK . The statement then follows from Lemma 3.36.   Let us look again at the unit ball M, equipped with the weak topology, and the inclusion ι : M → V . Recall that the final topology with respect to ι is the finest topology making ι continuous. Notice that the finest locally convex topology making ι continuous is not the final topology with respect to ι. The final topology is usually much finer, with many more open sets. The description of the bounded-weak topology in terms of the final topology is given below. Lemma 3.38 Let V be a normed K-vector space and M the unit ball in V . For k ∈ N, let E−k M carry the weak topology, and let ιk be the embedding ιk : E−k M → V . Then the bounded-weak topology on V is the final topology with respect to the −k = family of maps (ιk )k∈N . In other words, Vbs k∈N E M is the topological union.  

Proof Exercise.

, so the bounded-weak topology If L is an open lattice in Vs , then L is open in Vbs is finer than the weak topology. We will show below that it is strictly finer. Still, Vs have same dual spaces, as we can see from the following lemma. and Vbs

Lemma 3.39 Let V be K-Banach space. Then (Vs ) = (Vbs ).

Proof Follows form Proposition 3.2 and Corollary 3.3 of [58]. (Also, see Corollary 2.2 (i) of [57].) For a comment on normpolar spaces, see Remark 4.21.   Corollary 3.40 Let V be a K-Banach space. The duality map ε : V → (Vbs )b

given by ε(v) = evv is a topological isomorphism. Proof Follows from Lemma 3.39 and Proposition 3.25.

 

3.3 The Bounded-Weak Topology

53

3.3.1 The Bounded-Weak Topology is Strictly Finer than the Weak Topology In this section, we present an example showing that the bounded-weak topology is not same as the weak topology. Let K = Qp . Let V = c0 be the space of null sequences in Qp , that is V = {(an )n ∈



Qp | lim an = 0}. n→∞

N

The norm is given by (an )n  = maxn |an |. We know from Lemma 3.19 that the dual of V is isomorphic to the space of bounded sequences ∞ , V ∼ = {(bn )n ∈



Qp | |bn | bounded},

N

 where the pairing V × V → Qp is given by (bn )n , (an )n  → n an bn . The operator norm on V is then (bn )n  = maxn |bn |, and the unit ball in V is M = {(bn )n ∈



Qp | |bn | ≤ 1 ∀n} ∼ =



Zp .

N

n∈N

The Weak Topology on V  For i ∈ N, we define ei = (δi,n )n ∈ V and fi = (δi,n )n ∈ V , where δi,n is the Kronecker delta. We have L({ei }, k) = {b ∈ V | ei , b ∈ pk Zp } = {b = (bn )n ∈ V | bi ∈ pk Zp } ⊂



Qp × p k Z p ×

ni

For i ∈ N and k ∈ Z, define Ui,k = L({ei }, k). Lemma 3.41 Let A be a finite subset of V and  ∈ Z. Then there exist at most finitely many i ∈ N such that L(A, ) ⊂ Ui,k for some k ∈ Z. Proof We may assume that A is linearly independent. Set I (A) = {i ∈ N | ai = 0 for some a = (an )n ∈ A}.

54

3 Distributions

For j ∈ / I (A) we have a, fj  = 0 for all a ∈ A. It follows c fj ∈ L(A, ) for any c ∈ Qp . Hence, L(A, ) ⊂ Uj,k for any j ∈ / I (A) and any k ∈ Z. Thus, the statement is clearly true if I (A) is finite. Suppose I (A) is infinite. Denote by J (A) the set of indices j ∈ I (A) such that A contains a multiple of ej . Fix i ∈ I (A) \ J (A). We can select a finite set {i = i1 , i2 , . . . , im } ⊂ I (A) such that rank{(ai1 , ai2 , . . . , aim ) | a = (an )n ∈ A} = rank{(ai2 , . . . , aim ) | a = (an )n ∈ A}. Then we can find c2 , . . . , cm ∈ Qp such that ai1 + c2 ai2 + · · · + cm aim = 0 for all a = (an )n ∈ A. Set b = fi1 + c2 fi2 + · · · + cm fim . Then a, b = 0, for all a ∈ A. It follows that c b ∈ L(A, ) for any c ∈ Qp . Hence, L(A, ) ⊂ Ui,k for any k ∈ Z. Since we can repeat the process for any i ∈ I (A) \ J (A) and J (A) is finite, this completes the proof.   The Bounded-Weak Topology on V  We consider the unit ball M with the weak topology. Similarly, for any k ∈ N, we is open if and consider p−k M equipped with the weak topology. Then U ⊂ Vbs −k only if U ∩ p M is open for all k ≥ 0. Define    L0 = M = Zp , L1 = Zp × p−1 Zp , L2 = Zp × p−1 Zp × p−2 Zp , n∈N

n>1

n>2

and for any k ≥ 0 Lk = Zp × p−1 Zp × · · · × p−k+1 Zp ×



p−k Zp .

n>k

Set L =

k≥0 Lk .

Note that for every k ≥ 0, L ∩ p−k M = Lk =

k  n=1

Un,−n+1 ∩ p−k M

3.4 Locally Convex Topology on K[[G0 ]]

55

. On the other hand, is open in p−k M. It follows that L is open in Vbs

L⊂



Un,−n+1 .

n∈N

Lemma 3.41 implies that L is not open in Vs .

3.4 Locally Convex Topology on K[[G0 ]] Recall that D c (G0 , K) is a Banach space with respect to the operator norm. The unit ball D c (G0 , oK ) is a lattice in D c (G0 , K) and hence D c (G0 , K) =



K−k D c (G0 , oK ).

k∈N

From Theorem 3.32, we know that D c (G0 , oK ) is isomorphic to the Iwasawa algebra oK [[G0 ]], where D c (G0 , oK ) carries the weak topology and oK [[G0 ]] carries the projective limit topology. Our next step is to define the Iwasawa algebra corresponding to D c (G0 , K). Since the oK -module oK [[G0 ]] is torsion-free, the map µ → 1 ⊗ µ gives us an inclusion oK [[G0]] → K ⊗oK oK [[G0 ]]. Definition 3.42 Let G0 be a profinite group. Define K[[G0 ]] = K ⊗oK oK [[G0 ]], equipped with the finest locally convex topology such that the inclusion oK [[G0 ]] → K[[G0 ]] is continuous. Notice that the definition of K[[G0]] specifies the topology. By Theorem 3.44 below, we can identify K[[G0 ]] with D c (G0 , K). Then, we could consider other topologies on K[[G0 ]], defined by their D c (G0 , K)-counterparts. An important example can be found in Sect. 4.4.1. In this book, however, we always consider K[[G0 ]] with the topology as in Definition 3.42. In particular, Lemma 1.4 of [64] implies Lemma 3.43 K[[G0]] is complete. If a ∈ K and µ ∈ oK [[G0 ]], we write simply aµ for the element a ⊗ µ ∈ K[[G0 ]]. Theorem 3.44 The map g → δg ,

g ∈ G0 ,

56

3 Distributions

extends K-linearly and by continuity to a topological isomorphism of K-vector spaces K[[G0]] ∼ = D c (G0 , K), where D c (G0 , K) carries the bounded-weak topology. Thus, we can identify K[[G0 ]] and D c (G0 , K) by identifying g ∈ G0 with the Dirac distribution δg . Proof Let µ ∈ K[[G0 ]]. Then µ = aη, for some a ∈ K and η ∈ oK [[G0 ]]. If a ∈ oK , then µ ∈ oK [[G0 ]]. Otherwise, we can write a = K−k u, with k ∈ N and × u ∈ oK . Then µ ∈ aoK [[G0 ]] = K−k oK [[G0 ]]. It follows K[[G0]] =



K−k oK [[G0 ]].

k∈N

From Theorem 3.32, we know that oK [[G0 ]] ∼ = D c (G0 , oK ) as topological rings and oK -modules, where the isomorphism is obtained by extending oK -linearly and by continuity the map g → δg . Then also K−k oK [[G0 ]] ∼ = K−k D c (G0 , oK ) for any k ∈ N. Lemma 3.36 implies that the map in question is an isomorphism of locally convex K-vector spaces.  

3.4.1 The Canonical Pairing The dual pairing  ,  : oK [[G0]] × C(G0 , oK ) → oK can be extended K-bilinearly to  ,  : K[[G0 ]] × C(G0 , K) → K.

(3.6)

Alternatively, we identify K[[G0 ]] with D c (G0 , K) as in Theorem 3.44 and use the pairing  ,  : D c (G0 , K)×C(G0 , K) → K. The resulting pairing  ,  : K[[G0 ]]× C(G0 , K) → K is same as in (3.6). If we fix µ ∈ K[[G0 ]], then f → µ, f  defines a map µ,  : C(G0 , K) → K.

3.4 Locally Convex Topology on K[[G0 ]]

57

If we identify µ with an element of D c (G0 , K), then µ, f  = µ(f ) and hence µ,  = µ. On the other hand, if we fix f ∈ C(G0 , K), then µ → µ, f  defines a map  , f  : K[[G0 ]] → K. This is precisely the evaluation map evf : V → K, for the Banach space V = C(G0 , K) and its dual V = K[[G0 ]]. The evaluation map is continuous if we equip V with the weak topology, or any topology finer than the weak one. It follows that  , f  : K[[G0 ]] → K is continuous. Moreover, the map f →  , f  ) and it is a topological isomorphism by is precisely the duality map  : V → (Vbs b Corollary 3.40. We can use integral notation and write

 µ, f  =

f (x)dµ(x), G0

as it is done in [67, §12]. More details about the integration pairing in the context of locally analytic distributions can be found in [65, Section 2]. This is maybe a good place to say something about p-adic integration and also about the great obstacle: nonexistence of a p-adic Haar measure on Zp and on other p-adic groups.

3.4.2 p-adic Haar Measure The problem with defining a K-valued Haar measure on a pro-p-group comes from the properties of p-adic absolute value. Let us look for instance at Zp . Suppose that we have a p-adic measure µ on Zp which is translation invariant. If µ = 0, we may scale it to get µ(Zp ) = 1. We can write Zp as a disjoint union of cosets of pZp , Zp =

p−1 

(a + pZp ).

a=0

Then from µ(a + pZp ) = µ(pZp ) and µ(Zp ) =

p−1  a=0

µ(a + pZp )

58

3 Distributions

we get µ(pZp ) = 1/p. This looks reasonable, unless we think how “big” are the numbers 1 and 1/p. These are p-adic numbers, with |µ(Zp )|p = 1

and |µ(pZp )|p = |1/p|p = p.

Now, this does not look right, because the subset pZp is measured with a “bigger” number than Zp . We start rightfully suspecting that a p-adic Haar measure on Zp does not exist. Before proving our suspicions, of course, we have to define a Haar measure, so that we know what it is that does not exist (see [59, 30.4] and [77, pages 248 and 305]). Definition 3.45 Let G be a locally compact group. Denote by B(G) the set of all compact open subsets of G. A K-valued left Haar measure on G is a map µ : B(G) → K satisfying (i) additivity: µ(A ∪ B) = µ(A) + µ(B), for any two disjoint sets A, B ∈ B(G), (ii) translation invariance: µ(gA) = µ(A) for any A ∈ B(G) and g ∈ G, and (iii) boundedness: for every A ∈ B(G), the set {µ(B) | B ∈ B(G), B ⊂ A} is bounded. Proposition 3.46 (Nonexistence of a p-adic Haar Measure on Zp ) If µ is a Kvalued Haar measure on Zp , then µ = 0. Proof Let µ : B(Zp ) → K be a K-valued Haar measure on Zp . Set c = sup{|µ(A)|p | A ∈ B(Zp )}. By the boundedness requirement, we have c < ∞. For any n ∈ N, we can write Zp as a disjoint union of cosets of pn Zp , Zp =

n −1 p

(a + pn Zp ).

a=0

By additivity and translation invariance, we have µ(Zp ) = pn µ(pn Zp ). Then |µ(Zp )|p = p−n |µ(pn Zp )|p ≤ p−n c. It follows |µ(Zp )|p ≤ lim p−n c = 0. n→∞

Hence, µ(Zp ) = 0.

 

Not only Zp , but also any of the groups G(L), where L is a finite extension of Qp and G is a reductive L-group, do not posses a nontrivial K-valued Haar measure. More generally, Monna and Springer proved in [47] that a locally compact group G has a K-valued Haar measure if and only if it contains a p-free compact open subgroup H . Here, H being p-free means that H does not contain an open subgroup whose index is divisible by p.

3.4 Locally Convex Topology on K[[G0 ]]

59

3.4.3 The Ring Structure on D c (G0 , K) The ring structure on oK [[G0 ]] induces the ring structure on K[[G0]], which can be transferred to D c (G0 , K) using identification from Theorem 3.44. We denote by ∗ the corresponding multiplication map ∗ : D c (G0 , K) × D c (G0 , K) → D c (G0 , K) and call it the convolution product. Notice that for g, h ∈ G0 we have δgh = δg ∗ δh . There is also a direct approach for defining the convolution product on D c (G0 , K). Continuous distributions are contained in the ring of locally analytic distributions. For details about the convolution product of locally analytic distribution, built using the integration notation, see Section 2 in [65]. Recall that oK [[G0 ]] is a topological ring, and therefore the multiplication oK [[G0 ]] × oK [[G0 ]] → oK [[G0 ]] is continuous, where oK [[G0 ]] × oK [[G0 ]] carries the product topology. Proposition 3.47 Let G0 be a profinite group. The multiplication K[[G0]] × K[[G0]] → K[[G0 ]] is separately continuous (meaning that the map η → µη is continuous for fixed µ, and µ → µη is continuous for fixed η). Proof Fix µ ∈ K[[G0 ]]. We will prove that the map fµ : K[[G0 ]] → K[[G0]] given by fµ (η) = µη is continuous. Since fµ (η + ξ ) = fµ (η) + fµ (ξ ), it is enough to prove continuity at zero. For that, take an open lattice L in K[[G0 ]]. Notice that for every m, n ∈ Z, the multiplication gives a continuous map K−m oK [[G0 ]] × K−n oK [[G0 ]] → K−(m+n) oK [[G0 ]]. Let m be an integer such that µ ∈ K−m oK [[G0 ]]. For n ∈ N, let fn be the restriction of fµ to K−n oK [[G0]]. Then −(m+n)

fn : K−n oK [[G0 ]] → K −(m+n)

is continuous. The lattice L ∩ K hence

oK [[G0 ]] −(m+n)

oK [[G0 ]] is open in K

Ln := fn−1 (L ∩ K−(m+n) oK [[G0 ]])

oK [[G0 ]] and

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is an open lattice in K−n oK [[G0 ]]. We claim that Ln = fµ−1 (L) ∩ K−n oK [[G0]]. To prove the claim, take ξ ∈ fµ−1 (L) ∩ K−n oK [[G0 ]]. Then fn (ξ ) = fµ (ξ ) ∈ L,

so fn (ξ ) ∈ L and fn (ξ ) ∈ K−(m+n) oK [[G0 ]]. It follows ξ ∈ Ln . Conversely, if ξ ∈ Ln , then ξ ∈ fµ−1 (L) and ξ ∈ K−n oK [[G0 ]], proving the claim. Define L =



Ln .

n∈N

Then L = fµ−1 (L). The lattice L is open in K[[G0 ]] because L ∩K−n oK [[G0]] =   Ln is open in K−n oK [[G0 ]] for every n ∈ N. Corollary 3.48 With D c (G0 , K) carrying the bounded-weak topology, the convolution product ∗ : D c (G0 , K) × D c (G0 , K) → D c (G0 , K) is separately continuous.

A Big Projective Limit It is tempting to try to express K[[G0 ]] as a projective limit. We describe below something that at first may appear as a natural candidate, but we will recognize immediately that it is not. For m, n ∈ N, m < n, we have the natural projection ϕn,m : K[G0 /Gn ] → K[G0 /Gm ]. Then (K[G0 /Gn ], ϕn,m )N is an inverse system, and we can define lim K[G0 /Gn ]. ← −

n∈N

Exercise 3.49 (a) Let µ ∈ K[[G0]]. Then there exist a ∈ K and η ∈ oK [[G0 ]] such that µ = aη. Write η = (ηn )n∈N , ηn ∈ oK [G0 /Gn ]. Define µn = aηn ∈ K[G0 /Gn ]. Prove that the definition of µn does not depend on the choice of a and η.

3.4 Locally Convex Topology on K[[G0 ]]

61

(b) From (a), we have a well-defined map ϕn : K[[G0 ]] → K[G0 /Gn ] given by ϕn (µ) = µn . Prove that the maps ϕn , n ∈ N, are compatible and continuous, and hence induce in the projective limit the continuous map ϕ : K[[G0 ]] → lim K[G0 /Gn ]. ← − n∈N

Prove that ϕ is injective. (c) Prove that lim K[G0 /Gn ] is bigger than K[[G0 ]] by constructing an ← −n∈N element of lim K[G0 /Gn ] that is not in K[[G0]]. ← −n∈N (d) After solving (c), we realize that lim K[G0 /Gn ] is much bigger than ← −n∈N K[[G0]]. Describe how they are related. Discuss topological properties.

Chapter 4

Banach Space Representations

Throughout the book, K and L are finite extensions of Qp . For the most of the chapter, the group G0 is a profinite group. From Sect. 4.4 on, we require that G0 is a compact p-adic Lie group. Then we have the fundamental result of Lazard that the Iwasawa algebra oK [[G0 ]] is noetherian. This property, together with Schikhof’s duality, is a basis of the theory of admissible Banach space representations by Schneider and Teitelbaum [64]. In this chapter, we present results of [64]. Our writing is complementary to their. We introduce definitions and prove some basic properties, and then refer to [64] for the proofs of main results. These results are for compact Lie groups and can be simply extended to arbitrary Lie groups, which is done in Section 4.4.2.

4.1 p-adic Lie Groups We refer to [63] for the theory p-adic Lie groups. We will not use this theory directly—the dependance on it is hidden in the proof that oK [[G0 ]] is noetherian. For convenience of the reader, we review briefly the notion of p-adic Lie groups. Let L be a finite extension of Qp . A Lie group (over L) is a manifold (over L) which also carries the structure of a group such the multiplication map G × G → G is locally analytic (see [63, Section 13]). Such groups are also called locally Lanalytic groups. A p-adic Lie group is a Lie group over Qp . In Part II, we will consider a split reductive Z-group G. Then G = G(L) and G0 = G(oL) are locally L-analytic groups. If L = Qp , we can use the restriction of scalars to realize G and G0 as locally Qp -analytic groups. Hence, G and G0 are p-adic Lie groups, with G0 in addition being compact, and the results of Sect. 4.4 apply to them.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Ban, p-adic Banach Space Representations, Lecture Notes in Mathematics 2325, https://doi.org/10.1007/978-3-031-22684-7_4

63

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4.2 Linear Operators on Banach Spaces Recall that we define a K-Banach space as a complete locally convex vector space whose topology can be defined by a norm (see Definition 3.8). We denote by Ban(K) the category of all K-Banach spaces with morphisms being all continuous K-linear maps. In our definition of a Banach space, the norm is not considered to be part of the structure. Hence, if V is a K-Banach space, there exists a norm   on V such that the balls B (v0 ) = {v ∈ V | v − v0  < } for v0 ∈ V and  ∈ R+ , form a base of the topology. However, this norm is not fixed. One of the advantages of this approach is that we can change the norm, as in the following lemma. Lemma 4.1 Let V be a K-Banach space. Then there exists a norm   on V inducing the Banach space topology such that V  ⊂ |K|. Proof Let   be a norm inducing the Banach space topology on V . For v ∈ V , define v = inf{r ∈ |K| | r ≥ v}. Then   is another norm on V . For every nonzero v ∈ V , we have |K | ≤ v/v ≤ 1. It follows that the norms   and   are equivalent, so   also induces the original Banach space topology on V . This norm satisfies V  ⊂ |K|. 

4.2.1 Spherically Complete Spaces The concept of spherical completeness plays a fundamental role in the p-adic functional analysis. It is defined for ultrametric spaces. As explained in Appendix A, an ultrametric space is a metric space (X, d) such that the metric d satisfies the strong triangle inequality. Definition 4.2 An ultrametric space (X, d) is said to be spherically complete if every nested sequence of balls in X has nonempty intersection.

4.2 Linear Operators on Banach Spaces

65

Lemma 4.3 Let (X, d) be an ultrametric space. The following conditions are equivalent (i) X is spherically complete. (ii) If B is a collection of balls such that no two elements of B are disjoint, then B∈B B = ∅. (iii) Every sequence of balls B(x1 , 1 ) ⊃ B(x2 , 2 ) ⊃ · · · for which 1 > 2 > · · · has nonempty intersection. Proof Exercise.



Lemma 4.4 If (X, d) is a complete ultrametric space and if every strictly decreasing sequence of values of d converges to 0, then (X, d) is spherically complete. Proof Let B(x1 , 1 ) ⊃ B(x2 , 2 ) ⊃ · · · be a sequence of balls in X such that 1 > 2 > · · · In every ball B(xn , n ), we select yn ∈ B(xn , n ). Thesequence (yn )n∈N is Cauchy, so it converges in X to y = limn→∞ yn . Then {y} = n∈N B(xn , n ).

 Corollary 4.5 (i) K is spherically complete. (ii) If V is a K-Banach space, there exists a norm   on V such that V is spherically complete with respect to the metric induced by  . Proof (i) By Lemma A.8, K satisfies the requirements of Lemma 4.4 and it is therefore spherically complete. (ii) By Lemma 4.1, there exists a norm   on V inducing the Banach space topology such that V  ⊂ |K|. Then V is spherically complete by Lemma 4.4. 

As an example of a complete nonarchimedean field which is not spherically complete, we mention Cp , the completion of Qp , as defined in Appendix A.2.3. The proof that Cp is not spherically complete can be found in [60, §1].

4.2.2 Some Fundamental Theorems in Functional Analysis We give several important theorems from functional analysis, for Banach spaces, as in [77]. More general versions can be found in [60]. In the proof, we use Baire spaces. A topological space X is called a Baire space if it satisfies the following condition. Given any countable collection {An } of closed  sets of X each of which has empty interior in X, their union n An also has empty interior in X [48, page 295]. From the Baire category theorem, complete metric spaces are Baire spaces [48, Theorem 48.2]. We also need the notion of the sum of a series. It is defined in the standard way, as follows. Let V be a K-Banach space, and v1 , v2 , . . . a sequence in V . If the

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4 Banach Space Representations

 sequence of partial sums sn = ni=1 vi converges to s = limn→∞ sn , we say that ∞ the series n=1 vn is convergent, with sum s, and we write s=

∞ 

vn .

n=1

A wonderful thing in p-adic analysis is that the convergence of such series can be determined easily. Lemma 4.6 Let V be a K-Banach space, and v1 , v2 , . . . a sequence in V . The series ∞ n=1 vn converges if and only if lim vn = 0.

n→∞

Proof Suppose limn→∞ vn = 0. We will prove that the sequence of partial sums s1 , s2 , . . . is Cauchy. Let  > 0. Then there exists N ∈ N such that for every n > N we have vn  < . Then for m > n > N, using the strong triangle inequality, we get sm − sn  = 

m 

vj  ≤ max(vn+1 , . . . , vm ) < .

j =n+1

It follows that the sequence (sn )n is Cauchy, hence convergent. The converse is obvious.

 Theorem 4.7 (Closed Graph Theorem) Let f : V → W be a K-linear map of K-Banach spaces. If its graph {(v, f (v) | v ∈ V } is a closed subset of V × W , then f is continuous. Proof We will prove that there exists δ > 0 such that for all v ∈ Bδ (0) we have f (v) ≤ 1. By linearity, this implies that f is continuous. Let A be the closure in V of the set {v ∈ V | f (v) ≤ 1}. Then V =



K−n A

n≥0

with each K−n A closed. As a Banach space, V is also a Baire space, so one of the sets K−n A has nonempty interior. Therefore, there exist m ∈ N,  > 0, and u ∈ V such that B (u) ⊂ K−m A. It follows B (0) ⊂ u + K−m A ⊂ K−m A + K−m A = K−m A. Set δ = |K |m . Then Bδ (0) ⊂ A. Take any v ∈ Bδ (0). Then, there exists v0 such that f (v0 ) ≤ 1 and v − v0  ≤ |K |δ. Then K−1 v − K−1 v0  ≤ δ, and so

4.2 Linear Operators on Banach Spaces

67

K−1 v − K−1 v0 ∈ Bδ (0). Inductively, we can construct a sequence v0 , v1 , . . . such that f (vn ) ≤ 1, K−n−1 v − K−n−1 v0 − K−n v1 − · · · − K−1 vn  ≤ δ, for all n. Then v − (v0 + K v1 + · · · + Kn vn ) ≤ |K |n+1 δ, for all n. It follows v=



Kn vn .

n

Since f (vn ) ≤ 1, we have lim Kn f (vn ) ≤ lim Kn  = 0.

n→∞

n→∞



n Lemma 4.6 implies  thatn n K f (vn ) converges. Since the graph of f is closed, it follows f (v) = n K f (vn ). Then f (v) ≤ 1, completing the proof. 

Corollary 4.8 If f : V → W is a continuous linear bijection of K-Banach spaces, then f is a homeomorphism. Proof Since the graph of f is closed in V × W , and f is a bijection, it follows that the graph of f −1 is closed in W × V . Hence, f −1 is continuous. 

Recall that a map f : X → Y between topological spaces is said to be open if it maps open sets to open sets. Theorem 4.9 (Open Mapping Theorem) Suppose that V and W are K-Banach spaces. Then every surjective continuous linear map f : V → W is open. Proof Let U = ker f and denote by F the induced continuous linear bijection F : V /U → W. By Corollary 4.8, F is a homeomorphism. If O is an open set in ¯ = f (O) V , we denote by O¯ its image in V /U . Then O¯ is open in V /U , and F (O) is open in W . 

Theorem 4.10 (Uniform Boundedness Theorem (Banach-Steinhaus)) Let V be a Banach space and W a normed vector space over K. Let S ⊆ L(V , W ). If sup f (v) < ∞

f ∈S

for all v ∈ V , then sup f  < ∞.

f ∈S

Here, f  is the operator norm.

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Proof Let S ⊆ L(V , W ) be as in the statement of the theorem. For every n ∈ N, the set Vn = {v ∈ V | f (v) ≤ n for all f ∈ S}  is a closed oK -submodule of V . By the assumption on S, V = n∈N Vn . As mentioned before, V is a Baire space. It follows that one of the Vn has nonempty interior, so it must contain an open neighborhood of zero. Fix such an n and  > 0 such that B (0) ⊂ Vn . Take an arbitrary v ∈ V . Let m = m(v) be the least integer such that |K |m v ≤ . Then |K |m−1 v ≥ , so |K | ≤ |K |m v ≤ . The second inequality implies Km v ∈ B (0), so f (Km v) ≤ n for all f ∈ S. 1 v. Then for From the first inequality, taking reciprocal, we get |K−m | ≤ |K | any f ∈ S f (v) = K−m f (Km v) ≤ |K−m |n ≤

n v. |K |

n Notice that this shows f (v) ≤ v, for all v ∈ V and therefore f  ≤ |K | n n . . This holds for any f ∈ S and consequently, supf ∈S f  ≤ 

|K | |K | Corollary 4.11 Suppose that f1 , f2 , . . . is a sequence in L(V , W ) such that limn→∞ fn (v) exists for every v ∈ V . Define f : V → W by f (v) = lim fn (v). n→∞

Then f ∈ L(V , W ). Proof The map f is clearly K-linear. We have to show that it is continuous. By the uniform boundedness theorem, there exists a number c such that fn  ≤ c for all n ∈ N. Then f (v) ≤ cv for all v ∈ V . It follows that f is bounded, and hence it is continuous. 

4.2.3 Banach Space Representations: Definition and Basic Properties If V is a K-Banach space, we denote by Autc (V ) the group of all continuous automorphisms of V . It is a subspace L(V , V ), and it can carry a subspace topology.

4.2 Linear Operators on Banach Spaces

69

In particular, we write Autcs (V ) when it is equipped with the weak topology coming from the embedding Autc (V ) ⊂ Ls (V , V ) Lemma 4.12 Let G be a locally profinite group and let V be a K-Banach space. (i) Suppose that G acts on V by continuous linear automorphisms such that the map G × V → V describing the action is continuous. Define π : G → Autcs (V ) ⊂ Ls (V , V ) by π(g)v = gv. Then π is a continuous homomorphism. (ii) Conversely, suppose that we have a continuous homomorphism π : G → Autcs (V ). Define the action of G on V by gv = π(g)v. Then the map G × V → V describing the action is continuous. Proof Giving an action of G on V by continuous linear automorphisms is equivalent to giving a homomorphism π : G → Autc (V ). We have to prove continuity. For g ∈ G, denote π(g) by fg , so fg (v) = π(g)v = gv,

for all v ∈ V .

(i) This direction is the easy one, but let us write the details anyway. Suppose G × V → V is continuous. Take g ∈ G and a neighborhood of fg in Ls (V , V ). Then it contains a neighborhood of the form fg + L({v1 , . . . , vs }, ), for some v1 , . . . , vs ∈ V and  > 0. By continuity of the action, for every i ∈ {1, . . . , s} there exists anopen neighborhood Ui of g such that hvi − gvi  < , ∀h ∈ Ui . Let U = si=1 Ui . This is an open neighborhood of g. For any h ∈ U , fh (vi ) − fg (vi ) < ,

∀i ∈ {1, . . . , s}.

It follows fh −fg ∈ L({v1 , . . . , vs }, ), and hence fh ∈ fg +L({v1 , . . . , vs }, ). This proves that π : G → Ls (V , V ) given by g → fg is continuous. (ii) Suppose π : G → Ls (V , V ) is continuous. For every v ∈ V , the evaluation map evv : Ls (V , V ) → V given by evv (f ) = f (v) is continuous (see Exercise 3.16). Since π and   are continuous, the map   ◦ evv ◦ π : G → R≥0 h → fh (v)

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is continuous. If U is a compact subset of G, then for every v ∈ V the set {fh (v) | h ∈ U } is bounded, because the continuous image of a compact set is compact. Now, let g ∈ G and v0 ∈ V , and take an open neighborhood of gv0 . It contains an open ball B(gv0 , ) for some  > 0. Let us consider fg +L({v0 }, ), which is a neighborhood of fg in Ls (V , V ). By continuity of π, there exists a compact open neighborhood U of g such that fh ∈ fg + L({v0 }, ),

∀h ∈ U.

As observed above, the set {fh (v) | h ∈ U } is bounded. The BanachSteinhaus theorem (Theorem 4.10) implies that {fh  | h ∈ U } is bounded by a constant c > 0. Then for every h ∈ U and every v ∈ B(v0 , /c), we have hv − gv0  = fh (v) − fh (v0 ) + fh (v0 ) − fg (v0 ) ≤ max{fh (v) − fh (v0 ), fh (v0 ) − fg (v0 )} ≤ max{fh v − v0 , } = , thus proving that G × V → V is continuous.



Definition 4.13 Let G be a locally profinite group and V a K-Banach space. A K-Banach space representation of G on V is a continuous homomorphism π : G → Autcs (V ) ⊂ Ls (V , V ), or, equivalently, a G-action on V by continuous linear automorphisms such that the map G × V → V describing the action is continuous. We denote this Banach space representation by (π, V ), or just V , or just π. We need some standard terminology from representation theory, such as intertwining operators, equivalence of representations, and subrepresentations. We introduce it below for Banach space representations, but similar definitions work for other types of representations, such as smooth representations considered in Chap. 6. If (π, V ) and (τ, W ) are two K-Banach space representations of the same group G, we denote by HomcG (π, τ )

or

HomcG (V , W )

the space of all continuous K-linear maps f : V → W satisfying f ◦ π(g) = τ (g) ◦ f,

for all g ∈ G.

4.2 Linear Operators on Banach Spaces

71

We call such maps intertwining operators. The condition above can be also written as f (gv) = gf (v),

for all g ∈ G, v ∈ V ,

and we say that f is G-equivariant. If f ∈ HomcG (π, τ ) is bijective, we know from Lemma 4.8 that f is a homeomorphism. If so, say that π and τ are isomorphic or equivalent, and we write π ∼ = τ or V ∼ = W. We denote by BanG (K) the category of all K-Banach space representations of G with morphisms being all G-equivariant continuous linear maps. Let (π, V ) be a K-Banach space representation of G. A subspace U of V is said to be G-invariant (or G-stable) if π(g) v ∈ U,

∀g ∈ G, v ∈ U.

We also say that U is a subrepresentation of V . The representation (π, V ) is called topologically irreducible if V = 0 and V has no G-invariant closed subspaces except 0 and itself. Exercise 4.14 Suppose G0 is compact. Then V = C(G0 , K), with the sup norm, is a Banach space. There are two natural actions of G0 on V : (a) g ∈ G0 acts on f ∈ V by left translation Lg , where Lg f (x) = f (g −1 x),

∀x ∈ G0 .

(b) g ∈ G0 acts on f ∈ V by right translation Rg , where Rg (x) = f (xg),

∀x ∈ G0 .

Prove that each action defines a Banach space representation. Prove that these two representations are equivalent. Exercise 4.15 Let V = C(G0 , K) with the G0 -action as in the previous exercise (either (a) or (b)). Let V ∞ = C ∞ (G0 , K) be the subspace of C(G0 , K) consisting of smooth (i.e., locally constant) functions. Prove that V ∞ is G0 -invariant. Hence, it is a dense subrepresentation of V . Exercise 4.16 Suppose V and W are two Banach space representation of G. Let f ∈ HomcG (V , W ). Prove that ker f is a closed G-invariant subspace of V and that im f is a G-invariant subspace of W .

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Exercise 4.17 Let V be a K-Banach space representation of G. Suppose U is a closed G-invariant subspace of V . We consider the exact sequence of spaces 0 → U → V → V /U → 0. The action of G on V induces the canonical action of G on U/V given by g(v + U ) = gv + U,

g ∈ G, v ∈ V .

Prove that this is a Banach representation of G on V /U . It is called a quotient representation. Exercise 4.18 Let V be a K-Banach space representation of G. From Exercise 3.10, we know that V  , equipped with the operator norm, is also a K-Banach space. We define the action of g ∈ G on λ ∈ V  by gλ = gλ, where λ(v) = λ(g −1 v),

g

∀v ∈ V .

Prove that this defines a Banach space representation of G on V  , called the dual representation of V . Taking the dual V → V  defines a functor BanG (K) → BanG (K). Of course, for a functor, we also have to define it on morphisms. This is done in a standard way. Given ϕ ∈ HomcG (V , W ) and λ ∈ W  , using the following commutative diagram

we define ϕ  (λ) = λ ◦ ϕ. It is easy to show that ϕ  ∈ HomcG (W  , V  ) and that the map ϕ → ϕ  satisfies the properties in the definition of a contravariant functor. Exercise 4.19 Prove that V → V  is a contravariant functor BanG (K) → BanG (K). Proposition 4.20 The functor BanG (K) → BanG (K) V → V  is exact.

4.3 Schneider-Teitelbaum Duality

73

Proof We start with an exact sequence of K-Banach space representations of G ϕ

ψ

0 −→ U −→ V −→ W −→ 0. Using standard arguments (such as those in the proof of Theorem 28 in Section 10.5 of [28]), we can show that ψ

ϕ

0 −→ W  −→ V  −→ U  is an exact sequence of K-Banach space representations of G. To prove that ϕ  is surjective, we identify U with its image in V . Then ϕ  : V  → U  is the restriction λ → λ|U . By Hahn-Banach theorem, any continuous linear functional on U extends to V (see Corollary 9.4 in [60]). It follows that the restriction λ → λ|U is surjective. Going back to original U , we see that ϕ  : V  → U  is surjective. 

4.3 Schneider-Teitelbaum Duality The duality theory of Schneider and Teitelbaum is built on Schikhof’s duality between p-adic Banach spaces and compactoids [58].

4.3.1 Schikhof’s Duality Following Section 1 in [64], we will describe two functors M → M d and V → V d . The first one is on the category Modflcomp(oK ) of all torsionfree and compact Hausdorff linear-topological oK -modules, with morphisms being all continuous oK -linear maps. For any module M in Modflcomp (oK ), define M d = HomcoK (M, K), the space of all continuous oK -linear maps : M → K. This is a K-Banach space with respect to the norm   = maxm∈M | (m)|. If M and N are modules in Modflcomp(oK ) and ϕ : M → N is a continuous oK -linear map, then ϕ d : N d → M d is given by ϕ d ( ) = ◦ ϕ. Notice that for any ∈ N d , ϕ d ( ) = max | (ϕ(m))| ≤ max | (n)| =  . m∈M

n∈N

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It follows that ϕ d is norm decreasing. Let Norm(K)≤1 denote the category of all complete normed spaces (V ,  ) such that V  ⊂ |K|, with morphisms being all norm-decreasing K-linear maps. Then M → M d is a functor from Modflcomp(oK ) to Norm(K)≤1 . Remark 4.21 In [58], Schikhof works with normpolar normed spaces over nonarchimedean fields. Since the valuation of K is discrete, a complete normed space (V ,  ) is normpolar if and only if V  ⊂ |K| (see Section 2.3 in [58]). Lemma 4.22 For V ∈ Norm(K)≤1 , let V d = {λ ∈ V  | λ ≤ 1} be the unit ball of the dual space V  equipped with the weak topology. Then V d ∈ Modflcomp(oK ). Proof Let V0 = {v ∈ V | |v ≤ 1} be the unit ball in V . Then for every λ ∈ V d and v ∈ V0 , we have |λ(v)| ≤ 1. Hence, we can define a map ι : Vd →



oK

v∈V0

λ → (λ(v))v . It is clearly oK -linear. For injectivity, take λ1 , λ2 ∈ V d such that λ1 = λ2 . Then, there exists v ∈ V such that λ1 (v) = λ2 (v). Let v0 be a nonzero scalar multiple of v which belongs to V0 . Then λ1 (v0 ) = λ2 (v0 ), and consequently ι(λ1 ) =ι(λ2 ). To prove that ι is continuous, take an open neighborhood U of zero in v∈V0 oK .  It contains an open neighborhood of the form v∈V0 Uv where Uv = oK for all but n finitely many v1 , . . . , vs , and for those vi , Uvi = pKi . Let m = maxi ni . Then ι(L({v1 , . . . , vs }, pm K )) ⊂ U, proving that ι is continuous.  We claim that ι is closed. To prove the claim, take a = (av )v∈V0 ∈ v∈V0 oK such that a ∈ / ι(V d ). Then there exists S = {v0 , v1 , . . . , vn } ⊂ V0 such that v0 = b1 v1 + · · · + bn vn

but

av0 = b1 av1 + · · · + bn avn .

4.3 Schneider-Teitelbaum Duality

75

Take  > 0 such that |av0 − (b1 av1 + · · · + bn avn )| > . For i = 0, 1, . . . , n, let Ui be an open neighborhood of avi such that for any x ∈ Ui we have |bi (x − avi )| <  (with b0 = 1). Let U=

 vi ∈S

Ui ×



oK .

v∈ /S

 This is an open neighborhood of a = (av )v∈V0 in v∈V0 oK . We will show that U is disjoint from ι(V d ). Take c = (cv )v∈V0 ∈ ι(V d ). Then c = ι(λ) for some λ, so cv = λ(v), for all v. Suppose cvi ∈ Ui , for i = 1, . . . , n. By linearity of λ, we have cv0 = b1 cv1 + · · · + bn cvn . Then cv0 − av0 = b1 (cv1 − av1 ) + · · · + bn (cvn − avn ) + (b1 av1 + · · · + bn avn ) − av0 . Since |b1 (cv1 − av1 ) + · · · + bn (cvn − avn )| ≤

max |bi (cvi − avi )| < 

i∈{1,...,n}

and |av0 −(b1 av1 +· · ·+bn avn )| > , it follows from Lemma A.11 that |cv0 −av0 | > . Then cv0 ∈ / U0 and c ∈ / U. This proves that  ι is closed. As a direct product of compact spaces, v∈V0 oK is compact. It follows that V d is compact. 

Recall that for two categories C and D, an equivalence of categories consists of a functor F : C → D and a functor G : D → C such that GF is naturally isomorphic to IC (the identity functor of C) and F G is naturally isomorphic to ID . In this situation, F and G are said to be quasi-inverses. If the functors F and G are contravariant, we will use the term anti-equivalence of categories, also called a duality of categories. Here is another characterization of the equivalence of categories (see [45, IV.4, Theorem 1] or [55, Theorem 1.5.9]). Lemma 4.23 A functor F : C → D yields an equivalence of categories if and only if it is simultaneously: (i) full, i.e. for any two objects A and B of C, the map HomC (A, B) → HomD (FA, FB) induced by F is surjective; (ii) faithful, i.e. for any two objects A and B of C, the map HomC (A, B) → HomD (FA, FB) induced by F is injective; and (iii) essentially surjective (dense), i.e. each object A in D is isomorphic to an object of the form FB, for B in C. Lemma 4.24 If V ∈ Norm(K)≤1 then V dd ∼ = V. Proof By definition, V d is the unit ball of the dual space V  equipped with the weak topology and V dd = HomcoK (V d , K). We claim that  , K). HomcoK (V d , K) = HomcK (Vbs

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4 Banach Space Representations

 , K), then clearly f | One inclusion is trivial: if f ∈ HomcK (Vbs ∈ Vd c d HomoK (V , K). For the converse inclusion, take f0 ∈ HomcoK (V d , K). We can  , K). We have to prove that f extend it linearly to a K-linear map f ∈ HomK (Vbs is continuous. Let U be an open lattice in K. Then L0 = f0−1 (U ) is an open lattice in V d . Moreover, for every n ≥ 0, the set Ln = f0−1 (Kn U ) is an open lattice in V d . Define  K−n Ln . L= n≥0

Take ∈ L. Then = K−n 0 for some n and some 0 ∈ Ln , and we have f ( ) = K−n f0 ( 0 ) ∈ U. This proves that f ( ) ∈ U , for all ∈ L. Notice that L ∩ K−n V d = K−n Ln because if ∈ L ∩ K−n V d , then f ( ) ∈ U implies f0 (Kn ) = Kn f ( ) ∈ Kn U and ∈ K−n Ln . The other containment is  , thus proving that f is trivial. Lemma 3.36 tells us that the lattice L is open in Vbs continuous at zero. By linearity, it is continuous. Hence,    , K) = (Vbs ). V dd = HomcoK (V d , K) = HomcK (Vbs

The norm on V dd is given by f0  = max ∈V d |f0 ( )|. By Lemma 3.12, this is  ) . It follows V dd = (V  ) . This is isomorphic to V by the operator norm on (Vbs bs b Corollary 3.40. 

Theorem 4.25 The functor Modflcomp(oK ) → Norm(K)≤1 M → M d is an anti-equivalence of categories, with quasi-inverse V → V d . Proof This follows from the proof of Theorem 1.2 in [64]. The authors prove that the oK -linear map ιM : M → (M d )s given by m → [λ → λ(m)] has image M dd and that ιM : M → M dd is a topological isomorphism. On the other hand, V dd ∼

 = V by Lemma 4.24. Let A be an additive category. Then all its hom-sets are abelian groups and composition of morphisms is bilinear. It follows that we can define the additive category AQ with the same objects as A and such that HomAQ (X, Y ) = HomA (X, Y ) ⊗Z Q

4.3 Schneider-Teitelbaum Duality

77

for any two objects X, Y in A. The composition of morphisms in AQ is the Q-linear extension of the composition in A. Lemma 4.26 The categories (Norm(K)≤1 )Q and Ban(K) are equivalent. forget

Proof Clearly, we have a forgetful functor Norm(K)≤1 −→ Ban(K). Also, if f is a morphism in Norm(K)≤1 and a ∈ Q, then af is a morphism in Ban(K). This gives us the functor F : (Norm(K)≤1 )Q −→ Ban(K). Set A = (Norm(K)≤1 )Q and B = Ban(K). We will show that F : A → B is fully faithful and essentially surjective, and hence an equivalence of categories (see Lemma 4.23). From Lemma 4.1, we know that for every K-Banach space V there exists a norm   on V inducing the Banach space topology such that V  ⊂ |K|. We abuse the notation and write V = F V . This shows that F is essentially surjective. Take V , W ∈ Obj(A) and consider the map HomA (V , W ) −→ HomB (F V , F W )

(4.1)

induced by F . Given f ∈ HomA (V , W ), we have Ff = f , considered as a map in HomB (F V , F W ). Therefore, (4.1) is injective and F is faithful. To show that F is full, take f ∈ HomB (F V , F W ) = L(V , W ). If f  > 1, there exists k ∈ N such that pk f  ≤ 1. If f  ≤ 1, take k = 0. Set f0 = pk f. Then f0 is a morphism in the category Norm(K)≤1 and f = p−k f0 ∈ HomA (V , W ). This proves that (4.1) is surjective, so F is full, completing the proof. 

Corollary 4.27 The functor Modflcomp(oK )Q → Ban(K) M → M d is an anti-equivalence of categories. When applying Corollary 4.27 on Banach space representations, we will also consider topologies on hom-sets. If M and N are two modules in Modflcomp(oK ), then the natural topology to consider on HomcoK (M, N) is the compact-open topology. Suppose V and W are K-Banach spaces. As in the proof of Lemma 4.26, we can consider V and W as objects in (Norm(K)≤1 )Q and define V d and W d . By the above equivalence of categories we have a natural linear isomorphism ∼

L(V , W ) −→ HomcoK (W d , V d ) ⊗ Q.

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4 Banach Space Representations

Lemma 4.28 The bounded weak topology on L(V , W ) induces the compact-open topology on HomcoK (W d , V d ). 

Proof This is Proposition 8.1 in [62].

4.3.2 Duality for Banach Space Representations: Iwasawa Modules In this section, G0 is a profinite group. We want to apply Schikhof duality on Banach space representations of G0 . We start with Lemma 2.1 and Corollary 2.2 from [64]. The proofs are omitted and can be found in [64]. Lemma 4.29 Let M be a complete Hausdorff linear-topological oK -module. Then the restriction map f → f |G0 defines a bijection ∼

HomcoK (oK [[G0 ]], M) −→ C(Go , M). A locally convex vector space V is called quasi-complete if every bounded closed subset of V is complete (see [60, §7]). Corollary 4.30 For any quasi-complete Hausdorff locally convex K-vector space V we have the K-linear isomorphism ∼

L(K[[G0 ]], V ) −→ C(Go , V ) f → f |G0 . Example 4.31 (Dual Pairing) Recall that oK [[G0 ]] ∼ = D c (G0 , oK ) and K[[G0]] ∼ = c D (G0 , K), with both isomorphisms obtained from g → δg , for g ∈ G0 (Theorems 3.32 and 3.44). Then we can identify K[[G0 ]] and D c (G0 , K). The canonical pairing  ,  : D c (G0 , K) × C(G0 , K) → K induces  ,  : K[[G0 ]] × C(G0 , K) → K. In Sect. 3.2.3, we explained how to compute µ, f , for µ ∈ oK [[G0 ]] and C(G0 , oK ). This extends linearly to get µ, f  for µ ∈ K[[G0 ]] and f ∈ C(G0 , K). Now, we look at the dual pairing from the point of view of Corollary 4.30. Let f ∈ C(G0 , K). Set F =  , f . For g ∈ G0 , we have F (g) = g, f  = f (g), and therefore F |G0 = f . Then F =  , f  is the unique continuous linear map F : K[[G0 ]] → K such that F |G0 = f .

4.3 Schneider-Teitelbaum Duality

79

K[[G0 ]]-module structure on V  Let V be a K-Banach space representation of G0 . By Definition 4.13, we have a continuous homomorphism π : G0 → Ls (V , V ), so π ∈ C(G0 , Ls (V , V )). From [60, Corollary 7.14], we know that Ls (V , V ) is quasi-complete and Hausdorff. Then, by Corollary 4.30, π extends uniquely to a continuous K-linear map : K[[G0 ]] → Ls (V , V ),

|G0 = π.

On the other hand, we can extend π K-linearly to K[G0 ]. We denote the resulting map again by π. Then π : K[G0 ] → Ls (V , V ) is a K-algebra homomorphism. By K-linearity, we have |K[G0 ] = π. Since K[G0 ] is dense in K[[G0 ]], it follows that is also a K-algebra homomorphism. Hence, we have a continuous algebra homomorphism K[[G0]] → Ls (V , V ). Since oK [[G0 ]] is compact, the corresponding map K[[G0]] → Lbs (V , V )

(4.2)

is also a continuous algebra homomorphism. Next, we need a G0 -invariant norm inducing the topology on V . Since G0 is compact, such a norm always exists. We remark that for non-compact Lie groups the lemma below is no longer true. Lemma 4.32 Let V be a K-Banach space representation of G0 . Then there is a G0 -invariant norm inducing the topology on V . Proof We follow [62, Remark 8.2]. As a profinite group, G0 has a neighborhood basis of the identity consisting of compact open normal subgroups (see Proposition 2.32). Take a bounded open lattice L in V . Since the action G0 × V → V is continuous, there exist a compact open normal subgroup H of G0 and an open lattice L0 ⊆ L such that H · L0 ⊆ L. Set L1 =



hL.

h∈H

For any h ∈ H we have h−1 · L0 ⊆ L and hence L 0 ⊆ h · L. It follows L0 ⊆ L1 , so L1 is also an open lattice in V . The lattice L2 = g∈G0 gL is G0 -invariant. It is

80

4 Banach Space Representations

open because L2 =

g∈G0



gL =

gL1 ,

g∈G0 /H

so L2 is a finite intersection of open lattices gL1 . Having constructed an open G0 -invariant lattice, the corresponding norm is also G0 -invariant. More specifically, we define v2 = inf |a|. v∈aL2

Then  2 is a G0 -invariant norm on V which defines the topology [60, Proposition 4.4]. 

Lemma 4.33 Let V be a K-Banach space representation of G0 . Let   be a G0 invariant norm on V inducing the Banach space topology and let M = V d be the unit ball in V  equipped with the weak topology. (i) The G0 -action on M induces a continuous map oK [[G0 ]] × M → M.  induces a separately continuous map K[[G ]] × V  → (ii) The G0 -action on Vbs 0 bs  Vbs . Proof As explained above, the map K[[G0 ]] → Lbs (V , V ) from Eq. (4.2) is a continuous homomorphism. Since   is G0 -invariant, Lemma 4.28 gives us a continuous homomorphism oK [[G0 ]] → HomcoK (M, M), where HomcoK (M, M) carries the compact-open topology. Then Theorem 3 in [10, Ch. X, §3.4] implies that oK [[G0 ]] × M → M is continuous. This proves (i). Assertion (ii) follows from (i). The proof is left as an exercise. 

Let M be an oK [[G0 ]]-module. Then M also carries the underlying oK -module structure coming from the embedding oK ⊂ oK [[G0 ]]. Definition 4.34 Let G0 be a profinite group. (i) An oK [[G0 ]]-Iwasawa module is a topological oK [[G0 ]]-module M such that the underlying oK -module belongs to Modflcomp(oK ). (ii) A K[[G0 ]]-Iwasawa module is a locally convex K-vector space U equipped with a separately continuous action K[[G0 ]] × U → U . If M is an oK [[G0 ]]-Iwasawa module then, by definition of a topological oK [[G0 ]]-module, the action oK [[G0 ]] × M → M is continuous. We denote by Modflcomp (oK [[G0 ]])

4.4 Admissible Banach Space Representations

81

the category of all oK [[G0 ]]-Iwasawa modules with morphisms being all continuous oK [[G0 ]]-module homomorphisms. Then Theorem 2.3 in [64] states the following. Theorem 4.35 The functor Modflcomp(oK [[G0 ]])Q → BanG0 (K) M → M d is an anti-equivalence of categories. Remark 4.36 In [64], Schneider and Teitelbaum mention the following two pathologies of the category BanG0 (K): (i) There exist non-isomorphic topologically irreducible K-Banach space representations V and W of G0 such that HomcG0 (V , W ) = 0. (ii) The group Zp , which is a compact abelian group, still has a huge number of infinite dimensional topologically irreducible Banach space representations (see Diarra [27]). To avoid such pathologies, they introduce an additional finiteness condition called admissibility. In the next section, we will define the category Banadm G0 (K) of admissible K-Banach space representations of G0 . This category avoids both of the above listed pathologies. For the first one, this follows from Corollary 4.47. For the second one, we refer to the discussion on page 375 of [64].

4.4 Admissible Banach Space Representations In this section, G0 is a compact Lie group. Admissible Banach space representations of G0 , as in Definition 4.41, form an important subcategory of BanG0 (K). This subcategory is algebraic in nature and its definition is based on the following fundamental property of Iwasawa algebras. Theorem 4.37 Let G0 be a compact Lie group. Then (i) The ring oK [[G0 ]] is left and right noetherian. (ii) The ring K[[G0 ]] is left and right noetherian. Proof Assertion (i) was proved by Lazard in [44, V.2.2.4] (also, cf. [63, Theorem 33.4]). Assertion (ii) follows from (i). 

Since K[[G0]] is noetherian, the category Modfg (K[[G0]]). of all finitely generated K[[G0 ]]-Iwasawa modules has nice algebraic properties.

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Proposition 4.38 Let G0 be a compact Lie group. Then (i) Any finitely generated oK [[G0]]-module M carries a unique Hausdorff topology—its canonical topology—such that the action oK [[G0 ]] × M → M is continuous. (ii) Any submodule of a finitely generated oK [[G0 ]]-module is closed in the canonical topology. (iii) Any oK [[G0 ]]-linear map between two finitely generated oK [[G0]]-modules is continuous in the canonical topologies. Proof Follows from the fact that oK [[G0 ]] is compact and noetherian.



Exercise 4.39 Prove Proposition 4.38. Lemma 4.40 Let G0 be a compact Lie group, and let M be a K[[G0 ]]-module. Then for any compact open subgroup H of G0 , M is finitely generated as a K[[H ]]module if and only if it is finitely generated as a K[[G0 ]]-module. Proof From Proposition 2.54 (ii), we know that K[[H ]] is a closed subalgebra of K[[G0 ]]. If M is finitely generated as a K[[H ]]-module, then it follows trivially that M is finitely generated as a K[[G0 ]]-module. For the converse statement, let {g1 , . . . , gs } be a set of coset representatives of H \ G0 . From Proposition 2.54 (iv), tensoring with K, we obtain K[[G0]] = K[[H ]]g1 ⊕ · · · ⊕ K[[H ]]gs . If M is finitely generated as a K[[G0 ]]-module, with generators m1 , . . . , mr , then {gi mj | i = 1, . . . , s, j = 1, . . . , r} generate M as a K[[H ]]-module. 

Definition 4.41 Let G be a p-adic Lie group. A K-Banach space representation V of G is called admissible if the dual space V  is finitely generated as a K[[H ]]module for some compact open subgroup H of G. Suppose that V is an admissible K-Banach space representation of G. It follows from Lemma 4.40 that V  is finitely generated as a K[[H ]]-module for any compact open subgroup H of G. We remark that Definition 4.41 is not the original definition of admissibility, but an equivalent property given in Lemma 3.4 of [64]. We denote by Banadm G (K) the full subcategory in BanG (K) of all admissible representations. Let G0 be a compact p-adic Lie group and V an admissible K-Banach space representation of G0 . By definition of admissibility, V corresponds under duality to the finitely generated K[[G0]]-module V  . Lemma 4.42 Suppose U and W are finitely generated K[[G0]]-Iwasawa modules. If f ∈ HomK[[G0 ]] (U, V ), then f is continuous.

4.4 Admissible Banach Space Representations

83

Proof Exercise.



Theorem 4.43 Let G0 be a compact p-adic Lie group. The functor Banadm G0 (K) → Modfg (K[[G0 ]]) V → V  is an anti-equivalence of categories. 

Proof This follows from Theorem 3.5 in [64]. Corollary 4.44 The categories Modfg (K[[G0 ]]) and

Banadm G0 (K)

are abelian.

Proof Since K[[G0 ]] is noetherian, it follows that Modfg (K[[G0]]) is abelian. Theorem 4.43 implies that Banadm 

G0 (K) is also abelian. Lemma 4.45 Let V be an admissible K-Banach space representation of G0 . Then V is topologically irreducible if and only if V  is a simple K[[G0]]-module. Proof If U = 0 is a proper closed G0 -invariant subspace, then we have the exact sequence of Banach space representations 0 → U → V → V /U → 0 and the corresponding dual sequence of K[[G0 ]]-modules 0 → (V /U ) → V  → U  → 0 (see Proposition 4.20). If V  is a simple K[[G0]]-module, then V must be topologically irreducible representation of G0 . For the converse, suppose that V  contains a proper K[[G0 ]]-submodule S = 0. Let   be a G0 -invariant norm on V inducing the Banach space topology such that V  ⊂ |K|. Let M be the unit ball in V  equipped with the weak topology. Then M = V d , and hence M is compact by Lemma 4.22. Let λ1 , . . . , λs be a set of generators of V  , so V  = K[[G0 ]]λ1 + · · · + K[[G0 ]]λs . By scaling, using V  ⊂ |K|, we may assume λi  = 1, for all i = 1, . . . , s. Define N = S ∩ (oK [[G0 ]]λ1 + · · · + oK [[G0 ]]λs ). This is a proper oK [[G0 ]]-submodule of M. It is closed in M by Proposition 4.38, and M/N = 0. It follows from Proposition 1.3 in [64] that the kernel of the dual map M d → N d is a nonzero proper closed G0 -invariant subspace of M d = V dd . Since V dd ∼ = V by Lemma 4.24, it follows that V contains a proper closed G0 invariant subspace. 

From Lemma 4.45, we obtain the following simple and operative formulation of the Schneider-Teitelbaum duality: Corollary 4.46 (Schneider-Teitelbaum Duality) Let G0 be a compact p-adic Lie group. The functor V → V 

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4 Banach Space Representations

induces a bijection between the set of isomorphism classes of topologically irreducible admissible K-Banach space representations of G0 and the set of isomorphism classes of simple K[[G0]]-modules. The following corollary addresses Remark 4.36 (i). Corollary 4.47 Any nonzero G0 -equivariant map between two topologically irreducible admissible K-Banach space representations of G0 is an isomorphism. The duality given in Corollary 4.46 is powerful. It enables us to work with Banach space representations using (mostly) purely algebraic methods. This approach will be taken in Part II, for studying principal series representations. We will also need a formulation for non-compact groups, and it is given in Sect. 4.4.2.

4.4.1 Locally Analytic Vectors: Representations in Characteristic p In this section, we mention briefly two types of representations related to Banach space representations. Our discussion is informal.

Locally Analytic Vectors Let V be a K-Banach space. If G is a Lie group over L ⊆ K, we denote by C an (G, V ) the space of locally L-analytic functions f : G → V . Let (π, V ) be a K-Banach space representation of G. A vector v ∈ V is called locally L-analytic if the map g → π(g)v lies in C an (G, V ). A vector v ∈ V is called smooth if there exists an open subgroup H of G such that π(h)v = v for all h ∈ H (see Definition 6.12). Denote by V sm the subspace of smooth vectors of V and by V an the subspace of locally L-analytic vectors of V . Then we have the following sequence of G-invariant subspaces V sm ⊆ V an ⊆ V , where both V sm and V an may be zero. In the case when L = Qp and V is admissible, the space of locally analytic vectors V an is dense in V [66, Theorem 7.1]. The proof is based on algebraic properties of distribution algebras. Namely, for a compact padic Lie group G0 , the strong dual of C an (G0 , K) is denoted by D(G0 , K) = C an (G0 , K)b and it is called the algebra of locally Qp -analytic distributions on G0 . Then D c (G0 , K) is dense in D(G0 , K) [65, Lemma 3.1] and the natural ring homomorphism D c (G0 , K) → D(G0 , K) is faithfully flat [66, Theorem 5.2].

4.4 Admissible Banach Space Representations

85

Unitary Representations and Reduction Modulo pK Let G be a p-adic Lie group and let (π, V ) be a Banach space representation of G. We say that π is unitary if the topology on V is defined by a G-invariant norm  . Then π(g)v = v

for all g ∈ G, v ∈ V .

(4.3)

Remark 4.48 It would be more appropriate to call such representations isometric. The term unitary representation usually refers to a representation of G on a complex vector space V such that there is a Hermitian inner product on V that is Ginvariant (see [70, §1.3] or [53, IV.2.2.]). However, since the term unitary is used for norm-preserving representations in many important works on p-Banach space representations, we will use the latter terminology. Suppose (π, V ) is an admissible unitary K-Banach space representation of G. Let κ = oK /pK be the residue field of K, as defined in Appendix A.2.2. Then κ is a finite extension of Fp . Consider the lattice V0 = {v ∈ V | v ≤ 1}. By Eq. (4.3), V0 is G-invariant, and so is K V0 . Then V = V0 /K V0 is a smooth κ-linear representation of G.

4.4.2 Duality for p-adic Lie Groups Let G be a p-adic Lie group, and G0 a compact open subgroup of G. Suppose that V is an admissible K-Banach space representation of G. Then V  carries two actions: it is a G-module and also a finitely generated K[[G0 ]]-module. These two actions coincide on G0 , leading to the following definition (similar to augmented representations defined by Emerton in [31]). Definition 4.49 A (K[[G0 ]], G)-module is a K[[G0 ]]-Iwasawa module U equipped with a G-action G × U → U such that the two actions coincide on G0 . Recall that by definition of a K[[G0 ]]-Iwasawa module, U is a locally convex K-vector space and the action K[[G0 ]] × U → U is separately continuous. Theorem 4.50 (Schneider-Teitelbaum Duality II) Let G be a p-adic Lie group, and G0 a compact open subgroup of G. Let V be an admissible Banach space representation of G. The map U → U 

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4 Banach Space Representations

defines a bijection between the set of G-invariant closed vector subspaces of V and the set of G-invariant K[[G0 ]]-quotient modules of V  . Proof If U = 0 is a proper closed G-invariant subspace, then we have the exact sequence of Banach space representations 0 → U → V → V /U → 0. From Proposition 4.20, we have the corresponding dual sequence 0 → (V /U ) → V  → U  → 0 which is K[[G0 ]]-linear and G-equivariant. Then U  is a G-invariant K[[G0 ]]quotient module of V  . From the proof of Lemma 4.45, U → U  is a bijection between the set of G0 -invariant closed vector subspaces of V and the set of K[[G0 ]]quotient modules of V  . Then we have the corresponding bijection of G-invariant objects on both sides, proving the theorem. 

Remark 4.51 The finiteness condition in the definition of an admissible Banach space representation may seem very restrictive, as pointed to us by David Vogan. There are two aspects to consider: (i) the size of an admissible K-Banach space representation, and (ii) the size of the category Banadm G (K), where we use the term size loosely. In Chap. 7, we will define the continuous principal series V = IndG P (χ), where χ is a continuous character χ : T → K × . Then V sm = 0 if χ is smooth and V an = 0 if χ is locally L-analytic. For any χ, V is admissible (Corollary 7.13). Suppose that χ is smooth. Then V sm , considered as a G0 -representation, contains countably many finite-dimensional irreducible subrepresentations (see Sect. 8.3). Since they are finite-dimensional, they are also topologically irreducible. Then V contains countably many finite-dimensional topologically irreducible G0 subrepresentations, and they comprise only a tiny part of V . Still, V  is a finitely generated K[[G0 ]]-module, and V is admissible. Continuous principal series are just a special type of admissible Banach space representations. The size and diversity of the category Banadm G (K) may be illustrated by the p-adic Langlands correspondence for GL2 (Qp ). To give a bigger picture, we will briefly abandon our good mathematical housekeeping of properly defining everything we talk about. The concepts discussed below are not used in the rest of the book. Let (V , π) be an irreducible admissible K-Banach space representation of GL2 (Qp ). We say that V is ordinary if it is a subquotient of a continuous principal series induced from a unitary character. Let ψ : GQp = Gal(Qp /Qp ) → GL2 (K) be an absolutely irreducible Galois representation. We denote by (ψ) the unitary Banach space representation of GL2 (Qp ) attached to ψ by the p-adic Langlands correspondence (Colmez [20, 0.17]). The functor ψ → (ψ)

4.4 Admissible Banach Space Representations

87

induces a bijection between the set of equivalence classes of absolutely irreducible continuous two-dimensional K-representations of GQp and absolutely irreducible non-ordinary admissible unitary K-Banach space representations of GL2 (Qp ) [22, 1.1]. Moreover, the p-adic Langlands correspondence encapsulates the classical Langlands correspondence; any irreducible admissible smooth representation σ of GL2 (Qp ) different from a character is a subrepresentation of (ψ), for some ψ. Such ψ is necessarily de Rham, and from its Fontain-Deligne module we can obtain the L-parameter attached to σ by the classical Langlands correspondence. More generally, (ψ) may contain locally algebraic vectors. The space of locally algebraic vectors of (ψ) is non-zero if and only if ψ is de Rham with distinct Hodge-Tate weights [20, Theorem 0.20]. As noted by Taylor in [75], “most” p-adic representations of GQp are not de Rham. So, “most” admissible unitary K-Banach space representations of GL2 (Qp ) do not contain smooth vectors or locally algebraic vectors. In summary, Banadm G (K) contains the subcategory of unitary representations V with V sm = 0, comparable in size with the category of all smooth representations of G, then the bigger subcategory of unitary representations V with nonzero locally algebraic vectors, and then the much bigger subcategory of unitary representations.

Part II

Principal Series Representations of Reductive Groups

Notation in Part II In Part II, Qp ⊆ L ⊆ K is a sequence of finite extensions. More general fields appear in Chap. 5 and Appendix: k is an algebraically closed field and F is a subfield of k. We denote by G a split connected reductive Z-group. Let G = G(L) be the group of L-points of G. We study principal series representations of G on K-Banach spaces. Our methods rely on the structure theory of G. Chapter 5 gives an overview of the structure theory of reductive groups. Below, we summarize the notation for some of the standard objects associated to G. These objects will be introduced in a systematic way in Chap. 5. We take P a Borel subgroup of G and T ⊂ P a maximal torus, and denote the unipotent radical of P by U. The unipotent radical of the opposite parabolic is denoted by U− . We write  for the roots of T in G and + (respectively, − ) for the set of positive (respectively, negative) roots determined by the choice of P. For each α ∈ , we have the root subgroup Uα and an isomorphism xα : Ga → Uα from the additive group Ga to Uα . We denote by oL the ring of integers of L and by pL its unique maximal ideal. Let qL be the cardinality of the residue field of L. For each algebraic subgroup H of G we let H = H(L) and H0 = H(oL ). We write prn for the canonical projection oL → oL /pnL and also for the induced map H0 → H(oL /pnL ) for any H. The kernel of prn in H0 is denoted Hn . Finally, we set ¯ H¯ = H (oL/pL ). Then B = pr−1 1 (P ) is the standard Iwahori subgroup of G. We denote by W = W (G, T) the Weyl group of G relative to T. For each w ∈ W we select a representative w˙ ∈ G(Z).

Chapter 5

Reductive Groups

Throughout Part II, Qp ⊆ L ⊆ K is a sequence of finite extensions and G = G(L) is the group of L-points of a split connected reductive Z-group G. In this chapter, we give an overview of the structure theory of split reductive Zgroups. We avoid the language of algebraic geometry and base our presentation on the down-to-earth definition of an affine variety given in Appendix B. The purpose of this chapter is to help a learner navigate through the literature and to explain different objects associated to G, such as roots, unipotent subgroups, and Iwahori subgroups. We also review important structural results, such as Bruhat decomposition, Iwasawa decomposition, and Iwahori factorization. We do not try, however, to present all important results—we simply prepare for what will be needed in Chaps. 7 and 8. For a systematically developed theory of reductive groups, we refer to the books by Borel [8], Humphreys [38], or Springer [72], all entitled Linear Algebraic Groups. We follow them in the first three sections. Sections 5.1 and 5.2 are over an algebraically closed field k, while in Sect. 5.3 we talk about F -groups and F points, where F is a subfield of k. In Sect. 5.4, we introduce Z-groups, and in particular split reductive Z-groups. Our approach rely on intuitive explanations, and on proper definitions given in the literature (Jantzen [40]). In Sect. 5.5, we describe the structure of G(L). It is richer than the structure over an arbitrary field F because, in addition to the structure of reductive groups, it also relies on the properties of nonarchimedean fields. In this chapter, k is an algebraically closed field and F is a subfield of k.

5.1 Linear Algebraic Groups We refer to Appendix B for the definition and basic properties of affine varieties. We just mention that an affine variety V ⊆ An = k n is the set of common zeros of a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Ban, p-adic Banach Space Representations, Lecture Notes in Mathematics 2325, https://doi.org/10.1007/978-3-031-22684-7_5

91

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finite collection of polynomials in k[x1 , . . . , xn ]. It carries the Zariski topology (see Definition B.6). Also, if V ⊂ An and W ⊂ Am are two affine varieties, a morphism of affine varieties ϕ : V → W is defined by polynomial functions (see page 203). Definition 5.1 A linear algebraic group is an affine variety G which is also a group such that the maps µ : G × G → G, µ(a, b) = ab, i : G → G, i(a) = a −1 . are morphisms of affine varieties. A morphism of linear algebraic groups is a group homomorphism ϕ : G → G which is also a morphism of affine varieties. Linear algebraic groups G and G are isomorphic if there exists an isomorphism of varieties ϕ : G → G which is also an isomorphism of groups.

5.1.1 Basic Properties of Linear Algebraic Groups We start with basic examples of linear algebraic groups. 1. The additive group Ga is the affine line A1 = k with addition as group operation. 2. The multiplicative group Gm is k × with multiplication as group operation. To show that taking inverses is a polynomial function, we embed ι : Gm → A2 as follows: for x ∈ Gm , set ι(x) = (x, x −1 ). Let k[x, y] be the affine algebra of A2 . Then Gm is described as the set of zeros of the polynomial xy − 1 in k[x, y]. The map x → x −1 on Gm is now just the restriction to Gm of the projection map (x, y) → y. 3. The general linear group GLn is the group of all n×n matrices with coefficients in k and determinant = 0. If n = 1, then GL1 = Gm . 2 To show that GLn is a linear algebraic group, we embed ι : GLn → An +1 as follows: if g = (gij ) ∈ GLn , set ι(g) = (g11 , . . . , gnn , (det g)−1 ) ∈ An

2 +1

Let k[x11, . . . , xnn , y] be the affine algebra of An as the set of zeros of the polynomial

2 +1

.

. Then we can describe GLn

y(det(xij )) − 1. It follows that GLn is a closed subset of An +1 . The multiplication in GLn is the product of matrices. It is given by polynomial functions, so it is a morphism of varieties. Also, the inverse on GLn can be computed as (xij )−1 = y adj(xij ), so it is a polynomial function. It follows that GLn is a linear algebraic group. 2

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93

A linear algebraic group is sometimes defined as a Zariski closed subgroup of GLn for some n. This definition is equivalent to Definition 5.1, as we can see from the following theorem (Springer [72, Theorem 2.3.7]): Theorem 5.2 Let G be a linear algebraic group. Then there is an isomorphism of G onto a closed subgroup of some GLn . An element g ∈ GLn is called semisimple if it is diagonalizable. It is called unipotent if g − 1 is nilpotent, that is, (g − 1)n = 0 for some n ∈ N. Theorem 5.3 (Jordan Decomposition) Let g ∈ GLn . There are unique elements gs ∈ G and gu ∈ G such that gs is semisimple, gu is unipotent, and g = gs gu = gu gs . More generally, Jordan decomposition holds in an arbitrary linear algebraic group (see Theorem 2.4.8 in [72]).

More Examples of Linear Algebraic Groups 1. We write diag(a1 , . . . , an ) for the diagonal n × n matrix with diagonal entries a1 , . . . , an , ⎞

⎛ a1 ⎜ .. diag(a1 , . . . , an ) = ⎝ .

⎟ ⎠. an

Let Dn be the subgroup of GLn consisting of diagonal matrices, Dn = {diag(a1 , . . . , an ) | ai ∈ k × } ∼ = Gm × · · · × Gm . Dn is a closed subgroup of GLn . It is the set of zeros of the polynomials xij , i = j . 2. The group Tn of upper triangular invertible n × n matrices is a closed subgroup of GLn . It is the set of zeros of the polynomials xij , i > j . 3. Let Un be the subgroup of Tn consisting of unipotent matrices. These are upper triangular matrices with all diagonal entries 1. Hence, if g ∈ Un , it is of the form ⎛

⎞ 1 a12 a13 . . . a1n ⎜ 1 a23 . . . a2n ⎟ ⎜ ⎟ g=⎜ .. ⎟ . .. ⎝ . . ⎠ 1 4. The special linear group SLn consists of the matrices of determinant 1 in GLn .

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5. The symplectic group Sp2n Sp2n = {g ∈ GL2n



 0 J 0 J | g g= }, −J 0 −J 0 t

⎛

where

tg

is the transpose of g and J =

⎜ ⎝

.

..

1

⎞ ⎟ ⎠.

1

6. The special orthogonal group SOn , char(k) = 2, SOn = {g ∈ SLn | t gJ g = J }.

Unipotent Subgroups The linear algebraic group H is called unipotent if all its elements are unipotent. The following is Proposition 2.4.12 from [72]: Proposition 5.4 Let H ≤ GLn be a unipotent linear algebraic group. Then there exists g ∈ GLn such that gHg −1 ≤ Un .

Identity Component Let G be an algebraic group. As an affine variety, G is disjoint union of irreducible varieties, G = V0 ∪ V1 ∪ · · · ∪ Vm . Denote by G0 the unique irreducible component containing the identity element. We call G0 the identity component of G. Then from Proposition 7.3 in [38], we have Proposition 5.5 Let G be an algebraic group. Then the identity component G0 is a normal subgroup of finite index in G. The cosets of G0 are irreducible components of G. An algebraic group G is called connected if G = G0 . The group GLn is connected because it is a principal open set in an affine space. The classical groups SLn , Sp2n , and SOn are also connected (see [38], page 53). Example 5.6 The even orthogonal group O2n , for char(k) = 2, is defined as O2n = {g ∈ GLn | t gJ g = J }. For any g ∈ O2n , we have t gJ g = J and det J = det(t gJ g) = det(t g) det J det g = det J (det g)2 . It follows (det g)2 = 1 and det g = ±1. The group O2n has two irreducible components, O2n = SO2n ∪ sSO2n , where s is an element of O2n of determinant -1.

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95

Tori An algebraic group is called a torus if it is isomorphic to some Dn . Let G be a connected linear algebraic group. Then the maximal tori of G are all conjugate ([38], Section 21.3). The common dimension of the maximal tori of G is called the rank of G. Example 5.7 A maximal torus in GLn is Dn and so rank(GLn ) = n. Also, SLn ∩Dn is a maximal torus in SLn . Since dim(SLn ∩ Dn ) = n − 1, it follows rank(SLn ) = n − 1.

5.1.2 Lie Algebra of an Algebraic Group Given a linear algebraic group G, we denote by Lie(G) the Lie algebra of G. We may think of it as the tangent space Te G at e ∈ G. The definition of the tangent space of an affine variety at a given point can be found in [72, Section 4.1.3]. The vector space isomorphism between Lie(G) and Te G is given in [72, Proposition 4.4.5].

Lie Algebras A Lie algebra g is a vector space with a bilinear multiplication [x, y] such that [x, x] = 0 and such that the Jacobi identity holds [[x, y], z] + [[y, z], x] + [[z, x], y] = 0. An important example is the general linear algebra gln , which is the vector space Mn of all n × n matrices, with the bracket operation [g, h] = gh − hg.

Lie Algebra of an Algebraic Group A derivation D of a ring R is a mapping D : R → R which is linear and satisfies the ordinary rule for derivatives, i.e., D(x + y) = Dx + Dy

and D(xy) = xDy + yDx.

Let G be an algebraic group and A = k[G]. Let Der A denote the set of all derivations of A. Then Der A is a Lie algebra with respect to the bracket operation [x, y] = xy − yx (Exercises 5.8).

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Recall that for x ∈ G, the left translation Lx and the right translation Rx are given by (Lx f )(y) = f (x −1 y) and (Rx f )(y) = f (yx), where f is a function on G and y ∈ G. Define Lie(G) = {δ ∈ Der A | δLx = Lx δ, ∀x ∈ G}. Then Lie(G) is a Lie algebra (Exercises 5.8). We call Lie(G) the Lie algebra of the algebraic group G (see [38], Section 9.1). Exercise 5.8 Let G be an algebraic group and A = k[G]. Prove that Der A is a Lie algebra, with respect to the bracket operation [x, y] = xy − yx. Prove that Lie(G) is a Lie subalgebra. Let g ∈ G. Define the action Ad g on g = Lie(G) by Ad g(δ) = Rg δRg−1 . Then Ad g ∈ Aut(g). We define Ad : G → Aut(g)

by

Ad(g) = Ad g.

Then Ad : G → Aut(g) is a morphism of algebraic groups, called the adjoint representation [72, Proposition 4.4.5]. Example 5.9 Let G = GLn , with the Lie algebra g = gln . If g ∈ G, then Ad g acts on X ∈ g by Ad g(X) = gXg −1 .

(5.1)

5.2 Reductive Groups Over Algebraically Closed Fields Let G be a linear algebraic group. The radical of G, denoted by R(G), is the largest closed connected normal solvable subgroup of G. The semisimple rank of G, denoted by rankss G, is the rank of G/R(G). The subgroup of R(G) consisting of all its unipotent elements is normal in G; we call it the unipotent radical and denote it by Ru (G). Then Ru (G) is the largest closed connected normal unipotent subgroup of G. A connected algebraic group G = 1 is called semisimple if R(G) is trivial. A connected algebraic group G = 1 is called reductive if Ru (G) is trivial.

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97

Example 5.10 (a) Any semisimple group is reductive. (b) Let T be a torus. Then T is solvable, so R(T) = T. Since all elements of T are semisimple, the Jordan decomposition implies that the only unipotent element in T is 1. It follows that Ru (T) = 1, and hence T is reductive. Proposition 5.11 Let G be a reductive group. (i) R(G) is a torus and R(G) = Z(G)0 , the identity component of the center of G. (ii) The derived subgroup (G, G) is semisimple. (iii) G = R(G)(G, G). (iv) rankss G = rank(G, G). (v) The intersection R(G) ∩ (G, G) is finite. Proof (i) and (v) follow from [38], page 125. (ii) and (iii) are Corollary 8.1.6 in [72]. (iv) follows from (iii).   Example 5.12 (a) The groups SLn , Sp2n , and SOn are semisimple, because each of them is equal to its derived subgroup. (b) The group G = GLn is reductive. Its derived subgroup is SLn . The radical R(G) consists of all scalar matrices in G. Theorem 5.13 Assume that G is connected, semisimple, of rank one. Then G is isomorphic either to SL2 or the projective group PSL2 .  

Proof [72], Theorem 7.2.4.

5.2.1 Rational Characters A rational character or simply a character of a linear algebraic group G is any morphism of algebraic groups χ : G → Gm . We denote by X(G) the set of all characters of G. It has a natural structure of an abelian group. It χ1 , χ2 ∈ X(G), we define the product χ1 χ2 by (χ1 χ2 )(g) = χ1 (g)χ2 (g). Example 5.14 det : GLn → Gm is a character. Example 5.15 To describe X(Gm ), we first recall that we consider Gm as a closed subset of A2 consisting of pairs (x, x −1 ). Let χ : Gm → Gm be a character. Then χ is a polynomial in x and x −1 , and also a group homomorphism. Hence, it must be of the form χ(x) = x n ,

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for some n ∈ Z. If ψ is another character, ψ(x) = x  , then χψ(x) = x n+ . It follows X(Gm ) ∼ = Z. Example 5.16 Since Dn ∼ = Gm × · · · × Gm (n copies), it follows from the previous example that X(Dn ) ∼ = Zn . The character χ of Dn corresponding to (m1 , . . . , mn ) ∈ Zn is given by χ(diag(a1 , . . . , an )) = a1m1 a2m2 · · · anmn . Let e1 , . . . , en be the standard basis of Zn . Then, under the above isomorphism X(Dn ) ∼ = Zn , we identify ei with the character of Dn given by ei (diag(a1 , . . . , an )) = ai .

5.2.2 Roots of a Reductive Group Let G be a reductive group and T a maximal torus in G. Let g be the Lie algebra of G. We consider the adjoint action of T on g. For α ∈ X(T), define gα = {X ∈ g | Ad t X = α(t)X, ∀t ∈ T}. Here, we identify X(T) with Zm . If gα = 0 and α = 0, we call α a root of G relative to T and gα a root space. The set of roots is denoted by (G, T) or just

and it is called the root system of G with respect to T. Corresponding to α = 0 is the fixed point space gT = {X ∈ g | Ad t X = X, ∀t ∈ T}. It is equal to t = Lie(T), the Lie algebra of T. The following is Corollary B(b) in Section 26.2 of [38]. Proposition 5.17 (Root Space Decomposition) We have g=t⊕



gα ,

α∈

where dim gα = 1 for all α ∈ . In particular, the set of roots is a finite subset of X(T). Exercise 5.18 Let G = GLn and T = Dn . Let g = gln , and let t be the subalgebra consisting of diagonal elements in g. The group T acts on g by the adjoint action given by Eq. (5.1). For i = j , denote by Eij the n × n matrix having coefficient 1 on the place (i, j ), and all other coefficients zero. Let t = diag(a1 , . . . , an ) ∈ T. Prove

5.2 Reductive Groups Over Algebraically Closed Fields

99

that (Ad t)Eij = ai aj−1 Eij .

(5.2)

It follows that the character α of T defined by α(diag(a1 , . . . , an )) = ai aj−1 is a root of G. If we denote by [Eij ] the linear span of Eij , then (5.2) implies [Eij ] ⊆ gα . Then [Eij ] = gα because dim gα = 1, From the previous exercise, we can find the roots of G = GLn . The group of characters X(T) ∼ = Zn is generated by {ei | i = 1, . . . , n}, where ei (a) = ai , for a = diag(a1 , . . . , an ) ∈ T. Define αij = ei − ej .  From (5.2), αij is a root of G. Since g = t ⊕ i=j [Eij ], it follows that we have found all the roots, so (G, T) = {αij | i, j = 1, . . . n, i = j }. Proposition 5.19 Let G be a reductive group. Let T be a maximal torus in G, with Lie algebra t = Lie(T). Let = (G, T). Then (i) − = . (ii) For α ∈ , let Tα be the connected component of ker α. Then Tα is a subtorus of T of codimension one. We have Z(G) = α∈ Tα . (iii) The centralizer Zα = ZG (Tα ) is a reductive group of semisimple rank one. The groups Zα , α ∈ , generate G. (iv) Let zα be the Lie algebra of Zα . Then zα = t ⊕ gα ⊕ g−α Proof This is Corollary B in Section 26.2 of [38].

 

Example 5.20 Let us write everything explicitly for G = GL3 and T = D3 . Let α = e1 − e2 . Then ker α = {diag(a, a, b) | a, b ∈ k × }. This is a connected group, so Tα = ker α. Similarly, for β = e2 − e3 , we have ker β = {diag(a, b, b) | a, b ∈ k × } = Tβ . Then Tα ∩ Tβ = ker α ∩ kerβ = {diag(a, a, a) | a ∈ k × } = Z(G), the center of G. Notice that for any z = diag(a, a, a) ∈ Z(G), we have αij (z) = 1, for all roots αij .

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5 Reductive Groups

To find Zα = ZG (Tα ), we have to do some linear algebra. Let z ∈ Zα . We can write it as a block-matrix  AB z= , CD where A is 2×2, B is 2×1, C is 1×2, and D is 1×1. For any t = diag(a, a, b) ∈ Tα , we have tz = zt. It is easy to show that we must have B = 0 and C = 0, while A and D can be arbitrary. It follows Zα = {

 g0 | g ∈ GL2 , a ∈ k × }. 0a

The Lie algebra of Zα is zα = {

 X0 | X ∈ gl2 , y ∈ k}. 0 y

Notice that ⎛ 0x gα = {⎝0 0 00

⎞ 0 0⎠ | x ∈ k} 0

⎛

and g−α

⎞ 000 = {⎝x 0 0⎠ | x ∈ k}. 000

Then zα = t ⊕ gα ⊕ g−α . Exercise 5.21 Let G = GLn and T = Dn . Take a root α ∈ (G, T). It is of the form α = ei − ej , i = j . Compute the groups Tα and Zα , and the Lie algebra zα .

Weyl Group Let G be a connected reductive algebraic group and T a maximal torus of G. The Weyl group of G with respect to T is defined as W (G, T) = NG (T)/ZG (T), where NG (T) is the normalizer of T in G and ZG (T) is the centralizer. By Corollary 3.2.9 in [72], W (G, T) is finite. Since G is reductive and T is maximal, we have ZG (T) = T. Since all maximal tori in G are conjugate, their Weyl groups are isomorphic, and we call such a group the Weyl group of G. The roots of G relative to T form an abstract root system, as defined below.

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101

Abstract Root Systems Let E be a Euclidean space: a finite dimensional real vector space with an inner product ( , ). Let α ∈ E. Then Hα = {x ∈ E | (x, α) = 0} is the hyperplane orthogonal to α. Define sα (x) = x −

2(x, α) α. (α, α)

Then sα (α) = −α and sα (x) = x for all x ∈ Hα . It follows that sα is the reflection with respect to Hα . We define a pairing  ,  on E by x, y =

2(x, y) . (y, y)

Then sα (x) = x − x, αα. Definition 5.22 A (reduced) abstract root system is a subset of E such that (R1) (R2) (R3) (R4)

is finite, spans E and does not contain 0. If α ∈ , the only multiples of α in are ±α. If α ∈ , then sα ( ) ⊂ . If α, β ∈ , then α, β ∈ Z.

The rank of is dim E. We will give in Theorem 5.27 a classification of root systems. In the following example, we give explicitly root systems of rank 1 and 2. Example 5.23 There is only one root system of rank 1: A1 .

The (reduced) root systems of rank 2 are A1 × A1 , A2 , B2 , and G2 .

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5 Reductive Groups

Let be a root system in E. Denote by W ( ) the subgroup of GL(E) generated by the reflections sα , α ∈ . By (R3), W ( ) permutes the set , which is by (R1) finite and spans E. This allows us to identify W ( ) with a subgroup of the group of permutations of ; in particular, W ( ) is finite. We call W ( ) the Weyl group of . Theorem 5.24 Let G be a reductive group. Let T be a maximal torus of G, X = X(T) and = (G, T). Denote by Q the subgroup of X generated by . Then, (i) is an abstract root system in E = R ⊗Z Q, (ii) rank = rankss G, (iii) W ( ) ∼ = W (G, T). Proof If G is semisimple, this is Theorem 27.1 in [38]. In general, the theorem follows from Section 7.4 in [72].   Example 5.25 Let G = GLn and T = Dn . Let = (G, T). As explained earlier,

= {ei − ej | i, j = 1, . . . n, i = j }. We have W ( ) ∼ = Sn , with Sn acting on

by permuting e1 , . . . , en . The standard inner product on R ⊗Z X(T), given by (ei , ej ) = δij , is W -invariant. The derived subgroup of G = GLn is SLn . We have rank G = dim T = n and rank = rankss G = n − 1. The radical R(G) is equal to the center Z(G) and it consists of all scalar matrices in G.

Simple Roots Let be an abstract root system. A subset of is called a base if (B1) is a basis of E.

(B2) each root β can be written as β = kα α (α ∈ ) with kα ∈ Z and all kα ≥ 0 or all kα ≤ 0. The following is Theorem 10.1 from [39].

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103

Theorem 5.26 Any abstract root system has a base. Fix a base of . From (B1), | | = rank . The roots in are called simple. If α ∈ , we call sα a simple reflection. Let β ∈ and write β = kα α. If all kα ≥ 0 (respectively, kα ≤ 0), we call β positive (respectively, negative). We denote by + the set of all positive roots and by − the set of all negative roots. The Weyl group W ( ) is generated by simple reflections and hence any w ∈ W ( ) can be written as a product of simple reflections w = s1 s2 . . . sk . We call the above decomposition reduced if k is as small as possible. If so, k is called the length of w relative to and denoted by (w). It turns out that the length of w is equal to the cardinality of the set {α ∈ + | w(α) ∈ − } (see Lemma 10.3 A in [39]). In particular, if w = sα is a simple reflection, then the above set contains just one element, α, and sα permutes the elements of the set

+ − {α}. On the other hand, W ( ) contains the unique longest element. It is denoted by w and it is characterized by the property that w (α) < 0 for every positive root α.

5.2.3 Classification of Irreducible Root Systems A root system is called irreducible if it cannot be partitioned into the union of two proper subsets such that each root in one set is orthogonal to each root in the other. Equivalently, is irreducible if (a base of ) cannot be partitioned into the union of two proper subsets such that each root in one set is orthogonal to each root in the other. Let α, β ∈ . Let θ be the angle between α and β. Then (α, β) = α β cos θ, β, α =

β 2(β, α) =2 cos θ (α, α) α

and α, ββ, α = 4 cos2 θ . This number is an integer, by (R4), and it lays in the interval [0, 4]. It follows α, ββ, α = 0, 1, 2 or 3 (if α = β). Let = {α1 , . . . , α } be a base of . Define the Coxeter graph of as the graph with  vertices, with αi , αj αj , αi  edges between the ith and the j th vertex (i = j ). Whenever a double or triple edge occurs in the Coxeter graph of , we can add an arrow pointing to the shorter of the two roots. We call the resulting figure the Dynkin diagram of . The classification of root systems in terms of

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Dynkin diagrams is given by the following two theorems (Theorems 11.4 and 12.1 in Humphreys [39]). Theorem 5.27 If is an irreducible system of rank , its Dynkin diagram is one of the following ( vertices in each case):

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105

Theorem 5.28 For each Dynkin diagram of type A − G, there exists an irreducible root system having the given diagram.

5.2.4 Classification of Reductive Groups A semisimple algebraic group is called simple if its root system is irreducible. Such a group has no closed connected normal subgroups other than itself and the trivial group [38, 27.5, 32.1]. The centre Z(G) of a simple group G is finite, and G/Z(G) is simple as an abstract group [38, Corollary 29.5]. For each irreducible root system , there exists a simple algebraic group G having the given root system. The root system , however, does not determine the group G uniquely—nonisomorphic groups can have the same root system. Already in rank one we have two simple groups, SL2 and PSL2 , both with root system A1 (Theorem 5.13). The following is Theorem 32.1 from [38]. It classifies simple algebraic groups in terms of root systems and fundamental groups. (For the definition of the fundamental group of G, see page 127.) Theorem 5.29 Suppose G and G are simple algebraic groups over an algebraically closed field k. If they have isomorphic root systems and isomorphic fundamental group, then G and G are isomorphic (as algebraic groups). For a classification of reductive groups, we need more data. In addition to characters and roots, we need cocharacters and coroots, as defined below.

Cocharacters Let T be a torus, T ∼ = Dn ∼ = Gm × · · · × Gm . Any morphism of algebraic groups λ : Gm → T is called a cocharacter of T. We denote by X∨ (T) the set of cocharacters of T. It is an abelian group, with the product (λµ)(a) = λ(a)µ(a). If χ ∈ X(T) and λ ∈ X∨ (T), then χ ◦ λ : Gm → T → Gm and hence χ ◦ λ ∈ X(Gm ). Using the isomorphism X(Gm ) ∼ = Z, we can define a natural pairing  ,  : X(T) × X∨ (T) → Z, (χ, λ) → χ, λ ∈ Z given by χ(λ(a)) = a χ,λ , for all a ∈ Gm .

(5.3)

106

5 Reductive Groups

Root Datum of a Reductive Group Let G be a reductive group and T a maximal torus of G. We describe here the associated root datum (G, T), as outlined in [71]. Details can be found in [72, Chapter 7]. Let X = X(T) and X∨ = X∨ (T) (characters and cocharacters). Let  ,  : X(T) × X∨ (T) → Z be the natural pairing defined in Eq. (5.3). Take = (G, T), the root system of G with respect to T. Then is a finite subset of X. For α ∈ , we denote by Tα the identity component of the kernel of α. This is a subtorus of codimension one. Take the centralizer Zα = ZG (Tα ) and let Gα be the derived subgroup of Zα . Then Gα is a rank one semisimple group, so it is isomorphic to either SL2 or PSL2 . There is a unique co-character α ∨ : Gm → T such that α, α ∨  = 2, im α ⊂ Gα and T = (im α ∨ )Tα . The set ∨ = {α ∨ | α ∈ } is a finite subset of X∨ . It is also a root system. Its elements are called coroots. The quadruple (G, T) = (X, , X∨ , ∨ ) is called the root datum of G with respect to T. Exercise 5.30 Let G = GLn and T = Dn . Take a root α ∈ (G, T) of the form −1 α = ei − ei+1 , for some i ∈ {1, . . . , n − 1}. Then α(diag(a1 , . . . , an )) = ai ai+1 . Compute the groups Tα , Zα , and Gα defined above. (The groups Tα and Zα were already computed in Exercise 5.21.) Prove that α ∨ : Gm → T is defined by α ∨ (a) = diag(1, . . . , 1, a, a −1 , 1, . . . , 1), where the entries a and a −1 are at the ith and (i + 1)th position.

Abstract Root Datum Definition 5.31 A root datum is a quadruple  = (X, , X∨ , ∨ ) such that (i) X and X∨ are free abelian groups of finite type, in duality by a pairing X × X∨ → Z denoted by  , , (ii) and ∨ are finite subsets of X and X∨ and there is a bijection

→ ∨ ,

α → α ∨ .

(iii) For α ∈ we define endomorphisms sα and sα∨ of X and X∨ , respectively, by sα (x) = x − x, α ∨ α, sα ∨ (u) = u − α, uα ∨ .

5.2 Reductive Groups Over Algebraically Closed Fields

107

Then the following two axioms are imposed: (RD1) (RD2)

For all α ∈ we have α, α ∨  = 2; For all α ∈ we have sα ( ) ⊂ , sα ∨ ( ∨ ) ⊂ ∨ .

Theorem 5.32 Let k be an algebraically closed field. Let  = (X, , X∨ , ∨ ) be a root datum. Then there exists a connected reductive linear algebraic group G over k with a maximal torus T such that the root datum (G, T) is isomorphic to . The group G is unique up to an isomorphism of algebraic groups. Proof Follows from Springer [72], Theorems 10.1.1 and 9.6.2 (Existence Theorem and Isomorphism Theorem).  

5.2.5 Structure of Reductive Groups In this section, G is a connected reductive algebraic group, T is a maximal torus of G, and = (G, T) is the root system of G with respect to T. Let W = W (G, T).

Root Subgroups Let g = Lie(G) be the Lie algebra of G and t = Lie(T). Recall the root space decomposition g=t⊕



gα .

α∈

From Theorem 26.3 in [38] and Section 8.1 in [72] we have the following. Theorem 5.33 Let α ∈ . (i) There exists a unique connected T-stable subgroup Uα of G having Lie algebra ga . (ii) There exists an isomorphism xα : Ga → Uα such that for any t ∈ T and a ∈ Gm , txα (a)t −1 = xα (α(t)a). (iii) For any w ∈ W , wUα w−1 = Uw(α) . (iv) The group G is generated by the groups Uα (α ∈ ), along with T. (v) If G is semisimple, it is generated by Uα (α ∈ ). The groups Uα are called root subgroups.

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5 Reductive Groups

Example 5.34 Let G = GLn and T = Dn . Let α = ei − ej , i = j , be a root of G. We proved in Exercise 5.18 that gα = [Eij ], where Eij is the n × n matrix having coefficient 1 on the place (i, j ), and all other coefficients zero. Then Uα = 1 + gα = {1 + cEij | c ∈ k}.

Borel Subgroups and Parabolic Subgroups A Borel subgroup of G is a maximal closed connected solvable subgroup. Any two Borel subgroups of G are conjugate ([38], page 134). Fix a Borel subgroup P of G containing T. There is a bijection between the set of bases of and the set of Borel subgroups of G containing T ([38], page 166). Therefore, our choice of P determines the base of and also the set of positive roots + . We have the decomposition P = TU,

U=



Uα .

α∈ +

The group U is the unipotent radical of P. The Borel supgroup opposite P is P− = TU− ,

U− =



Uα .

α∈ −

Recall that w denotes the longest element in W , characterized by the property w ( + ) = − . Then w Uw−1 = U− and w U− w−1 = U. For w ∈ W = NG (T)/T, take a representative w˙ in G. Since T ⊂ P, the double coset PwP ˙ does not depend on the choice of a representative, and we write simply PwP. Then we have the following theorem (see Theorem 8.3.8 in [72]): Theorem 5.35 (Bruhat Decomposition) G is the disjoint union G=



PwP.

w∈W

Related to the Bruhat decomposition is a partial order on W . For x, y ∈ W , we define x≤y

if

PxP ⊆ PyP,

where PyP denotes the closure of PyP in G. This partial order can also be defined using the reduced decomposition of the elements of W into product of simple reflections, and it is called Bruhat order. More specifically, if w = s1 s2 . . . sk is a reduced decomposition of w ∈ W , then x ≤ w if and only if x can be written as

5.3 F -Reductive Groups

109

x = si1 si2 . . . sim with m ≥ 0 and 1 ≤ i1 < · · · < im ≤ k (see Proposition 8.5.5 in [72]). The longest element w is maximal with respect to Bruhat order. Directly from our definition of the partial order on W , we have PwP =



PxP.

x≤w

Since w ≤ w for all w ∈ W , it follows that the closure of Pw P is the whole group G. In other words, Pw P is dense in G. Lemma 5.36 Let w ∈ W and fix a representative w˙ ∈ G. (i) Every g ∈ PwP can be written in a unique way as g = uwp, ˙

u ∈ U ∩ wU− w−1 , p ∈ P.

(ii) The map (u, p) → uwp ˙ is an isomorphism of varieties (U ∩ wU− w−1 ) × P ∼ = PwP. (iii) dim(PwP) = (w) + dim P. Proof Follows from [38, Theorem 28.4] and [72, Lemma 8.3.6]. For w = w , we have w U− w−1 = U. Then by Lemma 5.36 isomorphism U × P ∼ = Uw P = Pw P. Conjugating by w ,

an multiplication induces an isomorphism of varieties

  (ii) we have we see that

∼

U− × P −→ U− P. The set U− P is open and dense in G [38, 28.5]. It is called the big cell. A closed subgroup Q of G is called parabolic if G/Q is a projective variety. A closed subgroup of G is parabolic if and only if it includes a Borel subgroup ([38], page 135). A Borel subgroup is a minimal parabolic subgroup.

5.3 F -Reductive Groups So far in this chapter, we have worked with an algebraically closed field k. Now, we consider an arbitrary subfield F of k. A closed set V in An is said to be F -closed if V is the set of zeros of some collection of polynomials having coefficients in F . If I(V ) is generated by F polynomials, we say that V is defined over F . If V is F -closed in An , then V is defined over a finite, purely inseparable extension of F (Humphreys [38], Lemma 34.1). When F is perfect (e.g., of characteristic 0), the two notions coincide. If V is a closed subset of An defined over F , the set V (F ) = V ∩ F n

110

5 Reductive Groups

is called the set of F -rational points of V . In the affine case, when U ⊂ An and V ⊂ Am are defined over F , we say that a morphism ϕ : U → V is defined over F (or is an F -morphism) if the coordinate functions all lie in F [x1 , . . . , xn ]. Let G be an algebraic group. If G, along with µ : G × G → G and ι : G → G, are defined over F , we say that G is defined over F or is an F -group. Set G = G(F ). A torus T defined over F is called an F -torus. We call T F -split if T is F isomorphic to the product of r-copies Gm × · · · × Gm , where r = dim T = rank G. If so, then T(F ) ∼ = F× × ··· × F×

(r copies).

Let G be a connected reductive F -group. It is said to be quasi-split if it contains a Borel subgroup which is defined over F . We say that G is F -split if it has a maximal torus T which is defined over F and F -split. An example of an F -split group is GLn (see page 111). Let G be a reductive F -group and G = G(F ). For the structure theory of G, see Borel [8, Capter V] and Borel-Tits [9]. We will work with F -split groups, and for them, the theory is simpler. Here, we just mention some of the objects associated to G in the more general context of reductive F -groups. The role of a maximal torus is replaced by a maximal F -split torus. Let T be a maximal F -split torus. Its dimension is called the F -rank of G. Define

F = (G, T)

and WF = NG (T)/ZG (T).

The elements of F are the roots of G relative to T and they are called the F -roots of G. The group WF is called the Weyl group of G relative to F (see [72], Section 15.3). The set F is a root system, but not necessarily reduced. Given α ∈ F , it is possible to have 2α ∈ F (see [8, Remark 21.7]). By a parabolic subgroup of G = G(F ), we mean the group of F -rational points Q = Q(F ) of a parabolic subgroup Q defined over F . The group G may not posses a Borel subgroup defined over F . In the general theory of F -reductive groups, Borel subgroups are replaced by minimal parabolic subgroups. Let P be a minimal parabolic subgroup of G defined over F and P = P(F ). Then, from [8, Theorem 21.15], we have the Bruhat decomposition G=



P wP .

w∈WF

Assume G is F -split. Then F = , WF = W , and the structure theory for G transfers to G. In particular, for α ∈ , the root group Uα is defined over F , as well as the isomorphism xα : Ga → Uα . We use the same symbol for the corresponding

5.4 Z-Groups

111

isomorphism xα : F → Uα = Ua (F ). We follow the same convention with roots and coroots of T, and more generally for characters and cocharacters. Corresponding to λ ∈ X(T) is the character of T which we denote by the same letter λ. For a root α, we have α : T → F×

and α ∨ : F × → T .

A classification of connected reductive split F -groups by root data can be found in Section 16.3 of [72]. Similarly to Theorem 5.32, there are versions over F of the Isomorphism Theorem and the Existence Theorem [72, Theorems 16.3.2 and 16.3.3].

5.4 Z-Groups We start with the example of general linear groups. So far, we considered GLn (k) and GLn (F ), but general linear groups are also defined over rings. For a commutative ring with identity R, we define GLn (R) as the group of n × n matrices with coefficients in R and determinant in R × . Let L be a finite extension of Qp , with the ring of integers oL and the field of fractions Fq . Then we have the following groups GLn (Q), GLn (Qp ), GLn (L), GLn (Z), GLn (Zp ), GLn (oL ), GLn (Fp ), GLn (Fq ).

(5.4)

All these groups can be treated simultaneously with the notion of a Z-group. Let us look back at the description of GLn on page 92. All the polynomials defining GLn as an algebraic group have coefficients 1 and -1. Hence, they are defined over any subfield F of k. Moreover, they are defined over Z. Let T = Dn be the subgroup of diagonal matrices in GLn . This is a maximal torus in GLn . It is defined as the set of zeros of the polynomials xij , i = j , and these polynomials are defined over F and also over Z. The torus T is F -split and we have T(F ) = Dn (F ) ∼ = F × × · · · × F ×. Having a maximal F -split torus implies that GLn is F -split. The Z-points of T are all diagonal matrices t = diag(a1 , . . . , an ) with coefficients in Z such that det t ∈

112

5 Reductive Groups

Z× = {±1}. It follows T(Z) = {diag(a1 , . . . , an ) | ai ∈ {±1}} ∼ = GL1 (Z) × · · · × GL1 (Z).

(5.5)

Before we can talk about Z-split tori, we have to introduce Z-groups. We just mention that T being Z-split implies not only (5.5), but a more general property (5.6).

5.4.1 Algebraic R-Groups Let R be a commutative ring with unity. An R-algebra is assumed to be commutative and associative, with unity. We refer to Jantzen [40], Sections 1 and 2 in Part I, for the definition of an algebraic R-group, which follows the functorial approach of Demazure and Gabriel [25]. It is important to point out that an algebraic R-group GR is a functor from the category of R-algebras to the category of groups. Hence, GR : A → GR (A), where A is an R-algebra and GR (A) is a group. Below, we give basic examples. Example 5.37 Let R = Z. 1. The additive group over Z is the Z-group functor Ga,Z with Ga,Z (A) = (A, +) for all Z-algebras A. It is an algebraic Z-group. Its polynomial ring Z[Ga,Z] is isomorphic to Z[x]. 2. The multiplicative group over Z is the Z-group functor Gm,Z with Gm,Z (A) = A× for all Z-algebras A. It is an algebraic Z-group with Z[Gm,Z ] ∼ = Z[x, x −1 ]. 3. Let GLn (A) be the group of n × n invertible matrices over A. Then GLn,Z : A → GLn (A). is a Z-group functor from the category of all Z-algebras to the category of groups. It is an algebraic Z-group. Then for A = Q, Qp , L, Z, Zp , oL , Fp , Fq we obtain the groups listed in Eq. (5.4).

5.4 Z-Groups

113

We will work with algebraic Z-groups, and it will be useful to know that such groups can be realized as groups of matrices. Similarly to Theorem 5.2 over algebraically closed fields, we have the following theorem over Z: Theorem 5.38 Let HZ be an algebraic Z-group. Then there is a Z-isomorphism of HZ onto a closed subgroup of some GLn,Z . Proof This is Proposition 13.2 (i) in Expose VIB of [35] in the special case when S = Spec(Z).   We remark that, by the same reference, Theorem 5.38 holds more generally over regular rings of dimension at most 2.

5.4.2 Split Z-Groups In this section, we will call a split and connected reductive algebraic Z-group simply a split Z-group. Some authors call such groups Chevalley groups [23], but a Chevalley group is more commonly assumed to be semisimple, as in [19]. Let GZ be a split Z-group, with a split maximal torus TZ . Let r = dim TZ = rank GZ . Then TZ ∼ = Gm,Z ×· · ·×Gm,Z , the product of r copies of the multiplicative group over Z [40, II.1.1]. For any Z-algebra A, TZ (A) ∼ = Gm,Z (A) × · · · × Gm,Z (A) = A× × · · · × A× .

(5.6)

Root Subgroups Let = (GZ , TZ ) be the set of roots of GZ with respect to TZ . For each α ∈

there is a root homomorphism xα : Ga,Z → GZ with txα (a)t −1 = xα (α(t)a) for any Z-algebra A and all t ∈ TZ (A), a ∈ A, as described in [40, II.1.2]. The functor Uα : A → xα (Ga,Z (A)) = xα (A) is a closed subgroup of GZ called the root subgroup of GZ corresponding to α. Similarly to reductive groups over an algebraically closed field, split Z-groups are classified by their root data. The proof of Existence and Isomorphism theorems can be found in [26] or [23]. The following is Proposition 6.4.1 from [23].

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5 Reductive Groups

Theorem 5.39 A split Z-group is determined up to isomorphism by its associated reduced root datum, and every such root datum arises in this way. Two split Z-groups are isomorphic over Z if and only if they are isomorphic over C. From now on, we will write simply G instead of GZ .

5.5 The Structure of G(L) In this section, L is a finite extension of Qp , with ring of integers oL and unique maximal ideal pL , and G is a split connected reductive Z-group.

5.5.1 oL -Points of Algebraic Z-Groups We start with some general observations about algebraic Z-groups. Let H be an algebraic Z-group. Define H = H(L)

and H0 = H(oL ).

We write prn for the canonical map oL → oL /pnL and also for the induced map H0 → H(oL /pnL ) for any H. The kernel of prn in H0 is denoted Hn . Hence, Hn = {h ∈ H0 | h ≡ 1 mod pnL }. Finally, H (oL /pL ) is denoted H¯ . When we consider H0 and Hn as subgroups of H , we do not work with algebraic subgroups (defined using polynomial equations); we are abandoning the territory of algebraic groups and work with topological groups. There is a natural topology on H and we will describe it here. First, recall that by Theorem 5.38, there is a Z-isomorphism of H onto a closed subgroup of some GLm (closed in Zariski topology). The group GLm (L) is equipped with the standard topology coming from the norm ||g|| = max |gij |, i,j

for g = (gij ) ∈ GLm (L). We may consider H as a subgroup of GLm (L) and equip it with the subspace topology. Notice that H ⊆ GLm is defined as the set of zeros of some set of polynomials. Consequently, H is closed in GLm (L) with respect to the standard topology.

5.5 The Structure of G(L)

115

Lemma 5.40 Let H be an algebraic Z-group. (i) The groups Hn , n ∈ N, are normal subgroups of H0 , with H0 /Hn finite, and H0 ∼ = lim H0 /Hn . ← − n∈N

(ii) The projective limit topology on H0 coincides with the standard topology coming from any embedding H ⊆ GLm of Z-groups. (iii) The groups Hn , n ∈ N, are compact and open, and form a neighborhood basis of 1 in H0 . Proof Fix an embedding H ⊆ GLm of Z-groups and equip H with the subspace topology coming from the standard topology on GLm (L). As the kernel of prn : H0 → H(oL /pnL ), Hn is normal in H0 . For G = GLm , the quotients G0 /Gn are clearly finite. It follows that H0 /Hn are finite. The groups Gn , n ≥ 0 are compact and open in G, and they form a neighborhood basis of 1 in G. Then Hn = H ∩ Gn are compact and open in H , and form a neighborhood basis of 1 in H . The canonical projections H0 → H0 /Hn are compatible and give in the projective limit the continuous homomorphism ϕ : H0 → lim H0 /Hn . ← − n∈N

By Corollary 2.20, ϕ is surjective. It is clearly injective. Also, it is easy to see using Proposition 2.32 that ϕ is open. It follows that ϕ is an isomorphism of topological groups, thus identifying the standard topology on H0 with the projective limit topology.  

5.5.2 oL -Points of Split Z-Groups In the split connected reductive Z-group G, we fix a maximal Z-split torus T. We denote by W the Weyl group of G relative to T. For each w ∈ W we select a representative w˙ ∈ G(Z). Fix a Borel subgroup P containing T. We write for the roots of T in G and + (respectively, − ) for the set of positive (respectively, negative) roots determined by the choice of P. For each α ∈ , we have the root subgroup Uα and the morphism xα from the additive Z-group Ga to Uα . Let U be the unipotent radical of P. Then P = TU and U=

 α∈ +

Uα .

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5 Reductive Groups

− − The  unipotent radical of the opposite parabolic is denoted by U . We have U = α∈ − Uα .

Lemma 5.41 For n > 0, the product map is a homeomorphism ∼

Un− × Tn × Un −→ Gn . Proof Follows from [17, Proposition 1.4.4].

 

A nonzero unital ring is called a local ring if it has a unique maximal left ideal (equivalently, a unique maximal right ideal) [42, §19]. Proposition 5.42 For n ≥ 1, (i) The group Gn is a pro-p group. (ii) The ring oK [[Gn ]] is local, with the unique maximal ideal m(Gn ) = {µ ∈ oK [[Gn ]] | aug(µ) ∈ pK }. (iii) The group of units of oK [[Gn ]] is × oK [[Gn ]]× = oK [[Gn ]] \ m(Gn ) = {µ ∈ oK [[Gn ]] | aug(µ) ∈ oK }.

(iv) The ideals m(Gn ) , for  ≥ 1, form a fundamental system of neighborhoods of zero. (v) The ring oK [[Gn ]] has no zero divisors. Proof (i) First, we prove the assertion for G = GLm . Let M be the additive group of m × m matrices with coefficients in oL . Then any element g ∈ Gn can be written in a unique way as g = 1 + Ln X with X ∈ M. Define ϕ : Gn → Ln M/Ln+1 M by ϕ(1 + Ln X) = Ln X mod Ln+1 . This is a group homomorphism, with kernel Gn+1 . It follows Gn /Gn+1 ∼ = M/L M. = Ln M/Ln+1 M ∼ From this, we can show easily that Gn is a pro-p group. Now, take a general G. By Theorem 5.39, there is a Z-isomorphism of G onto a closed subgroup H of GLm , for some m. Then, we can identify Gn with Hn . As a subgroup of a pro-p group, Hn is also a pro-p group, proving the statement for Gn . (ii) and (iv) follow from Proposition 19.7 in [63]. For (iii), we use the property that in a local ring, every element is either a unit or it belongs to the maximal ideal [42, Theorem 19.1 and Proposition 19.2]. Assertion (v) follows from Theorem 4.3 of [2], because Gn has no torsion elements.  

5.5 The Structure of G(L)

117

The Bruhat decomposition holds over any field and so we have G=



P wP ˙

¯ = and G

w∈W



P¯ w˙ P¯ .

w∈W

For G0 = G(oL ), however, the Bruhat decomposition is no longer true and the ¯ This will require correct one is obtained by pulling back the decomposition for G. ¯ replacing the Borel subgroup with the standard Iwahori subgroup B = pr−1 1 (P ). Proposition 5.43 (i) The standard Iwahori subgroup B factors as B = U1− T0 U0 = U1− P0 . (ii) G0 is the disjoint union G0 =



B wB ˙ =

w∈W



B wP ˙ 0.

w∈W

Proof Since G1 is the kernel of the projection map pr1 : G0 → G(oL /pL ) and pr1 (P0 ) = P¯ , we have − − ¯ B = pr−1 1 (P ) = G1 P0 = U1 T1 U1 P0 = U1 P0 .

Here, we use the decomposition G1 = U1− T1 U1 from Lemma 5.41. Similarly, for any w ∈ W ¯ ˙ P¯ ) = B wB pr−1 ˙ = B wU ˙ 1− P0 = B wP ˙ 0. 1 (P w ¯ = Pulling back the Bruhat decomposition G decomposition of G0 .



w∈W

P¯ w˙ P¯ , we obtain the desired  

Proposition 5.44 The group G has the Iwasawa decomposition G = G0 P , which refines as the disjoint union G=



B wP ˙ .

w∈W

Proof See [18, Page 392] or [13, Proposition 4.4.3].     The decompositions G = ˙ and G = ˙ may look w∈W P wP w∈W B wP similar, but they are fundamentally different. The pieces B wP ˙ are approximately of the same size, while the dimension of P wP ˙ depends on the length of w (see Lemma 5.36). Also, if w = 1, the double coset P wP ˙ is not closed in G (see page 109). From [18, Proposition 1.3], we have the following:  (i) B wB ˙ ⊆ x≥w P xP ˙ ; (ii) B wB ˙ ∩ P wP ˙ = P0 wU ˙ 0.

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5 Reductive Groups

5.5.3 Coset Representatives for G/P We will give a set of coset representatives for G/P as in [3, Section 4.1]. We start by defining some unipotent subgroups of G0 . For w ∈ W , let Vw± = wU ˙ − w˙ −1 . Note that Vw± is the product of all the root subgroups Uα attached to roots α such that wα < 0. We define U − 1 = w˙ −1 B w˙ ∩ U0− = (w˙ −1 U0 w˙ ∩ U0− )(w˙ −1 U1− w˙ ∩ U0− ), w, 2

V ± 1 = wU ˙ − 1 w˙ −1 = (U0 ∩ wU ˙ − w˙ −1 )(U1− ∩ wU ˙ − w˙ −1 ). w, 2

w, 2

Then ± ± ⊂ V ± 1 ⊂ Vw,0 . Vw,1 w, 2

The subscript

1 2

indicates that V ± 1 is a mixture of Uα,1 ’s and Uα,0 ’s, while the w, 2

superscript ± indicates that some roots α are positive and some are negative. For U − 1 , we have w, 2

U1− ⊆ U − 1 ⊆ U0− w, 2

and U− 1 = w, 2

decomposition

Uα,0 ×

α0



Uα,1 .

α 0 for all α ∈ . We denote by + the set of all dominant weights. If  = {α1 , . . . , α }, define λ1 , . . . , λ ∈ E by λi , αj = δij . Then λ1 , . . . , λ ∈ + . They are called the fundamental dominant weights (relative to ). Recall that a Z-lattice in E is the Z-span of an R-basis of E. Lemma 6.1  is a lattice in E. It is spanned by the fundamental dominant weights λ1 , . . . , λ . Proof Notice that {λ1 , . . . , λ } is the dual basis to {2αi /(αi , αi ) | i = 1, . . . , } (relative to the inner product on E). It follows that {λ1 , . . . , λ } is a basis of E. Then any λ ∈ E can be written as λ=

 

ai λi ,

i=1

where a1 , . . . , a ∈ R. Applying  , αj on the equation above gives aj = λ, αj . In particular, if λ ∈ , we have ai ∈ Z, for all i = 1, . . . , . Hence,  is the Z-span of {λ1 , . . . , λ }.



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6 Algebraic and Smooth Representations

Example 6.2 Suppose  is the root system of type A2 . Then  = {α1 , α2 }, + = {α1 , α2 , α1 + α2 },

− = {−α1 , −α2 , −α1 − α2 }.

We can compute the fundamental weights λ1 and λ2 directly, using λi , αj = δij 1 1 (see [39, 13.1]). We get λ1 = (2α1 +α2 ) and λ2 = (α1 +2α2 ). Note that λ1 ⊥ α2 3 3 and λ2 ⊥ α1 .

Recall that the Weyl group W = W () is generated by reflections sα , α ∈ . It preserves the inner product on E and it maps  to itself. It follows that W also maps  to itself. We can define a partial order on E as follows: µ ≤ λ ⇐⇒ λ − µ ∈

 α∈

Nα =



Nα,

(6.2)

α∈+

where N = {0, 1, 2, . . . }. This is Lemma A from Humphreys [39, 13.2]: Lemma 6.3 Each weight is conjugate under W to one and only one dominant weight. If λ is dominant, then wλ ≤ λ for all w ∈ W , and if λ is strongly dominant, then wλ = λ only when w = 1. Denote by r the Z-span of . Then r is equal to the Z-span of any base of . It follows that r is a Z-lattice. It is called the root lattice. Since r ⊂ , we know from the theory of Z-lattices that /r is a finite group (called the fundamental group of ).

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127

Weights of a Reductive Group Let X(T) be the group of characters of T. Suppose that G is semisimple. We know from Theorem 5.24 that  is an abstract root system in E = R ⊗Z X(T), and rank  = rank G. We have r ⊆ X(T) ⊆  [38, 31.1]. The group /X(T) is called the fundamental group of G. If X(T) =  (so the fundamental group is trivial), we say that G is simply connected. If X(T) = r , we say that G is adjoint. In general, if we do not assume that G is semisimple, we know from Proposition 5.11 that the derived subgroup (G, G) is semisimple and rank  = rankss G = rank(G, G). We define the set of dominant weights of X(T) as X(T)+ = {λ ∈ X(T) | λ, α ∨ ≥ 0 for all α ∈ } = {λ ∈ X(T) | λ, α ∨ ≥ 0 for all α ∈ + }. where  , : X(T) × X∨ (T) → Z is the dual pairing as in (5.3) and α ∨ is the coroot corresponding to α. Similarly as in Eq. (6.2), we define a partial order on X(T)⊗Z R as follows:   µ ≤ λ ⇐⇒ λ − µ ∈ Nα = Nα, α∈

α∈+

where N = {0, 1, 2, . . . }. Example 6.4 Let G = GLn , with T, , and  as in Sect. 5.6. Let λ ∈ X(T). Then λ(diag(a1 , . . . , an )) = a1m1 . . . anmn , and λ = m1 e1 + · · · + mn en . For a simple root αi = ei − ei+1 , we have λ, αi∨ = mi − mi+1 (see Exercise 5.30). It follows λ ∈ X(T)+ ⇐⇒ m1 ≥ m2 ≥ · · · ≥ mn .

Dominant Bases of X(T) If G is simply connected, then the fundamental weights λ1 , . . . , λr belong to X(T) and form a Z-basis of X(T). If G is not simply connected, we still can find a Z-basis of X(T) which consists of dominant elements. The following is Lemma 2.1 from [3]. Lemma 6.5 The lattice X(T) has a Z-basis which consists of dominant elements.

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Proof Fix a Z-basis ω = {ω1 , . . . , ωr } of X(T). If λ = {λ1 , . . . , λr } is a set of linearly independent elements in X(T), let P(λ) denote the fundamental parallelepiped {t1 λ1 + · · · + tr λr : t1 , . . . , tr ∈ [0, 1)}. Let M(λ) be the change of basis matrix from ω to λ and define d(λ) = | det M(λ)| (the volume of P(λ)). Then d(λ) ∈ N and λ is a Z-basis of X(T) if and only if d(λ) = 1. Write D for the set of dominant elements in X(T) ⊗Z R. Then 1. D has nonempty interior, 2. aD = D for any positive real number a, and 3. D is convex. Any subset of X(T) ⊗Z R with the first two properties has the additional property that D∩X(T) contains a basis for X(T)⊗Z R. Fix some such basis λ = {λ1 , . . . , λr }. If P(λ) ∩ X(T) = {0}, then λ is a Z-basis for X(T). Otherwise, take a nontrivial element λ = t1 λ1 + · · · + tr λr of P(λ) ∩ X(T). Select a nonzero coefficient ti . Let λ be the basis obtained from λ by replacing λi by λ . As D is convex, the new basis is still contained in D ∩ X(T). Moreover, d(λ ) = |ti |d(λ) < d(λ). After a finite number of steps, we obtain a basis λ such that d(λ) = 1.



Weights of a Module Suppose that M is a T-module. Since T is F -split, Proposition 3.2.12 of [72] tells us that M decomposes as a direct sum of weight spaces M=



Mλ

λ∈X(T)

where Mλ = {m ∈ M | g(m ⊗ 1) = m ⊗ λ(g) for all g ∈ G(A) and all A} (also, see [40, I.2.10-11]). All λ such that Mλ = 0 are called the weights of M. Notice that the roots of G with respect to T are the weights of g = Lie(G) with respect to the adjoint action of T on g. If M is a G-module, then it is of course a T-module, and it decomposes as above into a direct sum of weight spaces. We can look at the action of the Weyl group. Let w˙ ∈ NG (T)(F ) be a representative of w ∈ W. It is easy to show that wM ˙ λ = Mw(λ) ,

for all λ ∈ X(T).

Lemma 6.6 If λ is maximal among the weights of M, then Mλ ⊆ M U . Proof We discuss the proof briefly and refer to [40, II.1.19] for details. Recall that for each root α ∈  we have an isomorphism xα : Ga → Uα . The tangent map dxα induces an isomorphism dxα : Lie(Ga ) → gα . Let Xα = dxα (1) and Xα,n = Xαn /(n!) ⊗ 1. Then, it can be shown that Xα,n Mλ ⊂ Mλ+nα .

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129

On the other hand, Xα determines the action of Uα on M: for m ∈ M and a ∈ A, we have  xα (a)(m ⊗ 1) = (Xα,n m) ⊗ a n . n≥0

If λ is maximal among the weights of M, then for any α ∈ + we have Mλ+nα = 0, for any n > 0. It follows Xα,n m = 0 for any m ∈ Mλ and n > 0. Hence, xα (a)(m ⊗ 1) = m ⊗ 1 so Mλ is fixed by Uα . This holds for all α ∈ + . for any m ∈ M and a ∈ A, Consequently, Mλ is fixed by α∈+ Uα = U.

Algebraic Induction Let H be a subgroup scheme of G. Each G-module has a natural structure of an Hmodule given by the restriction of the action of G(A) to H(A) for each F -algebra A. We obtain a functor resG H : {G-modules} → {H-modules}. On the other hand, let M be an H-module. Define −1 indG H (M) = {f ∈ Mor(G, Ma ) | f (gh) = h f (g)

for all g ∈ G(A), h ∈ H(A) and all A}, where Ma is as in Eq. (6.1). Then indG H (M) is a G-module, with G action by left translations. It is called the induced module of M from H to G. Then M →  indG (M) defines a functor H indG H : {H-modules} → {G-modules}. G For the properties of the functors resG H and indH , see section I.3 in [40]. We are interested in the parabolic induction. Let λ ∈ X(T). We extend λ trivially to U− and obtain a character of P− = TU− , which we denote again by λ. Then λ defines a one-dimensional P− -module denoted by F (λ) or simply λ. The induced module is −1 indG P− (λ) = {f ∈ F [G] | f (gp) = λ(p) f (g)

for all g ∈ G(A), p ∈ P− (A) and all A}.

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6 Algebraic and Smooth Representations

The Frobenius reciprocity [40, Section I.3.4] for indG gives us P− G ∼ HomG (V , indG P− (λ)) = HomP− (resP− V , λ)

(6.3)

for each G-module V . Lemma 6.7 Let λ ∈ X(T). Then indG P− (λ) = 0 if and only if λ is dominant.



Proof Jantzen [40], Proposition II 2.6. P− ,

Remark 6.8 We may switch the roles of P and but we have to be careful about the dominance condition. With the same definition of a dominant weight as before, we proceed as follows. Let µ ∈ X(T) and −1 indG P (µ) = {f ∈ F [G] | f (gp) = µ(p) f (g) for all g ∈ G(A), p ∈ P(A) and all A}.

Then indG P (µ) = 0 if and only if λ = −µ is dominant. Suppose λ ∈ X(T) is dominant. Let V = indG P− (λ). We would like to say more about the structure of V . As a T-module, it decomposes as V =



Vµ .

µ∈X(T)

From the Frobenius reciprocity (6.3), we have G G ∼ HomG (V , V ) = HomG (V , indG P− (λ)) = Homp− (resP− V , λ) ⊂ HomT (resT V , λ).

Then HomG (V , V ) = 0 implies that HomT (resG T V , λ) = 0 as well. It follows that λ is a weight of V . Lemma 6.9 Suppose λ ∈ X(T) is dominant, and V = indG (λ). Then P− (i) dim V U = 1 and V U = Vλ . (ii) λ is the unique maximal weight of V . (iii) Each weight µ of V satisfies w λ ≤ µ ≤ λ, where w is the longest element in W . Recall that the longest element w in W may be characterized by the property that w (α) < 0 for every positive root α, or equivalently w (α) < 0 for every α ∈ . Proof By property (6) on page 124, V U = 0. If f ∈ V U , then u.f = f for all u ∈ U, where the action is by left translation. Then f (utu− ) = λ(t)−1 f (1),

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131

for all u ∈ U(A), t ∈ T(A), u− ∈ U− (A), and all A. Hence, f (x) is determined by f (1), for all x ∈ UP− . Since UP− is dense in G, it follows that f ∈ V U is completely determined by f (1). Hence, dim V U = 1. The evaluation map ε : V → λ, f → f (1) is a homomorphism of P− -modules and is injective on V U . This implies V U ⊆ Vλ . Suppose µ is a maximal weight of V . By Lemma 6.6 and the above equation, we have Vµ ⊂ V U ⊆ Vλ . It follows µ = λ and V U = Vλ . Now, suppose µ is an arbitrary weight of V . Then µ is less than or equal to a maximal weight. Since V has the unique maximal weight λ, it follows µ ≤ λ. As observed earlier, wµ is a weight of V , for any w ∈ W . In particular, w µ is a weight of V , and hence w µ ≤ λ. By the definition of the partial order on X(T), λ − w µ ∈ α∈+ Nα. Then w λ − µ ∈



N(w α) =

α∈+



Nβ.

β∈−

Since − = −+ , it follows w λ ≤ µ.



Simple Modules Let V be a simple G-module. By property (1), V is finite-dimensional. Since U and U− are unipotent, from property (6) we have V U = 0

−

and V U = 0. −

Since T normalizes U and U− , it follows that V U and V U are T-submodules of V and so they decompose as direct sums of their weight spaces. Let λ be a weight of V U . Then VλU is also a P-module and by property (4) it decomposes as a direct sum of one-dimensional subspaces. On each of these subspaces, T acts as λ and U acts trivially, so each is isomorphic to λ. The same reasoning applies to P− . It follows that there exist λ, µ ∈ X(T) such that HomP− (V , λ) = 0 and

HomP (V , µ) = 0.

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6 Algebraic and Smooth Representations

Here, V is considered as a P− -module (respectively, P-module), and we could also write resG V (respectively, resG P V ). Then the Frobenius reciprocity (6.3) implies P− HomG (V , indG P− (λ)) = 0

and

HomG (V , indG P (µ)) = 0.

(6.4)

Lemma 6.7 implies that such λ is dominant. If λ ∈ X(T) is dominant, we define M(λ) = indG P− (λ). Proposition 6.10 (i) M(λ) is a simple G-module. (ii) Any simple G-module is isomorphic to exactly one M(λ) with λ dominant. Proof (i) This follows from [40], II.2.4 and II.5.6. We remark that we assume char F = 0. In general, the simple G-module associated to the dominant weight λ is M(λ) = socG indG P− (λ). (ii) Suppose V is a simple G-module. From (6.4), we have HomG (V , M(λ)) = 0, for some λ ∈ X(T). Since V and M(λ) are both simple, they must be isomorphic.



6.2 Smooth Representations In this section, G is a locally profinite group. We study smooth (that is, locally constant) representations of G. There is a well-established theory of smooth representations of G on complex vector spaces. We would like to know what parts of the complex theory carry over if we replace C with a finite extension K of Qp . There are two obstacles. First, K is not algebraically closed. Second, there is no K-valued Haar measure on G (see Sect. 3.4.2). We work over an arbitrary field F of characteristic zero. Vignéras in Chapter I of [78] presents a basic theory of representations of a locally profinite group over a commutative ring. We say that a function f : G → F is smooth if it is locally constant. Denote by Cc∞ (G, F ) the space of smooth compactly supported functions f : G → F . Combining [78, Ch.I, 2.3] and [14, 3.1], we define a left Haar integral on G with values in F as a nonzero linear functional I : Cc∞ (G, F ) → F which is invariant under left translations by G. If S is a compact open subset of G, we define µ(S) = I (1S ), where 1S is the characteristic function of S. We also use

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133

the integral notation and write  f (x)dµ(x) G

for I (f ). Notice that µ satisfies properties (i) and (ii) of Definition 3.45, but not necessarily the boundedness condition (iii). This causes no troubles for com∞ pactly supported smooth functions. Namely, for f ∈ Cc (G, F ), the integral G f (x)dµ(x) is in fact a finite sum and it is well-defined over F , as we can see from the proof of the proposition below. Proposition 6.11 There exists a nontrivial left Haar integral on G with values in F . It is unique up to a multiplicative constant. Proof Take a nonzero scalar c ∈ F and define µ(G0 ) = c. If f ∈ Cc∞ (G, F ), then f is compactly supported and locally constant. Hence, there exist a compact open subgroup  H of G0 and a finite number of elements g1 , . . . , gn ∈ G such that supp(f ) ⊆ ni=1 gi H and f (gi h) = f (gi ) for all h ∈ H and all i. Define  f (x)dµ(x) = G

n  i=1

c f (gi ). |G0 : H |

It is easy to show that this defines a nontrivial left Haar integral on G with values in F . Also, it is easy to show that any Haar integral must be of the same form, so it is unique up to a choice of c ∈ F .



6.2.1 Absolute Value The definition of an absolute value on a field is given in Definition A.6 in Appendix A.2.1. The absolute value on L can be computed as follows (see Appendix A.2.2). Let oL be the ring of integers of L and pL the unique maximal ideal of oL . The ideal pL is principal. Let L be a generator of pL . Denote by qL order of the residue field oL /pL . Then | L |L = qL−1 . Any nonzero x ∈ L can be written as x = Lm u, where × m = vL (x) ∈ Z and u ∈ oL . Then |x|L = | Lm u|L = qL−m . Notice that for any x ∈ L, |x|L is a rational number. Since char F = 0, we have Q ⊆ F . Then |x|L , which by Definition A.6 is an element of R≥0 , can also be considered as an element of F . Still, when we write oL = {x ∈ L | |x|L ≤ 1}, i.e., if we consider oL to be the unit ball in L, we take |x|L to be a real number. On the other hand, if we talk about representations built using | |L , then |x|L ∈ F (see Example 6.13).

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6.2.2 Smooth Representations and Characters Let V be an F -vector space. Recall that a representation (π, V ) of G is a group homomorphism π : G → Aut(V ). A character of G is a one-dimensional representation of G. Equivalently, a character χ of G is a group homomorphism χ : G → F ×. So far, we talked about Banach space representations (Chap. 4) and algebraic representations (Sect. 6.1). Now, we define smooth representations. Definition 6.12 (i) A representation (π, V ) is called smooth if for every v ∈ V , there is a compact open subgroup H of G such that π(h)v = v for all h ∈ H. (ii) A smooth representation (π, V ) of G is called admissible-smooth if the space of H -fixed vectors V H is finite-dimensional for every compact open subgroup H of G. Here, V H = {v ∈ V | π(h)v = v for all h ∈ H }. Notice that (π, V ) is smooth if and only it is continuous with respect to the discrete topology on V . The definition of a smooth representation does not require a topology on V (and does not see it, if such topology exists). Similarly, there are no continuity requirements on intertwining operators: if (π, V ) and (τ, W ) are smooth representations of G, we denote by HomG (π, τ ) or HomG (V , W ) the space of Gequivariant linear maps f : V → W . We denote by Repsm F (G) the category of smooth representations of G on F -vector spaces. Here are first examples of smooth representations: some smooth characters. Example 6.13 (a) Let G = GL1 , so G = G(L) = L× . Then | |L : L× → F × is a smooth character of L× . (b) Let G = GLn (L). Define ν : G → F × by ν(g) = | det(g)|L . Then ν is a smooth character of G. (c) Let G = GLn (L) and let χ : G → F × be an arbitrary character. Let H = SLn (L). Then H is the derived subgroup of G, that is, the group generated by all commutators [g, h] = ghg −1 h−1 , g, h ∈ G. Notice that χ([g, h]) = χ(g)χ(h)χ(g −1 )χ(h−1 ) = 1. It follows that χ is trivial on H and hence χ = η ◦ det where η : L× → F × is a character of L× . In particular, if χ is smooth, then χ = η ◦ det, for a smooth character η.

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135

6.2.3 Basic Properties Some basic results about smooth representations of G on F -vector spaces, where F is an arbitrary field of characteristic zero, can be found in Section 2 of Casselman [17]. In this section, we present some of these results. Also, cf. Vignéras [78, Ch.I]. We start with a comment on isomorphic fields.

Isomorphic Fields Let K be a finite extension of Qp and let Qp be an algebraic closure of Qp containing K. As we will see below, some results on smooth representations hold only if the coefficient field is algebraically closed. Because of that, it is sometimes useful to work first over Qp , and then deduce results for K. Hence, we would like to understand the category Repsm (G). Interestingly, it is Qp

The reason is because the fields Qp and C are isomorphic equivalent to as abstract fields (see Appendix A.2.3). This may sound contra-intuitive because we usually consider them equipped with standard metrics. However, for smooth representations, the coefficient field is considered as an abstract field. Repsm C (G).

Proposition 6.14 If the fields E and F are isomorphic as abstract fields, then the sm categories Repsm E (G) and RepF (G) are equivalent. Proof We apply restriction of scalars. Fix an isomorphism ι : F → E. Any Evector space V becomes an F -vector space via (a, v) → ι(a)v,

a ∈ F, v ∈ V .

Also, if f ∈ HomE (V , W ), then clearly f ∈ HomF (V , W ). Now, take a smooth representation (π, V ) ∈ Repsm E (G). The homomorphism π : G → AutE (V ) can be seen as a homomorphism π : G → AutF (V ) and therefore sm (π, V ) ∈ Repsm F (G). It is now easy to show that the resulting map RepE (G) → sm RepF (G) is an equivalence of categories.

Absolutely Irreducible Representations A representation V is irreducible if V does not contain G-invariant subspaces except zero and itself. Since we are not assuming that F is algebraically closed, we have to distinguish irreducibility from absolute irreducibility. Let (π, V ) be a representation of G on the F -vector space V and let E/F be a field extension. We denote by π ⊗ E or by πE the representation of G on the E-vector space VE = V ⊗F E given by (π ⊗ E)(g)(v ⊗ c) = π(g)v ⊗ c

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and call it the scalar extension of π to E. Some results about the extension of smooth representations from F to E and the descent from E to F can be found in [37]. We say that π is absolutely irreducible if π ⊗ E is irreducible for any field extension E/F . Lemma 6.15 Let (π, V ) be a representation of G over F and let E/F be a field extension. Then, (i) (π, V ) is smooth if and only if (π ⊗ E, V ⊗F E) is smooth. (ii) (π, V ) is smooth-admissible if and only if (π ⊗ E, V ⊗F E) is smoothadmissible. Proof If H is a subgroup of G, we claim that (V ⊗ E)H = V H ⊗ E. Clearly, V H ⊗ E ⊆ (V ⊗ E)H . For the converse inclusion, take  vi ⊗ ci ∈ (V ⊗ E)H . µ= f init e

We may assume that ci are linearly independent over F . By the assumption on µ, for any h ∈ H , we have (π ⊗ E)(h)µ = µ, that is,    π(h)vi ⊗ ci = vi ⊗ ci . (π ⊗ E)(h)( vi ⊗ ci ) =  It follows (π(h)vi − vi ) ⊗ ci = 0, and therefore π(h)vi = vi , because ci are linearly independent. Thus, vi ∈ V H , for all i, and hence µ ∈ V H ⊗ E, proving the claim. From the claim, the lemma follows easily.

Proposition 6.16 If F is algebraically closed and (π, V ) is a smooth irreducible representation of G, then it is absolutely irreducible. Proof This is Proposition 2.2.6 in Casselman [17].



Contragredient Let (π, V ) be a smooth representation of G. We denote by π ∗ the representation of G on the algebraic dual V ∗ = HomF (V , F ) given by (π ∗ (g))(v) = (π(g −1 )v), the smooth part of V ∗ and by for g ∈ G, v ∈ V , and  ∈ V ∗ . We denote by V π the ∗ . We call ( ) the contragredient of (π, V ). The following restriction of π to V π, V are Propositions 2.1.10 and 2.1.11 from [17]. Proposition 6.17 The following are equivalent (i) π is admissible; (ii) π is admissible; π ∼ (iii) the contragredient of π is isomorphic to π, that is, = π.

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137

Proposition 6.18 The functor π → π is contravariant and exact.

Tensor Product of Representations Assume that we have another locally profinite group H . Then G × H is also locally profinite. If (π, U ) is a representation of G and (τ, V ) a representation of H , then π ⊗ τ is the representation of G × H on the space U ⊗ V given by (π ⊗ τ )(g, h)(u ⊗ v) = π(g)u ⊗ τ (h)v for g ∈ G, h ∈ H , u ∈ U , and v ∈ V . Proposition 6.19 (i) If (π1 , V1 ) and (π2 , V2 ) are irreducible admissible-smooth representations of G and H , respectively, then (π1 ⊗ π2 , V1 ⊗ V2 ) is an irreducible admissiblesmooth representation of G × H . (ii) If (π, V ) is an irreducible admissible-smooth representation of G × H , then there exist irreducible admissible-smooth representations (π1 , V1 ) and (π2 , V2 ) of G and H , respectively, such that π ∼ = π1 ⊗ π2 . Proof (i) is Proposition 2.6.3 of [17] and (ii) is Proposition 2.6.4 of [17].



If η is a character of G and (π, V ) a representation of G, then it is customary to denote by η ⊗ π the representation of G on V defined by (η ⊗ π)(g) = η(g)π(g),

g ∈ G.

6.2.4 Admissible-Smooth Representations The following two results are well-known for complex representations; we reprove them here for representations over F .

the set of Lemma 6.20 Let H be a compact open subgroup of G. Denote by H equivalence classes of irreducible smooth representations of H . Then

is countable; (i) H

is finite-dimensional. (ii) every ρ ∈ H

N the set Proof For a compact open normal subgroup N of H , let us denote by H

such that the restriction of ρ to N is trivial. If V is a representative of all ρ ∈ H

N , then we can consider V as a representation of the of an equivalence class in H finite group H /N. This representation is also irreducible, so it is finite-dimensional. By Maschke’s theorem [24, Theorem 10.8], the group ring F [H /N] is semi-

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simple. Theorem 4.3 of [43, XVII, §4] implies that there is only a finite number

N is finite. of equivalence classes of simple F [H /N]-modules. Hence, H Define Hn = H ∩ Gn . Then {Hn | n ∈ N} is a neighbourhood basis of identity in

is contained in H

Hn , for H consisting of compact normal subgroups. Every ρ ∈ H



some n. On the other hand, each HHn is finite. This implies that H is countable.

Proposition 6.21 Let (π, V ) be an admissible-smooth representation of G on an F -vector space V . Let H be a compact open subgroup of G. Then V is isomorphic to a countable direct sum  m(ρ)ρ V ∼ = ρ

, and the multiplicity m(ρ) < ∞. where ρ runs over a set of representatives of H Proof Take an arbitrary nonzero vector v ∈ V . Then v ∈ V N , for some compact open subgroup N of H . We may assume N is normal in H ; if not, we can replace it by h∈H/N hNh−1 . Denote by Uv the H -subrepresentation of V generated by v. We can consider Uv as a representation of the finite group H /N. Then Uv is finite-dimensional, because it is spanned over K by the finite set {π(h)v | h ∈ H /N}. Maschke’s theorem [24, Theorem 10.8] tells us that Uv is completely reducible as an H /N-representation. Then it is completely reducible as an H -representation as well.  Now, V = v∈V Uv is a sum of irreducible H -representations. Theorem 15.3 of [24] implies that V is a direct sum of irreducible H -subrepresentations, so we can write V = i∈I Ui . For an irreducible representation ρ of H , set V (ρ) =



Ui .

Ui ∼ =ρ

This is the ρ-isotypic component of V . There exists a compact open normal subgroup N of H such that ρ|N is trivial. Then Ui ⊆ V N for all Ui ∼ = ρ. It follows V (ρ) ⊆ V N , and hence dim V (ρ) < ∞. The set {i ∈ I | Ui ∼ = ρ} is finite and V (ρ) ∼ = m(ρ)ρ, where m(ρ) < ∞. Finally, V =



V (ρ) ∼ =



ρ

is countable. The sum is countable because H

m(ρ)ρ.

ρ



6.2.5 Smooth Principal Series From now on, L is a finite extension of Qp and G = G(L) is the group of L-points of a split connected reductive Z-group G.

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139

Let χ : T → F × be a smooth character of T . We extend it trivially to U and obtain a smooth character of P = T U , which we denote by the same letter χ. We denote by sm IndG P (χ)

the space of all functions f : G → F satisfying (I) f (gp) = χ(p)−1 f (g) for all p ∈ P , g ∈ G, and (II) there exists a compact open subgroup Hf of G such that f (hg) = f (g), for all h ∈ Hf , g ∈ G. sm by left translation (L f ) = f (g −1 x). Notice that The group G acts on IndG g P (χ) condition (II) can be written as Lh f = f , for all h ∈ Hf . This means that f belongs sm is a smooth representation to the space of Hf -fixed vectors, and hence IndG P (χ) of G. We call it the smooth principal series induced by χ from P to G. There is a more general concept of parabolically induced representations, where the minimal parabolic subgroup P is replaced by an arbitrary parabolic subgroup Q, and the character χ is replaced by a smooth representation of the Levi factor of Q. For details, see Casselman [17, Section 3]. sm is with respect to the action of G by left Remark 6.22 The definition of IndG P (χ) translation. If we instead want to use the right translation, we proceed as follows. Let V be the space of functions f : G → F satisfying

(I ) f (pg) = χ(p)f (g) for all p ∈ P , g ∈ G, and (II ) there exists a compact open subgroup Hf of G such that f (gh) = f (g), for all h ∈ Hf , g ∈ G. Then G acts on V by (Rg f )(x) = f (xg). Similarly, the algebraic induction could be defined using the right translation. sm To explain how the two forms of induction relate, let U = IndG P (χ) . For f ∈ −1 U , define ϕ(f ) : G → F by ϕ(f )(g) = f (g ). Then (II) for f implies (II ) for ϕ(f ). Also, ϕ(f )(pg) = f (g −1 p−1 ) = χ(p)f (g −1 ) = χ(p)ϕ(f )(g), for any p ∈ P and g ∈ G, proving (I ). This shows ϕ(f ) ∈ V . The corresponding map ϕ : U → V is clearly a linear bijection. It is easy to check that ϕ ◦ Lg = Rg ◦ ϕ. Hence, ϕ is an intertwining operator, and the representations U and V are equivalent. sm Although the character χ is trivial on U , the induced representation IndG P (χ) depends on the choice of P = T U ⊃ T . We start with the following observation.

Exercise 6.23 Suppose σ : G → G is an automorphism of G. If π is a representation of G, we denote by σ π the representation of G defined by σ π(g) =

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6 Algebraic and Smooth Representations

π(σ −1 (g)). Let χ : T → F × be a smooth character of T . Prove that sm ∼ sm IndG = σ IndG P (χ) . σ (P ) (σ χ)

If we apply the above exercise to the conjugation by w ∈ W , we get sm (wχ)sm ∼ IndG = IndG P (χ) , wP w−1

where wχ denotes the character of T defined by wχ(t) = χ(w−1 tw),

t ∈ T.

Then, if we want to compare the induction from P with the induction from the opposite Borel subgroup P − , we have to conjugate by the longest element w ∈ W . Proposition 6.24 Let χ : T → F × be a smooth character of T and η : G → F × a smooth character of G. Then sm sm IndG = η ⊗ IndG P (η ⊗ χ) P (χ) . sm and by V the Proof Notice that η ⊗ χ = ηχ. Denote by U the space of IndG P (χ) G G sm sm space of IndP (ηχ) . Then π = η ⊗ IndP (χ) is the representation of G on U given by π(g) = η(g)Lg , or

π(g)f (x) = η(g)f (g −1 x) for g, x ∈ G and f ∈ U . For f ∈ U , define ϕ(f ) : G → F by ϕ(f )(x) = η(x −1 )f (x). Then ϕ(f )(gp) = η((gp)−1 )f (gp) = (ηχ)(p−1 )ϕ(f )(g). It follows ϕ(f ) ∈ V , and f → ϕ(f ) defines a map ϕ : U → V . The map is clearly linear and bijective. We claim ϕ ∈ HomG (π, V ). To prove it, we have to show that ϕ(π(g)f ) = Lg ϕ(f ) for all f ∈ U , g ∈ G. Evaluating the left hand side at x ∈ G, we get ϕ(π(g)f )(x) = η(x −1 )π(g)f (x) = η(x −1 )η(g)f (g −1 x). On the other hand, (Lg ϕ(f ))(x) = ϕ(f )(g −1 x) = η(x −1 g)f (g −1 x). This proves the claim, thus proving that the representations π and V are equivalent.



6.2 Smooth Representations

141

Exercise 6.25 Let χ : T → F × be a smooth character of T . If E/F is a field extension, prove that sm sm IndG = IndG ⊗ E. P (χ ⊗ E) P (χ)

Notice that χ ⊗ E is simply χ taken as χ : T → E × . Using alternate notation, the sm = IndG (χ)sm . above equality is written as IndG E P (χE ) P G sm Let us compare the algebraic induction indG P (µ) to IndP (χ) . Assume L ⊆ F , because otherwise, there are no algebraic representations of G = G(L) on F -spaces except constants. G We write indG P (µ) for indP (µ)(F ). The algebraic induction starts with an algebraic character µ, while the smooth induction starts with a smooth character χ. The only character of T which is both algebraic and locally constant is the trivial character 1. Recall that the group X(T) of algebraic characters of T is written additively. Then the trivial character 1 corresponds to 0 ∈ X(T). The simple module V = L(0) has the highest weight 0. From Lemma 6.9, we see that 0 is the only weight of V . Then V = V0 , the weight space corresponding to the weight 0. It follows L(0) = 1 and

indG P (1) = 1, where 1 on the right is the trivial one-dimensional representation of G. On the other sm consists of smooth functions. Only constant functions are both hand, IndG P (1) smooth and algebraic. We have G sm indG P (1) → IndP (1) .

Normalized Induction Normalization is very important for complex smooth representations because normalized induction preserves complex unitarity (see [17, Proposition 3.1.4]). Here, we introduce normalization so that certain statements have nice and symmetric form (see Propositions 6.26, 6.28, 6.31, and 6.33). For normalization, we have to assume 1/2 that F contains a square root of qL , which we fix and denote by qL . Let u = Lie(U ) be the Lie algebra of the unipotent radical U of P . Then P acts on u by adjoint action. For p ∈ P , define δP (p) = | det Adu (p)|L as in [17, Section 1.5] or [69, (2.3.6)]. Then δP : P → F × is a smooth character of P called the modulus character. (See Exercise 6.30 for an explicit description of δP for GL2 (L).) It is trivial on U , hence essentially a character of T . If χ is a

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6 Algebraic and Smooth Representations

smooth character of T , we define 1/2

sm iG,P (χ) = IndG P (δP χ) .

This is called normalized induction. Some authors define the modulus character as −1/2 in the definition of normalized induction, thus the inverse of our δP and put δP getting the same iG,P (χ) (see [53, Remarque II.3.7]). Proposition 6.26 The contragredient of iG,P (χ) is isomorphic to iG,P (χ −1 ). Proof Follows from Theorem 2.4.2 of [17], since the contragredient of χ is equal to χ −1 .



Composition Factors of Principal Series Let (π, V ) be a smooth representation of G. Suppose there exists a finite sequence 0 = V0 ⊂ V1 ⊂ · · · ⊂ V = V

(6.5)

of G-invariant subspaces of V such that Vi /Vi−1 is irreducible for i = 1, . . . , . The sequence (6.5) is called a composition series or a Jordan-Hölder series of V . Exercise 6.27 Suppose 0 = V0 ⊂ V1 ⊂ · · · ⊂ V = V and 0 = U0 ⊂ U1 ⊂ · · · ⊂ Um = V are two composition series of a smooth representation (π, V ) of G. Prove that  = m and that there is some permutation σ of {1, . . . , } such that Uσ (i) /Uσ (i)−1 ∼ = Vi /Vi−1 ,

1 ≤ i ≤ .

Based on Exercise 6.27, if (π, V ) is a smooth representation of G with composition series (6.5), we say that  is the length of V , and Vi /Vi−1 are the composition factors. Proposition 6.28 Assume F = C or Qp . Let χ be a smooth character of T . Then, (i) The length of iG,P (χ) is at most the order of the Weyl group. (ii) Let w ∈ W . Then, the composition factors of iG,P (χ) and iG,P (wχ) are same. (iii) If π is a composition factor of iG,P (χ), then there exists w ∈ W and an embedding of π into iG,P (wχ). Proof For F = C, the assertions follow from [17], Corollary 6.3.9 and Theorem 6.3.11. The result for F = Qp then follows from the proof of Proposition 6.14. Namely, if we fix an isomorphism of abstract fields ι : C → Qp , then every representation (π, V ) ∈ Repsm (G) can be considered as a representation in Repsm C (G).

Qp



Corollary 6.29 Let F = K, a finite extension of Qp . Let χ : T → K × be a smooth character. Then the length of iG,P (χ) is at most the order of the Weyl group.

6.2 Smooth Representations

143

Proof The length of iG,P (χ) is less than or equal to the length of iG,P (χ) ⊗F Qp . The latter representation is isomorphic to iG,P (χ ⊗F Qp ) by Exercise 6.25, and its length is at most the order of the Weyl group by Proposition 6.28.



6.2.6 Smooth Principal Series of GL2 (L) and SL2 (L) In this section, G = GL2 (L) and H = SL2 (L). We consider representations on K-vector spaces, where K is a finite extension of Qp . We assume that K contains a 1/2

square root of qL , which we fix and denote by qL . Denote by T the diagonal torus in G and by P ⊂ G the group of upper triangular matrices. Exercise 6.30 We defined the modulus character on P as δP (p) = | det Adu (p)|L . Prove that it is given by

δP

ab 0c



a    =  . c L

Given a smooth character χ : T → K × , we consider the normalized principal series iG,P (χ). The groups G and H have the same Weyl group. It is isomorphic to the symmetric group S2 , so it has two elements. Corollary 6.29 then implies that the length of a principal series representation of G or H is at most two. Write W = {1, w}. A smooth character χ : T → K × can be written as χ = χ1 ⊗ χ2 , where χ1 and χ2 are characters of L× . Then wχ = χ2 ⊗ χ1 . −1

We first consider the special case when χ = δP 2 . Then −1

sm iG,P (δP 2 ) = IndG P (1) .

As explained earlier, this representation contains the trivial one-dimensional repre−1

sentation 1. By the above discussion, the length of iG,P (δP 2 ) is two. It follows that −1

the quotient iG,P (δP 2 )/1 is irreducible. It is called the Steinberg representation and it is denoted by St. We have the following exact sequence −1

0 → 1 → iG,P (δP 2 ) → St → 0.

(6.6)

The sequence does not split over Qp by Zelevinsky [80, Proposition 1.11 (b)] and consequently, it does not split over K. Furthermore, taking contragredient and using Propositions 6.18 and 6.26, we obtain 1

→ iG,P (δ 2 ) → 1 → 0. 0 → St P

(6.7)

144

6 Algebraic and Smooth Representations −1

1

Notice that w(δP 2 ) = δP2 . By Proposition 6.28, the composition factors of

− ∼ iG,P (δP 2 )Qp and iG,P (δP2 )Qp are isomorphic. Consequently, St Qp = StQp . Then ∼ Lemma 5.1 of [52] tells us that St = St. 1

1

Proposition 6.31 Let χ1 ⊗χ2 : T → K × be a smooth character. The representation iG,P (χ1 ⊗ χ2 ) is reducible if and only if χ1 χ2−1 = | |±1 L . (i) If χ1 χ2−1 = | |−1 L , then there is a smooth character η such that (χ1 , χ2 ) = −1

−1

1

1

(η| |L 2 , η| |L2 ). The representation iG,P (η| |L 2 ⊗ η| |L2 ) fits in the following exact sequence −1

1

0 → η ◦ det → iG,P (η| |L 2 ⊗ η| |L2 ) → (η ◦ det) ⊗ St → 0. The above sequence does not split. (ii) If χ1 χ2−1 = | |L , then there is a smooth character η such that (χ1 , χ2 ) = 1

−1

1

−1

(η| |L2 , η| |L 2 ). The representation iG,P (η| |L2 ⊗ η| |L 2 ) fits in the following exact sequence 1

−1

0 → (η ◦ det) ⊗ St → iG,P (η| |L2 ⊗ η| |L 2 ) → η ◦ det → 0. The above sequence does not split. Proof If we consider representations over Qp , then we know from [80, Proposition 1.11 (a)] that iG,P (χ1 ⊗ χ2 )Qp is reducible if and only if χ1 χ2−1 = | |±1 L . Consequently, over K, if χ1 χ2−1 = | |±1 L , then iG,P (χ1 ⊗ χ2 ) is irreducible. −1

1

Assume (χ1 , χ2 ) = (η| |L 2 , η| |L2 ). We can write −1

χ1 ⊗ χ2 = (η ◦ det) ⊗ δP 2 . If we twist the exact sequence (6.6) by η ◦ det and apply Proposition 6.24, we obtain assertion (i). Similarly, assertion (ii) follows from the exact sequence (6.7).

For future reference, we state the same result using unnormalized induction. The statement can also be found in Section 9.6 of [14]). Corollary 6.32 Let χ1 ⊗ χ2 : T → K × be a smooth character. The representation sm is reducible if and only if χ = χ or χ χ −1 = | |2 . In particular, IndG 1 2 1 2 P (χ1 ⊗ χ2 ) L sm and IndG (δ )sm = IndG (| | ⊗ | |−1 )sm are reducible. IndG L P (1) P P P L Next, we consider H = SL2 (L). Let PH = P ∩ H . Let TH = T ∩ H be the torus of diagonal matrices in H , which is isomorphic to L× via the map a → diag(a, a −1). Given a character χ of TH , we now consider iH,PH (χ). Since

6.2 Smooth Representations

145

1

δ 2 (diag(a, a −1)) = |a|L , we see that iH,PH (χ) does not depend on the choice of the square root of q that we fixed in the beginning. Restricting the exact sequence (6.6) to H , we get 0 → 1 → iH,PH ( | |−1 L ) → St → 0. Similarly, restricting (6.7) to H , we get 0 → St → iH,PH ( | |L ) → 1 → 0. Proposition 6.33 Let χ : TH → K × be a smooth character. Suppose that the smooth normalized induced representation iH,PH (χ) is reducible. Then either χ = 2 | |±1 L or χ = 1, χ = 1. Proof For representations over Qp , we know that iH,PH (χ)Qp is reducible if and

2 only if either χ = | |±1 L or χ = 1, χ = 1. This follows from the discussion in [34, ch. 2, §3–5]. It is assumed in [34] that the residue characteristic is odd, but the properties of these representations (in particular, their reducibility) also hold for residue characteristic two (see also [34, ch. 2, §8]). Irreducibility over Qp implies irreducibility over K, proving the proposition.



Suppose that χ is a nontrivial quadratic character. Then iH,PH (χ)Qp is a direct sum of two components, and these components are inequivalent by Tadi´c [74, 1.2]. It is remarked in Section 4 of [68] that iH,PH (χ) over K could be irreducible (but, of course, not absolutely irreducible).

Chapter 7

Continuous Principal Series

In this chapter, we use the notation listed in Notation in Part II (see page 89). In particular, Qp ⊆ L ⊆ K is a sequence of finite extensions, G is a split connected reductive Z-group and P = TU is a Borel subgroup of G. We study the continuous principal series of G = G(L) and G0 = G(oL ). Our approach is based on the Schneider-Teitelbaum duality from Chap. 4. More specifically, we use Theorem 4.43, which tells us that the duality map V → V  is an anti-equivalence between the category of admissible Banach space representations of G0 and the category of finitely generated K[[G0]]-Iwasawa modules. Let χ : P → K × be a continuous character and χ0 its restriction to P0 = P(oL ). In Sect. 7.1, we establish some basic properties of the continuous principal series −1 0 G −1 IndG P0 (χ0 ) and IndP (χ ). In particular, we prove that they are Banach. After that, we work on the dual side and study the corresponding (K[[G0]], G)-modules. 0 −1 In Theorem 7.12, we prove that the dual of IndG P0 (χ ) is isomorphic to M (χ) = K[[G0]] ⊗K[[P0 ]] K (χ) . In the rest of the chapter, we describe the structure of M (χ) . In Proposition 7.14, we introduce a decomposition M (χ) =

 w∈W

Mw(χ)

(χ)

into components Mw indexed by the Weyl group W of G. In Proposition 7.20, we (χ) (χ) give a projective limit realization of M0 = oK [[G0]] ⊗oK [[P0 ]] oK , namely (χ)

M0

  (χ) ∼ = lim oK [G0 /Gn ] ⊗oK [P0 ] oK . ← − n∈N

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Ban, p-adic Banach Space Representations, Lecture Notes in Mathematics 2325, https://doi.org/10.1007/978-3-031-22684-7_7

147

148

7 Continuous Principal Series (χ)

In Sect. 7.4, we describe Mw as a tensor product, thus obtaining a K[[B]]-module decomposition M (χ) ∼ =

 w∈W

K[[B]] ⊗K[[P w,± ]] K (wχ) 1 2

7.1 Continuous Principal Series Are Banach Let χ : T0 → K × be a continuous character of T0 . Since T0 is compact, the image × of χ is compact, and hence it must be contained in oK . We consider the continuous principal series representation G

IndP00 (χ −1 ) = {f ∈ C(G0 , K) | f (gp) = χ(p)f (g) for all p ∈ P0 , g ∈ G0 }, with the action of G0 by left translation (Lg f )(x) = f (g −1 x). The space G IndP00 (χ −1 ) is a closed subspace of the Banach space C(G0 , K), so it is itself a Banach space. As introduced in Sect. 3.2, the Banach topology on C(G0 , K) is induced by the supremum norm

f = sup |f (g)|. g∈G0

By Exercise 4.14, the action of G0 on C(G0 , K) by left and right translations is 0 −1 continuous. It follows that the action of G0 on IndG P0 (χ ) is also continuous. Similarly, we define the continuous principal series of G. If χ : T → K × is a continuous character, we define −1 IndG P (χ ) = {f : G → K continuous | f (gp) = χ(p)f (g) for all p ∈ P , g ∈ G}

with the action of G by left translations. G While the Banach space structure on IndP00 (χ −1 ) comes from the embedding 0 G −1 −1 IndG P0 (χ ) ⊂ C(G0 , K), for IndP (χ ), we do not have a natural ambient Banach −1 space. Instead, we proceed as follows. Set χ0 = χ|T0 . If f ∈ IndG P (χ ), then G0 −1 clearly f |G0 ∈ IndP0 (χ0 ). We will prove that the map f → f |G0 is a bijection

G0 −1 −1 from IndG P (χ ) to IndP0 (χ0 ) and we will use this bijection to define a Banach −1 space structure on IndG P (χ ) (Proposition 7.3). The proof that this structure, with the action of G by left translations, is a Banach space representation of G is given in Proposition 7.5.

7.1 Continuous Principal Series Are Banach

149

G

7.1.1 Direct Sum Decomposition of IndP 0 (χ0−1 ) 0

r Exercise 7.1 Suppose r G0 = i=1 Si is a disjoint union of compact open subsets. ∼ Prove r that f → i=1 f |Si defines an isomorphism of Banach spaces C(G0 , K) = i=1 C(Si , K), where the topology of the direct sum comes from the norm defined by Eq. (A.3) in Appendix A.3. Let χ : T → K × be a continuous character of T and χ0 = χ|T0 . Recall the disjoint union G0 =



wU ˙ − 1 P0

w∈W

w, 2

as in (5.7). If we apply Exercise 7.1 to the above disjoint union, we see that −1 0 IndG P0 (χ0 ) decomposes as G

IndP00 (χ0−1 ) =

 w∈W

G

IndP00 (χ0−1 )w ,

(7.1)

where G0 −1 −1 − 0 IndG P0 (χ0 )w = {f ∈ IndP0 (χ0 ) | supp(f ) ⊂ wU 1 P0 }. w, 2

Proposition 7.2 (i) For every w ∈ W , we have the following isomorphism of topological vector spaces: −1 − 0 ∼ IndG P0 (χ0 )w = C(Uw, 1 , K). 2

(ii) G

IndP00 (χ0−1 ) =

 w∈W

G IndP00 (χ0−1 )w ∼ =

 w∈W

C(U − 1 , K). w, 2

Proof (i) Recall that we fixed a set W˙ of representatives for the elements of W. Let w˙ be −1 0 the representative for w. Notice that an element of IndG P0 (χ0 )w is determined − ˙ by its restriction to wU 1 , which is continuous. Moreover, given a continuous w, 2

150

7 Continuous Principal Series

function h : U − 1 → K, we may define w, 2

fh (g) =

⎧ ⎨h(u)χ0 (p),

g = wup, ˙ u ∈ U − 1 , p ∈ P0

⎩0,

g∈ / wU − 1 P0 .

w, 2

(7.2)

w, 2

G

Then h → fh is a K-linear bijection C(U − 1 , K) → IndP00 (χ0−1 )w . Since w, 2

× χ(P0 ) ⊆ oK , it follows fh = h , and hence h → fh is an isomorphism of topological vector spaces. (ii) Follows from (i) and Eq. (7.1).



From Proposition 5.45, we know that G/P 0 have the same set of coset  and G0 /P −1 representatives. The disjoint union G = w∈W wU ˙ − 1 P implies that IndG P (χ ) w, 2

decomposes as the direct sum −1 IndG P (χ ) =

 w∈W

−1 IndG P (χ )w ,

(7.3)

where − −1 G −1 IndG P (χ )w = {f ∈ IndP (χ ) | supp(f ) ⊂ wU 1 P }. w, 2

−1 So far, we have not defined the topology on IndG P (χ ). From the above decompoG −1 sition, it is natural to equip IndP (χ ) with the topology coming from identifying − −1 IndG P (χ )w with C(U 1 , K). w, 2

Proposition 7.3 Let χ : T → K × be a continuous character. −1 (i) IndG P (χ ) is a Banach space with norm

f = sup |f (g)|.

(7.4)

g∈G0

−1 (ii) The restriction f → f |G0 is a topological isomorphism from IndG P (χ ) to G0 −1 IndP0 (χ0 ). (iii) −1 IndG P (χ ) =

 w∈W

−1 ∼ IndG P (χ )w =

 w∈W

C(U − 1 , K). w, 2

7.1 Continuous Principal Series Are Banach

151

Proof Similarly as in the proof of Proposition 7.2, we can define a bijection h → Fh −1 from C(U − 1 , K) to IndG P (χ )w . We use this bijection to define the topology on w, 2

−1 IndG P (χ )w . We will prove that it is equivalent to the Banach topology induced by the norm (7.4). We have G0 − −1 −1 ∼ ∼ IndG P (χ )w = C(Uw, 1 , K) = IndP0 (χ0 )w , 2

where the second isomorphism follows from Proposition 7.2 (i). Moreover, given −1 h ∈ C(U − 1 , K), the corresponding functions Fh ∈ IndG P (χ )w and fh ∈ w, 2 G0 −1 IndP0 (χ0 )w satisfy

Fh |G0 = fh . It follows that the restriction f → f |G0 is G

−1 0 −1 a topological isomorphism from IndG P (χ )w to IndP0 (χ0 )w . Then Eqs. (7.1) and (7.3) imply G0 −1 −1 ∼ IndG P (χ ) = IndP0 (χ0 ),

with the isomorphism given by the restriction f → f |G0 . Then the supremum norm −1 G −1 0

on IndG P0 (χ0 ) induces the norm (7.4) on IndP (χ ). Parts (i) and (ii) in Proposition 7.3 still hold if we replace G0 by an arbitrary compact subgroup H of G satisfying the Iwasawa decomposition G = H P . We −1 define the corresponding norm on IndG P (χ ) by

f H = sup |f (h)|. h∈H

If H = G0 , this norm is equal to the norm introduced earlier. In the proof of the lemma below, we follow the approach from an unpublished note by Peter Schneider. Lemma 7.4 Let H be an arbitrary compact subgroup of G satisfying the Iwasawa decomposition G = H P . −1 (i) The norm H on IndG P (χ ) is equivalent to . −1 (ii) The restriction f → f |H is a topological isomorphism from IndG P (χ ) to H −1 IndH ∩P (χ ).

Proof (i) Since H is totally disconnected, the projection H → H /P has a continuous section. It follows that there exists a compact open subset A ⊂ H such that ∼ the multiplication map induces a homeomorphism A × H ∩ P −→ H. The Iwasawa decomposition G = H P then induces a homeomorphism ∼

A × P −→ G.

(7.5)

Let H  be another maximal compact subgroup of G such that G = H  P . We will prove that the norms H and H  are equivalent. By (7.5), for any h ∈

152

7 Continuous Principal Series

H  , there exist unique hP ∈ P and hA ∈ A such that h = hA hP . Define c = sup |χ(hP )|. h ∈H 

−1 Then c = 0 and moreover c < ∞, by compactness of H  . Take f ∈ IndG P (χ ).        Then for any h ∈ H , |f (h )| = |f (hA hP )| = |χ(hP )| · |f (hA )| ≤ c · |f (hA )|. Since A ⊂ H , it follows

f H  = sup |f (h )| ≤ c · sup |f (h)| = c · f H . h ∈H 

h∈H

Switching H and H  , we see that there exists c > 0 such that f H ≤ c · −1

f H  for all f ∈ IndG P (χ ). This proves that the norms H and H  are equivalent, also implying assertion (i). −1 (ii) By (i), we may consider IndG P (χ ) equipped with the norm H . Define −1 ϕ : IndG P (χ ) → C(A, K) −1 by f → f |A. Then ϕ is a linear map. Recall that f ∈ IndG P (χ ) satisfies ∼ f (gp) = χ(p)f (g). This together with the homeomorphism A × P −→ G implies that ϕ is bijective. As usual, we consider C(A, K) equipped with the supremum norm. Notice that χ|H ∩P : H ∩ P → K × has compact image, so × the image must be contained in oK . It follows |χ(p)| = 1 for all p ∈ H ∩ P . G −1 Consequently, for f ∈ IndP (χ ) we have

f H = sup |f (h)| = h∈H

sup

(a,p)∈A×(P ∩H )

|f (ap)| = sup |f (a)| = ϕ(f ) . a∈A

It follows that ϕ is an isometry, and hence it is a topological isomorphism. We have −1 ∼ IndG P (χ ) = C(A, K)

and

−1 ∼ IndH H ∩P (χ ) = C(A, K)

where both isomorphisms are given by restrictions and fit in the following commutative diagram of restriction maps

Then the top arrow is also an isomorphism, proving (ii).



7.1 Continuous Principal Series Are Banach

153

−1 Proposition 7.5 The space IndG P (χ ), with norm (7.4) and the G-action by left translations, is a K-Banach space representation of G. −1 Proof We know from Proposition 7.3 that IndG P (χ ) is a Banach space. By Definition 4.13, we have to show that G acts on V by continuous linear automorphisms such that the map G × V → V describing the action is continuous. Fix g ∈ G and write it as g = g0 p, with g0 ∈ G0 and p ∈ P . Set H = p−1 G0 p. −1 Then for any f ∈ IndG P (χ ), we have

Lg (f ) = sup |f (p−1 g0−1 x)| = sup |f (hp−1 )| x∈G0

h∈H

= |χ(p−1 )| · f H .

This together with Lemma 7.4 shows that Lg is a continuous linear automorphism on V . We know that G0 × V → V is continuous. To show that G × V → V is continuous, take g ∈ G, f ∈ V , and an open neighborhood U of Lg f in V . By continuity of G0 × V → V , there exist a neighborhood U0 of 1 in G0 and a neighborhood UV of Lg f in V such that Lx h ∈ U

for all x ∈ U0 , h ∈ UV .

By continuity of Lg , there exists a neighborhood W of f such that Lg h ∈ UV for all h ∈ W . Then for any y = xg ∈ U0 g and any h ∈ W we have Ly h = Lxg h = Lx (Lg h) ∈ U, completing the proof.



7.1.2 Unitary Principal Series As defined in Sect. 4.4.1, a Banach space representation (π, V ) of G is called unitary if the group action is norm-preserving, that is, π(g)v = v for all g ∈ G and v ∈ V . A character χ : T → K × is unitary if |χ(t)| = 1 for all t ∈ T . −1 Lemma 7.6 If χ : T → K × is a continuous unitary character, then IndG P (χ ) is unitary. −1 Proof Since |χ(p)| = 1 for all p ∈ P and G = G0 P , for any f ∈ IndG P (χ ) we have

f = sup |f (g)| = sup |f (g)|. g∈G0

g∈G

Then for any h ∈ G we have Lh f = supg∈G |f (h−1 g)| = supg∈G |f (g)| = f .



154

7 Continuous Principal Series

7.1.3 Algebraic and Smooth Vectors Any algebraic character of T is continuous. Also, any smooth character of T is continuous. We want to see how algebraic induction and smooth induction relate to continuous induction.

Algebraic Characters Let χ : T → K × be an algebraic character. Suppose χ is dominant. We apply Remark 6.8 to μ = χ −1 . (In the additive notation of Sect. 6.1, we would write μ = −χ.) Then we have the following algebraic representation −1 indG P (χ ) = {f : G → K algebraic | f (gp) = χ(p)f (g) for all g ∈ G, p ∈ P }. −1 This is a finite-dimensional subspace of IndG P (χ ) consisting of all polynomial G −1 −1 functions. As a finite-dimensional subspace, indP (χ ) is closed in IndG P (χ ). It −1 follows that IndG P (χ ) is topologically reducible Banach space representation of G G −1 and indP (χ ) is a finite-dimensional G-subrepresentation.

Smooth Characters Suppose χ : T → K × is a smooth character. The smooth induced representation −1 sm is usually considered just as an abstract vector space. Since we have a IndG P (χ ) natural embedding −1 sm −1 ⊂ IndG IndG P (χ ) P (χ ), −1 sm with the subspace topology. we can equip IndG P (χ ) ∞ Recall that C (G0 , K) is the subspace of C(G0 , K) consisting of smooth (i.e., locally constant) functions. By Exercise 3.20, C ∞ (G0 , K) is dense in C(G0 , K).

Lemma 7.7 Suppose χ : T → K × is a smooth character. Then (i) −1 sm ∼ IndG = P (χ )

 w∈W

C ∞ (U − 1 , K). w, 2

−1 sm is dense in IndG (χ −1 ). (ii) IndG P (χ ) P −1 Proof Any element of IndG P (χ ) is a sum over w ∈ W of elements fh , where − h ∈ C(U 1 , K) and fh is defined as in the proof of Proposition 7.2. If h is w, 2

7.1 Continuous Principal Series Are Banach

155

smooth, then fh is also smooth, by smoothness of χ. Since C ∞ (U − 1 , K) is dense w, 2

−1 sm is dense in IndG (χ −1 ). in C(U − 1 , K), it follows that IndG P (χ ) P w, 2



−1 sm is G-invariant, but it is not closed in IndG (χ −1 ). The space IndG P (χ ) P −1 an Remark 7.8 There is also the space of locally analytic vectors IndG P (χ ) , −1 sm −1 an G −1 IndG ⊂ IndG P (χ ) P (χ ) ⊂ IndP (χ ). −1 sm is closed in IndG (χ −1 )an , under an appropriate topology, and Then IndG P (χ ) P −1 both spaces are dense in IndG P (χ ) with respect to the Banach topology [61, Section 3].

7.1.4 Unitary Principal Series of GL2 (Qp ) Let G = GL2 (Qp ). Let P = T U be the group of upper triangular matrices, with T the group of diagonal matrices in G and U the unipotent radical of P . Recall the exact sequence of smooth representations sm 0 → 1 → IndG → St → 0. P (1)

The trivial one-dimensional representation 1 is of course closed in IndG P (1). Denote the quotient IndG (1)/1. Then we have the following exact sequence by St P

0 → 1 → IndG P (1) → St → 0. We know from Lemma 7.6 that IndG P (1) is unitary. The representation St is topologically irreducible and admissible as a representation of G [30, Lemma 5.3.3]. Remark 7.9 Emerton in [29] introduced the notion of a universal unitary Banach is the universal unitary completion of St [30, Lemma space completion. Then St 5.3.3]. The following proposition describes the reducibility of continuous principal series of GL2 (Qp ) induced from unitary characters. Proposition 7.10 Let χ1 ⊗ χ2 : T → K × be a unitary character. (i) If χ1 =  χ2 , then IndG P (χ1 ⊗ χ2 ) is topologically irreducible. (ii) If χ1 = χ2 = χ, then IndG P (χ ⊗ χ) fits in the following exact sequence 0 → χ ◦ det → IndG P (χ ⊗ χ) → χ ◦ det ⊗St → 0. The above sequence does not split.

156

7 Continuous Principal Series



Proof This follows from Proposition 5.3.4 in [30].

If we compare the above proposition to the smooth case described in Corollary 6.32, we see that both detect reducibility for χ1 = χ2 . However, Proposition 7.10 does not cover the case χ1 χ2−1 = | |2p because it treats only unitary characters. Thus, it does not cover the induction from the modulus character δP . We will show in Example 8.18 that the representation IndG P (δP ) is topologically G sm irreducible. On the other hand, IndP (δP ) is reducible, as we know from Corollary 6.32.

7.2 Duals of Principal Series We will study continuous principal series using the Schneider-Teitelbaum duality described in Chap. 4. Our first step is to describe explicitly duals of principal series (Theorem 7.12). × Let χ : P0 → oK be a continuous character. By Lemma 4.29 and Corollary 4.30, it extends uniquely to a continuous homomorphism of oK -modules χ : oK [[P0 ]] → oK and a continuous homomorphism of K-algebras χ : K[[P0 ]] → K. The extension is achieved by ν, χ, where  ,  : K[[P0 ]] × C(P0 , K) → K is the canonical pairing described in Sect. 3.2.3. Hence, for ν ∈ K[[P0 ]] we have χ(ν) = ν, χ. (χ)

We denote by oK (respectively, K (χ) ) the corresponding one dimensional oK [[P0 ]]-module (respectively, K[[P0 ]]-module).

(χ)

7.2.1 Module M0

With K (χ) the one dimensional K[[P0 ]]-module defined above, we define M (χ) = K[[G0 ]] ⊗K[[P0 ]] K (χ)

(χ)

and M0

(χ)

= oK [[G0 ]] ⊗oK [[P0 ]] oK .

Notice that any element of M (χ) can be written as μ ⊗ 1 for some μ ∈ K[[G0 ]]. Hence, M (χ) is isomorphic to a quotient of K[[G0 ]]. To describe it explicitly, denote by N (χ) the left ideal in K[[G0]] generated by the elements of the form η − χ(η),

η ∈ K[[P0 ]].

(χ) (χ) (χ) Then M (χ) ∼ = K[[G0 ]]/N (χ) . Similarly, M0 ∼ = oK [[G0 ]]/N0 , where N0 is the appropriately defined left ideal in oK [[G0 ]].

7.2 Duals of Principal Series

157

Lemma 7.11 G

(i) If ν ∈ N (χ) and f ∈ IndP00 (χ −1 ), then ν, f  = 0.

G

(ii) We have a well-defined pairing  ,  : M (χ) × IndP00 (χ −1 ) → K given by μ ⊗ 1, f  = μ, f , where the pairing on the right is the canonical pairing  ,  : K[[G0]] × C(G0 , K) → K. Proof Assertion (ii) follows from (i). For (i), we have to show that for all μ ∈ G K[[G0 ]], η ∈ K[[P0 ]], and f ∈ IndP00 (χ −1 ), μ(η − χ(η)), f  = 0. G

To do so, we first observe that for all g ∈ G0 , p ∈ P0 , and f ∈ IndP00 (χ −1 ), g(p − χ(p)), f  = f (gp) − χ(p)f (g) = 0.

(7.6)

G

Next, fix p ∈ P0 and f ∈ IndP00 (χ −1 ) and define a map ϕp,f : K[[G0 ]] → K by ϕp,f : μ → μ(p − χ(p)), f . Then ϕp,f is a continuous K-linear map. Equation (7.6) tells us that the restriction of ϕp,f to G0 is zero. Corollary 4.30 then implies ϕp,f is also zero, so μ(p − χ(p)), f  = 0

(7.7)

0 −1 for all μ ∈ K[[G0 ]], p ∈ P0 , and f ∈ IndG P0 (χ ). Next, fix μ ∈ K[[G0 ]] and 0 −1 f ∈ IndG P0 (χ ), and define a map ϕμ,f : K[[P0 ]] → K by

ϕμ,f (η) = μ(η − χ(η)), f . This is again a continuous K-linear map and from Corollary 4.30 and Eq. (7.7) we get ϕμ,f = 0, completing the proof.

Define m(χ, n) = sup{m ∈ N ∪ {0} | χ(p) ∈ 1 + pm K for all p ∈ Pn }, where we follow the convention that p0K = oK and p∞ K = 0. Note that m(χ, n) = ∞ if and only if χ|Pn = 1. In any case, limn→∞ m(χ, n) = ∞ by continuity of χ.

158

7 Continuous Principal Series G

Theorem 7.12 The continuous dual of IndP00 (χ −1 ) is isomorphic to M (χ) = K[[G0]] ⊗K[[P0 ]] K (χ) . G

Proof Set V = IndP00 (χ −1 ). From the short exact sequence 0 −→ V −→ C(G0 , K) −→ C(G0 , K)/V −→ 0 we obtain, taking the continuous dual, 0 −→ H (χ) −→ K[[G0 ]] −→ V  −→ 0, ∼ where H (χ) = {μ ∈ K[[G0 ]] | μ, f  = 0 ∀f ∈ V }. Recall that M (χ) = K[[G0 ]]/N (χ) . From Lemma 7.11(i), N (χ) ⊆ H (χ) and we have a well-defined surjective map ϕ : K[[G0 ]]/N (χ) −→ V  = K[[G0 ]]/H (χ) ϕ(μ)(f ¯ ) = μ, f , where μ ∈ K[[G0 ]] is any representative of μ¯ ∈ K[[G0 ]]/N (χ) and f ∈ V . To prove that ϕ is injective, we will show that N (χ) = H (χ) , and this will be done by (χ) (χ) (χ) showing N0 = H0 , where H0 = H (χ) ∩ oK [[G0 ]]. Recall from Proposition 2.50 that oK [[G0 ]] can be expressed as the projective limit oK [[G0 ]] = lim oK /pm(n) K [G0 /Gn ]. ← − n∈N

m(χ,n)

Let us denote by prn the natural projection prn : oK [[G0 ]] → oK /pK Then (χ)

H0 (χ)

(χ)

[G0 /Gn ].

(χ)

= lim prn (H0 ). ← − n∈N

(χ)

(χ)

Define n : N0 → prn (H0 ) by n = prn ◦ ι, where ι : N0 → H0 is the embedding. We claim that n is surjective. (χ) (χ) To prove the claim, take μ¯ ∈ prn (H0 ). Then μ¯ = prn (μ), for some μ ∈ H0 . (χ) (χ) For w ∈ W , we denote by Hw,0 the set of elements of H0 supported on B wB ˙ =

7.2 Duals of Principal Series (χ)

wU ˙ − 1 P0 . Then H0 w, 2

=

159



w∈W

μ=

(χ)

Hw,0 and we can write

μw ,

w∈W

(χ)

μw ∈ Hw,0 .

 (χ) Then also μ¯ = w∈W μ¯ w , where μ¯ w ∈ prn (Hw,0 ).  ˙ − 1 P0 from Proposition 5.45. Fix Recall the decomposition G0 = w∈W wU w, 2

w ∈ W . Let {u1 , . . . , ur } be a set of coset representatives of U − 1 /Un− and {p1 , . . . , ps } a set of coset representatives of P0 /Pn . We can write μ¯ w = w˙

i

w, 2

ai,j ui pj Gn ,

j

m(χ,n)

, for all i, j . Fix i ∈ {1, . . . r}. Recall the isomorphism where ai,j ∈ oK /pK G − − IndP00 (χ0−1 )w ∼ , , K) to be = C(Uw, 1 K) form Proposition 7.2. Define hi ∈ C(U w, 1 2

2

the characteristic function of ui Un− and let fi ∈ V be the function attached to hi as in the proof of Proposition 7.2. It is defined as fi (x) =

 χ(p), 0,

x = wu ˙ i vp, v ∈ Un− , p ∈ P0 − P . x∈ / wu ˙ i Uw,n 0

G

Notice that for v = w we have μv , fi  = 0, because fi ∈ IndP00 (χ0−1 )w and μv is supported on vU ˙ − 1 P0 . Then μ, fi  = μw , fi  and this is equal to zero because (χ)

v, 2

μ ∈ H0 . The function fi is oK -valued, and we can define m(χ,n) f¯i = fi mod pK .

We claim that f¯i is Gn -invariant. If m(χ, n) = ∞, the statement is true because fi is locally constant. Assume m(χ, n) < ∞. Take g ∈ Gn and x ∈ G0 . If x ∈ / B wB, ˙ then f (g −1 x) = 0 = f (x). If x ∈ B wB, ˙ we can write x = wu ˙ j vp with v ∈ Un− and p ∈ P0 . Let h = (wu ˙ j v)−1 g −1 wu ˙ j v. Since Gn is normal in G0 , h is also an element of Gn and we know from Lemma 5.41 that we can write it as h = v2 p2 where v2 ∈ Un− and p2 ∈ Pn . Then Lg fi (x) = fi (g

−1

x) = fi (wu ˙ j vv2 p2 p) =

 χ(p2 )χ(p), 0,

if j = i, if j = i.

160

7 Continuous Principal Series m(χ,n)

The definition of m(χ, n) implies that χ(p2 ) ∈ 1 + pK

m(χ,n)

Lg fi (x) ≡ fi (x) mod pK

. Then

,

proving that f¯i is Gn -invariant. It follows that we have a well-defined μ¯ w , f¯i  ∈ m(χ,n) , and it is equal to zero because μw , fi  = 0. Moreover, since fi (uj ) = oK /pK δij , we have μ¯ w , f¯i  =

ai,j χ(p ¯ j ) = 0,

j m(χ,n)

where χ¯ = χ mod pK

. This holds for any i. It follows

μ¯ w = μ¯ w − = =



i

j

i

j



⎛ ⎞

ui ⎝ ai,j χ(p ¯ j )⎠ G n

i

j

ai,j ui pj Gn −

i

ai,j ui χ(p ¯ j )Gn

j

ai,j ui (pj − χ(p ¯ j ))Gn . (χ)

This is clearly an element of n (N0 ), thus proving that n is surjective. (χ) (χ) Hence, we have a family of compatible surjections n : N0 → prn (H0 ). By the universal property of projective limits, the corresponding continuous linear (χ) (χ) (χ) (χ) (χ) map  : N0 → H0 is equal to the embedding ι : N0 → H0 . Since N0 is compact and n are surjective, it follows from Corollary 2.20 that  = ι is surjective.

0 −1 Corollary 7.13 The continuous principal series IndG P0 (χ ) is an admissible Banach space representation. 0 −1 (χ) = K[[G ]] ⊗ Proof Since the dual of IndG 0 K[[P0 ]] P0 (χ ) is isomorphic to M (χ) K , we see that it is finitely generated as a K[[G0 ]]-module (it is generated by 0 −1 1 ⊗ 1 ∈ M (χ) ). By Definition 4.41, it follows that IndG

P0 (χ ) is admissible.

From Proposition 7.3, we have the following decomposition −1 IndG P (χ ) =

 w∈W

−1 ∼ IndG P (χ )w =

 w∈W

C(U − 1 , K). w, 2

7.2 Duals of Principal Series

161

Taking dual and applying Theorem 7.12, we get 

M (χ) ∼ =

D c (U − 1 , K) ∼ = w, 2

w∈W



K[[U − 1 ]]. w, 2

w∈W

(7.8)

Here, we use the fact that the continuous dual D c (U − 1 , K) is isomorphic to w, 2

K[[U − 1 ]] (see Theorem 3.44). The isomorphism (7.8) is abstract; we would like to w, 2

describe the summand K[[U − 1 ]] as a subspace of M (χ) . Define w, 2

G

Mw(χ) = {μ ⊗ 1 ∈ M (χ) | μ ⊗ 1, f  = 0, f ∈ IndP00 (χ −1 )w , w = w} (χ)

(χ)

(7.9)

(χ)

and Mw,0 = Mw ∩ M0 . Proposition 7.14 (i) We have a K[[B]]-module decomposition M (χ) =

 w∈W

Mw(χ) .

(ii) For every w ∈ W , define ϕw : K[[U − 1 ]] → M (χ) = K[[G0]] ⊗K[[P0 ]] K (χ) w, 2

by ϕw : μ → wμ ˙ ⊗ 1. Then ϕw induces an isomorphism of K-spaces ∼

K[[U − 1 ]] −→ Mw(χ) . w, 2

Moreover, ϕ=



ϕw :

w



∼

K[[U − 1 ]] −→ M (χ) w, 2

w∈W

is an isomorphism of K-spaces. (χ) (iii) As K[[V ± 1 ]]-modules, Mw ∼ = K[[V ± 1 ]]. More specifically, we have an w, 2

w, 2

isomorphism

∼

ψw : K[[V ± 1 ]] −→ Mw(χ) w, 2

given by ψw : μ → μw˙ ⊗ 1. Proof  (χ) (χ) (i) Directly from the definition of Mw , we obtain M (χ) = w∈W Mw , the G decomposition of M (χ) as a K-space. Since each subspace IndP00 (χ −1 )w is (χ)

B-invariant, Mw is also B-invariant, and hence a K[[B]]-module.

162

7 Continuous Principal Series G

(ii) Any f ∈ IndP00 (χ0−1 ) can be written as f =

G

fw ∈ IndP00 (χ0−1 )w .

fw ,

w∈W

Each summand fw is of the form fw = fhw , where hw ∈ C(U − 1 , K) and fhw w, 2

is the function attached to hw as in the proof of Proposition 7.2. Fix w ∈ W . Then the inclusion U − 1 → G0 induces an inclusion w, 2

oK [[U − 1 ]] → oK [[G0 ]] and hence K[[U − 1 ]] → K[[G0]]. Let η ∈ w, 2

w, 2

K[[U − 1 ]]. We claim that wη ˙ ⊗ 1, fw  = η, hw . Indeed, for general w, 2

f ∈ C(G0 , K), the value of wη ˙ ⊗ 1, f  = wη, ˙ f  is obtained by applying η to the function u → f (wu), ˙ (u ∈ U − 1 ). And, if f = fw this is precisely hw . w, 2

Moreover, for v = w, we have wη ˙ ⊗ 1, fv  = 0. Combining with (7.9) − and (i), we see that for each w, the subspace wK[[U ˙ 1 ]] ⊂ K[[G0 ]] maps (χ) (χ) isomorphically onto Mw . This proves Mw

w, 2

(χ) ∼ − ∼ = K[[Uw, = 1 ]]. Then also Mw 2

K[[V ± 1 ]], because the groups U − 1 and V ± 1 are conjugate. w, 2

w, 2

(iii) Let us look again at the disjoint unions G0 =

 w∈W

w, 2

wU ˙ − 1 P0 = w, 2

 w∈W

V ± 1 wP ˙ 0 w, 2

 (χ) (χ) and the direct sum M (χ) = w∈W Mw . If η ⊗ 1 ∈ Mw , then there exists a unique μ ∈ K[[U − 1 ]] such that η = wμ ˙ ⊗ 1. Similarly, there exists a unique ν ∈

w, 2 ± K[[V 1 ]] such w, 2

(χ)

that η = ν w˙ ⊗ 1. Then Mw

K[[V ± 1 ]]-modules.

± ∼ = K[[Vw, 1 ]] as 2

w, 2



Clearly, the isomorphism of K-spaces ϕw :

K[[U − 1 ]] w, 2

→

(χ) Mw

given by

ϕw : μ → wμ ˙ ⊗1 is an isomorphism of K[[U − 1 ]]-modules only for w = 1.

w, 2 (χ) (χ) Recall that M0 = oK [[G0 ]] ⊗oK [[P0 ]] oK . By Theorem 3.32, μ ∈ K[[G0 ]] lies in oK [[G0 ]] if and only if it maps elements of C(G0 , oK ) ⊂ C(G0 , K) into oK . It follows that, for any such μ, the image μ ⊗ 1 ∈ M (χ) maps oK -valued elements −1 0 of IndG P0 (χ0 ) into oK . Using Proposition 7.14 we show that this characterizes the image of oK [[G0 ]] in M (χ) .

7.2 Duals of Principal Series

163

Proposition 7.15 (χ)

(i) M0 is the set of elements of M (χ) which map oK -valued elements of G IndP00 (χ −1 ) into oK . (ii) We have an oK [[B]]-module decomposition (χ)

M0

=

 w∈W

(χ)

Mw,0 .

Proof (i) We have to prove that any class [μ] in M (χ) which maps oK -valued elements to oK has a representative in oK [[G0 ]]. By Proposition 7.14, we can take a representative of the form η = w∈W wη ˙ w with ηw ∈ K[[U − 1 ]]. Now, fix w, 2

w and take h ∈ C(U − 1 , K). Clearly, the function fh defined by (7.2) is oK w, 2

valued whenever h is. It follows that ηw maps C(U − 1 , oK ) into oK and hence w, 2

lies in oK [[U − 1 ]]. As this holds for all w and the representatives w˙ were taken w, 2

from G0 , it follows that η ∈ oK [[G0 ]]. (ii) Exercise.





Exercise 7.16 Let ϕ = w ϕw be the isomorphism from Proposition 7.14 (iii), where ϕw : μ → wμ ˙ ⊗ 1. From the embedding oK [[G0 ]] → K[[G0]] we obtain f =



fw :

w

 w∈W

(χ)

oK [[U − 1 ]] −→ oK [[G0 ]] ⊗oK [[P0 ]] oK , w, 2

˙ ⊗ 1. Prove that f is an isomorphism. where fw : μ → wμ (χ)

The space Mw is a K[[B]]-module, and so the isomorphism from Proposition 7.14 induces a K[[B]]-module structure on K[[V ± 1 ]]. The action of T0 w, 2

can be described explicitly. We start with the standard action of T0 on V ± 1 (by w, 2

conjugation), which induces the action of T0 on C(V ± 1 , K) given by t · h(u) = w, 2

h(t −1 ut). Then, the standard action of T0 on K[[V ± 1 ]] is given by w, 2

t · μ, h = μ, t −1 · h for μ ∈ K[[V ± 1 ]], t ∈ T0 , and h ∈ C(V ± 1 , K). Considering μ and t as elements w, 2

w, 2

of K[[G0]], the above formula implies t · μ = tμt −1 . Next, we want to describe the action of T0 on K[[V ± 1 ]] from Proposition 7.14. ∼

(χ)

w, 2

We use the isomorphism ψw : K[[V ± 1 ]] −→ Mw given by ψw : μ → μw˙ ⊗ 1. w, 2

164

7 Continuous Principal Series (χ)

If t ∈ T0 and μ ∈ K[[V ± 1 ]], then in Mw ⊂ M (χ) = K[[G0 ]] ⊗K[[P0 ]] K (χ) we w, 2

have

tμw˙ ⊗ 1 = tμt −1 w˙ ⊗ (w˙ −1 t w) ˙ · 1 = χ(w−1 tw)tμt −1 w˙ ⊗ 1 = wχ(t)tμt −1 w˙ ⊗ 1, where wχ is the character of T0 defined by wχ(t) = χ(w−1 tw). Then the action of t on μ is given by (t, μ) → wχ(t)tμt −1 . Let us denote this action by Awχ . It is equal to the standard action twisted by the character wχ. Then Awχ (t).μ = wχ(t)t · μ, and Awχ (t).μ, h = wχ(t)t · μ, h = wχ(t)μ, t −1 · h for all t ∈ T0 , μ ∈ K[[V ± 1 ]], and h ∈ C(V ± 1 , K). Combined with the action w, 2

w, 2

of K[[V ± 1 ]] on itself by left translation, this action of T0 makes K[[V ± 1 ]] into a w, 2

w, 2

K[[Q± 1 ]]-module, where w, 2

Q±

w, 12

= T0 V ± 1 = B ∩ wP0− w−1 . w, 2

Write K[[V ± 1 ]](wχ) for this K[[Q± 1 ]]-module structure on K[[V ± 1 ]]. Then we w, 2

w, 2

have proved:

w, 2

(χ) ± (wχ) . Lemma 7.17 As K[[Q± 1 ]]-modules, Mw ∼ = K[[Vw, 1 ]] w, 2

2

(χ)

7.3 Projective Limit Realization of M0

In the rest of the chapter, we follow [4]. (χ) In Proposition 7.20, we give a realization of M0 as the projective limit over (χ) n ∈ N of tensor products oK [G0 /Gn ]⊗oK [P0 ] oK . We start by proving two technical lemmas about those tensor products. Recall that m(χ, n) = sup{m ∈ N ∪ {0} | χ(p) ∈ 1 + pm K for all p ∈ Pn }. × Lemma 7.18 Let χ : P0 → oK be a continuous character and let n ∈ N. (χ)

m(χ,n)

(i) In oK [G0 /Gn ] ⊗oK [P0 ] oK , for any ξ ∈ oK [G0 /Gn ] and any b ∈ pK have ξ ⊗ b = 0.

we

(χ )

7.3 Projective Limit Realization of M0

165 (χ)

(ii) The oK -module oK [G0 /Gn ] ⊗oK [P0 ] oK is isomorphic to  w∈W

m(χ,n)

oK /pK

[U − 1 /Un− ] w, 2

Proof (i) If m(χ, n) = ∞, then there is nothing to prove. Assume m(χ, n) < ∞. For any p ∈ Pn and any ξ ∈ oK [G0 /Gn ], we have ξ = ξp, and hence ξ ⊗ (1 − χ(p)) = (ξ ⊗ 1) − (ξ ⊗ χ(p)) = (ξ ⊗ 1) − (ξp ⊗ 1) = 0. Now, take p0 ∈ Pn such that ordK (χ(p0 ) − 1) = m(χ, n). Then any b ∈ m(χ,n) can be written as b = b0 (1 − χ(p0 )) for some b0 ∈ oK . It follows pK ξ ⊗ b = ξ ⊗ b0 (1 − χ(p0 )) = b0 (ξ ⊗ (1 − χ(p0 ))) = 0. (ii) We first recall the disjoint union decomposition G0 =



w∈W

wU ˙ − 1 P0 . Define w, 2

(χ)

hw : oK [U − 1 /Un− ] → oK [G0 /Gn ] ⊗oK [P0 ] oK w, 2

μ → wμ ˙ ⊗ 1. Then



w

hw :



− − w oK [Uw, 1 /Un ] 2

(χ)

→ oK [G0 /Gn ] ⊗oK [P0 ] oK is easily seen

to be surjective. (χ) Next, we want to realize oK [G0 /Gn ] ⊗oK [P0 ] oK as the dual of a suitable space of functions. We consider the oK -module m(χ,n)

i(χ, n) := {f : G0 /Gn → oK /pK

| f (gp) = prm(χ,n) χ(p)f (g), for g ∈ G0 /Gn and p ∈ P0 /Pn }, m(χ,n)

where prm(χ,n) is the canonical projection oK → oK /pK (g, a) → λg,a , where λg,a (f ) = af (g),

. The mapping

a ∈ oK , g ∈ G0 /Gn , f ∈ i(χ, n)

extends to a surjective oK [P0 ]-middle linear map from oK [G0 /Gn ] × oK to the oK -module m(χ,n)

i(χ, n)∗ := HomoK (i(χ, n), oK /pK

).

166

7 Continuous Principal Series

This middle linear map then induces a linear map oK [G0 /Gn ] ⊗oK [P0 ] (χ) oK → i(χ, n)∗ . It is then easy to see that the kernel of the map from   m(χ,n) − − ∗ [U − 1 /Un− ]. w woK [Uw, 1 /Un ] into the i(χ, n) is w wpK w, 2 2

Lemma 7.19 Let μ ∈ oK [[G0 ]] and ν ∈ oK [[P0 ]]. Write μ = (μn )∞ n=1 and ν = (χ) ∞ (νn )n=1 as in Sect. 2.3. Then in oK [G0 /Gn ] ⊗oK [P0 ] oK we have μn νn ⊗ a = μn ⊗ χ(ν)a. Proof Let {cn }∞ n=1 be a sequence of functions as in Sect. 3.2.3: each cn : P0 → oK is right Pn -invariant and χ = limn→∞ cn . Let us make a reasonable and explicit choice of {cn }∞ n=1 . For each n, we select cn : P0 → oK which is constant on cosets of Pn , such that inside each coset there is at least one point p0 where cn (p0 ) = χ(p0 ). m(χ,n) Now, let m(χ, n) be the maximal integer such that χ(Pn ) ⊂ 1 + pK . If m(χ,n) p1 Pn = p2 Pn , then χ(p1 ) − χ(p2 ) ∈ pK . It follows m(χ,n)

cn (p) − χ(p) ∈ pK

,

for all p ∈ P0 .

m(χ,n)

Consequently, ξ, cn − χ ∈ pK for all ξ ∈ oK [[P0 ]]. Let {p1 , . . . , ps } be a set of coset representatives of P0 /Pn consisting of points satisfying cn (pi ) = χ(pi ) for all i. (By our construction of cn , such points exist.) Then we can write νn = a1 p1 Pn + · · · + as ps Pn , where ai ∈ oK . Define η = a1 p1 + · · · + as ps . This is an element of oK [P0 ] ⊂ oK [[P0 ]] such that ηn = νn . Since χ(η) = a1 χ(p1 ) + · · · + as χ(ps ) = a1 cn (p1 ) + · · · + as cn (ps ) = η, cn , it follows m(χ,n)

χ(η) = η, cn  = ηn , cn  = νn , cn  = ν, cn  ∈ χ(ν) + pK

.

(χ)

Now, in oK [G0 /Gn ] ⊗oK [P0 ] oK , we have μn νn ⊗ a = μn ηn ⊗ a = μn ⊗ χ(η)a. To show that the above expression is equal to μn ⊗ χ(ν)a, we observe that χ(η) − m(χ,n) , and apply Lemma 7.18(i).

χ(ν) ∈ pK

(χ )

7.3 Projective Limit Realization of M0

167

× Proposition 7.20 Let χ : P0 → oK be a continuous character trivial on U0 . Then (χ)

M0

  (χ) ∼ = lim oK [G0 /Gn ] ⊗oK [P0 ] oK . ← − n∈N

Proof As explained in Sect. 2.3, any μ ∈ oK [[G0 ]] can be written as μ = (μn )∞ n=1 , where μn ∈ oK [G0 /Gn ]. For each n ∈ N, we define a map (χ)

(χ)

ψn : oK [[G0 ]] × oK → oK [G0 /Gn ] ⊗oK [P0 ] oK (μ, a) → μn ⊗ a.

It follows from Lemma 7.19 that ψn is oK [[P0 ]]-middle linear. Hence, it gives rise to a linear map (χ)

(χ)

n : oK [[G0 ]] ⊗oK [[P0 ]] oK → oK [G0 /Gn ] ⊗oK [P0 ] oK . Now, (n )n∈N is a family of compatible continuous linear maps which map   (χ) (χ) oK [[G0 ]] ⊗oK [[P0 ]] oK to the inverse system oK [G0 /Gn ] ⊗oK [P0 ] oK . By n∈N the universal property of projective limits, there exists a continuous linear map (χ)

 : M0

  (χ) (χ) = oK [[G0 ]] ⊗oK [[P0 ]] oK → lim oK [G0 /Gn ] ⊗oK [P0 ] oK . ← − n∈N

(χ)

(χ)

This map is surjective because M0 = oK [[G0 ]] ⊗oK [[P0 ]] oK is compact and n are surjective (Corollary 2.20). For injectivity, we first recall from Proposition 7.14 and Exercise 7.16 the isomorphism f =

 w

fw :

 w∈W

∼

(χ)

oK [[U − 1 ]] −→ oK [[G0 ]] ⊗oK [[P0 ]] oK , w, 2

where fw : μ → wμ ˙ ⊗ 1 for μ ∈ oK [[U − 1 ]]. w, 2

For every w ∈ W, we have the following commutative diagram

168

7 Continuous Principal Series

The map hw is built from the natural projections m(χ,n)

oK [[U − 1 ]] → oK /pK w, 2

[U − 1 /Un− ], w, 2

using the universal property of projective limits. The map gw is defined as follows. We know from the proof of Lemma 7.18(ii) that the maps gn,w : m(χ,n) (χ) oK /pK [U − 1 /Un− ] → oK [G0 /Gn ] ⊗oK [P0 ] oK , given by gn,w : μ → wμ ˙ ⊗ 1, w, 2  are injective, and that gn = w gn,w is an isomorphism of oK -modules. Define gw = lim gn,w . Then gw is injective. Thus, we reduce our proof to proving the ← −n injectivity of hw , for all w ∈ W . Suppose η is a nonzero element of oK [[U − 1 ]] and write η = (ηn )∞ n=1 where w, 2

ηn ∈ oK [U − 1 /Un− ]. Then for each n we have w, 2

ηn =

cu u

u∈U − 1 /Un− w,

2

and for some n0 , u0 cu0 = 0. Then for all n ≥ n0 there exists u ∈ U − 1 /Un− such that |cu | ≥ |cu0 |. Then for all n sufficiently large we will have cu0 m(χ,n) hence the image of ηn in oK /pK [U − 1 /Un− ] is nonzero.

w, 2 m(n,χ) ∈ / pK ,

w, 2

and



7.4 Direct Sum Decomposition of M (χ) G

G

Recall the space IndP00 (χ −1 )w = {f ∈ IndP00 (χ −1 ) | supp(f ) ⊂ B wB}, ˙ and its (χ)

(χ)

dual Mw . The purpose of this section is to give a realization of Mw as a tensor product, analogous to the realization of M (χ) itself as K[[G0 ]] ⊗K[[P0 ]] K (χ0 ) . It is motivated by the case of G0 = GL2 (Zp ) described by Schneider and Teitelbaum in [64].

7.4.1 The Case G0 = GL2 (Zp ) Let G = GL2 . As in Chap. 5, we denote by P the Borel subgroup of upper triangular matrices, with Levi decomposition P = TU. The opposite parabolic subgroup P− = TU− consists of lower triangular matrices. The Weyl group W = W (G, T) =   {1, w}, and we select the representative w˙ =

01 . Let G0 = GL2 (Zp ). Let B be 10

the Iwahori subgroup of all matrices which are upper triangular modulo p. Then B

7.4 Direct Sum Decomposition of M (χ )

169

× decomposes as B = U1− T0 U0 . Let χ : T0 → oK be a continuous character. Define

Nχ = K[[B]] ⊗K[[P0 ]] K (χ)

− and Nwχ = K[[B]] ⊗K[[T0 U − ]] K (wχ) . 1

Then from [64, Section 4] we have the following isomorphism of K[[B]]-modules − M (χ) ∼ = Nχ ⊕ Nwχ

 The statement follows from the disjoint union decomposition G0 = B B wP ˙ 0 and the proof is left as an exercise. We will prove a general statement for an arbitrary G (see Corollary 7.24). − . Exercise 7.21 Let G0 = GL2 (Zp ). Prove M (χ) ∼ = Nχ ⊕ Nwχ

7.4.2 General Case G

We will prove that IndP00 (χ −1 )w is isomorphic as a B-module to a representation ˙ 0 w˙ −1 , and obtain the corresponding tensor product expression induced from B ∩ wP (χ) for Mw . To prepare for the proof, we introduce the following technical result.  Lemma 7.22 Let F be any field. Let + = S1 S2 be any partition of the positive roots into two disjoint sets. Take any numbering of S1 as {β1 , . . . , βn } and any numbering of S2 as {γ1 , . . . , γm }. Then 

 (b1 , . . . , bn ), t, (c1 , . . . , cm ) →  xβ1 (b1 ) . . . xβn (bn ) · t · xγ1 (c1 ) . . . xγm (cm )

is a bijection F n × T(F ) × F m → P(F ). Proof By§14.4 of [8], multiplying root subgroups gives an isomorphism of varieties α Uα → U, for any ordering of the roots. Since G is Z-split, ((b1 , . . . , bn ), (c1 , . . . , cn )) → xβ1 (b1 ) . . . xβn (bn )xγ1 (c1 ) . . . xγm (cm ) is a bijection F n × F m → U. On the other hand, P = T U, and we can conjugate t ∈ T to the middle.

  ± ˙ = w∈W V 1 wP ˙ 0 , where Recall the decomposition G0 = w∈W B wB w, 2

V ± 1 = B ∩ wU ˙ 0− w˙ −1 = (U0 ∩ wU ˙ − w˙ −1 )(U1− ∩ wU ˙ − w˙ −1 ). w, 2

170

7 Continuous Principal Series

Its “multiplicative complement” in B is P 1w,± = B ∩ wP ˙ 0 w˙ −1 = T0 (U0 ∩ wU ˙ 0 w˙ −1 )(U1− ∩ wU ˙ 0 w˙ −1 ), 2

as we will show in Lemma 7.23 below. The unipotent radical of P 1w,± is 2

V 1w,± = B ∩ wU ˙ 0 w˙ −1 = (U0 ∩ wU ˙ 0 w˙ −1 )(U1− ∩ wU ˙ 0 w˙ −1 ). 2

Lemma 7.23 (i) Multiplication induces a homeomorphism V ± 1 × P 1w,± → B. w, 2

(ii) As representations of B,

2

−1 B −1 0 ∼ IndG P0 (χ )w = IndP w,± wχ . 1 2

(iii) As K[[B]]-modules, Mw(χ) ∼ = K[[B]] ⊗K[[P w,± ]] K (wχ) . 1 2

Proof (i) We know that the multiplication induces an isomorphism of varieties U− × ∼ P −→ U− P. Conjugating with w, we get ∼

wU− w−1 × wPw−1 −→ (wU− w−1 )(wPw−1 ). Let us look at the oL -points contained in B. Recall that we write P¯ for P0 /P1 = P(oL /pL ) = B/G1 . Similarly, we define T¯ , U¯ and U¯ α for each root α. Given b ∈ B, let b¯ be the image in P¯ . We factor it as b¯ = u¯ 1 t¯u¯ 2 ,

u¯ 1 ∈



Uα (oL /pL ),

α>0, w−1 α0, w−1 α>0

Choosing representatives in Uα (oL ) and T (oL ), we obtain u1 , u2 and t such −1 −1 that g1 := u−1 ∈ G1 ⊂ B. Now using the Iwahori factorization of 1 bu2 t B we can write w˙ −1 g1 w˙ as v1 v2 with v1 ∈ U1− and v2 ∈ P1 = T1 U1 . Put vi = wv ˙ i w˙ −1 . Then g1 = v1 v2 with v1 ∈ wU ˙ 1− w˙ −1 and v2 ∈ T1 wU ˙ 1 w˙ −1 . w,± ±     Now b = u1 v1 v2 tu2 , and u1 v1 ∈ V 1 and v2 tu2 ∈ P 1 . w, 2

2

7.4 Direct Sum Decomposition of M (χ )

171

G

(ii) For f ∈ (IndP00 χ −1 )w , define ϕ(f ) : B → K by ϕ(f )(b) = f (bw). ˙ Then for b ∈ B and p ∈ P 1w,± , we have 2

ϕ(f )(bp) = f (bpw) ˙ = f (bw˙ w˙ −1 pw) ˙ = χ(w˙ −1 pw)f ˙ (bw) ˙ = wχ(p)ϕ(f ), because w˙ −1 pw˙ ∈ P0 by definition of P 1w,± . This proves that ϕ(f ) ∈ 2

IndB w,± (wχ −1 ). The resulting map P1 2

G

ϕ : (IndP00 χ −1 )w → IndB (wχ −1 ) P w,± 1 2

is clearly linear and injective. To show that it is surjective, we use the homeomorphism V ± 1 × P 1w,± → B. Then IndB w,± (wχ −1 ) ∼ = C(V ± 1 , K), w, 2

with f ∈

IndB w,± (wχ −1 ) P1

P1

2

w, 2

2

given by

2

(v ∈ V ± 1 , p ∈ P 1w,± ),

f (vp) = h(v)wχ(p),

w, 2

2

for some h ∈ C(V ± 1 , K). From this, it easily follows that ϕ is surjective. w, 2

Finally, we see directly from the definition that ϕ is B-equivariant under left translations, thus proving that ϕ is an isomorphism of B-representations. (iii) It follows readily from the definitions that μπ, f  = wχ(π)μ, f  for all μ ∈ K[[B]], π ∈ K[[P 1w,± ]], and f ∈ IndB w,± (wχ −1 ). It follows that the map P1 2 2   (μ, a) → aμ B is a middle-linear map from K[[B]]×K (wχ) to the −1 w,± (wχ P1 2

Ind

)

dual of IndB w,± (wχ −1 ), and hence gives rise to a map from K[[B]] ⊗K[[P w,± ]] P1

K (wχ)

1 2

2

to this dual. ± ± Since IndB w,± (wχ −1 ) ∼ ]] maps = C(Vw, 1 , K), it follows that K[[V P w, 1 1 2

2

2

isomorphically onto the dual, and from this it follows that the map from K[[B]] ⊗K[[P w,± ]] K (wχ) is surjective. 1 2

− ˙ The same reasoning used in Proposition 7.14 to show that wK[[U 1 ]] (χ)

surjects onto Mw

w, 2

may be used here to prove that K[[V ± 1 ]] surjects onto w, 2

K[[B]] ⊗K[[P w,± ]] K (wχ) . Since the map from K[[V ± 1 ]] to the dual of 1 2

w, 2

172

7 Continuous Principal Series

IndB w,± (wχ −1 ) is an isomorphism, the map from K[[B]] ⊗K[[P w,± ]] K (wχ) P1 2

onto the dual of

IndB w,± (wχ −1 ) P1

1 2

must be injective, which completes the proof.

2



Corollary 7.24 As K[[B]]-modules, M (χ) ∼ =

 w∈W

K[[B]] ⊗K[[P w,± ]] K (wχ) . 1 2

Proof From Proposition 7.14, we have a K[[B]]-module decomposition M (χ) =  (χ)

w∈W Mw . Lemma 7.23(iii) then gives the statement.

Chapter 8

Intertwining Operators

Throughout Part II, Qp ⊆ L ⊆ K is a sequence of finite extensions, G is a split connected reductive Z-group and P = TU is a Borel subgroup of G. Furthermore, G = G(L) and G0 = G(oL ). Additional notation is listed on page 89. In Sects. 8.1 and 8.2, we present the main results and proofs from [4]. The purpose is to describe the space of continuous G0 -intertwining operators G

G

HomcG0 (IndP00 (χ1 ), IndP00 (χ2 )) × for two continuous characters χ1 , χ2 : T0 → oL . As before, we apply the SchneiderTeitelbaum duality and work with the corresponding K[[G0]]-modules M (χ1 ) and M (χ2 ) . We show that  0 if χ1 = χ2 , HomK[[G0 ]] (M (χ1 ) , M (χ2 ) ) = K · id if χ1 = χ2 ,

(see Corollary 8.13). By duality, this gives us  G0 0 HomcG0 (IndG P0 (χ1 ), IndP0 (χ2 ))

=

0

if χ1 = χ2 ,

K · id

if χ1 = χ2 ,

(Corollary 8.14). So, in the world of continuous principal series on K-Banach spaces, the intertwining operators are essentially non-existent: there are no intertwiners between different principal series, and only possible self-intertwining operators are scalar multiples of the identity map. This stands in striking contrast to the world of smooth principal series, where intertwiners are abundant. As a consequence, the question of reducibility of continuous principal series cannot be solved using intertwining operators. In Sect. 8.4, we present some methods for addressing this question. In Sect. 8.4.1, we state Schneider’s conjecture © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Ban, p-adic Banach Space Representations, Lecture Notes in Mathematics 2325, https://doi.org/10.1007/978-3-031-22684-7_8

173

174

8 Intertwining Operators

and describe some irreducibility results which come from locally analytic vectors. In Sect. 8.4.2, we present the criterion for irreducibility from [3].

8.1 Invariant Distributions (χ )

In this section, we will prove that HomK[[P0 ]] (K (χ1 ) , Mw 2 ) = 0 for all w other than the identity. In fact, what we prove in Proposition 8.4 is a more general statement (χ ) (χ ) which allows us, in Corollary 8.6 to show that HomK[[B]] (Mw 1 , Mv 2 ) = 0 implies w = v.

8.1.1 Invariant Distributions on Vector Groups Recall that the space of continuous distributions D c (G0 , K) is isomorphic to K[[G0 ]]. In this section, we will study distributions on unipotent subgroups. We will prove that the only invariant distribution on a group isomorphic to several copies of Zp is the trivial one. If N is an abelian unipotent group, this applies to the groups Nn , n ∈ N. These are the “vector groups” of the title. r for some positive integer r and that μ ∈ K[[V ]] Lemma 8.1 Suppose V ∼ = oL satisfies

v · μ = μ,

∀v ∈ V .

Then μ = 0. That is, the space K[[V ]]V of V -invariant distributions on V is 0. Proof Let V0 = V and for n = 1, 2, 3 . . . let Vn be the image of the r-copies of r → V . So, [V : V ] = q r(n−m) for any nonnegative pnL under an isomorphism oL m n L integers m, n with m < n. Now, there is a constant c such that |μ(f )| ≤ c for all f ∈ C(V , oK ). This follows from the fact that μ = aμ0 for some a ∈ K and μ0 ∈ oK [[G0 ]], and |μ0 (f )| ≤ 1 for all f ∈ C(V , oK ). Then |μ(1Vm )| = |qLr(n−m)μ(1Vn )| ≤ c|qLr(n−m)| for all n, m. Since c|qLr(n−m)| → 0 as n → ∞ for each fixed m, we deduce that μ(1Vm ) = 0 for all m. But the space spanned by translates of these functions is C ∞ (V , K) and, as observed in Sect. 3.2.3, it is dense in C(V , K). 

Corollary 8.2 Take V as in Lemma 8.1. Regard K as a K[[V ]] module with trivial action. Then HomK[[V ]] (K, K[[V ]]) = 0.

8.1 Invariant Distributions

175

Proof The image of any element of HomK[[V ]] (K, K[[V ]]) is an element of K[[V ]]V .



8.1.2 “Partially Invariant” Distributions on Unipotent Groups Lemma 8.3 Let V0 ⊂ G0 be a subgroup. Let V1 be a closed subgroup of V0 which r for some r, and V a closed subset of V such that multiplication is isomorphic to oL 2 0 is a homeomorphism V1 × V2 → V0 . Then K[[V0 ]]V1 = 0 and hence HomK[[V1 ]] (K, K[[V0 ]]) = 0. Proof Since multiplication is a homeomorphism V1 × V2 → V0 , we have an injective map C(V1 , K) × C(V2 , K) → C(V0 , K). For each fixed nonzero h ∈ C(V2 , K) we get an injective map ih : C(V1 , K) → C(V0 , K). Explicitly, ih (f )(uv) = f (u)h(v),

(u ∈ V1 , v ∈ V2 ).

Assume that μ ∈ K[[V0]] is a V1 -invariant element. Then μ ◦ ih is an invariant element of K[[V1 ]]. Thus μ ◦ ih = 0, by Lemma 8.1. But this means that μ vanishes on ih .f for all f and all h. That is μ vanishes on the image of C(V1 , K) × C(V2 , K) in C(V0 , K). But the span of this image is dense, so μ must vanish identically.

 × Proposition 8.4 Suppose χ, ξ : T0 → oK are continuous characters (we allow χ = ξ ). If w, v ∈ W , w = v, then

HomK[[P w,± ]] (K (wχ) , Mv(ξ ) ) = 0. 1 2

Proof With V 1w,± as in Sect. 7.4.2, we have P 1w,± = T0 V 1w,± . We select a root γ 2

such that w−1 γ > 0 and v −1 γ < 0, and define  =

2

2

0, if γ > 0, 1, if γ < 0.

Then Uγ , ⊂ V 1w,± ∩ V ±1 . Clearly 2

v, 2

HomK[[P w,± ]] (K (wχ) , Mv(ξ ) ) ⊂ HomK[[Uγ , ]] (K, Mv(ξ ) ). 1 2

176

8 Intertwining Operators

∼ = K[[Vv,±1 ]] as a K[[Vv,±1 ]] and hence as a K[[Uγ , ]]-module, and it

(ξ )

But Mv

2

2

follows from Lemma 8.3 that HomK[[Uγ , ]] (K, K[[V ±1 ]]) = 0.



v, 2

In the proof of Corollary 8.6, we use the following result from abstract algebra: Theorem 8.5 (Adjoint Isomorphism) Suppose R and S are rings with unity. (i) If A is a left R-module, B is an (S, R)-bimodule, and C is a left S-module, then HomS (B ⊗R A, C) ∼ = HomR (A, HomS (B, C)). (ii) If A is a right R-module, B is an (R, S)-bimodule, and C is a right S-module, then HomS (A ⊗R B, C) ∼ = HomR (A, HomS (B, C)). 

Proof Rotman [56], Theorem 2.11. Corollary 8.6 Suppose χ, ξ : T0 → ξ ). If w, v ∈ W , w = v, then

× oK

are continuous characters (we allow χ =

HomK[[B]] (Mw(χ) , Mv(ξ ) ) = 0. (χ) Proof Since Mw ∼ = K[[B]] ⊗K[[P w,± ]] K (wχ) , we have 1 2

HomK[[B]] (Mw(χ) , Mv(ξ ) ) ∼ = HomK[[B]] (K[[B]] ⊗K[[P w,± ]] K (wχ) , Mv(ξ ) ). 1 2

We can regard K[[B]] as a bimodule with K[[B]] acting on the left and K[[P 1w,± ]] 2

acting on the right, and apply adjoint associativity (Theorem 8.5(i)). It follows that HomK[[B]] (K[[B]] ⊗K[[P w,± ]] K (wχ) , Mv(ξ ) ) 1 2

∼ = HomK[[P w,± ]] (K (wχ) , HomK[[B]] (K[[B]], Mv(ξ ))). 1 2

(ξ )

And HomK[[B]] (K[[B]], Mv ) has the structure of a left K[[P 1w,± ]]-module iso2

(ξ )

morphic to Mv by Theorems 1.15 and 1.16 of [56], so HomK[[P w,± ]] (K (wχ) , HomK[[B]] (K[[B]], Mv(ξ ))) ∼ = HomK[[P w,± ]] (K (wχ) , Mv(ξ ) ), 1 2

which is zero by Proposition 8.4.

1 2



8.1 Invariant Distributions

177

8.1.3 T0 -Equivariant Distributions on Unipotent Groups Let V be a T-stable unipotent Z-subgroup of G. The natural action of T on V by conjugation induces actions of T0 on Vn /Vm for all positive integers m, n with m > n. To describe T0 -equivariant distributions on V , we will consider their canonical pairing with characteristic functions of the form 1u0 Vn . Fix n > 0 and u0 ∈ V0 such that u0 ∈ / Vn . We will further decompose 1u0 Vn as the sum of characteristic functions of the form 1uVm , for m > n. To simplify notation, we define, for m > n, Qm = Qum0 ,n = {u0 vVm | v ∈ Vn }. This is a subset of the quotient group V0 /Vm . Lemma 8.7 The set u0 Vn = {u0 v | v ∈ Vn } is preserved under the action of Tn . Consequently, the set Qm is also preserved under the action of Tn . Proof From the hypothesis that V is T-stable, we deduce that there is a set S of roots such that taking any fixed order on the elements of S and multiplying in that order gives an isomorphism of Z-schemes  α∈S Uα → V. (Cf. [40, §1.7, p. 159]) In particular, we get a homeomorphism α∈S Uα (oL ) → V (oL ), which induces a  n ), for each n, from which we deduce that bijection α∈S Uα (oL /pnL ) → V (o /p L  L the preimage of Vn in α∈S Uα is α∈S Uα,n . Hence, we can write u0 as u0 =



xα (rα ),

α∈S

where rα ∈ oL . Let t ∈ Tn . For each root α ∈ S txα (rα )t −1 = xα (rα t α ) = xα (rα )xα (rα (t α − 1)). But t α ≡ 1 (mod pnL ), so xα (rα (t α − 1)) ∈ Vn .



Given u¯ ∈ Qm , we denote by OrbTn (u) ¯ its orbit under the action of Tn . Then OrbTn (u) ¯ ⊂ Qm ⊂ V0 /Vm is a finite set and we denote its cardinality by |OrbTn (u)|. ¯ Lemma 8.8 Let V be a T-stable unipotent Z-subgroup of G. Take u0 ∈ V0 and n > 0 such that u0 ∈ / Vn . Then ¯ = ∞, lim min ordp (|OrbTn (u)|)

m→∞ u∈Q ¯ m

where ordp is the p-adic valuation on Z.

178

8 Intertwining Operators

 Proof Let u¯ = u0 vVm ∈ Qm and write u0 v = α∈S xα (uα ). Denote by StabTn (u) ¯ the stabilizer of u¯ in Tn . Let t ∈ Tn . We may then note that  t · u¯ = t



 xα (uα ) t −1 Vm =

α∈S



txα (uα )t −1 Vm =

α∈S



xα (t α uα )Vm ,

α∈S

and hence t · u¯ = u¯ ⇐⇒



xα (t α uα )Vm =

α∈S



xα (uα )Vm

α∈S

⇐⇒ (t α − 1)uα ∈ pm L , for all α ∈ S. It follows StabTn (u) ¯ = {t ∈ Tn | ordL (t α − 1) ≥ m − ordL (uα ), for all α ∈ S}. Of course, uα ’s can be zero for some roots α. In this case ordL (uα ) = ∞ and the condition becomes vacuous. However, the requirement that u0 ∈ / Vn implies that for any u ∈ u0 Vn there will be at least one root α0 such that ordL(uα0 ) < n. For m > n+ordL (uα0 ), the condition ordL (t α0 −1) ≥ m−ordL(uα0 ) determines m−ordL(uα )−n

0 a subgroup of Tn of index qL . Indeed α0 is a homomorphism Tn → n α 1 + pL . The condition ordL(t 0 − 1) ≥ m − ordL (uα0 ) is equivalent to t α0 ∈

m−ordL(uα0 )

1 + pL

m−ordL (uα0 )

. If m − ordL (uα0 ) > n, then 1 + pL

m−ordL (uα0 )−n index qL

is a subgroup of

m−ordL (uα0 ) in and {t ∈ Tn : ∈ 1 + pL } is a subgroup of the same index in Tn . The actual stabilizer StabTn (u) ¯ is then a subgroup of this m−ordL (uα0 )−n subgroup, and its index is a multiple of qL . Hence,

1 + pnL

t α0

ordp (|OrbTn (u)|) ¯ ≥ (m − ordL(uα0 ) − n) ordp (qL ) which tends to ∞ with m.



The action of T0 on V0 (by conjugation) induces the action of T0 on C(V0 , K) given by t · h(u) = h(t −1 ut). Then we define an action of T0 on K[[V0 ]] by t · μ, h = μ, t −1 · h

(8.1)

for μ ∈ K[[V0]], t ∈ T0 , and h ∈ C(V0 , K). Lemma 8.9 Let μ be a nonzero element of K[[V0 ]]. Suppose there exists a character ξ of T0 such that t · μ = ξ(t)μ for all t ∈ T0 . Then ξ is trivial and μ = c · 1 for some scalar c. Proof Take a nonzero μ ∈ K[[V0]] and assume that μ, t −1 · h = ξ(t)μ, h for all t ∈ T0 .

8.1 Invariant Distributions

179

Suppose first that ξ is not smooth and consider the characteristic function 1vVn , for some v ∈ V0 and some n ∈ N. Then there exists t ∈ Tn such that ξ(t) = 1. Notice that t −1 · 1vVn = 1vVn . This follows from Lemma 8.7 if v ∈ / Vn and it holds trivially if v ∈ Vn . Then μ, t −1 · 1vVn  = μ, 1vVn  = ξ(t)μ, 1vVn  implies μ, 1vVn  = 0. This condition forces μ, h = 0 for all smooth h, and then for all h. Thus we are reduced to the case when ξ is smooth. To treat the case when ξ is smooth, we will show that μ, 1u0 Vn  = 0 for any u0 ∈ V0 and positive integer n such that u0 Vn does not contain the identity. There exists n0 such that the restriction of ξ to Tn0 is trivial. Fix u0 and n as above and assume n ≥ n0 . For m > n, 1 u0 V n =



1u¯ .

u∈Q ¯ m

Hence μ, 1u0 Vn  =



μ, 1u¯ .

u∈Q ¯ m

Now, let Tn act on V0 /Vm . We know from Lemma 8.7 that Qm is preserved under this action. Write [Tn \ Qm ] for a set of representatives of the distinct orbits in Qm . Then   μ, 1u0 Vn  = μ, t · 1u¯  u∈[T ¯ n \Qm ]



=

t ·u∈Orb ¯ ¯ Tn (u)

|OrbTn (u)|μ, ¯ 1u¯ 

u∈[T ¯ n \Qm ]

because μ, t · 1u¯  = ξ(t −1 )μ, 1u¯  = μ, 1u¯  for any t ∈ Tn . Since μ is a scalar multiple of some μ0 ∈ oK [[V0 ]] and μ0 , 1u¯  ∈ oK , it follows that minu∈Q ¯ m ordp (μ, 1u¯ ) is bounded independently of m. By Lemma 8.8, ¯ = ∞. lim min ordp (|OrbTn (u)|)

m→∞ u∈Q ¯ m

It follows that μ, 1u0 Vn  = 0, for any u0 ∈ V0 and positive integer n such that u0 Vn does not contain the identity. Thus μ is supported at the identity, so it is c · 1 for some c. It now follows easily that ξ is trivial. 

180

8 Intertwining Operators

× Corollary 8.10 Let ξ : T0 → oK be a continuous character. Let ι : K → K[[V0]] be the natural embedding defined by ι(a) = a · 1. Then

HomK[[T0 ]] (K

(ξ )

, K[[V0 ]]) =

 K · ι, 0,

if ξ is trivial, otherwise.

Here, the action of K[[T0 ]] on K[[V0]] is defined by equation (8.1). Proof Take ψ ∈ HomK[[T0 ]] (K (ξ ) , K[[V0 ]]) and set μ = ψ(1). Then for any t ∈ T0 , t · μ = ψ(t · 1) = ξ(t)ψ(1) = ξ(t)μ. By Lemma 8.9, if ξ = 1, then μ = 0, while for ξ = 1 we have μ = c · 1 for some scalar c. 

Recall from Sect. 7.2 that K[[V ± 1 ]](wχ) denotes the space K[[V ± 1 ]] considw, 2

ered as a K[[Q± 1 ]]-module, where Q±

w, 2

w, 12

w, 2

= T0 V ± 1 , with the action of T0 twisted w, 2

by the character wχ (see the discussion before Lemma 7.17). Similarly, we denote by K[[V0]](χ2 ) the space K[[V0]] equipped with the action of K[[T0]] such that t · μ, h = χ2 (t)μ, t −1 · h

(8.2)

for t ∈ T0 , μ ∈ K[[V0]], and h ∈ C(V0 , K). As before, t −1 · h(u) = h(tut −1 ). × Corollary 8.11 Suppose χ1 , χ2 : T0 → oK are continuous characters. Let ι : K → K[[V0]] be the natural embedding defined by ι(a) = a · 1. Then

HomK[[T0 ]] (K

(χ1 )

, K[[V0]]

(χ2 )

)=

 K · ι, 0,

if χ1 = χ2 , otherwise.

Proof The statement follows from Corollary 8.10, using the equality −1

HomK[[T0 ]] (K (χ1 ) , K[[V0 ]](χ2 ) ) = HomK[[T0 ]] (K (χ1 χ2 ) , K[[V0 ]]). To prove the equality, write A for the action of T0 on K[[V0]] given by Eq. (8.1) and Aχ2 for the action of T0 on K[[V0 ]](χ2 ) given by Eq. (8.2). Then Aχ2 (t).μ = −1

χ2 (t)A(t).μ. Hence, if ψ ∈ HomK[[T0 ]] (K (χ1 χ2 ) , K[[V0 ]]) then χ1 χ2−1 (t)ψ(x) = ψ(χ1 χ2−1 (t).x) = A(t).ψ(x),

8.2 Intertwining Algebra

181

whence χ1 (t)ψ(x) = χ2 (t)A(t).ψ(x) = Aχ2 (t).ψ(x). So, ψ is also in HomK[[T0 ]] (K (χ1 ) , K[[V0 ]](χ2 ) ). A similar argument shows containment in the other direction. 

8.2 Intertwining Algebra Composing the embedding ι : K → K[[G0 ]] with the projection K[[G0 ]] → M (χ) , we obtain ϕ : K → M (χ) given by ϕ(a) = (a · 1) ⊗ 1. Since K[[U1− ]](χ) embeds in M (χ) as a direct summand and contains 1 ∈ G0 , we see that ϕ is injective. Theorem 8.12 For any two continuous characters χ1 and χ2 of T0 , we have  HomK[[P0 ]] (K

(χ1 )

,M

(χ2 )

)=

0

if χ1 = χ2 ,

K ·ϕ

if χ1 = χ2 ,

where ϕ : K → M (χ2 ) sends a ∈ K to (a · 1) ⊗ 1 in M (χ2 ) .  (χ ) Proof Since M (χ2 ) = v∈W Mv 2 , as a K[[B]]-module and hence as a K[[P0 ]] module, we obtain HomK[[P0 ]] (K (χ1 ) , M (χ2 ) ) = HomK[[P0 ]] (K (χ1 ) , Mv(χ2 ) ). v∈W

We apply Proposition 8.4, taking w to be the identity element of W, which we (χ ) denote e. Then P 1w,± = P0 , and Proposition 8.4 implies HomK[[P0 ]] (K (χ1 ) , Mv 2 ) 2

(χ )

= 0 for all v = e. Thus HomK[[P0 ]] (K (χ1 ) , M (χ2 ) ) = HomK[[P0 ]] (K (χ1 ) , Me 2 ). (χ ) Now, by Lemma 7.17, Me 2 is isomorphic to K[[U1− ]](χ2 ) , as a K[[Q± 1 ]]-module and in particular as a K[[T0 ]]-module. And, by Corollary 8.11,  HomK[[T0 ]] (K (χ1 ) , K[[U1− ]](χ2 ) ) =

e, 2

K · ϕ,

if χ1 = χ2 ,

0,

otherwise.

One readily confirms that ϕ is a K[[P0 ]]-module map, completing the proof. Corollary 8.13 For any two continuous characters χ1 and χ2 of P0 , we have HomK[[G0 ]] (M

(χ1 )

,M

(χ2 )

)=

 0 K · id

if χ1 = χ2 , if χ1 = χ2 .



182

8 Intertwining Operators

Proof Similarly to the proof of Corollary 8.6, using adjoint associativity (Theorem 8.5) we have HomK[[G0 ]] (M (χ1 ) , M (χ2 ) ) = HomK[[G0 ]] (K[[G0]] ⊗K[[P0 ]] K (χ1 ) , M (χ2 ) ) ∼ = HomK[[P0 ]] (K (χ1 ) , HomK[[G0 ]] (K[[G0 ]], M (χ2 ) )) ∼ = HomK[[P0 ]] (K (χ1 ) , M (χ2 ) ). 

The statement now follows from Theorem 8.12. Since

M (χ)

is the dual of

G IndP00 (χ −1 ),

we obtain the following result.

Corollary 8.14 For any two continuous characters χ1 and χ2 of P0 , we have G0 0 HomcG0 (IndG P0 (χ1 ), IndP0 (χ2 ))

=

 0

if χ1 = χ2 ,

K · id if χ1 = χ2 .

An analogous result for principal series representations of G was proved by Peter Schneider in an unpublished note. We deduce it from Corollary 8.14 as follows: Proposition 8.15 For any two continuous characters χ1 and χ2 of P , we have G HomcG (IndG P (χ1 ), IndP (χ2 ))

=

 0 K · id

if χ1 = χ2 , if χ1 = χ2 .

Proof If χ1 = χ2 or χ1 |P0 = χ2 |P0 , the statement follows immediately from Corollary 8.14. It remains to consider the case when χ1 and χ2 are not equal but they have the same restriction to P0 . Set χ0 = χ1 |P0 = χ2 |P0 . Let V0 = IndP00 (χ0−1 ) G

−1 and Vi = IndG P (χi ), i = 1, 2.

Recall that Vi = {f : G → K continuous | f (gp) = χi (p)f (g) for all p ∈ P , g ∈ G}, for i = 1, 2, and similarly for V0 . We know from Proposition 7.3 that the restriction f → f |G0 is a topological isomorphism from Vi to V0 , i = 1, 2. We prove that HomcG (V1 , V2 ) = 0. Suppose, on the contrary, that there exists a nonzero intertwining operator ϕ ∈ HomcG (V1 , V2 ). By Corollary 8.14, ϕ|V0 is a nonzero scalar multiple of the identity operator, so by scaling we may assume ϕ|V0 = id. Take f1 ∈ V1 such that f1 (1) = 1 and let f0 = f1 |G0 . Select t ∈ T such that χ1 (t) = χ2 (t). Let F1 = Lt f1 , the action by left translation. That is,

8.2 Intertwining Algebra

183

F1 (g) = f1 (t −1 g), g ∈ G. Notice that ϕ(F1 )(1) = F1 (1) = f1 (t −1 ) = χ1 (t −1 ). Next, let f2 = ϕ(f1 ). This is the element of V2 uniquely determined by the requirement f2 |G0 = f0 . Since ϕ is an intertwining operator, we must have ϕ(Lt f1 ) = Lt f2 . Evaluating the above functions at 1 gives us ϕ(F1 )(1) = χ1 (t −1 ) on the left and f2 (t −1 ) = χ2 (t −1 ) on the right. This contradicts χ1 (t −1 ) = χ2 (t −1 ). 

Suppose χ : P → K × is a continuous character, and set χ0 = χ|P0 . As discussed in Sect. 7.1, restriction to G0 gives an isomorphism of Banach spaces G0 G IndG P (χ) → IndP0 (χ0 ). Not surprisingly, the G-representation IndP (χ) differs G

significantly from the G0 -representation IndP00 (χ0 ). For example, we know from G

[76] that in the case of G = GL2 , the GL2 (Zp )-representation IndP00 (χ0 ) can have infinitely many finite dimensional subrepresentations, while the GL2 (Qp )representation IndG P (χ), if reducible, has a unique irreducible subrepresentation. With this in mind, the result of Corollary 8.14 for G0 seems surprising. Examples of 0 IndG P0 (χ0 ) with infinitely many finite dimensional subrepresentations for a general group G are constructed in Sect. 8.3.

8.2.1 Ordinary Representations of GL2 (Qp ) Let G = GL2 (Qp ). As before, T is the group of diagonal matrices and P = T U is the group of upper triangular matrices. Recall that an irreducible admissible KBanach space representation of GL2 (Qp ) is called ordinary if it is a subquotient of a continuous principal series induced from a unitary character. Proposition 8.16 Ordinary representations of G = GL2 (Qp ) are

where χ is a unitary character of Q× (i) χ ◦ det and χ ◦ det ⊗St, p , and G (ii) IndP (χ1 ⊗ χ2 ), where χ1 , χ2 are unitary characters of Q× such that χ1 = χ2 . p All these representations are pairwise inequivalent. Proof The irreducible components of all principal series induced from unitary characters are listed in Propositions 7.10. We have to show that they are pairwise inequivalent. This clearly holds for the representations listed under (i). In addition, Proposition 8.15 tells us that the representations listed under (ii) are pairwise

inequivalent. Finally, if χ1 = χ2 then IndG (χ1 ⊗χ2 ) is not isomorphic to χ ◦det ⊗St P

184

8 Intertwining Operators

for any χ. Otherwise, the exact sequence

0 → χ ◦ det → IndG P (χ ⊗ χ) → χ ◦ det ⊗St → 0 G would give us a nonzero intertwining operator IndG P (χ ⊗ χ) → IndP (χ1 ⊗ χ2 ), thus contradicting Proposition 8.15. 

8.3 Finite Dimensional G0 -Invariant Subspaces In this section we discuss finite dimensional G0 -invariant subspaces of the repre−1 × sentation V = IndG P (χ ) in the case when χ : T → K is locally algebraic. This means that χ = χalg χsm , with χsm smooth and χalg L-algebraic. A similar, but more general discussion, with χalg replaced by a Qp -rational character, can be found in Appendix to [4]. Recall that we introduced several types of parabolic induction. Then, we can consider the smooth induction by χsm and the algebraic induction by χalg . Set −1 sm U = IndG P (χsm ) . Suppose in addition that χalg is dominant. Then we have −1 a nonzero algebraically induced representation W = indG P (χalg ), which is finite dimensional and irreducible. Given f ∈ U and h ∈ W , let f h be the pointwise product, (f h)(x) = f (x)h(x). We see immediately that the product f h : G → K is continuous and satisfies (f h)(gp) = f (gp)h(gp) = χsm (p)f (g)χalg (p)h(g) = χ(p)(f h)(g) for all p ∈ P and g ∈ G, so f h ∈ V . The tensor product U ⊗K W is a locally algebraic representation (see Appendix to [68] or Section 4.2 in [32] for the definition and basic properties of such representations). Pointwise multiplication of functions gives us a natural map U ⊗K W → V . This map is injective, which follows from [51], using exactness of the functor FG P for the split group G. Now, we consider the corresponding G0 -representations. The algebraic representation W remains irreducible when restricted to G0 . By Proposition 6.21, U decomposes as a countable direct sum of finite dimensional representations ρ with

8.3 Finite Dimensional G0 -Invariant Subspaces

185

finite multiplicities U∼ =



m(ρ)ρ.

ρ

Then V contains U ⊗K W ∼ =



m(ρ)(ρ ⊗K W ).

(8.3)

ρ

Note that every subspace ρ ⊗K W is finite-dimensional, and hence closed in V . Alternatively, we can use Corollary 4.2.9 of [32] to show that U ⊗K W decomposes as a direct sum of irreducible finite-dimensional representations. In conclusion, the G0 -representation V contains countably many finite-dimensional topologically irreducible subrepresentations. Still, by Lemma 8.14, HomG0 (V , V ) = K · id. On the other hand, the decomposition (8.3) implies that U ⊗K W , as a G0 -representation, has numerous self-intertwining operators that are not scalar multiples of the identity. These operators, however, cannot be extended continuously to V . Here is an example by Matthias Strauch.

8.3.1 Induction from the Trivial Character: Intertwiners Let G = GL2 (Qp ) and G0 = GL2 (Zp ). Let P = T U be the group of upper triangular matrices, with T the group of diagonal matrices in G and U the unipotent G sm radical of P . We consider the principal series V = IndG P (1). Let W = IndP (1) , which is a dense subspace of V . As discussed on page 155, we have the following exact sequences of representations of G

→ 0. 0 → 1 → W → St → 0 and 0 → 1 → V → St The trivial representation 1 is realized on the one-dimensional subspace W1 ⊂ W consisting of all constant functions G0 → K. To find its  complement in W , we fix the left Haar integral on G0 with values in K such that G0 dx = 1. Define

W2 = f ∈ W |

 f (x)dx = 0 . G0

This is a G0 -invariant subspace of W , and W = W1 ⊕ W2 . Define ϕ : W → W by

ϕ(f ) =

f (x)dx, G0

186

8 Intertwining Operators

where the number ϕ(f ) ∈ K is interpreted as the constant function G0 → K with the value ϕ(f ). Then ϕ is the projection onto W1 and also an intertwining operator. Hence, ϕ is an intertwining operator on W which is not a scalar multiple of idW . Next, we consider W equipped with the sup norm (the norm inherited from V ). We will show that ϕ is not bounded, and hence it is not continuous. For that, we take the following decomposition (see Proposition 5.45) G0 = U1− P0 where w˙ =





wU ˙ 0− P0 ,

 01 . Then, W ∼ = C ∞ (U1− , K) ⊕ C ∞ (U0− , K). We identify U0− with 10

Zp . For n ∈ N, define h ∈ C ∞ (U0− , K) by  hn (x) =

1, x ∈ pn Zp 0, otherwise.

Let fn be the function in W corresponding to hn . It is given by  fn (x) =

1, x ∈ wU ˙ n− P0 0, otherwise.

Then fn  = supx∈G0 |fn (x)|K = 1. On the other hand, let c =  −n and wU ˙ n− P0 dx = cp



ϕ(fn ) =

f (x)dx = G0

wU ˙ n− P0



wU ˙ 0− P0

dx. Then

dx = cp−n .

The norm | |K satisfies |p|K = p−d , where d is the degree of the field extension K/Qp (see Appendix A.2.2). It follows that ϕ(fn ) = |c|K pdn , fn  proving that ϕ is unbounded. As such, it cannot be extended continuously to V . Exercise 8.17 With notation as above, prove that W2 is dense in V .

8.4 Reducibility of Principal Series One of the basic questions in the representation theory is to decide whether an induced representation is reducible. Among smooth representations, there is a nice

8.4 Reducibility of Principal Series

187

and important class called discrete series representations. For them, the reducibility of a parabolically induced representation is completely determined by intertwining operators. They define the R-group, which governs not only reducibility, but also the decomposition of the induced space. As we see from Proposition 8.15, we cannot hope for anything similar for principal series representations on p-adic Banach spaces. Away from scalar multiples of the identity map, there are no intertwiners. So, we need other methods for determining reducibility. In this section, we give an overview of some reducibility results. The case L = Qp is best understood.

8.4.1 Locally Analytic Vectors So far, locally analytic representations were mentioned in passing but we didn’t use them in proofs. Now, to tell more about the reducibility of continuous principal series, we will tap into the knowledge on locally analytic representations. And this knowledge is extensive, so we will just mention what we need. Take a continuous character χ : T → K × . Then χ is locally Qp -analytic, and G −1 Qp −an −1 the induced representation IndG P (χ ) contains the dense subset IndP (χ ) of locally Qp -analytic vectors. As in [50], there is a canonically associated Verma −1 Qp −an is topologically irreducible. module, and whenever it is simple, IndG P (χ ) This was proved by Frommer in [33] for L = Qp and G split. The general case, with L a finite extension of Qp and G a connected reductive algebraic group over L, was proved by Orlik and Strauch in [50]. Reducibility Question for G(Qp ) The theory we developed so far was for G(L), where Qp ⊆ L ⊆ K is a sequence of finite extensions. Our methods worked equally well for an arbitrary L as for L = Qp . However, there are parts of the theory of Banach space representations that work only for G(Qp ). Assume in this section that L = Qp and G = G(Qp ). Then, as mentioned in Sect. 4.4.1, if V is an admissible Banach space representation of G, the space of locally analytic vectors V an = V Qp −an is dense in V . Moreover, the functor −1 an is the locally V → V an is exact. We apply this to V = IndG P (χ ). Then, V analytic principal series whose reducibility is well-understood. By the exactness of V → V an and the density of V an in V , if V an is irreducible, then V is irreducible as well. However, if V an reduces, V may be either reducible or irreducible. Assume that G is semisimple  and simply connected. Let λ1 , . . . , λr be the fundamental weights and δ = ri=1 λi . The assumption that G is simply connected implies that λi ∈ X(T), for all i, and so δ ∈ X(T). As explained on page 111, we

188

8 Intertwining Operators

use the same symbol δ for the corresponding character δ : T → Q× p . Similarly, for a root α, we have the coroot α ∨ : Q× → T . p The character χ : T → K × is called anti-dominant if χδ ◦ α ∨ = ( )m

(8.4)

for any integer m ≥ 1 and any positive root α. In [61, Conjecture 2.5], Schneider −1 conjectures that the G-representation IndG P (χ ) is topologically irreducible if χ is anti-dominant. It is explained there that the conjecture holds for GL2 (Qp ). (The group GL2 (Qp ) is not semisimple, but the anti-dominance condition is welldefined.) Example 8.18 Let G = GL2 (Qp ). As before, P = T U is the group of upper triangular matrices. Let δP : T → K × be the modulus character of P , δP (diag(a, b)) = |ab−1|p . Then (δp−1 ◦ α ∨ )(a) = |a −2|p , which is not an algebraic character. It follows that δP−1 is anti-dominant, and IndG P (δP ) is irreducible. The proposition below describes not only G-irreducibility, but also the much stronger G0 -irreducibility. Proposition 8.19 Suppose G is semisimple and simply connected, and G = G(Qp ). If the anti-dominance condition for χ continues to hold after restriction to an arbitrary small open subgroup of Q× p then (i) M (χ) is simple as a K[[G0]]-module, and G (ii) IndP00 (χ −1 ) is topologically irreducible as a G0 -representation. Proof This is Proposition 2.6 (ii) from [61]. Notice that (i) and (ii) are equivalent by duality (Corollary 4.46). The proof follows from the corresponding facts for the locally analytic principal series and the density of analytic vectors in admissible Banach space representations. More details can be found in [61]. 

An example of a character not satisfying the anti-dominance condition is what else but a dominant character. Example 8.20 Suppose χ : T → K × is a dominant algebraic character. Take a positive root α and set n = χ, α ∨ . Then n is a non-negative integer and we ∨ have χ(α ∨ (a)) = a n , for all a ∈ Q× p . In addition, δ, α  ≥ 1. As we discussed G −1 in Sect. 7.1.3, the representation IndP (χ ) is reducible. It contains the algebraic −1 representation indG P (χ ). Reducibility Question for G(L) Let L ⊆ K be an arbitrary finite extension of Qp .

8.4 Reducibility of Principal Series

189

Example 8.21 As above, if χ : T → K × is a dominant algebraic character, then G −1 −1 indG P (χ ) is a closed finite-dimensional G-invariant subspace of IndP (χ ). Let −1 σ be an automorphism of K which is nontrivial on L. Given f ∈ indG P (χ ), the function F = σ ◦ f satisfies F (gp) = σ (f (gp)) = (σ ◦ χ)(p)F (g), −1 for all g ∈ G, p ∈ P . It follows that {σ ◦ f | f ∈ indG P (χ )} is a closed finiteG −1 −1 dimensional G-invariant subspace of IndG P ((σ ◦χ) ) and therefore IndP ((σ ◦χ) ) is reducible. On the other hand, if χ = 1, then σ ◦ χ satisfies the condition (8.4) because on the left hand side we have a non-algebraic character, and it cannot be equal to the algebraic character ( )m .

Thus, the anti-dominance condition (8.4) does not predict irreducibility if L = Qp . To understand the reason, we have to consider G as a locally Qp -analytic group. This is done by the restriction of scalars ResL/Qp . The group G can be identified with the group of Qp -points of ResL/Qp G, which is an algebraic group defined over Qp . However, this group is not split. For non-split groups, the anti-dominance condition (8.4) has to be modified. The following example shows how we can use locally Qp -analytic vectors to get the irreducibility of continuous principal series. Example 8.22 We consider G = GL2 (L) as in [50, Example 4.2.2]. Let  denote the set of Qp -embeddings of L into K. Suppose that the cardinality of  is equal to [K : L]. Let T be the set of diagonal matrices, and P the set of upper triangular matrices. Let χ : T → K × be a continuous character. There are c1,σ , c2,σ ∈ K such that, for t1 , t2 ∈ L× sufficiently close to 1, we can write χ

   t1 0 = σ (t1 )c1,σ σ (t2 )c2,σ . 0 t2 σ ∈

If for all σ ∈  we have (c1,σ − c2,σ ) ∈ / Z≥0 , then the locally Qp -analytic principal −1 series is irreducible. This implies that IndG P (χ ) is irreducible.

8.4.2

A Criterion for Irreducibility

Back to our general case, with G a split connected reductive Z-group and G = G(L), we will present an irreducibility criterion from [3]. First, we note that by Lemma 6.5, there is a basis λ1 , . . . , λr for X(T) consisting of dominant elements. × × Let η : L× → K × be a continuous character. As η must map oL into oK , it × × × × × induces a map L /oL → K /oK . That is, for a ∈ L , the valuation vK (η(a))

190

8 Intertwining Operators

depends only on vL (a). Let e(η) denote the integer such that vK ◦ η = e(η) · vL . L×

(8.5)

K×

Theorem 8.23 Let χ1 , . . . , χr : → be continuous characters such that  e(χi ) < 0 for 1 ≤ i ≤ r. Define χ : T → K × by χ(t) = ri=1 χi (t λi ). Then −1 is topologically irreducible. IndG P χ Notice that we can check whether e(χi ) < 0 simply by evaluating χi (L ); no information about χi near 1 is needed. On the other hand, the irreducibility results from the previous section are given in terms of the behavior of χi in a small neighborhood of 1. This is because they use locally analytic vectors (see Proposition 8.19 and Example 8.22). In any case, Proposition 8.19 and Theorem 8.23 give irreducibility for different sets of characters. For the proof of Theorem 8.23, we use the duality and prove the corresponding result for M (χ) (Theorem 8.27). As discussed in Sect. 7.2.1, M (χ) is isomorphic to a quotient of K[[G0]]. For μ ∈ K[[G0 ]], we denote by [μ] its image in M (χ) , that is, [μ] = μ ⊗ 1 ∈ M (χ) . By Proposition 7.14, any [μ] ∈ M (χ) has a representative of the form η=

 w∈W

wη ˙ w,

ηw ∈ K[[U − 1 ]]. w, 2

The idea of the proof of Theorem 8.27 can be explained by the following simple example. Example 8.24 Let G = GL2 (L), with P = T U the group of upper triangular matrices. Notice that U − 1 = U0− . Let w, 2



         a0 01 10 1 0 b0 − t= ∈ T , w˙ = ,u= ∈ U0 , v = , and s = . 0b 10 x1 ab−1 x 1 0a

Take ν = wu. ˙ To find the action of t on [ν], we use the following formula         a0 b0 01 10 01 1 0 = 0b 0a 10 x1 10 ab−1 x 1 Suppose that N is a (K[[G0 ]], G)-submodule of M (χ) which contains [1 + ν]. If ∈ oL , then

ab−1x

χ(t −1 )t · [1 + ν] = [1 + χ(t −1 s)η],

8.4 Reducibility of Principal Series

191

where η = wv ˙ ∈ wo ˙ K [[U0− ]]. If we impose the additional condition that χ(t −1 s) ∈ pK , then 1 + χ(t −1 s)η ∈ 1 + K · oK [[G0 ]]. Such an element is invertible in oK [[G0 ]], by Exercise 2.48. Consequently, N contains [1], and so N = M (χ) . We can apply a similar method in the general case. Let N be a nonzero (K[[G0 ]], G)-submodule of M (χ) . The requirement that N contains [1 + wu] ˙ is, of course, very strong, but it is not needed. It turns out that N always contains an element of the form [1 + ν], where ν satisfies the following two conditions: 1. ν ∈ oK [[G0 ]], and 2. the support of ν is disjoint from Gn P0 , for some n. Such [1 + ν] serves us well: with an appropriate choice of t ∈ T and an appropriate condition on χ, we can repeat the same trick as above to show that N = M (χ) . (χ) We will work with the image of oK [[G0 ]] in M (χ) , which is equal to M0 = (χ) oK [[G0 ]] ⊗oK [[P0 ]] oK . We need the following two technical lemmas from [3]. Lemma 8.25 Fix n ∈ N. With assumptions as in Theorem 8.23, there exists t ∈ T such that χ(t −1 )t · ν ∈ [K · oK [[G0 ]]] (χ)

for any ν ∈ M0

that vanishes on Gn P .

Proof This is Corollary 7.4 in [3]. It follows from Lemma 5.1 in [3], the main technical result of that paper, which describes the action of T on a convenient, explicit model for the space G/P . 

Lemma 8.26 Fix w0 ∈ W and u0 ∈ U −

w0 , 12

u−1 ˙ 0−1 · ( 0 w

 w

. Let n ≥ 1. Then

wU ˙ − 1 ) ∩ Gn P0 = Un− , w, 2

that is, if w ∈ W, u ∈ U − 1 and u−1 ˙ 0−1 wu ˙ ∈ Gn P0 , then w = w0 and u−1 0 w 0 u ∈ Un− .

w, 2

¯ (the points of G over the finite field Proof Consider the projection from G0 to G − l = oL /pL ). The sets wU ˙ 1 for w ∈ W all project to distinct Bruhat cells. Hence w, 2

w0 u0 G1 P0 ∩ wU ˙ − 1 = ∅ ⇒ w = w0 . w, 2

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8 Intertwining Operators

− − Assume w = w0 . Then u−1 ˙ 0−1 wU ˙ − 1 = u−1 0 w 0 U 1 ⊂ U0 . Hence, it is enough to w, 2

w, 2

prove U0− ∩ Gn P0 ⊂ Gn . To show this, we consider the projection to G0 /Gn . Since U0− ∩ P0 = {1}, the only element of G0 /Gn which is in the image of both P0 and U0− is the identity. 

The following is Theorem 7.5 from [3] and its proof.  Theorem 8.27 Define χ : T → K × by χ(t) = ri=1 χi (t λi ) where λ1 , . . . λr are dominant and form a basis for X(T). Assume that e(χi ) < 0 for 1 ≤ i ≤ r. Then M (χ) has no proper nontrivial G-invariant K[[G0]]-submodules. Proof Choose a nontrivial element of M (χ) , and construct a representative η =  ˙ w with ηw ∈ K[[U − 1 ]] for each w ∈ W. Let N be the (K[[G0]], G)w∈W wη w, 2

submodule generated by [η]. We want to show that N = M (χ) . − By scaling, we may assume that ηw = (ηw, )∞ =0 ∈ oK [[U 1 ]] for each w, and that there exists n ≥ 1, w0 ∈ W and u¯ 0 ∈ U − c0 of u¯ 0 in ηw0 ,n is a unit. Choose u0 ∈ U −

w0 , 21

w0 , 12

w, 2

/Un− such that the coefficient

which projects to u¯ 0 , and let μ =

˙ 0−1 η. If we write μ as an element of the projective limit μ = (μ )∞ u−1 =0 , then 0 w μn = c 0 +



cg¯ g, ¯

× c0 ∈ o K , cg¯ ∈ oK .

(8.6)

g∈G ¯ 0 /Gn g¯ =1

Now, observe that the partition of G0 as Gn ∪ (G0  Gn ) gives rise to direct sum decompositions of C(G0 , K), oK [[G0 ]] and K[[G0 ]]. Moreover {λ ∈ K[[G0 ]] : supp(λ) ⊂ Gn } is canonically identified with K[[Gn ]]. Let μ = μ + μ ,

μ ∈ oK [[Gn ]], supp(μ ) ⊂ G0  Gn ,

and note that, by Lemma 8.26, the support of μ is actually disjoint from Gn P0 . The image of μ under the augmentation map is precisely c0 , the coefficient of the identity coset of μn in Eq. (8.6). Since c0 is a unit, we know from Proposition 5.42 that μ is an invertible element of oK [[Gn ]]. Multiplying by its inverse, we obtain an element of the form 1 + ν where the support of ν is disjoint from Gn P0 . Note that the elements [η], [μ] and [1 + ν] generate the same submodule N of M (χ) . Now, choose t as in Lemma 8.25. If we act by χ(t)−1 t on [1 + ν], we see that N contains χ(t)−1 t · [1 + ν] = [1 + χ(t)−1 t · ν] ∈ [1 + K · oK [[G0 ]]]. From Exercise 2.48 or [3, Lemma 6.1], we know that the elements of 1 + K · oK [[G0 ]] are units. Hence, N = M (χ) . 

Finally, Theorem 8.23 follows from Theorem 8.27, using the duality between −1 and M (χ) . IndG P χ

Appendix A

Nonarchimedean Fields and Spaces

A.1 Ultrametric Spaces A metric space (X, d) is called an ultrametric space if d satisfies the strong triangle inequality (see Definition A.1). Many of the objects we consider in this book are ultrametric: the field of p-adic numbers Qp , any finite extension K/Qp , as well as any normed K-vector space. An ultrametric space has staggering topological properties that could look strange to the novices to the p-adic world. In this section, we prove some of these properties. Definition A.1 (i) Let X be a nonempty set. A metric, or distance, on X is a function d : X×X → R≥0 such that (1) d(x, y) = 0 if and only if x = y, (2) d(x, y) = d(y, x), (3) d(x, y) ≤ d(x, z) + d(z, y) (the triangle inequality), for all x, y, z ∈ X. A set X together with a metric d is called a metric space. (ii) A metric d on a set X is called an ultrametric or a nonarchimedean metric if (3’) d(x, y) ≤ max(d(x, z), d(z, y)) (the strong triangle inequality) for all x, y, z ∈ X. (iii) An ultrametric space is a pair (X, d) consisting of a set X together with an ultrametric d on X.

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Proposition A.2 (The Isosceles Triangle Principle) Let (X, d) be an ultrametric space. Let x, y, z ∈ X. If d(x, y) = d(y, z), then d(x, z) = max{d(x, y), d(y, z)}. Hence, every “triangle” is isosceles, and if it is not equilateral, then the legs are longer than the base. Proof We may assume without loss of generality that d(x, y) < d(y, z). Notice that d(y, z) ≤ max{d(y, x), d(x, z)} and this maximum must be equal to d(x, z) because it is strictly greater than d(x, y). It follows d(y, z) ≤ d(x, z) ≤ max{d(x, y), d(y, z)} = d(y, z), which implies d(x, z) = d(y, z).

 

Let (X, d) be a metric space. Then for a ∈ X and a real number r > 0, the closed ball (respectively open ball) of radius r centered at a is Br (a) = {x ∈ X | d(x, a) ≤ r},

respectively, Br− (a) = {x ∈ X | d(x, a) < r}.

The open balls form the base for a topology on X, making it a topological space. The fundamental property of ultrametric spaces is that all balls are both open and closed. Proposition A.3 Let (X, d) be an ultrametric space. Let a ∈ X and r > 0. (i) Every point inside a ball is its center. More specifically, if b ∈ Br (a), then Br (a) = Br (b) and if b ∈ Br− (a), then Br− (a) = Br− (b). (ii) Intersecting balls are contained in each other. That is, if Br (a) ∩ Bs (b) = ∅, then either Br (a) ⊆ Bs (b) or Bs (b) ⊆ Br (a). (iii) Br (a) is both open and closed in X. Similarly, Br− (a) is both open and closed in X. Proof (i) Let b ∈ Br (a). Then x ∈ Br (a) ⇒ d(x, a) ≤ r ⇒ d(x, b) ≤ max(d(x, a), d(a, b)) ≤ r ⇒ x ∈ Br (b). It follows Br (a) ⊆ Br (b). In the same way, Br (b) ⊆ Br (a). Therefore, Br (a) = Br (b). The proof for Br− (a) is analogous.

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195

(ii) Suppose Br (a)∩Bs (b) = ∅ and take x ∈ Br (a)∩Bs (b). By (i), Br (a) = Br (x) and Bs (x) = Bs (b). If r ≤ s, then Br (a) = Br (x) ⊆ Bs (x) = Bs (b). Otherwise, we show in the same way that Bs (b) ⊂ Br (a). (iii) If x ∈ Br (a), then Br− (x) ⊆ Br (x) = Br (a). This shows that Br (a) is open. To prove that Br− (a) is closed, we will show that its complement C = X \ − Br (a) is open. If C = ∅, then C is open. Otherwise, take x ∈ C. Since x∈ / Br− (a), it follows d(x, a) ≥ r. It is easy to show that the balls Br− (x) and − Br (a) are disjoint. It follows that Br− (x) ⊆ C is an open neighborhood of x in C. This shows that C is open.   Corollary A.4 Let (X, d) be an ultrametric space and r > 0. Then X is a disjoint union of open balls of radius r. Similarly, X is a disjoint union of closed balls of radius r. As a consequence, there are many locally constant functions on an ultrametric space. Another consequence is that ultrametric spaces are totally disconnected. Recall that a nonempty topological space X is totally disconnected if the only connected components of X are the singletons. Proposition A.5 Let (X, d) be an ultrametric space. Then X is totally disconnected. Proof Exercise.

 

A.2 Nonarchimedean Local Fields In this section, we give an overview of the properties of nonarchimedean local fields needed in this book. For details, see [79] or [49]. A locally compact non-discrete field is called a local field. Let F be a local field. If F is connected, we call it archimedean. Otherwise, we call it nonarchimedean. If a local field is archimedean, then it is isomorphic to R or C. Any non-archimedean local field of characteristic 0 can be described as a finite algebraic extensions of Qp , for some p. Here, Qp is the field of p-adic numbers defined below.

A.2.1 p-Adic Numbers For an introduction to the p-adic numbers, see Schikhof [59] or Koblitz [41]. Definition A.6 Let F be a field.

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(i) An absolute value on F is a map   : F → R≥0 such that, for all x, y ∈ F , (1) x = 0 if and only if x = 0, (2) xy = x · y, (3) ||x + y|| ≤ x + y (the triangle inequality). (ii) An absolute value   is called nonarchimedean if for all x, y ∈ F (3’) x + y ≤ max{x, y} (the strong triangle inequality). Let x be a nonzero rational number. Then there exists a unique integer ordp (x) a such that x = pordp (x) , where p  |ab. We define the p-adic absolute value of x b by |x|p = p− ordp (x). For x = 0, we define |x|p = 0. Then | |p is a nonarchimedean absolute value on Q. It induces the metric dp : Q × Q → R≥0 given by dp (x, y) = |x − y|p which satisfies the strong triangle inequality dp (x, y) ≤ max(dp (x, z), dp (z, y)) for all x, y, z ∈ Q. Hence, (Q, dp ) is an ultrametric space. We denote by Qp the completion of Q with respect to dp . Then Qp is also a field, called the field of padic numbers. The p-adic absolute value extends to Qp . Namely, if x ∈ Qp , then there exists a Cauchy sequence {xn } of rational numbers such that x = limn→∞ xn . Then |x|p = lim |xn |p . n→∞

Define Zp = {x ∈ Qp | |x|p ≤ 1}. It is easy to show, using the strong triangle inequality, that Zp is a ring, called the ring of p-adic integers. Note that Z ⊂ Zp . The ring Zp is a local ring, with the unique maximal ideal p = (p) = pZp = {x ∈ Zp | |x|p < 1}. The ring Zp is compact and open. The ideals pn = pn Zp , n ∈ N, form a neighborhood basis of zero consisting of compact open sets. For a ∈ Qp , the set {a + pn Zp | n ∈ N} is a neighborhood basis of a consisting of compact open sets.

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197

There is another construction of Qp , starting with the definition of the ring Zp . We consider the ring Z and the ideals (pn ) in Z. We define Zp as the projective limit Zp = lim Z/(pn ). ←−

n

Then Qp can be described algebraically as the field of fractions of Zp or as Qp = Zp ⊗Z Q

or as Qp =



p−n Zp .

n≥1

As an ultrametric space, Qp is totally disconnected. Let us mention that Qp is not discrete. (A topological space X is discrete if each point in X is an open subset.) We will prove that the set {0} is not open, which is equivalent to proving that S = Qp \{0} is not closed. Consider the Cauchy sequence {pn } in S. Then limn→∞ pn = 0∈ / S, so S is not closed.

A.2.2 Finite Extensions of Qp Let K be a finite extension of Qp with d = [K : Qp ]. For any x ∈ K, define |x|K = |NK/Qp (x)|p ,

(A.1)

where the right-hand side is the p-adic absolute value on Qp . Then | |K is a nonarchimedean absolute value on K. Note that this absolute value does not extend | |p . The absolute value which extends | |p is denoted again by | |p and it is computed as 1/d

|x|p = |NK/Qp (x)|p ,

x ∈ K.

(A.2)

The field K is totally disconnected and locally compact. Set oK = {x ∈ K | |x|K ≤ 1}. It is easy to show that oK is a ring, called the ring of integers of K. Its group of × units is oK = {x ∈ K | |x|K = 1}. The ring oK is local, with the unique maximal ideal pK = {x ∈ oK | |x|K < 1}. The ideal pK is principal. Every nonzero ideal of oK is principal and is a power of pK .

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Any element of pK of maximal norm is a generator of pK . Such an element is called a uniformizer. Fix a uniformizer K . Then any nonzero x ∈ K can be × written as x = Kn u, where n ∈ Z and u ∈ oK . The power n is denoted by vK (x) and is called the valuation of x. In addition, we define vK (0) = ∞. We have  K= K−n oK . n∈N

If we define dK (x, y) = |x − y|K , for x, y ∈ K, we obtain an ultrametric dK : K × K → K. Then (K, dK ) is an ultrametric space. The topology on K is algebraic in nature. It is easy to show that for any 0 < r < 1, the ball of radius r centered at zero Br (0) = {x ∈ K | |x|K ≤ r} is an ideal in oK , and hence it is of the form pnK for some n ∈ N. Then K has a neighborhood basis of zero consisting of the ideals pnK , n ∈ N. Moreover, an arbitrary element a ∈ K has a neighborhood basis of the form a + pnK , n ∈ N. × Similarly, the topology on oK is also algebraic in nature. For n ∈ N, the set × × n 1 + pK is a subgroup of oK , and also an open neighborhood of 1 in oK . Proposition A.7 The canonical mapping oK −→ lim oK /pnK ← − n∈N

is an isomorphism and a homeomorphism. The same is true for the mapping × × oK −→ lim oK /(1 + pnK ). ← − n∈N

Proof This is Proposition 4.5 in Chapter II of Neukirch [49].

 

The residue field of K is defined as κ = oK /pK . The inclusion of Zp into oK induces a map of Fp = Zp /pZp into κ and thus Fp is the prime field of κ. Let f = [κ : Fp ]. We call f the residue degree of K over Qp . −1 and Denote by qK the order of κ. Then qK = pf . We have |K |K = qK −d × |p|K = p , where d = [K : Qp ]. The absolute value of x ∈ K is −vK (x) . |x|K = qK

Since any ideal in oK is a power of pK , the same is true for the principal ideal (p) = poK ⊂ oK . It is of the form pe = (Ke ) for some e ∈ N. The number e is called the ramification degree of K over Qp . We have [K : Qp ] = ef since −e = p−ef . p−d = |p|K = | |eK = qK

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Lemma A.8 Let K be a finite extension of Qp . Every strictly decreasing sequence in |K| converges to zero. n Proof This is clear since |K| = |K × | ∪ {0} = {qK | n ∈ Z} ∪ {0}.

 

The above lemma is related to K being discretely valued. Namely, if F is a field with the absolute value  , we say that F is discretely valued if the set F ×  is discrete in R.

A.2.3 Algebraic Closure Qp We denote by Qp the algebraic closure of Qp . It is used in Chap. 6 where we consider it as an abstract field. It is known that two algebraically closed fields of the same characteristic and same uncountable cardinality are isomorphic [46, Proposition 2.2.5]. It follows that Qp and C are isomorphic as abstract fields. The p-adic absolute value on Qp extends to an absolute value on Qp , denoted again by | |p . The field Qp is not complete [41, Theorem 12]. We denote by Cp the completion of Qp with respect to | |p . Then Cp is algebraically closed and complete [41, Theorem 13]. By the same arguments as above, Cp is also isomorphic to C as an abstract field.

A.3 Normed Vector Spaces In this section, K is a finite extension of Qp and | | = | |K . Definition A.9 (i) Let V be a K-vector space. A (nonarchimedean) norm on V is a map   : F → R≥0 such that (1) v = 0 if and only if v = 0, (2) av = |a| · v, for a ∈ K, v ∈ V (3) v + w ≤ max{v, w}, for v, w ∈ V . (ii) A normed K-space is a pair (V ,  ) consisting of a K-vector space V together with a norm   on V . Let (V ,  ) be a normed K-space. The norm   induces a metric on V given by the formula d(v, w) = v − w. Then (V , d) is an ultrameteric space, as in Definition A.1.

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Example A.10 Let V = K n be the n-dimensional K-vector space. Define (a1 , . . . , an ) = max |ai |. 1≤i≤n

This is a norm on V . The space V = K n is complete with respect to the corresponding metric, and hence it has the structure of a Banach space. For more examples, see Sect. 3.1.3. The direct sum U ⊕ V of two normed vector spaces U and V is the vector space U × V equipped with the norm (u, v) = max{u, v}.

(A.3)

If U and V are Banach spaces, then so is U ⊕ V . We conclude this section with the following simple and very useful result which follows from the isosceles triangle principle for ultrametric spaces. Lemma A.11 Let (V ,  ) be a normed K-space. If v, w ∈ V satisfy v = w, then v + w = max{v, w}. Proof Follows directly from Proposition A.2, with x = v, y = 0, and z = −w.

 

Appendix B

Affine and Projective Varieties

In this section, k is an algebraically closed field.

B.1 Affine Varieties Let F be a field. The set F n = F × · · ·× F will be called affine n-space over F and denoted An . A subset V of An is called an affine variety if V is the set of common zeros of a finite collection S = {fα } of polynomials in F [X] = F [x1, . . . , xn ]. We write V = V(S). Example B.1 Some familiar objects in the real plane and the three-dimensional real space are affine varieties over R. (a) The circle x 2 +y 2 = 1 is the set of zeros of the polynomial f (x, y) = x 2 +y 2 −1 in the two-dimensional affine space. (b) The paraboloid z = x 2 + y 2 is the set of zeros of the polynomial f (x, y, z) = x 2 + y 2 − z in the three-dimensional affine space. (c) Let f (x, y, z) = x 2 + y 2 − z2 and g(x, y, z) = y + z − 1. The affine variety V(f, g) is the intersection of the cone z2 = x 2 + y 2 and the plane z = 1 − y, so it is a conic section (a parabola). If I is an ideal of F [X], we denote by V(I ) the set of its common zeros in An . By Hilbert’s Basis Theorem, F [X] is noetherian, so there exists a finite set of generators of I . This implies V(I ) is an affine variety. Let U ⊂ An . We denote by I(U ) the collection of all polynomials vanishing on U . Then I(U ) is an ideal. We have U ⊂ V(I(U )),

I ⊂ I(V(I )).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Ban, p-adic Banach Space Representations, Lecture Notes in Mathematics 2325, https://doi.org/10.1007/978-3-031-22684-7

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Exercise B.2 Let U, V ⊂ An and S, T ⊂ F [X]. Prove: U ⊆ V ⇒ I(U ) ⊇ I(V ), I(U ∪ V ) = I(U ) ∩ I(V ), S ⊆ T ⇒ V(S) ⊇ V(T ), V(S ∪ T ) = V(S) ∩ V(T ).

(B.1)

From now on, An is the affine n-space over an algebraically closed field k. The radical of the ideal I is defined as rad I = {f ∈ k[X] | f r ∈ I, for some r ≥ 0}. The ideal I is called a radical ideal if rad I = I . Theorem B.3 (Hilbert’s Nullstellensatz) If I is an ideal in k[x1 , . . . , xn ], then I(V(I )) = rad I. Moreover, the maps V and I in the correspondence I

{affine varieties}

−→ ←−

{radical ideals}

V

are bijections that are inverses of each other. Proof See Theorem 32 in Chapter 15 of [28].

 

Example B.4 (a) If c = (c1 , . . . , cn ) ∈ An , then I(c) = (x1 − c1 , . . . , xn − cn ). (b) Let I ⊂ k[X] be a maximal ideal and U = V(I ). From the Nullstellensatz, U is nonempty, so let c = (c1 , . . . , cn ) ∈ U . Then (B.1) I ⊆ I(c) = k[X], so by maximality I = I(c) and U = V(I ) = V(I(c)) = {c}. There is a bijective correspondence between points in An and maximal ideals in k[X].

B.1.1 Zariski Topology on Affine Space Recall that a topology on a set X can be defined by a collection of closed subsets satisfying: (T1) (T2) (T3)

The empty set and X are closed. The intersection of any collection of closed sets is also closed. The union of any finite number of closed sets is also closed.

We want to define the topology on An in which the closed sets are the affine varieties. We first observe that the empty set and An are affine varieties, so property (T1) holds. For (T2) and (T3), we apply the following exercise:

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Affine and Projective Varieties

203

Exercise B.5 (a) If I and J are ideals in k[X], prove that V(I ) ∪ V(J ) = V(I J ). (b) If Iα , α ∈ A is an arbitrary collection of ideals in k[X], prove that 

V(Iα ) = V(

α



Iα ).

α

Then (a) implies (T3) and (b) implies (T2). It follows that affine sets define a topology on An . Definition B.6 The Zariski topology on affine n-space is the topology in which the closed sets are the affine varieties in An . Every point in An is a closed set. But the Hausdorff separation axiom fails. There are relatively few closed (or open) sets. A nonempty affine variety V is called irreducible if it cannot be written as V = V1 ∪ V2 , where V1 and V2 are proper affine varieties. Proposition B.7 Let V be an affine variety. (i) V is irreducible if and only if I(V ) is a prime ideal. (ii) If V is not empty, it may be written uniquely in the form V = V1 ∪ V2 ∪ · · ·∪ Vq , where each Vi is irreducible and Vi ⊆ Vj for j = i. Proof Dummit and Foote [28, page 680].

 

B.1.2 Morphisms and Products of Affine Varieties Suppose V ⊂ An and W ⊂ Am are two affine varieties. A map ϕ : V → W is called a morphism of affine varieties if there are polynomials ϕ1 , . . . , ϕm ∈ k[x1, . . . , xn ] such that ϕ((a1 , . . . , an )) = (ϕ1 (a1 , . . . , an ), . . . ϕm (a1 , . . . , an )), for all (a1 , . . . , an ) ∈ V . The map ϕ : V → W is an isomorphism of affine varieties if there is a morphism ψ : W → V with ϕ ◦ ψ = 1W and ψ ◦ ϕ = 1V . Note that in general ϕ1 , . . . , ϕm are not uniquely defined. Assume  (a)), (ϕ1 (a), . . . ϕm (a)) = (ϕ1 (a), . . . ϕm

for all a = (a1 , . . . , an ) ∈ V . Then, for i = 1, . . . , m, (ϕi − ϕi )(a) = 0 for all a ∈ V , so ϕi − ϕi ∈ I(V ). We define k[V ] = k[X]/I(V )

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B Affine and Projective Varieties

and call it the affine algebra of V (or the algebra of polynomial functions on V ). The distinct polynomial functions on V are in one-to-one correspondence with k[V ]. A morphism ϕ : V → W is a mapping of the form ϕ(a) = (ψ1 (a), . . . , ψm (a)), where ψi ∈ k[V ]. A morphism ϕ : V → W is continuous for the Zariski topologies involved. Indeed, if U ⊂ V is the set of zeros of polynomial functions fi on W , then ϕ −1 (U ) is the set of zeros of the polynomial functions fi ◦ ϕ on V . Let V ⊂ An and U ⊂ Am be affine varieties. Then V is the common zeros of a finite set of polynomials {fi | i = 1, . . . , p} in k[x1 , . . . , xn ], V = V({fi }), and U = V({gj }) is the common zeros of a finite set of polynomials {gj | j = 1, . . . , q} in k[y1 , . . . , ym ]. Then, the cartesian product V × U is an affine variety in An+m . It is the set of common zeros of {fi gj | i = 1, . . . , p, j = 1, . . . , q} ⊂ k[x1 , . . . , xn , y1 , . . . , ym ], V × U = V({fi gj }) ⊂ An+m . We define the topology on An × Am , called the Zariski product topology, which identifies An × Am with An+m . This topology is different from the usual product topology. Exercise B.8 Let k = C. (a) Prove that the Zariski-closed sets in A1 are ∅, C, and all finite sets. (b) Prove that the Zariski product topology on A1 ×A1 (which identifies it with A2 ) is strictly finer than the product topology on A1 × A1 .

B.2 Projective Varieties Projective n-space Pn is the set of equivalence classes of k n+1 − {(0, . . . , 0)} relative to the equivalence relation (c0 , c1 , . . . , cn ) ∼ (d0 , d1 , . . . , dn ) if and only if (d0 , d1 , . . . , dn ) = α(c0 , c1 , . . . , cn ) = (αc0 , αc1 , . . . , αcn ), for some α ∈ k × . Each point in Pn can be described by homogeneous coordinates c0 , c1 , . . . , cn which are not unique but may be multiplied by any nonzero scalar.

B

Affine and Projective Varieties

205

A polynomial f ∈ k[X] = K[x0, . . . , xn ] is homogeneous of degree d if it is a linear combination of monomials of degree d. This is equivalent to the condition f (αx0 , . . . , αxn ) = α d f (x0 , . . . , xn ), α ∈ k × . If f ∈ k[X] is a homogeneous polynomial and f (c0 , c1 , . . . , cn ) = 0, then f (αc0 , αc1 , . . . , αcn ) = α d f (c0 , . . . , cn ) = 0. This justifies the following definition. A subset V of Pn is called a projective variety if V is the set of common zeros of a finite collection S = {fα } of homogeneous polynomials in K[X]. We write V = V(S). Let U ⊂ Pn . We denote by I(U ) the collection of all polynomials vanishing on U . Then I(U ) is an ideal. An arbitrary polynomial f ∈ k[X] can be written in the form f =



f (d),

where f (d) ∈ k[X] is a homogeneous polynomial of degree d. We call f (d) the homogeneous part of f of degree d. An ideal I of k[X] is called homogeneous if whenever f ∈ I then each homogeneous part f (d) also lies in I . Exercise B.9 Let U be a subset of projective space Pn . Prove that the set I(U ) of all polynomials vanishing on U is a homogeneous ideal. Now we define a topology on Pn by taking a closed set to be a projective variety (the common zeros of a collection of homogeneous polynomials, or equivalently of the ideal they generate). Let I0 denote the ideal generated by x0 , . . . , xn . Then I0 has no common zeros in Pn . Proposition B.10 The operators V, I set a one-to-one inclusion-reversing correspondence between the closed subsets of Pn and the homogeneous radical ideals of k[X] other than I0 . Proof This is Proposition 1.6 in Humphreys [38].

 

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Index

gα , 98 gln , 95 pK , 197  155 St, lim Xi , 14 ← −i∈I g λ, 72 aug, 33 Dn , 93 P− , 108 Tn , 93 Uα , 107 Un , 93 U− , 108 c0 (X), 40 diag(a1 , . . . , an ), 93 δg , 46 δP , 141 N, 7 Qp , 196 η ⊗ π, 137  ,  : oK [[G0 ]] × C(G0 , oK ) → oK , 50  ,  : D c (G0 , K) × C(G0 , K) → K, 43  ,  : K[[G0 ]] × C(G0 , oK ) → K, 56  ,  : X(T) × X ∨ (T) → Z, 105 (Ban(K)≤1 )Q , 77 (Xi , ϕij )I , 11 (Xi )I , 12 m(χ, n), 157 Modflcomp (oK )Q , 77 Modflcomp (oK ), 73 Modflcomp (oK [[G0 ]]), 81 oK [[G0 ]], 26 oK , 197 ordp (x), 196 K , 198

π ⊗ τ , 137 π ⊗ E, 135  π , 136 ∞ (X), 40 (w), 103 B (G), 58 Dn , 46 I(U ), 201 Jn , 29 L0 (F, m), 44 Lb (V , W ), 39 Ls (V , W ), 39 LB (V , W ), 39 L(B, ), 38 L(B, M), 38 L(V , W ), 37 N(G0 ), 23 V(I ), 201 V(S), 201 Ad, 96 Autcs (V ), 69 Autc (V ), 68 B (v), 36 B− (v), 36 BanG (K), 71 Banadm G (K), 82 Ban(K), 64 Cc∞ (G, F ), 132 C ∞ (G0 , K), 41 C(G0 , oK ), 43 C(G0 , K), 40 D c (G0 , oK ), 43 D c (G0 , K), 43 D Dir (G0 , oK ), 46 D(G0 , K), 84

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Ban, p-adic Banach Space Representations, Lecture Notes in Mathematics 2325, https://doi.org/10.1007/978-3-031-22684-7

211

212 GLn (oL ), 23 HomG (π, τ ), 134 HomG (V , W ), 134 HomcG (π, τ ), 70 HomcG (V , W ), 70 0 −1 IndG P0 (χ )w , 149 0 −1 IndG P0 (χ ), 148 G −1 IndP (χ ), 148 Jm,n (G0 ), 27 K[[G0 ]], 55 Lg , 71 (χ ) M0 , 156

(χ )

Mw , 161 M d , 73 M (χ ) , 156 Modfg (K[[G0 ]]), 81 Norm(K)≤1 , 74 (G, T), 98 (G, T), 106 Rg , 71 Repsm F (G), 134 R(G), 96 St, 143 U − 1 , 118 w, 2

V 1w,± , 118 2

V ± 1 , 118 w, 2 V d , 74

, 136 V  , 50 Vbs Vb , 39 Vs , 39 W (), 102 X ∨ (T), 105 X(G), 97 X(T)+ , 127 vK (x), 198 w , 103 wχ, 140 xα , 107 (K[[G0 ]], G)-module, 85 B-topology, 39 absolutely irreducible representation, 136 absolute value, 196 nonarchimedean, 196 adjoint group, 127 admissible Banach space representation, 82 admissible-smooth representation, 134 affine algebra, 204

Index n-space, 201 variety, 201 algebraic induction, 129 algebraic representation, 124 anti-equivalence of categories, 75 augmentation map, 33 Baire space, 65 ball, 36 Banach space, 37 Banach space representation, 70 Banach-Steinhaus theorem, 67 big cell, 109 Borel subgroup, 108 bounded subset, 36 bounded weak star topology, 51 bounded-weak topology, 50 Bruhat decomposition, 108 Bruhat order, 108 character anti-dominant, 188 rational, 97 closed ball, 36, 194 closed graph theorem, 66 cocharacter, 105 completed group ring, 26 composition factors, 142 composition series, 142 continuous dual, 37 contragredient, 136 convex subset, 35 convolution product, 59 coroot, 106 Dirac distribution, 46 directed partially ordered set (poset), 11 direct sum of normed spaces, 200 discretely valued field, 199 dominant weight of a root system, 125 of X(T), 127 duality map, 42 dual representation, 72 dual space, 37

equivalence of categories, 75

F -group, 110 field of p-adic numbers, 196

Index final topology, 52 F -rational points, 110 F -split reductive group, 110 torus, 110 fundamental group, 127

G-equivariant map, 71

Haar integral, 132 Haar measure, 58 Hilbert’s Nullstellensatz, 202

identity component, 94 intertwining operator, 71 inverse system, 11 of topological groups, 19 of topological rings, 20 of topological oK -modules, 20 Iwahori subgroup, 117 Iwasawa algebra, 26 Iwasawa module, 80 Iwasawa ring, 26

213 normalized induction, 142 norm-bounded subset of V  , 50 normed K-space, 199

open ball, 36, 194 open map, 67 open mapping theorem, 67 operator norm, 37 opposite Borel subgroup, 108 ordinary representation, 86

p-adic absolute value, 196 p-adic Lie group, 63 parabolic subgroup, 109 principal series representation continuous, 148 smooth, 139 profinite group, 21 projection homomorphism, 20 projections in projective limits, 14 projective limit, 13 projective variety, 205 pro-p group, 21

quasi-split group, 110 lattice, 35 left translation, 71 length, 103 length of a representation, 142 Lie group, 63 linear algebraic group, 92 linear-topological oK -modules, 25 local field, 195 archimedean, 195 nonarchimedean, 195 locally analytic vector, 84 locally convex K-vector space, 36 topology, 36 locally L-analytic group, 63 local ring, 116 longest element, 103

metric, 193 metric space, 193 modulus character, 141 morphism of inverse systems, 17

norm, 199

radical of an algebraic group, 96 of an ideal, 202 radical ideal, 202 rank of an algebraic group, 95 of a root system, 101 reduced decomposition, 103 reductive group, 96 reflexive Banach space, 42 residue field, 198 right translation, 71 ring of integers, 197 ring of p-adic integers, 196 root, 98 root datum, 106 root lattice, 126 root space, 98 root subgroup, 107

Schneider-Teitelbaum duality, 83 Schneider-Teitelbaum duality II, 85 semisimple

214 element, 93 group, 96 semisimple rank, 96 separately continuous map, 59 simple algebraic group, 105 simply connected group, 127 smooth function, 132 representation, 134 socle, 124 spherically complete space, 64 split Z-group, 113 strong topology, 39 subbase of the topology, 15 subrepresentation, 71 sup norm, 40 tensor product of representations, 137 topological group, 19 topologically irreducible representation, 71 topological ring, 19

Index torus, 95 totally disconnected, 195

ultrametric, 193 ultrametric space, 193 uniform boundedness theorem, 67 uniformizer, 198 unipotent element, 93 group, 94 unitary Banach space representation, 85

weak topology, 39 weight, 128 Weyl group of a reductive group, 100 of a root system, 102

Zariski topology, 203

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