Representations of Finite Groups of Lie Type (London Mathematical Society Student Texts) [2 ed.] 1108481485, 9781108481489

Table of contents :
Contents
Introduction to the Second Edition
From the Introduction to the First Edition
1 Basic Results on Algebraic Groups
1.1 Basic Results on Algebraic Groups
1.2 Diagonalisable Groups, Tori, X(T),Y(T)
1.3 Solvable Groups, Borel Subgroups
1.4 Unipotent Groups, Radical, Reductive and Semi-Simple Groups
1.5 Examples of Reductive Groups
2 Structure Theorems for Reductive Groups
2.1 Coxeter Groups
2.2 Finite Root Systems
2.3 Structure of Reductive Groups
2.4 Root Data, Isogenies, Presentation of G
3 (B,N)-Pairs; Parabolic, Levi, and Reductive Subgroups; Centralisers of Semi-Simple Elements
3.1 (B,N)-Pairs
3.2 Parabolic Subgroups of Coxeter Groups and of (B,N)-Pairs
3.3 Closed Subsets of a Crystallographic Root System
3.4 Parabolic Subgroups and Levi Subgroups
3.5 Centralisers of Semi-Simple Elements
4 Rationality, the Frobenius Endomorphism, the Lang–Steinberg Theorem
4.1 k0-Varieties, Frobenius Endomorphisms
4.2 The Lang–Steinberg Theorem; Galois Cohomology
4.3 Classification of Finite Groups of Lie Type
4.4 The Relative (B,N)-Pair
5 Harish-Chandra Theory
5.1 Harish-Chandra Induction and Restriction
5.2 The Mackey Formula
5.3 Harish-Chandra Theory
6 Iwahori–Hecke Algebras
6.1 Endomorphism Algebras
6.2 Iwahori–Hecke Algebras
6.3 Schur Elements and Generic Degrees
6.4 The Example of G2
7 The Duality Functor and the Steinberg Character
7.1 F-rank
7.2 The Duality Functor
7.3 Restriction to Centralisers of Semi-Simple Elements
7.4 The Steinberg Character
8 ℓ-Adic Cohomology
8.1 ℓ-Adic Cohomology
9 Deligne–Lusztig Induction: The Mackey Formula
9.1 Deligne–Lusztig Induction
9.2 Mackey Formula for Lusztig Functors
9.3 Consequences: Scalar Products
10 The Character Formula and Other Results on Deligne–Lusztig Induction
10.1 The Character Formula
10.2 Uniform Functions
10.3 The Characteristic Function of a Semi-Simple Class
11 Geometric Conjugacy and the Lusztig Series
11.1 Geometric Conjugacy
11.2 More on Centralisers of Semi-Simple Elements
11.3 The Lusztig Series
11.4 Lusztig’s Jordan Decomposition of Characters: The Levi Case
11.5 Lusztig’s Jordan Decomposition of Characters: The General Case
11.6 More about Unipotent Characters
11.7 The Irreducible Characters of GLnF and UFn
12 Regular Elements; Gelfand–Graev Representations; Regular and Semi-Simple Characters
12.1 Regular Elements
12.2 Regular Unipotent Elements
12.3 Gelfand–Graev Representations
12.4 Regular and Semi-Simple Characters
12.5 The Character Table of SL2(Fq)
13 Green Functions
13.1 Invariants
13.2 Green Functions and the Springer Correspondence
13.3 The Lusztig–Shoji Algorithm
14 The Decomposition of Deligne–Lusztig Characters
14.1 Lusztig Families and Special Unipotent Classes
14.2 Split Groups
14.3 Twisted Groups
References
Index

Citation preview

LONDON MATHEMATICAL SOCIETY STUDENT TEXTS Managing Editor: Ian J. Leary, Mathematical Sciences, University of Southampton, UK 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Introduction to Banach algebras, operators and harmonic analysis, GARTH DALES et al Computational algebraic geometry, HAL SCHENCK Frobenius algebras and 2-D topological quantum field theories, JOACHIM KOCK Linear operators and linear systems, JONATHAN R. PARTINGTON An introduction to noncommutative Noetherian rings (2nd Edition), K. R. GOODEARL & R. B. WARFIELD, JR Topics from one-dimensional dynamics, KAREN M. BRUCKS & HENK BRUIN Singular points of plane curves, C. T. C. WALL A short course on Banach space theory, N. L. CAROTHERS Elements of the representation theory of associative algebras I, IBRAHIM ASSEM, ´ DANIEL SIMSON & ANDRZEJ SKOWRONSKI An introduction to sieve methods and their applications, ALINA CARMEN COJOCARU & M. RAM MURTY Elliptic functions, J. V. ARMITAGE & W. F. EBERLEIN Hyperbolic geometry from a local viewpoint, LINDA KEEN & NIKOLA LAKIC Lectures on Kähler geometry, ANDREI MOROIANU Dependence logic, JOUKU VÄÄNÄNEN Elements of the representation theory of associative algebras II, DANIEL SIMSON & ´ ANDRZEJ SKOWRONSKI Elements of the representation theory of associative algebras III, DANIEL SIMSON & ´ ANDRZEJ SKOWRONSKI Groups, graphs and trees, JOHN MEIER Representation theorems in Hardy spaces, JAVAD MASHREGHI ´ An introduction to the theory of graph spectra, DRAGOŠ CVETKOVIC, PETER ROWLINSON & SLOBODAN SIMIC´ Number theory in the spirit of Liouville, KENNETH S. WILLIAMS Lectures on profinite topics in group theory, BENJAMIN KLOPSCH, NIKOLAY NIKOLOV & CHRISTOPHER VOLL Clifford algebras: An introduction, D. J. H. GARLING Introduction to compact Riemann surfaces and dessins d’enfants, ERNESTO GIRONDO & GABINO GONZÁLEZ-DIEZ The Riemann hypothesis for function fields, MACHIEL VAN FRANKENHUIJSEN Number theory, Fourier analysis and geometric discrepancy, GIANCARLO TRAVAGLINI Finite geometry and combinatorial applications, SIMEON BALL The geometry of celestial mechanics, HANSJÖRG GEIGES Random graphs, geometry and asymptotic structure, MICHAEL KRIVELEVICH et al Fourier analysis: Part I - Theory, ADRIAN CONSTANTIN ˇ Dispersive partial differential equations, M. BURAK ERDOGAN & NIKOLAOS TZIRAKIS Riemann surfaces and algebraic curves, R. CAVALIERI & E. MILES Groups, languages and automata, DEREK F. HOLT, SARAH REES & CLAAS E. RÖVER Analysis on Polish spaces and an introduction to optimal transportation, D. J. H. GARLING The homotopy theory of (∞,1)-categories, JULIA E. BERGNER The block theory of finite group algebras I, M. LINCKELMANN The block theory of finite group algebras II, M. LINCKELMANN Semigroups of linear operators, D. APPLEBAUM Introduction to approximate groups, M. C. H. TOINTON Representations of finite groups of Lie type (2nd Edition), F. DIGNE & J. MICHEL Tensor products of C*-algebras and operator spaces, G. PISIER

London Mathematical Society Student Texts 95

Representations of Finite Groups of Lie Type second edition F RANÇOIS DIGNE Université de Picardie Jules Verne, Amiens JEAN MICHEL Centre National de la Recherche Scientifique (CNRS), Paris

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06-04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108481489 DOI: 10.1017/9781108673655 First edition © Cambridge University Press 1991 Second edition © François Digne and Jean Michel 2020 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1991 Second edition 2020 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Digne, François, author. | Michel, Jean, author. Title: Representations of finite groups of Lie type / François Digne, Université de Picardie Jules Verne, Amiens, Jean Michel, Centre National de la Recherche Scientifique (CNRS), Paris. Description: Second edition. | Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2020. | Series: London mathematical society student texts ; 95 | Includes bibliographical references and index. Identifiers: LCCN 2019035714 | ISBN 9781108481489 (hardback) | ISBN 9781108673655 (ebook) Subjects: LCSH: Lie groups. | Representations of groups. Classification: LCC QA387 .D54 2020 | DDC 512/.482--dc23 LC record available at https://lccn.loc.gov/2019035714 ISBN 978-1-108-48148-9 Hardback ISBN 978-1-108-72262-9 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Introduction to the Second Edition From the Introduction to the First Edition 1

page 1 2

Basic Results on Algebraic Groups 1.1 Basic Results on Algebraic Groups 1.2 Diagonalisable Groups, Tori, X(T), Y (T) 1.3 Solvable Groups, Borel Subgroups 1.4 Unipotent Groups, Radical, Reductive and Semi-Simple Groups 1.5 Examples of Reductive Groups

13 15

2

Structure Theorems for Reductive Groups 2.1 Coxeter Groups 2.2 Finite Root Systems 2.3 Structure of Reductive Groups 2.4 Root Data, Isogenies, Presentation of G

19 19 24 29 35

3

(B, N)-Pairs; Parabolic, Levi, and Reductive Subgroups; Centralisers of Semi-Simple Elements 3.1 (B, N)-Pairs 3.2 Parabolic Subgroups of Coxeter Groups and of (B, N)-Pairs 3.3 Closed Subsets of a Crystallographic Root System 3.4 Parabolic Subgroups and Levi Subgroups 3.5 Centralisers of Semi-Simple Elements

v

5 5 8 11

39 39 42 45 51 56

vi

4

Contents

Rationality, the Frobenius Endomorphism, the Lang–Steinberg Theorem 4.1 k0 -Varieties, Frobenius Endomorphisms 4.2 The Lang–Steinberg Theorem; Galois Cohomology 4.3 Classification of Finite Groups of Lie Type 4.4 The Relative (B, N)-Pair

59 59 63 70 75

5

Harish-Chandra Theory 5.1 Harish-Chandra Induction and Restriction 5.2 The Mackey Formula 5.3 Harish-Chandra Theory

79 79 83 86

6

Iwahori–Hecke Algebras 6.1 Endomorphism Algebras 6.2 Iwahori–Hecke Algebras 6.3 Schur Elements and Generic Degrees 6.4 The Example of G2

91 91 97 104 108

7

The Duality Functor and the Steinberg Character 7.1 F-rank 7.2 The Duality Functor 7.3 Restriction to Centralisers of Semi-Simple Elements 7.4 The Steinberg Character

113 113 116 123 126

8

-Adic Cohomology 8.1 -Adic Cohomology

130 130

9

Deligne–Lusztig Induction: The Mackey Formula 9.1 Deligne–Lusztig Induction 9.2 Mackey Formula for Lusztig Functors 9.3 Consequences: Scalar Products

137 137 140 146

10

The Character Formula and Other Results on Deligne–Lusztig Induction 10.1 The Character Formula 10.2 Uniform Functions 10.3 The Characteristic Function of a Semi-Simple Class

148 148 153 157

Geometric Conjugacy and the Lusztig Series 11.1 Geometric Conjugacy 11.2 More on Centralisers of Semi-Simple Elements 11.3 The Lusztig Series

161 161 167 170

11

Contents

11.4 Lusztig’s Jordan Decomposition of Characters: The Levi Case 11.5 Lusztig’s Jordan Decomposition of Characters: The General Case 11.6 More about Unipotent Characters 11.7 The Irreducible Characters of GLFn and UFn 12

vii

175 183 188 191

Regular Elements; Gelfand–Graev Representations; Regular and Semi-Simple Characters 12.1 Regular Elements 12.2 Regular Unipotent Elements 12.3 Gelfand–Graev Representations 12.4 Regular and Semi-Simple Characters 12.5 The Character Table of SL2 (Fq )

196 196 201 207 214 220

13

Green Functions 13.1 Invariants 13.2 Green Functions and the Springer Correspondence 13.3 The Lusztig–Shoji Algorithm

225 225 231 236

14

The Decomposition of Deligne–Lusztig Characters 14.1 Lusztig Families and Special Unipotent Classes 14.2 Split Groups 14.3 Twisted Groups

242 242 244 247

References Index

249 255

Introduction to the Second Edition

We had two main aims in writing this edition: • Be more self-contained where possible. For instance, we have added brief overviews of Coxeter groups and root systems, and given some more details about the theory of algebraic groups. • While retaining the same level of exposition as in the first edition, we have given a more complete account of the representation theory of finite groups of Lie type. In view of the second aim, we have added the following topics to our exposition: • We cover Ree and Suzuki groups extending our exposition of Frobenius morphisms to the more general case of Frobenius roots. • We have added to Harish-Chandra theory the topic of Hecke algebras and given as many results as we could easily do for fields of arbitrary characteristic prime to q, in view of applications to modular representations. • We have added a chapter on the computation of Green functions, with a brief review of invariant theory of reflection groups, and a chapter on the decomposition of unipotent Deligne–Lusztig characters.

Acknowledgements In addition to the people we thank in the introduction to the first edition, we have benefitted from the input of younger colleagues such as Cédric Bonnafé, Olivier Dudas, Meinolf Geck, Daniel Juteau, and Raphaël Rouquier, and we give special thanks to Gunter Malle for a thorough proofreading of this edition. We also thank the many people who pointed out to us misprints and other errors in the first edition.

1

From the Introduction to the First Edition

These notes follow a course given at the Paris VII university during the spring semester of academic year 1987–88. Their purpose is to expound basic results in the representation theory of finite groups of Lie type (a precise definition of this concept will be given in the chapter “Rationality, Frobenius”). Let us start with some notations. We denote by Fq a finite field of characteristic p with q elements (q is a power of p). The typical groups we will look at are the linear, unitary, symplectic, orthogonal, . . . , groups over Fq . We will consider these groups as the subgroups of points with coefficients in Fq of the corresponding groups over an algebraic closure Fq (which are algebraic reductive groups). More precisely, the group over Fq is the set of fixed points of the group over Fq under an endomorphism F called the Frobenius endomorphism; this will be explained in the chapter “Rationality, Frobenius”. In the following paragraphs of this introduction we will try to describe, by some examples, a sample of the methods used to study the complex representations of these groups.

Induction from Subgroups Let us start with the example where G = GLn (Fq ) is the general linear group over Fq . Let T be the subgroup of diagonal matrices; it is a subgroup of the group B of upper triangular matrices, and there is a semidirect product decomposition B = U  T, where U is the subgroup of the upper triangular matrices which have all their diagonal entries equal to 1. The representation theory of T is easy, since it is a commutative group (actually isomorphic to a product of n copies of the multiplicative group F×q ). Composition with the natural homomorphism from B to T (quotient by U) lifts representations of T to representations of B. Inducing these representations from B to the whole of the general linear group gives representations of G (whose irreducible constituents are called 2

From the Introduction to the First Edition

3

“principal series representations”). More generally, we can replace T with a group L of block-diagonal matrices, B with the group of corresponding upper block-triangular matrices P, and we have a semi-direct product decomposition (called a Levi decomposition) P = V  L, where V is the subgroup of P whose diagonal blocks are identity matrices; we may as before induce from P to G representations of L lifted to P. The point of this method is that L is isomorphic to a direct product of general linear groups of smaller degrees than n. We thus have an inductive process to get representations of G if we know how to decompose induced representations from P to G. This approach has been developed in the works of Harish-Chandra, Howlett and Lehrer, and is introduced in the chapter “Harish-Chandra Theory”.

Cohomological Methods Let us now consider the example of G = Un , the unitary group. It can be defined as the subgroup of matrices A ∈ GLn (Fq2 ) such that tA[q] = A−1 , where A[q] denotes the matrix whose entries are those of A raised to the qth power. It is thus the subgroup of GLn (Fq ) consisting of the fixed points of the endomorphism F : A → ( tA[q] ) −1 . A subgroup L of block-diagonal matrices in Un is again a product of unitary groups of smaller degree. But this time we cannot construct a bigger group P having L as a quotient. More precisely, the group V of upper block-triangular matrices with entries in Fq and whose diagonal blocks are the identity matrix has no fixed points other than the identity under F. To get a suitable theory, Harish-Chandra’s construction must be generalised; instead of inducing from V  L to G, we construct a variety attached to V on which both L and G act with commuting actions, and the cohomology of that variety with -adic coefficients gives a (virtual) bi-module which defines a “generalised induction” from L to G. This approach, due to Deligne and Lusztig, will be developed in the chapters from “-adic Cohomology” to “Geometric Conjugacy and Lusztig Series”.

Gelfand–Graev Representations Using the above methods, a lot of information can be obtained about the characters of the groups G(Fq ), when G has a connected centre. The situation is not so clear when the centre of G is not connected. In this case one can use the Gelfand–Graev representations, which are obtained by inducing a linear character “in general position” of a maximal unipotent subgroup (in GLn the subgroup of upper triangular matrices with ones on the diagonal is such a subgroup). These representations are closely tied to the theory of regular unipotent

4

From the Introduction to the First Edition

elements. They are multiplicity-free and contain rather large cross-sections of the set of irreducible characters, and so give useful additional information in the nonconnected centre case. (In the connected centre case, they are linear combinations of Deligne–Lusztig characters.) For instance, in SL2 (Fq ) they are obtained by inducing a nontrivial linear   1 u character of the group of matrices of the form : such a character cor0 1 responds to a nontrivial additive character of Fq ; there are two classes of such characters under SL2 (Fq ), which corresponds to the fact that the centre of SL2 has two connected components (its two elements). The theory of regular elements and Gelfand–Graev representations is expounded in chapter “Regular Elements; Gelfand–Graev Representations”, with, as an application, the computation of the values of all irreducible characters on regular unipotent elements.

Acknowledgements We would like to thank the “équipe des groupes finis” and the mathematics department of the École Normale Supérieure, who provided us with a stimulating working environment, and adequate facilities for composing this book. We thank also the Paris VII university, which gave us the opportunity to give the course which started this book. We thank all those who carefully read the earlier drafts and suggested improvements, particularly Michel Enguehard, Guy Rousseau, Jean-Yves Hée and the editor. We thank Michel Broué and Jacques Tits, who provided us with various ideas and information, and above all George Lusztig, who invented most of the theory.

1 Basic Results on Algebraic Groups

A finite group of Lie type – sometimes also called a finite reductive group – is (in a first approximation) the group of points over a finite field Fq of a (usually connected) reductive algebraic group over an algebraic closure Fq . We begin by recalling the definition of these terms. See 4.2.6 for a precise definition of finite groups of Lie type. Let us first establish some notations and conventions we use throughout. If g is an automorphism of a set (resp. variety, group, . . . ) X, we will denote by X g the set of fixed points of g, and by gx or g(x) the image of the element x ∈ X by g. A group G acts naturally on itself by conjugation, and we will write gh for ghg−1 , where g and h are elements of G and will write ad g for the morphism h → gh. We will write Z(G) for the centre of G; if X is a subset of a set on which G acts, we put NG (X) = {g ∈ G | gX = X} = {g ∈ G | ∀x ∈ X, gx ∈ X} and CG (X) = {g ∈ G | ∀x ∈ X, gx = x}. If A is a ring and X a set (resp. a group), we denote by AX the free A-module with basis X (resp. the group algebra of X with coefficients in A). We write H  G for a semi-direct product of groups where H is normal in the product. We will generally use bold letters for algebraic groups and varieties.

1.1 Basic Results on Algebraic Groups An algebraic group is an algebraic variety G endowed with a group structure such that the multiplication and inverse maps are algebraic. In this book, we will need only affine algebraic groups over an algebraically closed field k (which will be taken to be Fp from Chapter 4 onwards): that is, affine algebraic varieties G = Spec A, where A is a reduced k-algebra; that is, without non-zero nilpotent elements. The group structure gives a coalgebra structure on A, or even a Hopf algebra structure. For such a group G, we will call elements of G 5

6

Basic Results on Algebraic Groups

the elements of the group G(k) of k-valued points of G; that is, the morphisms A → k. Since an algebraic variety over k is determined by the set of its k-points, we will sometimes identify G with the set of its elements. Examples 1.1.1 (of affine algebraic groups) (i) The additive group Ga , defined by the algebra k[t], with comultiplication t → t ⊗ 1 + 1 ⊗ t. We have Ga (k)  k+ . (ii) The multiplicative group Gm , defined by the algebra k[t, t−1 ], with comultiplication t → t ⊗ t. We have Gm (k)  k× . (iii) The general linear group GLn , defined by the algebra k[{ti,j }1≤i,j≤n ,  det(ti,j ) −1 ]. The comultiplication is given by ti,j → k ti,k ⊗ tk,j . The group GLn (k) identifies with the group of invertible n × n-matrices with entries in k. As an open subvariety of an affine space, GLn is connected. (iv) A finite group Γ is algebraic; its algebra is the algebra of k-valued functions on Γ, which can be identified to Homk (kΓ, k). A basis of orthogonal idempotents is formed of the Dirac functions δg (g ) = δg,g . The comulti plication is δg → {g ,g ∈Γ |g g =g} δg ⊗ δg . Actually every connected one-dimensional affine algebraic group is isomorphic to Gm or Ga – this is surprisingly difficult to prove; see for instance Springer (1998, 3.4.9). A morphism of algebraic groups is a morphism of varieties which is also a group homomorphism. A closed subvariety which is a subgroup is naturally an algebraic subgroup; that is, the inclusion map is a morphism of algebraic groups. For some problems about algebraic groups, working with Remark 1.1.2 varieties instead of schemes loses information. For instance, in our setting, when k = Fp , an algebraic closure of the prime field Fp , the kernel of the morphism x → xp : Ga → Ga is trivial. However, in the setting of schemes, it can be computed as Spec(k[t]/tp ), which is not allowed in the setting of varieties since it is a non-reduced algebra. Proposition 1.1.3 Let {Vi }i∈I be a family of irreducible subvarieties all containing the identity element of an algebraic group G; then the smallest closed subgroup H of G containing the Vi is equal to the product Wi1 . . . Wik for some finite sequence (i1 , . . . , ik ) of elements of I, where either Wi = Vi or Wi = {x−1 | x ∈ Vi }. Reference

See Borel (1991, I, 2.2).



1.1 Basic Results on Algebraic Groups

7

H is called the subgroup generated by the Vi . It is clear from 1.1.3 that the subgroup generated by a family of connected subvarieties is connected. An algebraic group is called linear if it is isomorphic to a closed subgroup of GLn . It is clear from the definition that a linear algebraic group is affine; the converse is also true; see Springer (1998, 2.3.7(i)). All algebraic groups considered in the sequel will be linear; an example of non-linear algebraic groups are the elliptic curves. Proposition 1.1.4 group; then:

Let G be a linear algebraic group and H a closed sub-

(i) The quotient G/H exists and is a quasi-projective variety (that is, an open subvariety of projective variety). (ii) If H is a normal subgroup, then G/H is an affine variety and a linear algebraic group for the induced group structure. References

See Springer (1998, 5.5.5 and 5.5.10).



The connected components of an algebraic group G are finite in number and coincide with its irreducible components; the component containing the identity element of G is called the identity component and denoted by G0 . It is a characteristic subgroup of G, and G/G0 identifies with the set of connected components of G. Conversely, every normal closed subgroup of finite index contains G0 – these properties are elementary; see, for example, Springer (1998, 2.2.1). An element of a linear algebraic group G is called semi-simple (resp. unipotent) if its image in some embedding of G as a closed subgroup of some GLn is semi-simple; that is, conjugate to a diagonal matrix (resp. unipotent, that is, conjugate to an upper unitriangular matrix). This property does not depend on the embedding. Every element has a unique decomposition, its Jordan decomposition, as the product of two commuting semi-simple and unipotent elements; see Springer (1998, 2.4.8). Proposition 1.1.5 Let G be a linear algebraic group over Fp , where p is a prime. Then every element of G has finite order: the semi-simple elements are the p -elements, and the unipotent elements are the p-elements. Proof This results from the fact that the above result holds for GLn (Fp ). × Indeed, the diagonal entries of a diagonal matrix are in Fp , thus of order prime to p, and a matrix X is unipotent if and only if N = X − In is nilpotent, equivaa a  lently X p = N p + In = In for a large enough.

8

Basic Results on Algebraic Groups

1.2 Diagonalisable Groups, Tori, X(T), Y(T) Definition 1.2.1 (i) A torus is an algebraic group T isomorphic to Grm for some r called the rank of T. (ii) A diagonalisable group is an algebraic group isomorphic to a closed subgroup of a torus. (iii) A rational character of an algebraic group G is a morphism of algebraic groups G → Gm . (iv) The character group X(G) of an algebraic group G is the group of rational characters of G. Proposition 1.2.2 A diagonalisable group D is equal to Spec(kX(D)). Proof We first prove the result for a torus T of rank r; its algebra is A = k[t1 , . . . , tr , t1−1 , . . . , tr−1 ]  k[t, t−1 ] ⊗r . An element of X(T) is given by a morphism of both algebras and coalgebras k[t, t−1 ] → A, which is defined by giving the image of t which must be invertible, thus must be a monomial; this monomial must be unitary for the morphism to be a coalgebra morphism. The group law on X(T), given by pointwise multiplication, corresponds to the multiplication of monomials; thus X(T) is isomorphic to the group of unitary monomials, and A  kX(T), the group algebra of the group X(T). Let now D be a closed subgroup of a torus T; its algebra is a quotient of the algebra kX(T) of T, hence is spanned by the images of the rational characters of T; that is, by the restrictions to D of these characters. By the linear independence of the characters of a group as functions to k – see, for example, Lang (2002, Chapter VI, Theorem 4.1) – these restrictions form a basis of the algebra of D, whence the result.  Proposition 1.2.3 (i) A diagonalisable group is the direct product of a torus by a finite abelian p -group, where p is the characteristic of k. (ii) The quotient of a torus by a closed subgroup is a torus. References

See Springer (1998, 3.2.7(i)) and Borel (1991, Corollary of 8.4). 

Proposition 1.2.4 Seen as a functor from diagonalisable groups to Z-modules, X is an exact functor. Proof Left exactness is a general fact for Hom functors. Now, a similar argument to that in the proof of 1.2.2 shows that if D ⊂ D are diagonalisable  groups, then the restriction X(D ) → X(D) is surjective.

1.2 Diagonalisable Groups, Tori, X(T), Y (T)

9

Exercise 1.2.5 Show that X is an antiequivalence of abelian categories from diagonalisable groups over k to Z-modules of finite type without p-torsion, where p is the characteristic of k, and that X restricts to an antiequivalence of additive categories from tori to finite rank lattices (here “lattice” means free Z-module). Use the fact that the inverse antiequivalence maps a Z-module M to Spec k[M]. Proposition 1.2.6 Any exact sequence of tori is split; in particular, a subtorus of a torus is a direct factor. Proof By the antiequivalence of categories X (see 1.2.5) an exact sequence of tori corresponds to an exact sequence of lattices. Such a sequence is always split.  Proposition 1.2.7 Given an algebraic action of a torus T on an affine variety X, there exists an element t ∈ T such that Xt = XT . Proof It is known that an algebraic group action can be linearised; that is, there exists an embedding of X into a finite-dimensional k-vector space V and an embedding of T into GL(V) such that the action of T factors through this embedding; see Slodowy (1980, I, 1.3). The space V has a decomposition into irreducible T-submodules, each defining some χ ∈ X(T). The kernel of a nontrivial such χ has codimension 1 in T, since its algebra is the quotient of the algebra of T by the ideal generated by χ (identifying the algebra of T with the group algebra of X(T)). It follows that there is some element t ∈ T which lies outside the finitely many kernels of the non-trivial characters of T occurring in V; the fixed points of t on V (and so on X) are the same as those of T, whence the result.  Proposition 1.2.8 (i) A torus consists of semi-simple elements. (ii) A connected linear algebraic group containing only semi-simple elements is a torus. Proof Proposition 1.2.8 (i) comes from the fact that Grm can be embedded as the group of diagonal matrices in GLr . For (ii) see Springer (1998, 6.3.6).  Definition 1.2.9 (i) A one-parameter subgroup of an algebraic group G is a morphism of algebraic groups Gm → G. (ii) The abelian group of one-parameter subgroups of a torus T is denoted by Y (T).

10

Basic Results on Algebraic Groups

An element of Y (T) is given by a morphism k[t1 , . . . , tr , t1−1 , . . . , tr−1 ] → k[t, t−1 ], determined by the images of the ti which must be invertible, thus monomials. These monomials must be unitary, thus powers of t, for the morphism to be a coalgebra morphism. The group law on one-parameter subgroups corresponds to multiplying these powers, so Y (T) is isomorphic to a product of r copies of the group of powers of t. There is an exact pairing between X(T) and Y (T) (that is, a bilinear map X(T) × Y (T) → Z making each one the Z-dual of the other) obtained as follows: given χ ∈ X(T) and ψ ∈ Y (T), the composite map χ ◦ ψ is a homomorphism from Gm to itself, so is the map t → tn for some n ∈ Z; the pairing is defined by ( χ, ψ) → n. Proposition 1.2.10 The map y ⊗ x → y(x) is a group isomorphism: ∼

Y (T) ⊗Z k× − → T(k). Proof Let (xi )i=1, ...,n and (yi )i=1, ...,n be two dual bases of X(T) and Y (T),  → Y (T) ⊗Z k× is respectively. It is easy to check that t → i=n i=1 yi ⊗ xi (t) : T(k)  the inverse of the map of the statement, using the fact that x∈X(T) Ker x = {1}. (This follows from the isomorphism between the algebra of the variety T and kX(T); see 1.2.2.)  We now study the relationship between closed subgroups of T and subgroups of X(T). Lemma 1.2.11 Given a torus T, let x1 , . . . , xn be linearly independent elements of X(T) and λ 1 , . . . , λ n arbitrary elements of k× ; then there exists s ∈ T such that xi (s) = λ i for i = 1, . . . , n. Reference

See, for example, Humphreys (1975, 16.2, Lemma C).



Definition 1.2.12 Given a torus T and a closed subgroup S of T, we define S⊥ X(T) = {x ∈ X(T) | ∀s ∈ S, x(s) = 1}; conversely, given a subgroup A of X(T), we define a subgroup of T by A⊥ T = {s ∈ T | ∀x ∈ A, x(s) = 1} (which is closed, since A is finitely generated). Proposition 1.2.13 If k has characteristic p, given a torus T and a subgroup ⊥ A of X(T), the group (A⊥ T )X(T) /A is the p-torsion subgroup of X(T)/A. Proof First notice that, for any closed subgroup S of T, the group X(T)/S⊥ X(T) has no p-torsion. Indeed, p ⊥ pn x ∈ S⊥ X(T) ⇔ ∀s ∈ S, x(s) = 1 ⇔ ∀s ∈ S, x(s) = 1 ⇔ x ∈ SX(T) , n

n

where the middle equivalence holds, since x → xp is an automorphism of k.

1.3 Solvable Groups, Borel Subgroups

11

⊥ ⊥ ⊥ Thus it is enough to see that (A⊥ T )X(T) /A is a p-group. Let x ∈ (AT )X(T) − A. It is a standard result on submodules of free Z-modules that there is a basis (x, x1 , . . . , xr ) of A, x such that (mx, x1 , . . . , xr ) is a basis of A, with m ∈ Z, possibly m = 0 (which means omit mx). The result follows if we can prove that m is a power of p. Let us assume otherwise: then there exists λ ∈ k× such that λ  1 and λ m = 1 (even if m = 0). By 1.2.11, there exists some s ∈ T such that x(s) = λ, x1 (s) = 1, . . . , xr (s) = 1. Thus mx(s) = 1, so s ∈ A⊥ T , but x(s)  1, ⊥ ⊥  which contradicts x ∈ (AT )X(T) .

1.3 Solvable Groups, Borel Subgroups We will denote by Gu the set of unipotent elements of an algebraic group G. Proposition 1.3.1

Let G be a connected solvable algebraic group; then:

(i) Gu is a normal connected subgroup of G. (ii) For every maximal torus T of G, there is a semi-direct product decomposition G = Gu  T. (iii) Let S be a subtorus of G – then NG (S) = CG (S). Proof For (i) and (ii) see Springer (1998, 6.3.3(ii) and 6.3.5(iv)); these assertions follow from the theorem of Lie–Kolchin, which states that every closed solvable subgroup of GLn (k) is conjugate to a subgroup of the group of upper triangular matrices. For (iii), if n ∈ NG (S), s ∈ S, then [n, s] ∈ [G, G] ∩ S ⊂ Gu ∩ S = 1, where the first inclusion comes from (ii).  Let us note that this proposition implies that every connected solvable algebraic group containing no unipotent elements is a torus. Definition 1.3.2 Maximal closed connected solvable subgroups of an algebraic group are called Borel subgroups. These groups have paramount importance in the theory. The next theorem states their basic properties. Theorem 1.3.3 (i) (ii) (iii) (iv) (v)

Let G be a connected algebraic group; then:

All Borel subgroups of G are conjugate. Every element of G is in some Borel subgroup. The centraliser in G of any torus is connected. A Borel subgroup is equal to its normaliser in G. Every maximal torus of G is in some Borel subgroup.

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Basic Results on Algebraic Groups

Outline of proof To show (i) – see Springer (1998, 6.2.7(iii)) – one first shows that G/B is a complete variety; a complete variety has the property that a connected solvable group acting on it always has a fixed point. Thus another Borel subgroup B acting on G/B by left translation has a fixed point; that is, there exists g ∈ G such that B gB = gB, so g−1 B g ⊂ B, whence the result. Property (ii) similarly results from properties of complete varieties; see Springer (1998, 6.4.5(i)). For (iii) see Springer (1998, 6.4.7(i)). For (iv) see Springer (1998, 6.4.9). Property (v) results from the definition of a Borel subgroup.  It follows from (iv) above and the remark that the closure of a solvable group is solvable – see Borel (1991, I, 2.4) – that the words “closed connected” can be omitted from the definition of a Borel subgroup when G is connected. Theorem 1.3.4

Let T be a torus of a connected algebraic group G, then:

(i) NG (T) 0 = CG (T) = CG (T) 0 . (ii) The Borel subgroups of CG (T) are the groups CG (T) ∩ B, where B runs over the Borel subgroups of G containing T. References

See, for example Springer (1998, 3.2.9 and 6.4.7).



It follows from (i) that the quotient NG (T)/CG (T) is finite. Definition 1.3.5 A closed subgroup of a connected algebraic group G which contains a Borel subgroup is called a parabolic subgroup. Corollary 1.3.6

Let G be a connected algebraic group:

(i) Any parabolic subgroup is equal to its normaliser in G and is connected. (ii) Two parabolic subgroups containing the same Borel subgroup and G-conjugate are equal. (iii) All maximal tori of G are conjugate; every semi-simple element of G is in some maximal torus. (iv) Two elements of a maximal torus T are G-conjugate if and only if they are NG (T)-conjugate. Proof Let us prove (i). As the Borel subgroups are connected, P0 contains a Borel subgroup B. As another Borel subgroup of G in P0 is P0 -conjugate to B, we have NG (P0 ) = P0 NG (B) = P0 B = P0 . As P ⊂ NG (P0 ) we have P = P0 . Let us prove (ii). Using again that Borel subgroups of P are P-conjugate, we get that two conjugate parabolic subgroups containing the same Borel subgroup are NG (B)-conjugate, thus are equal, since NG (B) = B. Since any maximal torus is in some Borel subgroup, the first assertion in (iii) results from 1.3.3 (i) and the same property for connected solvable groups;

1.4 Unipotent Groups, Radical, Reductive and Semi-Simple Groups

13

see Springer (1998, 6.3.5 (iii)). The second assertion of (iii) comes from 1.3.3 (ii) and the same property for connected solvable groups; see Springer (1998, 6.3.5 (i)). −1 Let us prove (iv). If s and gs both lie in T, then T and g T are two maximal tori containing s, thus are two maximal tori of the group CG (s), and since they are connected they lie in the identity component CG (s) 0 . By (iii) they are −1 conjugate by some element x ∈ CG (s) 0 ; that is, xT = g T so gx ∈ NG (T). As gxs = gs, we get the result.  Statement 1.3.6(iii) allows to give the following definition. Definition 1.3.7 maximal tori.

The rank of an algebraic group is the rank of its

Exercise 1.3.8 If a closed subgroup H of the connected algebraic group G contains a maximal torus T of G, then NG (H) ⊂ H0 · NG (T).

1.4 Unipotent Groups, Radical, Reductive and Semi-Simple Groups A unipotent algebraic group is a group containing only unipotent elements. Proposition 1.4.1 is nilpotent.

Every unipotent subgroup of an affine algebraic group G

Proof By embedding G into GLn , we can reduce to the case G = GLn ; any unipotent subgroup of GLn (k) is conjugate to a subgroup of the group of upper triangular matrices which have all their diagonal entries equal to 1 – see Springer (1998, 2.4.12) for a proof – whence the result.  Proposition 1.4.2 As an algebraic variety, a connected unipotent group is isomorphic to an affine space; in characteristic 0 a unipotent group is necessarily connected. Proof For the first part, see Springer (1998, 14.2.7 and 14.3.10). If the characteristic is 0 and U is a unipotent algebraic group, then U/U0 is a finite unipotent group. But a unipotent element in characteristic 0 has infinite order, since this  holds in GLn , whence the second part. Corollary 1.4.3 The maximal connected unipotent subgroups of an algebraic group G are the groups Bu where B is a Borel subgroup of G. Proof By 1.4.1 such a subgroup is nilpotent, thus in a Borel subgroup, whence the result by 1.3.1(i). 

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Basic Results on Algebraic Groups

Proposition 1.4.4 Let G be an algebraic group. (i) The product of the closed connected normal solvable subgroups of G is also a closed connected normal solvable subgroup of G, called the radical of G and denoted by R(G). (ii) Similarly, the set of all closed connected normal unipotent subgroups of G has a unique maximal element called the unipotent radical of G and denoted by Ru (G). (iii) Ru (G) = R(G)u (where R(G)u is defined as in 1.3.1(i)). Proof Using 1.1.3 it follows that the product in (i) is actually finite. This implies (i), since the product of two solvable groups normalising each other is still solvable. To see (ii) and (iii), we first remark that a closed connected normal unipotent subgroup is in R(G), thus in R(G)u . We then observe that R(G)u is normal in G, being characteristic in R(G), and is connected by 1.3.1(i).  Corollary 1.4.5 R(G) is the identity component of the intersection of all Borel subgroups. Proof Indeed, R(G) is contained in at least one Borel subgroup. Since it is normal and all Borel subgroups are conjugate, it is contained in their intersection. Since it is connected, it is contained in the identity component of this intersection. Conversely, this component is solvable and normal.  Definition 1.4.6 An algebraic group is called reductive if its unipotent radical is trivial, and semi-simple if its radical is trivial. Exercise 1.4.7 For an algebraic group G, the group G/R(G) is semi-simple and the group G/Ru (G) is reductive. Proposition 1.4.8 If G is connected and reductive, then R(G) = Z(G) 0 , the identity component of the centre of G. Proof By 1.3.1(iii), since Ru (G) is trivial, R(G) is a torus, normal in G. Since G is connected and for any torus T we have NG (T) 0 = CG (T) 0 (see 1.3.4), R(G) is central in G, and, being connected, is in Z(G) 0 . Conversely, Z(G) 0 , being normal solvable and connected, is contained in R(G).  Proposition 1.4.9 If G is reductive and connected, its derived group Der(G) is semi-simple and has a finite intersection with Z(G). Proof The first assertion results from the second one, since the radical R(Der(G)), being characteristic by 1.4.8, is in R(G), thus in Z(G) ∩ Der(G); it is trivial, since it is connected and this last group is finite.

1.5 Examples of Reductive Groups

15

To see the second assertion, we may embed G in some GL(V); the space V is a direct sum of isotypic spaces V χ for some χ ∈ X(Z(G)). The action  of G preserves this decomposition, so the image of G is in χ GL(V χ) and  that of Der(G) in SL(V χ), while that of Z(G) consists of products of scalar  matrices in each V χ , whence the result.

1.5 Examples of Reductive Groups Example 1.5.1 The general linear group GLn ; see 1.1.1(iii). The theorem of Lie–Kolchin (see the proof of 1.3.1) shows that the upper triangular matrices in −1 } GLn , defined by the algebra k[{ti,j }1≤i≤j≤n , {ti,i 1≤i≤n ], form a Borel subgroup B – this group is connected as open in an affine space. The group of diagonal matrices is a torus T. It is maximal in B since it is equal to the quotient of B by the upper unitriangular matrices, which are unipotent, and a torus contains only semi-simple elements; see 1.2.8(i). It is thus maximal in G by 1.3.3(v) and (i). The lower triangular matrices – conjugate to the upper triangular by the matrix of the permutation (1, n)(2, n − 1) . . . – form another Borel subgroup, whose intersection with B is T. Thus GLn is reductive by 1.4.5 and 1.4.4(iii). However, GLn is not semi-simple: the centre is the group of scalar matrices, which is connected, thus equal to R(G) by 1.4.8. The Borel subgroups are the conjugates of the upper triangular matrices, hence identify as the stabilisers of the complete flags: that is, the increasing sequences of vector subspaces 0 = F0  F1  · · ·  Fn = kn . This allows to identify the variety G/B with the flag variety. Example 1.5.2 The special linear group SLn = Spec k[ti,j ]/(det(ti,j ) − 1). Since a Borel subgroup of SLn is a subgroup of a Borel subgroup of GLn by maximality, and since the upper triangular matrices of SLn form a connected group, by the same argument as in GLn they form a Borel subgroup. A similar argument shows that the diagonal matrices of SLn form a maximal torus. The group SLn is reductive by the same argument as GLn , thus is semi-simple, as it has a finite centre (see 1.4.8). Example 1.5.3 PGLn is the quotient of GLn by Gm embedded as the centre of GLn . To see it is an affine variety, we can either refer to Proposition 1.1.4 or identify it with the subgroup of g ∈ GL(Mn (k)), which are algebra automorphisms; that is, such that g(Ei,j )g(Ek,l ) = δj,k g(Ei,l ) where Ei,j is the elementary matrix defined by {Ei,j }k,l = δi,j δk,l . The image of a maximal torus (resp. a Borel subgroup) of GLn is a maximal torus (resp. a Borel subgroup) of PGLn , since the centre of GLn is contained in all Borel subgroups and all maximal tori.

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Basic Results on Algebraic Groups

If char k = p, the centre Z(SLp ) is Spec k[t]/(tp − 1) = Spec k[t]/(t − 1) p , which as a variety has a single point and thus is the trivial group, but it is not trivial as a scheme! In characteristic p the natural morphism SLp → PGLp is not separable; it is a bijection on the points over k but is not an isomorphism of group schemes, an instance of the problem we mentioned in 1.1.2. Example 1.5.4 The symplectic group Sp2n . On V = k2n with ordered basis (e1 , . . . , en , en , . . . , e1 ), we define the symplectic bilinear form ei , ej  = ei , ej  = 0, ei , ej  = −ej , ei  = δi,j . The group Sp2n is the (clearly closed) subgroup of GL2n which preserves this form. If J is the n × n matrix J =   1   J .. . , we have v, v  = tvJv ; thus g is symplectic if and J =   −J 1  and only if tgJg = J. The torus T = {diag(t1 , . . . , tn , tn−1 , . . . , t1−1 )} is a maximal torus: it is in a unique maximal torus T1 of GL2n , the torus of diagonal matrices, since T1 = CGL2n (T) and T consists of all symplectic diagonal matrices. The symplectic upper triangular matrices form a subgroup B consisting of the   B BJ S , where B is upper triangular and S is symmetric. The matrices 0 J tB−1 J group B is connected since it is the product of the connected varieties of the upper triangular and the symmetric matrices. It is solvable since it is a subgroup of the upper triangular matrices of GL2n . Thus it is contained in some Borel subgroup of GL2n ; that is, in the stabiliser of a complete flag. But the only line stable by B is e1 , and by induction, considering e⊥ 1 /e1 , the only flag stabilised by B is the flag e1  ⊂ e1 , e2  ⊂ · · · ⊂ e1 , . . . , en  ⊂ e1 , . . . , en , en  ⊂ · · · ⊂ e1 , . . . , en , en , . . . , e1 . Thus the only Borel subgroup of GL2n containing B is the group B1 of (invertible) upper triangular matrices; thus B is a Borel subgroup of Sp2n , since any larger solvable connected subgroup would be in B1 , and B = B1 ∩ Sp2n . The group B is the stabiliser of a complete (that is, maximal) flag of isotropic subspaces e1  ⊂ e1 , e2  ⊂ · · · ⊂ e1 , . . . , en . By conjugation in Sp2n , we see that any stabiliser of a complete isotropic flag is a Borel subgroup. We deduce that Sp2n is connected: it is enough to see that every element is in a Borel subgroup. Any g ∈ Sp2n has at least one eigenvector x ∈ V; as g is symplectic, it induces a symplectic automorphism h of V1 = x⊥ /x (which is naturally endowed with a symplectic form induced by the initial form on V). By induction on dim V, we may assume that Sp(V1 ) is connected, and thus h is in some Borel subgroup of Sp(V1 ); that is, it stabilises a complete isotropic flag of V1 . The inverse image in V of this flag, completed by x, is a complete isotropic flag of V stabilised by g, whence the result. The induction starts with the group Sp2 , which is equal to SL2 , hence connected.

1.5 Examples of Reductive Groups

17

The permutation (1, 2n)(2, 2n − 1) . . . is symplectic and conjugates B to the symplectic lower triangular matrices; thus by the same argument as for GLn , we find that Sp2n is reductive. The centre of Sp2n is formed of the scalar symplectic matrices, which are only I2n and − I2n ; thus Sp2n is semi-simple (see also the end of Example 2.3.14 for a computation with roots). Example 1.5.5 The orthogonal and special orthogonal groups. We will assume the characteristic of k different from 2. On V = kn (n ≥ 1) with ordered basis (e1 , . . . , en ), we define the symmetric bilinear form ⎧ ⎪ 1 if i + j = n + 1, ei , ej  = ⎨ ⎪ 0 otherwise. ⎩ The group On is the (closed) subgroup of GLn which preserves this form. Similarly as for the symplectic group, we consider the n × n matrix 1   . t t .. J =  ; we have v, v  = vJv , and g is orthogonal if gJg = J. 1  A diagonal matrix diag(t1 , . . . , tn ) is orthogonal if and only if ti tn+1−i = 1 for all i. These matrices form a torus if n is even. If n is odd, they verify t(n+1)/2 = ±1 and form a non-connected group whose identity component is defined by t(n+1)/2 = 1. Using the same argument as for the symplectic group, we get that this identity component is a maximal torus of On . The orthogonal upper triangular form a subgroup B; if n = 2m is   matrices B BJ A , where A and B are matrices of even, it consists of the matrices 0 J tB−1 J size m × m with B upper triangular and A antisymmetric, and J is the matrix defined in Example 1.5.4. If n = 2m + 1 is odd, B consists of the matrices BJ A B −BJ tw   1 w , where A, B, and J are the same as in the even case 0 0 J tB−1 J  0 and w is a row vector of size m. The group B is connected since it is the product of the connected varieties of the upper triangular matrices and of antisymmetric matrices, and in addition, in the odd case, the product by Gm a. By the same argument as for the symplectic group, we get that B is a Borel subgroup of On . In the even case, the group B is also the stabiliser of the maximal flag of isotropic subspaces e1  ⊂ e1 , e2  ⊂ · · · ⊂ e1 , . . . , en/2 . By conjugating in On , we see that any stabiliser of a complete isotropic flag is a Borel subgroup. A similar characterisation can be made in the odd case, since B is then the

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Basic Results on Algebraic Groups

identity component of the stabiliser of the maximal flag of isotropic subspaces e1  ⊂ e1 , e2  ⊂ · · · ⊂ e1 , . . . , e(n−1)/2 . The matrix equation defining On shows that the determinant of an orthogonal matrix has to be ±1. There are orthogonal elements of determinant −1, for 1   example, if n ≥ 2,  In−2 . Hence the group On is not connected. We 1   define the special orthogonal group SOn as On ∩ SLn . The proof that SOn is connected goes along the same lines as for Sp2n ; see Exercise 1.5.6 below. In the even case, the inductionstarts with  SO2 , which is the one dimensional torus a 0 . In the odd case, the induction starts with consisting of the matrices 0 a−1 SO1 = {1}. The permutation (1, n)(2, n − 1) . . . is orthogonal and conjugates B to orthogonal lower triangular matrices; thus, as in the above examples, we deduce that SOn is reductive. The centre of SOn is formed of the scalar orthogonal matrices, which are {± In } if n is even and In if n is odd; thus SOn is semi-simple (see the end of Examples 2.3.15 and 2.3.16 for a computation with roots). Exercise 1.5.6 Show that any g ∈ SOn has an isotropic eigenvector x and that the restriction of g to V1 = x⊥ /x is in SO(V1 ). (Hint: decompose V = x ⊕ V1 ⊕ y with y ∈ V − x⊥ and x⊥ = x ⊕ V1 .) Deduce the connectedness of SOn by induction on dim V. Notes There are a number of books we recommend that provide a good exposition of the basic theory of algebraic groups; for instance, the “classical” books of Borel (1991), Springer (1998), or Milne (2017), or the less complete but more pedagogical approach of Geck (2003).

2 Structure Theorems for Reductive Groups

We give in this chapter the main structure theorems for reductive groups. In order to do that, we first recall the definition and some properties of Coxeter groups and root systems (refer to 2.3.1(i) and (iii) to see why root systems appear in this context).

2.1 Coxeter Groups Let W be a group generated by a set S of elements left stable by taking inverses. Let S∗ be the free monoid on S; that is, the set of words on S (finite sequences of elements of S). Let w ∈ W be the image of s1 . . . sk ∈ S∗ . The word s1 . . . sk ∈ S∗ is called a reduced expression for w if it has minimal length among the words representing w; we then write l(w) = k. We call l the length on W with respect to S. We assume now the set S which generates W consists of involutions; that is, each element of S is its own inverse (and is different from 1). Notice that reversing words is then equivalent to taking inverses in W. For s, s ∈ S we (m) ss . . .. If the product ss has finite order m, will denote by Δs,s the word ss  

m terms (m) we will just write Δs,s for Δs,s ; then the relation Δs,s = Δs ,s holds in W. m Writing the relation (ss ) = 1 this way has the advantage that transforming a word by the use of this relation does not change the length – this will be useful later. Relations of this kind are called braid relations because they define the braid groups, which are groups related to the Coxeter groups but also have a topological definition.

Definition 2.1.1 A pair (W, S) where S is a set of involutions generating the group W is a Coxeter system if s ∈ S | s2 = 1, Δs,s = Δs ,s for s, s ∈ S with ss of finite order is a presentation of W. 19

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Structure Theorems for Reductive Groups

We may ask if a presentation of the above kind always defines a Coxeter (m) (m) system. That is, given a presentation with relations Δs,s = Δs ,s , is m the order of ss in the defined group and is the order of s equal to 2? This is always the case – see Bourbaki (1968, V, §4.3); but it is not obvious. If (W, S) is a Coxeter system, we say that W is a Coxeter group and that S is a Coxeter generating set of W. According to, for example, Bourbaki (1968, V, §4.4 corollaire 2) a Coxeter group has a faithful representation in which the elements of S act as reflections (see 2.2.1), so we also call the elements of S the generating reflections of W, and the W-conjugates of elements of S the reflections of W, their set being denoted by Ref(W).

Characterisations of Coxeter Groups Theorem 2.1.2 Let W be a group generated by a set S of involutions. The following are equivalent: (i) (W, S) is a Coxeter system. (ii) There exists a (unique) map N from W to the set of subsets of Ref(W), such that N(s) = {s} for s ∈ S and for x, y ∈ W we have N(xy) = ˙ −1 N(x)y, where +˙ denotes the symmetric difference of two sets (the N(y) +y sum modulo 2 of the characteristic functions). (iii) (Exchange condition) If s1 . . . sl(w) is a reduced expression for w ∈ W and s ∈ S is such that l(sw) ≤ l(w), then there exists i such that sw = s1 . . . sˆi . . . sl(w) , where sˆi denotes a missing term. (iv) W satisfies l(sw)  l(w) for s ∈ S, w ∈ W, and (Matsumoto’s lemma) two reduced expressions of the same word can be transformed into each other by using just the braid relations. Formally, given any monoid M and any morphism f : S∗ → M such that f (Δs,s ) = f (Δs ,s ) when ss has finite order, then f is constant on the reduced expressions of any w ∈ W. We will see along the way that |N(w)| = l(w). Note that (iii) could be called the “left exchange condition”. By symmetry there is a right exchange condition where sw is replaced by ws. Proof We first show that (i)⇒(ii). The definition of N may look technical and mysterious, but the intuition is – see 2.2.11(i) below – that W has a reflection representation permuting a set of “root vectors” (there are two opposite roots attached to each reflection), that these roots are divided into positive and negative by a linear form which does not vanish on any root, and that N(w) records the reflections whose roots change sign by the action of w. Lemma 2.1.3 If N is as in (ii), then ˙ sk sk−1 }+˙ · · · +{ ˙ sk sk−1 ...s2 s1 }. N(s1 . . . sk ) = {sk }+{

2.1 Coxeter Groups

Proof This is just an inductive application of the formulae of (ii).

21 

We get (ii) if we show that the function defined by 2.1.3 on S∗ factors through W. To do that we need N to be compatible with the relations defining W, that is N(ss) = ∅ and N(Δs,s ) = N(Δs ,s ). This is straightforward. We now show (ii)⇒(iii). We will actually check the right exchange condition; by symmetry if (i) implies this condition it also implies the left condition. We first show that if s1 . . . sk is a reduced expression for w, then |N(w)| = k; that is, all the elements of Ref(W) which appear on the right-hand side of 2.1.3 are distinct. Otherwise, there would exist i < j such that sk . . . si . . . sk = sk . . . sj . . . sk ; then si si+1 . . . sj = si+1 si+2 . . . sj−1 which contradicts that the expression is reduced. We next observe that l(ws) ≤ l(w) implies l(ws) < l(w). Indeed N(ws) = ˙ −1 N(w)s, thus by the properties of +˙ we have l(ws) = l(w) ± 1. Also, if {s}+s l(ws) < l(w), we must have s ∈ s−1 N(w)s or equivalently s ∈ N(w). It follows that there exists i such that s = sk . . . si . . . sk , which multiplying on the left by w gives ws = s1 . . . sˆi . . . sk , whence (iii). We now show (iii)⇒(iv). The exchange condition implies l(sw)  l(w) because if l(sw) ≤ l(w) it gives l(sw) < l(w). Given f : S∗ → M as in (iv) we use induction on l(w) to show that f is constant on reduced expressions. Otherwise, let s1 . . . sk and s1 . . . sk be two reduced expressions for the same element w whose images by f differ. By the exchange condition there exists i such that s1 s1 . . . sk = s1 . . . sˆi . . . sk in W, thus s1 s1 . . . sˆi . . . sk is another reduced expression for w. If i  k we may apply induction to deduce that f (s1 . . . sk ) = f (s1 s1 . . . sˆi . . . sk ) and similarly apply induction to deduce that f (s1 . . . sk ) = f (s1 s1 . . . sˆi . . . sk ), a contradiction. Thus i = k and s1 s1 . . . sk−1 is a reduced expression for w such that f (s1 s1 . . . sk−1 )  f (s1 . . . sk ). Arguing the same way, starting this time from the pair of expressions s1 . . . sk and s1 s1 . . . sk−1 , we get that s1 s1 s1 . . . sk−2 is a reduced expression for w such that f (s1 s1 s1 . . . sk−2 )  f (s1 s1 . . . sk−1 ). Going on, this process will stop when we get two reduced expressions of (m) (m) the form Δs(m),s , Δs(m) ,s , such that f (Δs ,s )  f (Δs ,s ). We cannot have m greater 1 1 1 1 1 1 1 1 than the order of s1 s1 since the expressions are reduced, nor less than that order, because the order would be smaller. And we cannot have m equal to the order of s1 s1 because this contradicts the assumption. We finally show (iv)⇒(i). (i) can be stated as: given any group G and a morphism of monoids f : S∗ → G such that f (s) 2 = 1 and f (Δs,s ) = f (Δs ,s )

22

Structure Theorems for Reductive Groups

then f factors through a morphism g : W → G. Let us define g by g(w) = f (s1 . . . sk ) when s1 . . . sk is a reduced expression for w. By (iv) the map g is well defined. To see that g factors f we need to show that for any expression w = s1 . . . sk we have g(w) = f (s1 . . . sk ). This will follow by induction on the length of the expression if we show that f (s)g(w) = g(sw) for s ∈ S, w ∈ W. If l(sw) > l(w) this equality is immediate from the definition of g. If l(sw) < l(w) we use f (s) 2 = 1 to rewrite the equality g(w) = f (s)g(sw) and we apply the reasoning of the first case. Finally l(sw) = l(w) is excluded by assumption.  Exercise 2.1.4 Show, using the exchange property, that if (W, S) is a Coxeter system and for some s, t ∈ S and w ∈ W we have l(swt) = l(w) and l(sw) = l(wt), then sw = wt.

The Longest Element Proposition 2.1.5 Let (W, S) be a Coxeter system. Then the following properties are equivalent for an element w0 ∈ W: (i) l(w0 s) < l(w0 ) for all s ∈ S. (ii) l(w0 w) = l(w0 ) − l(w) for all w ∈ W. (iii) w0 has maximal length amongst elements of W. If such a w0 exists, it is unique, w20 = 1, W is finite, and conjugating by w0 is an automorphism of (W, S). Proof It is clear that (ii) implies (iii) and that (iii) implies (i). To see that (i) implies (ii), we will show by induction on l(w) that w0 as in (i) has a reduced expression ending by a reduced expression for w−1 . Write w−1 = vs where l(v) + l(s) = l(w). By induction we may write w0 = yv where l(w0 ) = l(y) + l(v). The (right) exchange condition, using that l(w0 s) < l(w0 ) but l(vs) > l(v), shows that w0 s = yˆ v where yˆ represents y with one letter omitted. It follows that yˆ vs is a reduced expression for w0 . For an element satisfying (ii) we have w20 = 1 since l(w20 ) = l(w0 ) − l(w0 ) = 0 and w0 is unique since another w1 satisfying (ii) has the same length by (iii) thus l(w0 w1 ) = l(w0 ) − l(w1 ) = 0 thus w1 = w−1 0 = w0 . If w0 as in (i) exists, then S is finite since S ⊂ N(w0 ) and W is then finite by (iii). Finally, if s ∈ S and w0 sw−1 0 = t, then from w0 s = tw0 and (ii) and its symmetric we get l(w0 ) − l(s) = l(w0 ) − l(t) thus l(t) = l(s) = 1 and t ∈ S. 

2.1 Coxeter Groups

23

Lemma 2.1.6 (i) For r ∈ Ref(W), we have r ∈ N(r). (ii) For w ∈ W, we have N(w) = {r ∈ Ref(W) | l(wr) < l(w)}. (iii) (Generalised exchange condition) If s1 . . . sk is a reduced expression for w ∈ W and r ∈ Ref(W) is such that l(wr) < l(w), then there exists i such that wr = s1 . . . sˆi . . . sk . (iv) If s1 . . . s2k+1 is a reduced expression for r ∈ Ref(W), then s1 . . . sk sk+1 sk . . . s1 is another one. (v) If W is finite, N(w0 ) = Ref(W). Proof Let r ∈ Ref(W) and choose an expression r = vsv−1 with s ∈ S and l(v) minimal; that is, considering a reduced expression for v, choose an expression r = sk sk−1 . . . s1 s2 . . . sk with si ∈ S and k minimal. Let r1 = sk , r2 = sk sk−1 , . . . ˙ 2k−1 }. be the terms in the right-hand side of 2.1.3, so that N(r) = {r1 }+˙ · · · +{r Then it is easy to check that r = ri for i < k would imply that k is not minimal, and since r2k−i = rri r, we also cannot have r = r2k−i for i < k since it would imply r = ri . Since r = rk it follows that r ∈ N(r). Now for (ii), if r ∈ N(w) then r = sk sk−1 ...si+1 si for some i ≤ k, where w = s1 . . . sk is a reduced expression for w. Then wr = s1 . . . si−1 si+1 . . . sk has a length strictly smaller than w. Conversely, if r  N(w), then r  r−1 N(w)r ˙ −1 N(w)r = N(wr), hence and since r ∈ N(r) by (i), we have r ∈ N(r) +r l(w) = l((wr)r) < l(wr) by the direct part of the proof. (iii) is a consequence of (ii), since if l(wr) < l(w) by (ii) r is equal to one of the terms on the right-hand side of 2.1.3, and r = sk ...si+1 si implies si si+1 . . . sk r = si+1 . . . sk . For (iv), let x = sk sk−1 . . . s1 and y = sk+2 . . . s2k+1 . It is sufficient to show x = y. Since sk+1 yr = x, by (iii) we have x = sk+1 . . . sˆi . . . s2k+1 for some i, and since l(x) = k this is a reduce expression. Since sk+1 x is part of another reduced expression (the reverse of the one given for r), we have l(sk+1 x) > l(x) which forces i = k + 1 thus x = y. Finally (v) is an immediate consequence of (ii). 

Coxeter Diagrams We will now describe finite Coxeter groups, thus we assume S = {s1 , . . . , sn } is finite. A Coxeter system (W, S) is encoded by a graph with vertices 1, . . . , n and edges encoding the order of ss when it is greater than 2. This order is encoded by a single edge when equal to 3, a double edge when equal to 4, a triple edge when equal to 6, and an edge decorated by the order when equal to 5 or greater than 6.

24

Structure Theorems for Reductive Groups

Classification 2.1.7 The finite irreducible (meaning that the diagram is connected) Coxeter groups are classified as follows (see Bourbaki (1968, VI, §4.1 Théorème 1)):

An : 

· · · 

1

n

2

Bn /Cn : 

· · ·  

1

2

n−1

n

n Dn : 

· · ·

1

2

n −2



n −1

2 E6 : 



1



3

E7 : 

4



3

4



F4 : 

  2

H3 : 

5

H4 : 

5

I2 (e) : 

e

1 1 1

4







5

6

6

7

7

 8

4







2



6



3

 2



5



3

1



2

E8 :  1



5

2



1



3 3

 4

 2

A finite Coxeter group is called a Weyl group if it is a reflection group in a Q-vector space. This selects in the above list exactly the diagrams where the order of ss is always in {2, 3, 4, 6}. The group I2 (6) is also denoted by G2 .

2.2 Finite Root Systems In this section V is a finite dimensional vector space on a subfield K ⊂ R and V ∗ is its dual. Definition 2.2.1 A reflection s ∈ GL(V) is an element of order 2 such that Ker(s − IV ) is a hyperplane. It follows that s has an eigenvalue −1 with multiplicity 1, and that if α ∈ V is an eigenvector for −1 and α ∨ ∈ V ∗ is a linear form of kernel Ker(s − IV ), chosen such that α ∨ (α) = 2, then s(x) = x − α ∨ (x)α.

2.2 Finite Root Systems

25

We call α a root attached to the reflection s and α ∨ the corresponding coroot. They are unique up to scalings inverse to each other. Conversely any pair of vectors α ∈ V, α ∨ ∈ V ∗ such that α ∨ (α) = 2 defines a reflection. Definition 2.2.2 • A (finite) root system is a finite set Φ ⊂ V with a bijection α → α ∨ : Φ → Φ∨ ⊂ V ∗ such that for any α ∈ Φ we have α ∨ (α) = 2 and Φ is stabilised by the reflection sα of root α and coroot α ∨ . • The system is crystallographic if α ∨ ( β) ∈ Z for all α, β ∈ Φ. • The system is reduced if for any α ∈ Φ we have Φ ∩ Kα = {α, −α}. • The system is irreducible if there is no partition into two non-empty subsets  Φ = Φ1 Φ2 such that α1∨ (α2 ) = 0 for all α1 ∈ Φ1 and α2 ∈ Φ2 . If the system is crystallographic, Φ and Φ∨ generate dual lattices. We omitted two conditions compared to the definition given in Bourbaki (1968, VI, §1), where it is added that Φ spans V and is crystallographic; we will see in 2.2.10 that a root system is associated with a Coxeter group; any finite Coxeter group has a root system as we defined it, but only the Weyl groups have crystallographic ones. In the following we fix a root system Φ and denote by W the group generated by {sα }α ∈Φ . It is finite since its elements are determined by the permutation of Φ they induce. Thus there exists a W-invariant scalar product (, ) on V. Lemma 2.2.3

Using (, ) to identify V ∗ with V, we have α ∨ =

2α (α,α) .

Proof Using the invariance of (, ) we get for all v ∈ V that (α, v) =  (sα α, sα v) = (−α, v − α ∨ (v)α) which gives α ∨ (v) = 2(α,v) (α,α) . Using the identification of Lemma 2.2.3 allows us to work in a Euclidean space and forget Φ∨ ; but keeping V ∗ allows the extension of the theory to infinite root systems. In the following we assume Φ is reduced, in order to simplify somewhat the statements and proofs – a non-reduced system BCn occurs in certain parts of reductive groups theory that we will not cover. We will need an order on Φ induced by a total order on V. It is sufficient for our purpose to determine which roots are positive or negative for this order, and for that it is sufficient to have a linear form λ on V which does not vanish on Φ: the positive roots are those on which λ takes positive values. Theorem 2.2.4 Given a linear form which does not vanish on Φ, denote by Φ+ the corresponding set of positive roots. Then there exists a unique minimal subset Π ⊂ Φ+ such that Φ+ = Φ ∩ K ≥0 Π. The set Π thus defined is a basis of the subspace V0 ⊂ V generated by Φ.

26

Structure Theorems for Reductive Groups

Proof Note first that there exists a minimal subset Π ⊂ Φ+ such that Φ+ = Φ ∩ K ≥0 Π: to obtain such a subset, starting from Φ+ , just iteratively remove elements that are a positive linear combination of others in the considered subset. Lemma 2.2.5 For a minimal Π as above (α, β) ≤ 0 for α, β ∈ Π, α  β. Proof Assume by contradiction that (α, β) > 0. Then sα ( β) = β − cα where β) + ∈ Φ+ . c = 2(α, (α,α) > 0. Either sα ( β) ∈ Φ or −sα ( β)  In the first case by assumption sα ( β) = γ ∈Π cγ γ with cγ ≥ 0; we rewrite  this as γ ∈Π−{β } cγ γ + cα + (c β − 1) β = 0. We cannot have c β − 1 ≥ 0 since a non-empty sum of positive vectors cannot be zero. Thus we expressed β as an element of K ≥0 (Π − { β}) which contradicts the minimality of Π.  In the second case we similarly rewrite −sα ( β) = γ ∈Π cγ γ with cγ ≥ 0 as  γ ∈Π−{α } cγ γ + β + (cα −c)α = 0, and similarly we must have cα −c < 0 giving an expression of α as an element of K ≥0 (Π − {α}) which again contradicts the minimality of Π.  Let us see now that Π is a basis. We know it spans V0 since Φ does. We have to exclude a linear dependence amongst its elements. Such a relation   can be written v = α ∈Π1 cα α = β ∈Π2 c β β where v is a non-zero vector, where cα , c β ≥ 0 and where Π = Π1  Π2 . But then we have 0 < (v, v) =   ( α ∈Π1 cα α, β ∈Π2 c β β) which contradicts Lemma 2.2.5. We finally show that Π is unique: if there are two such bases Π  Π let us  consider α ∈ Π − Π ; express it on Π as α = β ∈Π c β β then express each involved β on Π: since β  α these expressions will involve a root in Π − α (we use here that the system is reduced) and this root will remain when doing the sum, since the coefficients are positive; this is a contradiction to the minimality of Π.  A Φ+ as above is called a positive subsystem and a Π as above is called a basis of Φ. The cardinality of Π is called the rank of Φ. Note that in the basis Π the entries of the matrix sα are 1 or −α ∨ ( β), thus in this basis we have W ⊂ GLn (Z) if the root system is crystallographic, where n = |Π|. Proposition 2.2.6 Two positive subsystems (resp. bases) are W-conjugate. Proof

It is enough to consider positive subsystems since they determine bases.

Lemma 2.2.7 For α ∈ Π and any β ∈ Φ+ − {α} we have sα ( β) ∈ Φ+ .  Proof If β ∈ Φ+ − {α}, then β = γ ∈Π cγ γ where at least one cγ > 0 with γ  α, otherwise β ∈ Φ+ ∩ K ≥0 α = {α}. But then sα ( β) = β − α ∨ ( β)α has the same coefficient on γ, and, since any root has all non-zero coefficients on  Π of the same sign, the root sα ( β) is positive.

2.2 Finite Root Systems

27

We use the lemma to conjugate another positive subsystem Φ on Φ+ , using induction on |Φ+ ∩−Φ |. If this number is positive, then Π ∩−Φ  ∅, otherwise Π ⊂ Φ which implies Φ+ ⊂ Φ which implies Φ+ = Φ since all positive subsystems have same cardinality |Φ|/2. Choose thus α ∈ Π ∩ −Φ ; since  sα (Φ+ ) = (Φ+ − {α}) {−α}, the set sα (Φ+ ) is a positive subsystem such that  |sα (Φ+ ) ∩ −Φ | = |Φ+ ∩ −Φ | − 1. Corollary 2.2.8 Every root α is in the W-orbit of Π, and either α or −α is a sum of elements of Π. Proof It is enough to show it for every positive root since sα (α) = −α. Take   α = γ ∈Π cγ γ ∈ Φ+ − Π; as 0 < (α, α) = γ ∈Π cγ (α, γ) there exists γ ∈ Π such that (α, γ) > 0. Then α = sγ (α) is still positive by 2.2.7 and is obtained  by removing a positive multiple of γ from α. Thus if we set h(α) = γ cγ we have h(α ) < h(α). We can repeat this process as long as α  Π. As Φ+ is finite this process must eventually stop, at a root in Π.  The proof of the corollary shows more, that every root is conjugate to an element of Π by a sequence of sγ , γ ∈ Π. In particular every sα is in the group generated by {sγ }γ ∈Π , thus W itself is generated by {sγ }γ ∈Π . We show now that W is a Coxeter group using yet another characterisation of Coxeter groups. Lemma 2.2.9 Let W be group generated by a set S of involutions and let {Ds }s∈S be a set of subsets of W such that: • Ds  1. • Ds ∩ sDs = ∅. • If for s, s ∈ S we have w ∈ Ds , ws  Ds then ws = sw. Then (W, S) is a Coxeter system, and Ds = {w ∈ W | l(sw) > l(w)}. Proof We will show the exchange condition. Let s1 . . . sk be a reduced expression for w  Ds and let i be minimal such that s1 . . . si  Ds ; we have i > 0 since 1 ∈ Ds . From s1 . . . si−1 ∈ Ds and s1 . . . si  Ds we get ss1 . . . si−1 = s1 . . . si , whence sw = s1 . . . sˆi . . . sk thus l(sw) < l(w) and we have checked the exchange condition in this case. If w ∈ Ds then sw  Ds and by the first part l(w) < l(sw) so we have nothing to check.  Proposition 2.2.10 (i) (W, S) where S = {sα | α ∈ Π} is a Coxeter system; we have Ref(W) = {sα | α ∈ Φ+ }. (ii) A root system is irreducible if and only if the Coxeter group is irreducible.

28

Structure Theorems for Reductive Groups

Proof We apply 2.2.9 with Dsα = {w ∈ W | w−1 (α) > 0} for α ∈ Π. That Dsα ∩ sα Dsα = ∅ is clear. Now take w ∈ Dsα and α ∈ Π such that wsα  Dsα , that is w−1 (α) > 0 and sα w−1 (α) < 0. As by 2.2.7 sα changes the sign of only α , we must have w−1 (α) = α . As w preserves the scalar product, it conjugates sα to sα , whence (i). The last sentence of (i) reflects 2.2.8. (ii) comes from the fact that in the Coxeter diagram two nodes are linked if and only if the corresponding reflections do not commute which is equivalent to the corresponding roots not being orthogonal.  Lemma 2.2.11

For W as in 2.2.10:

(i) The set N(w) of 2.1.2(ii) is {sα | α ∈ Φ+ , w(α) < 0}. (ii) The element w0 of 2.1.5 is such that w0 (Φ+ ) = Φ− where we write Φ− for −Φ+ . Proof We set N (w) = {sα | α ∈ Φ+ , w(α) < 0} and show by induction on l(w) that N(w) = N (w): let w = vs with s ∈ S, l(w) > l(v); by 2.1.2(ii) and its  proof we have N(w) = s sN(v)s; by the right counterparts of 2.2.9 and of the  proof of 2.2.10, we have N (w) = s sN (v)s. This proves (i). (ii) is a consequence of 2.1.6(v).  The classification of finite crystallographic root systems is the same as the classification of finite Weyl groups, except that there are two root systems Bn and Cn corresponding to the same Weyl group. The list of irreducible crystallographic root systems is thus, An , Bn , Cn , Dn , E6 , E7 , E8 , F4 and G2 ; see Bourbaki (1968, VI, §4). We describe now the first four types. Example 2.2.12 Root system of type An . Let {e1 , . . . , en+1 } be an orthonormal basis of Rn+1 . Then Φ = {ei − ej }i,j∈[1, ...,n+1],ij is a root system of cardinality n(n + 1). The set {ei − ej | i < j} is a positive subsystem relative to the linear form x → (x, (n + 1)e1 + ne2 + · · · + en+1 ). We have Π = {ei − ei+1 }i=1, ...,n . If we set αi = ei − ei+1 , we have ei − ej = αi + αi+1 + · · · + αj for i < j. The group W is the symmetric group Sn+1 , permuting the ek : the reflection si of root αi interchanges ei and ei+1 and fixes the others ek ; these reflections form a Coxeter system defining the Coxeter graph An of 2.1.7. Example 2.2.13 Root system of type Cn . It consists of the 2n2 roots in Rn given by ±2ei and ±ei ± ej (i  j) with (ei ) as in Example 2.2.12. For the same linear form as above we have Φ+ = {2ei }i ∪ {ei ± ej }i0 uiα+jβ (cα, β,i,j xi yj ), where the cα, β,i,j are integral constants.

The cα, β,i,j depend on the choice of a subset of Φ+ × Φ+ ; when changing this subset their sign may vary, but their absolute value depends only on Φ. Notes A classic reference for properties of Coxeter groups and root systems is Bourbaki (1968). The construction and classification of reductive groups was first done around 1955, see Chevalley (2005).

3 (B, N)-Pairs; Parabolic, Levi, and Reductive Subgroups; Centralisers of Semi-Simple Elements

3.1 (B, N)-Pairs We review properties of reductive groups related to existence of a (B, N)-pair. For an abstract group, having a (B, N)-pair is a very strong condition; many of the theorems we will give for reductive groups follow from this single property. Definition 3.1.1 We say that two subgroups B and N of a group G form a (B, N)-pair (also called a Tits system) for G if: (i) B and N generate G and T := B ∩ N is normal in N. (ii) The group W := N/T is generated by a set S of involutions such that: (a) For s ∈ S, w ∈ W we have BsB.BwB ⊂ BwB ∪ BswB. (b) For s ∈ S, we have sBs  B. The group W is called the Weyl group of the (B, N)-pair. Note that we write elements of W – instead of representatives of them in N – in expressions representing subsets of G when these expressions do not depend upon the chosen representative. We will see in 3.1.3(v) that under the assumptions of 3.1.1 we have S = {w ∈ W − {1} | B ∪ BwB is a group}, thus S is determined by (B, N). Proposition 3.1.2 If G is a connected reductive group and T ⊂ B is a pair of a maximal torus and a Borel subgroup, then (B, NG (T)) is a (B, N)-pair for G. Proof We show first that B ∩ NG (T) = T. By 1.3.1(iii) we have NB (T) = CB (T) ⊂ CG (T) = T (see 2.3.1(iii)). By definition T is normal in NG (T). To prove (i) it remains to show that B and NG (T) generate G. Let Φ+ be the positive subsystem defined by B. By 2.3.1(vi), B contains all the Uα (α ∈ Φ+ ). Since sα conjugates Uα to Usα (α) = U−α , the group generated by B and NG (T) contains T and all the Uα (α ∈ Φ), thus by 2.3.1(v) this group is equal to G. 39

40

(B,N)-Pairs; Parabolic and Reductive Subgroups

If Π is the basis defined by the ordering Φ+ , (ii) is obtained by taking for S the {sα | α ∈ Π}. (ii)(b) reflects that sα Uα = U−α is not in B.  It remains to show (ii)(a). Let s = sα , and write B = T β ∈Φ+ U β . As s normalises T, as sU β = Usα (β) and as sα ( β) ∈ Φ+ if β ∈ Φ+ − {α}, we get BsBwB = BsUα wB. If w−1 (α) ∈ Φ+ the right hand side is equal to BswB. Otherwise we write it as BsUα sswB where this time (sw) −1 (α) ∈ Φ+ . Let Bα be the image by φα (see 2.3.1(ii)) of the Borel subgroup of SL2 of upper triangular matrices. If c  0 we have in SL2 : 

  a b −1/c = c d 0

−a −c



0 1 −1 0



1 0

d/c 1



which taking images shows that sUα s ⊂ Im φα = Bα ∪ Bα sUα , whence BsUα sswB ⊂ BsUα swB ∪ BswB where the first term in the right-hand side  is BwB since (sw) −1 (α) ∈ Φ+ . Theorem 3.1.3 If G has a (B, N)-pair, then  (i) G = w∈W BwB (Bruhat decomposition). (ii) (W, S) is a Coxeter group. (iii) Condition (ii)(a) of 3.1.1 can be refined to ⎧ ⎪ BswB BsB.BwB = ⎨ ⎪ BswB ∪ BwB ⎩

if l(sw) = l(w) + 1, otherwise.

(iv) For any t ∈ N(w) (see 2.1.2(ii)), we have BtB ⊂ Bw−1 BwB. (v) S = {w ∈ W − {1} | B ∪ BwB is a group}. (vi) We have NG (B) = B. Proof Let us show (i). As B and N generate G, we have G = ∪i (BNB) i . Since BNB = BWB we will get G = BWB if we show that BWBWB = BWB. For this it is enough to show that BwBWB ⊂ BWB for w ∈ W; writing w = s1 . . . sn with si ∈ S, since BwB ⊂ Bs1 B . . . Bsn B it is enough to show BsBWB ⊂ BWB for s ∈ S; but this results from 3.1.1(ii)(a). It remains to show that BwB  Bw B if w  w . We show this by induction on inf(l(w), l(w )), where l is the length with respect to S; assume for instance that l(w) ≤ l(w ). The start of the induction is l(w) = 0 and the result comes from w  B. Otherwise, taking s ∈ S such that l(sw) < l(w), by induction BswB is equal neither to Bw B nor to Bsw B thus BswB ∩ BsB.Bw B = ∅; as BswB ⊂ BsB.BwB it follows that BwB  Bw B. For (ii), we use 2.2.9 with Ds = {w ∈ W | BsBwB = BswB} (note that if  w  Ds then BsBwB = BswB BwB). Clearly Ds  1.

3.1 (B, N)-Pairs

41

If w, sw ∈ Ds , then from BsBwB = BswB and BsBswB = BwB we get BsBsBwB = BwB which is a contradiction since multiplying on the right by  BwB the equality BsBsB = BsB B (since sBs  B by 3.1.1(ii)(b)), we get  BsBsBwB = BswB BwB. It remains to show for (ii) that w ∈ Ds , ws  Ds implies ws = sw. The as sumption ws  Ds implies BsBws B = Bsws B Bws B; in particular BsBws meets Bws B; multiplying on the right by s B it follows that BsBwB meets  Bws Bs B ⊂ (BwB Bws B) (this last inclusion follows from 3.1.1(ii)(a) reversed, which is obtained by taking inverses). Thus BswB = BsBwB (since w ∈ Ds ) is equal to Bws B, or to BwB. The latter cannot happen since w  sw, thus sw = ws as was to be shown. We have also shown (iii) by the property of Ds given in the last sentence of 2.2.9. Let us show (iv). If w = s1 . . . sk is a reduced expression, for all i we can write by (iii) BwB = Bs1 . . . si−1 Bsi Bsi+1 . . . sk B and similarly for Bw−1 B whence Bw−1 BwB = Bsk . . . si+1 Bsi Bsi−1 . . . s1 Bs1 . . . si−1 Bsi Bsi+1 . . . sk B ⊃ Bsk . . . si+1 Bsi Bsi Bsi+1 . . . sk B ⊃ Bsk . . . si+1 Bsi Bsi+1 . . . sk B ⊃ Bsk . . . si+1 si si+1 . . . sk B whence the result. (v) follows immediately from (iv), which implies that B ∪ BwB can be a group only if |N(w)| ≤ 1, and from (iii) which implies that B ∪ BsB is a group. (vi) also follows from (iv). For g ∈ BwB we have gB = B ⇔ wB = B ⇔  BwBw−1 B = B which by (iv) happens only for w = 1. In a group G with a (B, N)-pair, we call Borel subgroups the conjugates of B and maximal tori the conjugates of T; this fits the terminology for algebraic groups. Corollary 3.1.4 In a group G with a (B, N)-pair, every pair of Borel subgroups is conjugate to a pair of the form (B, wB) with w ∈ W; the intersection of two Borel subgroups contains a maximal torus. Proof Up to conjugacy, we may assume the given pair of Borel subgroups of the form (B, gB). By the Bruhat decomposition we may write g = bwb where b, b ∈ B; thus the pair is equal to (B, bwB), which is conjugate to (B, wB). Since B and wB both contain T, the intersection of every conjugate pair also contains a maximal torus.  Example 3.1.5 For m a matrix in GLn , let mi,j be the submatrix on the last lines i, . . . , n and first columns 1, . . . , j. Let w be a permutation matrix; then

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m ∈ BwB, where B is the Borel subgroup of upper triangular matrices, if and only if the matrices mi,j and wi,j have same rank for all i, j. Indeed, • The ranks of mi,j are invariant by left or right multiplication of m by an upper triangular matrix. • A permutation matrix w for the permutation σ is characterised by the ranks of wi,j , given by |{k ≤ j | σ(k) ≥ i}|. If {Fi } and {Fi } are two complete flags whose stabilisers are the Borel subgroups B and B , then the permutation matrix w such that (B , B ) is conjugate to (B, wB) (the relative position of the two flags) is characterised by rank wi,j = dim

Fi ∩Fj ∩F )+(F ∩F ) . (Fi−1 j i j−1

3.2 Parabolic Subgroups of Coxeter Groups and of (B, N)-Pairs Lemma 3.2.1 Let (W, S) be a Coxeter system, let I be a subset of S, and let WI be the subgroup of W generated by I, called a standard parabolic subgroup of W. Then (WI , I) is a Coxeter system. An element w ∈ W is said to be reduced-I if it satisfies one of the equivalent conditions: (i) (ii) (iii) (iv) (v)

For any v ∈ WI , we have l(wv) = l(w) + l(v). For any s ∈ I, we have l(ws) > l(w). w has minimal length in the coset wWI . N(w) ∩ I = ∅. N(w) ∩ Ref(WI ) = ∅.

There is a unique reduced-I element in wWI . By exchanging left and right we have the notion of I-reduced element which satisfies the mirror properties. A subgroup of W conjugate to a standard parabolic subgroup is called a parabolic subgroup. Proof A reduced expression in WI is reduced in W by the exchange condition and then satisfies the exchange condition in WI , thus (WI , I) is a Coxeter system. (iii)⇒(ii) since (iii) implies l(ws) ≥ l(w) when s ∈ I. Let us show that “not (iii)”⇒ “not (ii)”. If w does not have minimal length in w WI , then w = wv with v ∈ WI and l(w) < l(w ); adding one by one the terms of a reduced expression for v to w and applying at each stage the exchange condition, we ˆ v where wˆ (resp. vˆ ) denotes find that w has a reduced expression of the shape wˆ

3.2 Parabolic Subgroups of Coxeter Groups and of (B, N)-Pairs

43

a subsequence of the chosen reduced expression. As l(w) ˆ ≤ l(w) < l(w ), we have l(ˆv) > 0, thus w has a reduced expression ending by an element of I, thus w does not satisfy (ii). (i)⇒(iii) is clear. Let us show “not (i)”⇒ “not (iii)”. If l(wv) < l(w) + l(v) then a reduced expression for wv has the shape wˆ ˆ v where l(w) ˆ < l(w). Then wˆ ∈ wWI and has a length smaller than that of w. By 2.1.6(ii) property (ii) is equivalent to (iv). It is clear that (v) implies (iv), and (i) applied to v ∈ Ref(W) implies (v) by 2.1.6(ii).  Finally, an element satisfying (i) is clearly unique in wWI . Lemma 3.2.2 Let I and J be two subsets of S. An element w ∈ W is I-reducedJ if it satisfies one of the equivalent properties: (i) w is both I-reduced and reduced-J. (ii) w has minimal length in WI wWJ . (iii) Every element of WI wWJ can be written uniquely xwy with x ∈ WI , y ∈ WJ , l(x) + l(w) + l(y) = l(xwy) and xw is reduced-J. (iii) implies that in a double coset in WI \W/WJ there is a unique I-reduced-J element, which has minimal length; by symmetry we can replace in condition (iii) the assumption that xw is reduced-J by the assumption that wy is I-reduced. Proof We first show that two elements w, w in the same double coset and satisfying (i) have the same length. Write w = xwy with x ∈ WI and y ∈ WJ ; then w y−1 = xw and x−1 w = wy; by the defining properties of I-reduced and reduced-J and using l(y−1 ) = l(y), l(x−1 ) = l(x) we get l(w ) +l(y) = l(x) +l(w) and l(x) + l(w ) = l(w) + l(y), whence l(x) = l(y) and l(w) = l(w ). As clearly (ii)⇒(i) this common length must be the minimal length, thus (i)⇔(ii). We now show (ii)⇒(iii). Assume w satisfies (ii); write an element v ∈ WI wWJ as xwy with x ∈ WI , y ∈ WJ and x of minimal possible length. By the exchange lemma a reduced expression for xwy is of the form xˆ wˆ ˆ y where xˆ (resp. w, ˆ yˆ ) is a subsequence of a reduced expression for x (resp. w, y). Necessarily w ˆ = w otherwise w would not be of minimal length in its double coset. Then the minimal length assumption on x implies xˆ = x, whence yˆ = y, thus l(x) + l(w) + l(y) = l(xwy). The element xw is reduced-J otherwise we can write xw = v y where v ∈ WI wWJ , y ∈ WJ − {1} and l(v ) + l(y ) = l(xw). Using what we just proved on w we can write v = x wy with l(x ) + l(w) + l(y ) + l(y ) = l(x) + l(w) which implies l(x ) < l(x), contradicting the minimality of l(x). Finally the decomposition xwy is unique since xw is the unique J-reduced element in its coset. Finally, (iii)⇒(ii) is clear. 

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Note that not every decomposition xwy where w is I-reduced-J satisfies (iii); consider for instance the case w = y = 1, I = J and x the longest element of WI ; thus the situation is not as good as in the I-reduced case. In a group with a (B, N)-pair, we use the term parabolic subgroups for the subgroups containing a Borel subgroup. Proposition 3.2.3 Let G be a group with a (B, N)-pair. Then (i) The (parabolic) subgroups containing B are the PI = BWI B for some I ⊂ S. (ii) Given two parabolic subgroups PI and PJ , we have a relative Bruhat de composition G = w PI wPJ where w runs over the I-reduced-J elements. ∼ → WI \W/WJ . It follows a natural bijection PI \G/PJ − Proof Let us show (i). Let P be a subgroup containing B and let w ∈ W be such that BwB ⊂ P. Since P is a group we get Bw−1 BwB ⊂ P, thus by 3.1.3(iv) we get BtB ⊂ P for any t ∈ N(w). If w = s1 . . . sk is a reduced expression we get in particular Bsk B ⊂ P, thus s1 . . . sk−1 ∈ P and by induction for each i we have si ∈ P. It follows that P = BWI B where I is the union of the elements of S appearing in any reduced expression of any w such that BwB ⊂ P. Conversely, for any I ⊂ S, using 3.1.1(ii)(a) we see that BWI B is a group. Let us show (ii). For any w ∈ W we have PI wPJ = BWI BwBWJ B = BWI wWJ B, the last equality by repeated application of 3.1.1(ii)(a) and of its right counterpart. Since, by Lemma 3.2.2 we can take I-reduced-J elements as representatives of the double cosets we see that the first assertion of (ii) is just the Bruhat decomposition. Conversely, any coset PI gPJ is of the form PI wPJ if g ∈ BwB whence the last assertion of (ii).  Remark 3.2.4 Using 3.2.3 we see that in the definition 1.3.5 of a parabolic subgroup the word “closed” can be omitted. Indeed a reductive group has a (B, N) pair, hence by 3.2.3 a subgroup containing a Borel subgroup is conjugate to some BWI B, hence it is closed. In general, if G is a connected group and P is a subgroup containing a Borel subgroup, then P/Ru (G) contains a Borel subgroup of the reductive group G/Ru (G) hence it is closed, thus P is closed by continuity of the quotient morphism. Example 3.2.5 In GLn , the parabolic subgroup PJ for J ⊂ S containing the Borel subgroup of upper triangular matrices is the subgroup of upper blocktriangular matrices where the blocks correspond to maximal intervals [i, k] in [1, n] such that si , . . . , sk−1 ∈ J. Example 3.2.6 For the symplectic group Sp2n , as the stabiliser B of any complete isotropic flag V1 ⊂ · · · ⊂ Vn in Sp2n is a Borel subgroup, the stabiliser of any subflag is a parabolic subgroup. We thus get 2n distinct parabolic subgroups containing B. Since there are also 2n subsets of S, they are the only parabolic

3.3 Closed Subsets of a Crystallographic Root System

45

subgroups containing B. As any isotropic flag may be completed to a complete one, we get the result that in general parabolic subgroups are the stabilisers of (complete or not) isotropic flags. Lemma 3.2.7 (unicity in Bruhat decomposition) Let G be a connected reductive group and B = U  T be a decomposition of a Borel subgroup B as in 1.3.1(ii), where U = Ru (B). Then BwB has a direct product decomposition  U × Tw × Uw where Uw := {α ∈Φ+ |w(α)0} Uα thus U ⊂ U; thus BwB = UTwU Uw = UTwUw . It remains to be shown that the decomposition is unique; that is, if uTwu = Tw with u ∈ U, u ∈ Uw then u = u = 1. The condition implies wu ∈ B. But wU ∩ B = 1 since all U in wU are for negative α. Thus u = 1, w α w whence u = 1.  The next proposition says that the decomposition of G in Bruhat cells BwB is a stratification (the closure of a stratum is a union of strata). Proposition 3.2.8 Let G be a connected reductive group and B = U  T be a decomposition of a Borel subgroup B as in 3.2.7. Then the Zariski closure of  BwB in G is given by BwB = v≤w BvB, where ≤ is the Bruhat–Chevalley order on w, given by v ≤ w if a reduced expression of v is a subsequence of a reduced expression of w. Reference

See Chevalley (1994, Proposition 6).



3.3 Closed Subsets of a Crystallographic Root System In this section, Φ will be a reduced crystallographic root system in the Q-vector space V, and Π will be a basis of Φ; we denote by Φ+ the corresponding positive subsystem and by (W, S) the corresponding Coxeter system, where S = {sα }α ∈Π . Definition 3.3.1

A subset Ψ ⊂ Φ is:

(i) closed if α, β ∈ Ψ, α + β ∈ Φ ⇒ α + β ∈ Ψ. (ii) symmetric if Ψ = −Ψ. The intersection of two closed subsets is clearly closed. Lemma 3.3.2 The reduced crystallographic root systems of rank 2 are A1 ×A1 , A2 , C2 = B2 and G2 .

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Here is a picture of their positive roots: 2β+3α β

β α

β

β+α α

β+α

β+2α

α

β

β+α

3α+β

2α+β α

Proof Let Φ be crystallographic of rank 2 with Weyl group W. Let Π = {α, β}. Choosing a W-invariant scalar product (, ) as in 2.2.3, we have α ∨ ( β) β ∨ (α) = (α, β) 2 2 4 (α,α)(β, β) = 4 cos θ where θ is the angle between α and β. Since α, β ∈ Π we have (α, β) ≤ 0 thus π/2 ≤ θ ≤ π and the integrality of 4 cos2 θ implies that 4 cos2 θ ∈ {0, 1, 2, 3} thus π − θ ∈ {π/2, π/3, π/4, π/6}. Except for A1 × A1 , the ratio of the lengths of α and β is implied by the equation α ∨ ( β) β ∨ (α) = 4 cos2 θ. For instance if 4 cos2 θ = 2 the only integral solution, up to exchanging α and β, is α ∨ ( β) = −1 and β ∨ (α) = −2 whence 2( β, β) = (α, α). For A1 × A1 the ratio of the lengths is not determined, we have chosen 1 in the picture.  Corollary 3.3.3 For a crystallographic root system Φ and a positive subsystem Φ+ , we have: (i) If α, β ∈ Φ, α  − β and (α, β) < 0, then α + β ∈ Φ. (ii) If α, β ∈ Φ and α + n β ∈ Φ for n ∈ N, then α + m β ∈ Φ for all 0 ≤ m ≤ n. (iii) If α1 , . . . , αk ∈ Φ+ and α = α1 + · · · + αk ∈ Φ+ , then if k > 1 we have α − αi ∈ Φ+ for some i. (iv) If Ψ ⊂ Φ is closed, α, β ∈ Ψ, α  − β and nα +m β ∈ Φ for some n, m > 0, then nα + m β ∈ Ψ. Proof For (i), by the argument in the proof of 3.3.2 about possible integral solutions, up to exchanging α and β we have α ∨ ( β) = −1, whence α + β = sα ( β) ∈ Φ. For (iii) as (α, α) > 0 we must have (α, αi ) > 0 for some i thus by (i) α − αi ∈ Φ. For (ii), by (iii) either α + (n − 1) β or n β is in Φ, and since Φ is reduced, n β  Φ if n  1, whence the result by induction on n. For (iv) we may assume both α and β positive (they are for some order since α  − β), and then we apply (iii) and induction on n + m.  Corollary 3.3.4

If Ψ ⊂ Φ is closed and symmetric, it is a root subsystem.

Proof For α, β ∈ Ψ, we have to show that sα ( β) ∈ Ψ. This is true if β = ±α since Ψ is symmetric. Otherwise, replacing α by −α if necessary we have sα ( β) = β + nα for some n ∈ N∗ ; then Corollary 3.3.3(iv) gives the result.  Proposition 3.3.5 If Ψ is closed and Ψ ∩ −Ψ = ∅, there exists a positive subsystem Φ+ such that Ψ ⊂ Φ+ .

3.3 Closed Subsets of a Crystallographic Root System

47

Proof We first show by induction on k > 0 that 0 is not the sum of k elements of Ψ. This is clear for k = 1. If 0 = α1 + · · · + αk then 0 < (α1 , α1 ) = (−α1 , α2 + · · · + αk ) thus there exists i  1 such that (α1 , αi ) < 0. Using α1  −αi (since −αi  Ψ by assumption) and 3.3.3(i) we get α1 + αi ∈ Φ thus α1 + αi ∈ Ψ, thus the sum is the sum of k − 1 elements, a contradiction. We now build by induction on k a sequence γk of elements of Ψ such that γk ∈ Ψ is the sum of k elements of Ψ. We start with γ1 equal to an arbitrary element of Ψ. If there is α ∈ Ψ such that (γk , α) < 0 we set γk+1 = α + γk ∈ Ψ. For i < j we have γi  γj , otherwise γj − γi would be a zero sum of elements of Ψ, thus by finiteness the sequence must stop on some γk such that (γk , α) ≥ 0 for any α ∈ Ψ. The linear form (γk , ) almost defines an order as in 2.2.4. We need to modify it on γk⊥ . But γk⊥ ∩ Ψ ⊂ γk⊥ ∩ Φ satisfies the same assumptions as the proposition and we may iterate the construction on this subspace.  For I ⊂ S we set ΠI := {α ∈ Π | sα ∈ I} and ΦI = Φ ∩ QΠI ; it is clearly a root subsystem with basis ΠI , since when decomposed on Π a root of ΦI involves only elements of ΠI . It is clear that ΦI is closed and symmetric and that Φ+ − ΦI and Φ+ ∪ ΦI are closed. Example 3.3.6 There exist closed and symmetric subsystems which are not of the form ΦI ; for instance the long roots in a system B2 form a system of type A1 × A1 , and the long roots in G2 form a system of type A2 . See also 11.2.7. Lemma 3.3.7

If sα ∈ WI for α ∈ Φ, then α ∈ ΦI .

Proof Elements of WI are the product of some s β for β ∈ ΠI , thus they fix  ΠI⊥ . Thus sα fixes ΠI⊥ , which implies that α ∈ QΠI ∩ Φ = ΦI . We say that Ψ is a parabolic subset of Φ if Ψ is closed and Ψ ∪ −Ψ = Φ. Proposition 3.3.8 (i) A parabolic subset is conjugate to a parabolic subset containing Φ+ ; such a subset is of the form Φ+ ∪ ΦI for some I ⊂ S. (ii) A parabolic subset is a set of the form {α | λ(α) ≥ 0} for some linear form λ on V. Proof For the first part of (i) it is equivalent to show that a parabolic subset Ψ contains some positive subsystem. Choose such a positive subsystem Φ+ such that |Ψ ∩ Φ+ | is maximal. We show by contradiction that Φ+ ⊂ Ψ. Otherwise let Π be the basis of Φ defining Φ+ ; there must exist α ∈ Π, α  Ψ, thus −α ∈ Ψ. Since α  Ψ we have sα (Ψ ∩ Φ+ ) ⊂ Φ+ ; applying sα again we get Ψ ∩ Φ+ ⊂ sα (Φ+ ). But then the positive subsystem sα (Φ+ ) contains −α thus satisfies |Ψ ∩ sα (Φ+ )| > |Ψ ∩ Φ+ |, a contradiction.

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We now assume that Ψ ⊃ Φ+ . Let I = {sα | −α ∈ {−Π ∩ Ψ}}. Let us show that Ψ ∩ Φ− = Φ−I . We first show that Φ−I ⊂ Ψ. Note that by 2.2.8 applied to the basis −ΠI of − ΦI any root in Φ−I is a sum of elements of −ΠI . We show by induction on k that a root in Φ−I sum of k roots in −ΠI is in Ψ. It is true by assumption when k = 1; in general by 3.3.3(iii) we may write the root as α + β where α ∈ −ΠI and β ∈ Φ−I sum of k − 1 roots in −ΠI ; by induction β ∈ Ψ and as α ∈ Ψ and Ψ is closed α + β ∈ Ψ. We finally show the reverse inclusion by induction. Let γ ∈ Ψ ∩ Φ− be the sum of k roots of −Π, and write it γ = α + β where α ∈ −Π and β ∈ Φ is the sum of k − 1 roots in −Π. As − β ∈ Φ+ ⊂ Ψ we get α = γ + (− β) ∈ Ψ whence α ∈ −Π ∩ Ψ = −ΠI . Thus −α ∈ Ψ whence β = γ + (−α) ∈ Ψ, and we conclude since by induction β ∈ Φ−I . Conversely the fact that for any I ⊂ S the set Φ+ ∪ ΦI is parabolic is a consequence of the proof of (ii) below. We now show (ii). It is clear that a subset of the form {α | λ(α) ≥ 0} is parabolic. It is thus sufficient to show that Φ+ ∪ ΦI is of this form. Take any x such that x, α = 0 if α ∈ ΦI and x, α > 0 if α ∈ Ψ − ΦI . Such an x exists: the projection of Φ+ on Φ⊥ I lies in a half-space, and we may take x in this half-space, orthogonal to the hyperplane which delimits it. It is clear that by construction x has the required properties.  A consequence of 3.3.8 is that the complement of a parabolic subset is closed.

Subgroups of Maximal Rank and Quasi-closed Sets In the remainder of this chapter G is a connected reductive algebraic group, T is a maximal torus of G, and Φ is the set of roots of G relative to T. For Ψ ⊂ Φ, we set G∗Ψ := Uα | α ∈ Ψ and GΨ := T, Uα | α ∈ Ψ. These are closed connected subgroups by 1.1.3 and Gψ∗ is a normal subgroup of GΨ . Definition 3.3.9 A subset Ψ ⊂ Φ is called quasi-closed if G∗Ψ does not contain any Uα with α ∈ Φ − Ψ. We get an equivalent definition by replacing G∗Ψ with GΨ , since GΨ /G∗Ψ is a quotient of T and thus is a torus. Hence any Uα ⊂ GΨ is in the kernel of this quotient, and is thus in G∗Ψ . Proposition 3.3.10 A closed and connected subgroup H ⊂ G containing T is equal to GΨ with Ψ = {α ∈ Φ | Ua ⊂ H}; the set Ψ is quasi-closed.

3.3 Closed Subsets of a Crystallographic Root System

49

Proof By 2.3.1(iv) H is generated by T and the Uα it contains. The subset Ψ ⊂ Φ of those α is quasi-closed by definition.  Let Ψ, Ψ be quasi-closed; it is clear that Ψ∩Ψ is quasi-closed (since GΨ∩Ψ is a subgroup of both GΨ and GΨ ); actually one sees that GΨ∩Ψ = (GΨ ∩GΨ ) 0 by applying 2.3.1(iv) to the right-hand side. Definition 3.3.11 A connected linear algebraic group P has a Levi decomposition if there is a closed subgroup L ⊂ P such that P = Ru (P)  L. The group L is called a Levi subgroup of P (or a Levi complement). A Levi complement is clearly reductive. Proposition 3.3.12 Let Ψ ⊂ Φ be quasi-closed, and let Ψs = {α ∈ Ψ | −α ∈ Ψ} and Ψu = {α ∈ Ψ | −α  Ψ}. Then Ψs and Ψu are quasi-closed and GΨ has a Levi decomposition GΨ = G∗Ψu  GΨs where G∗Ψu = Ru (GΨ ). In particular GΨ is reductive if and only if Ψ is symmetric. Proof We first show that Ψs is quasi-closed. As the intersection of two quasiclosed sets is quasi-closed, it is enough to show that −Ψ is quasi-closed. This results from the existence of the opposition automorphism of G which acts by −1 on X(T); see Example 2.4.9. As a connected group normalised by T the group Ru (GΨ ) is – by Theorem 2.3.1(v) – of the form G∗Ψ for a subset Ψ ⊂ Ψ that we may assume quasiclosed. We have Ψ ⊂ Ψu , otherwise there is α ∈ Ψs ∩ Ψ , thus U−α ⊂ GΨ thus normalises Ru (GΨ ) thus [U−α , Uα ] ⊂ Ru (GΨ ) which is a contradiction since this commutator set contains non-unipotent elements by Theorem 2.3.1(ii). To show Ψ = Ψu it is thus enough to show Ψ − Ψ ⊂ Ψs . If α ∈ Ψ − Ψ , then Uα ∩ Ru (GΨ ) = 1 since this intersection is normalised by T thus contains the whole Uα if not trivial. Thus, in the quotient GΨ → L , where L is the reductive group GΨ /Ru (GΨ ), the group Uα maps injectively to a root subgroup of L . Let U be the root subgroup of L corresponding to the opposed root and U its preimage. Any element of U is unipotent since, its image being unipotent, its semi-simple part is in Ru (GΨ ) so is trivial. Hence U is a unipotent subgroup normalised by T, so is a product of certain root subgroups and must contain U−α , thus −α ∈ Ψ and α ∈ Ψs . It also follows from the proof that GΨs maps injectively to L , thus GΨs is a  Levi complement of Ru (GΨ ). Proposition 3.3.13

A closed subset is quasi-closed.

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Proof Let Ψ ⊂ Φ be a closed subset and let Ψs and Ψu be as in 3.3.12. It is clear that Ψs is closed. Note that if α ∈ Ψ, β ∈ Ψu and α + β ∈ Φ then α + β ∈ Ψu , otherwise α + β ∈ Ψs whence −α − β ∈ Ψs thus α + (−α − β) = − β ∈ Ψ which contradicts β ∈ Ψu . In particular Ψu is closed. By 3.3.5 there exists a positive subsystem such that Ψu ⊂ Φ+ . Lemma 3.3.14 If Ψ is a closed subset of a positive subsystem Φ+ of Φ, then Ψ  is quasi-closed and G∗Ψ = α ∈Ψ Uα where the product is taken in an arbitrary order. In the situation of the lemma we will write UΨ for G∗Ψ . Proof

By 2.3.1(vii) and 3.3.3(iv)



α ∈Ψ

Uα is a group, thus equal to G∗Ψ .



We deduce that Ψu is quasi-closed. In addition, if α ∈ Ψs , β ∈ Ψu and α + β ∈ Φ, using the fact that α + β ∈ Ψu and 3.3.3(iv), we get nα + m β ∈ Ψu for n, m ≥ 1 such that nα + m β ∈ Φ. Thus G∗Ψu is normalised by GΨs . Since Ψs is closed and symmetric, it is a root subsystem by 3.3.4. Let Πs be its basis corresponding to the positive subsystem Ψs ∩ Φ+ . Note that GΨs is already generated by T and Uα such that α ∈ ±Πs ; indeed Uα , U−α  contains a representative of sα by 2.3.1(iii), thus GΨs contains WΨs , and every root of Ψs is in the orbit of Πs by 2.2.8, whence the result by the remark above 2.3.3. We show now that GΨs = UΨs+ WΨs TUΨs+ . For that it is enough to show that the right-hand side is a group. Since it is stable by left translation by T and by any Uα for α ∈ Ψs+ it is enough to see it is stable by left translation by U−α for α ∈ Πs . Decomposing UΨs+ = UΨs+ −{α } Uα , and using that by 2.3.1(vii) U−α normalises UΨs+ −{α } since α is simple, it is enough to see that Uα WΨs TUΨs+ is stable by left translation by U−α . The Bruhat decomposition T, Uα , U−α  = Uα T ∪ Uα sα TUα shows that U−α Uα ⊂ Uα T ∪ Uα sα TUα . We just need to consider the second term Uα sα Uα wTUΨs+ . Uα sα TUα WΨs TUΨs+ = w∈Ψs

If w−1 (α) ∈ Ψ+ , then Uα wT = wTUw−1 (α) and the term has the right form. Otherwise, letting β = −w−1 (α) ∈ Ψs+ we get Uα sα wTUw−1 (α) UΨs+ = Uα sα wTU−β U β UΨs+ −{β } ⊂ Uα sα wT(U β ∪ U β s β U β )UΨs+ −{β } = Uα sα wTUΨs+ ∪ Uα sα wTU β s β UΨs+ . We just need to consider the rightmost term. Since sα wU β T = sα U−α wT = Uα sα wT we get the result.

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51

Let us now show that Ψs is quasi-closed. Let γ be such that Uγ ⊂ GΨs ; since Ψs = −Ψs , we may assume γ ∈ Φ+ , thus Uγ ⊂ B. As each term UΨs+ wTUΨs+ is in a unique Bruhat cell of G, we must have Uγ ⊂ TUΨs+ . By Lemma 3.3.14 Ψs+ is quasi-closed, thus γ ∈ Ψs+ . We have seen that GΨ has a semi-direct product decomposition G∗Ψu  GΨs . It follows that Ψ is quasi-closed since if α  Ψu and Uα ⊂ GΨ then Uα maps  isomorphically to the quotient GΨs thus α ∈ Ψs . Conversely any quasi-closed subset of Φ is closed apart from some exceptions in characteristics 2 and 3; see Borel and Tits (1965, 3.8). This can be shown by proving that in other characteristics the group Uα , U β  contains all Unα+mβ for n, m ∈ N such that nα + m β ∈ Φ, using the explicit values of the coefficients in the proof of 2.3.1(vii). For a combinatorial description of quasi-closed subsets in these characteristics, see Malle and Testerman (2011, Corollary 13.7).

3.4 Parabolic Subgroups and Levi Subgroups Proposition 3.4.1 The parabolic subgroup PI = BWI B (see 3.2.3) has a Levi decomposition PI = Ru (PI )  LI where Ru (PI ) = UΦ+ −ΦI and LI = T, {Uα }α ∈ΦI  is reductive. We have PI = NG (Ru (PI )). Proof The set Ψ = Φ+ ∪ ΦI is quasi-closed since it is closed by Proposition 3.3.8. The proposition is then a consequence of 3.3.12 if we show that PI = GΨ . We have PI ⊃ GΨ since PI ⊃ Uα for α ∈ Φ+ since PI ⊃ B, and by 3.2.7 PI contains all Uα for α ∈ Φ− which change sign by some element of WI , thus contains all Uα for α ∈ Φ−I . Conversely GΨ contains Uα and U−α for all α ∈ ΠI , hence GΨ contains a representative of sα in Uα , U−α  hence GΨ contains WI , thus contains PI . Finally, NG (Ru (PI )) contains PI thus B, thus is some parabolic subgroup PJ . If J  I we have Ru (PJ ) = UΦ+ −ΦJ  Ru (PI ) which contradicts that PJ normalises Ru (PI ) since Ru (PJ ) is the largest normal connected unipotent sub group of PJ . We will say that PI (resp. LI ) is a standard parabolic subgroup (resp. Levi subgroup) of G. Proposition 3.4.2

Let P be a parabolic subgroup of G containing T.

(i) There is a unique Levi subgroup of P containing T. (ii) Two Levi subgroups of P are conjugate by a unique element of Ru (P).

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Proof The existence of a Levi subgroup containing T results from 3.4.1, since P is conjugate to some PI and all maximal tori of PI are conjugate in PI . Conversely, we may assume P = PI ; any Levi subgroup of PI containing T is a GΨ for some Ψ ⊂ Φ+ ∪ ΦI by Proposition 3.3.10. Since any Uα where α ∈ Φ+ − ΦI is in Ru (PI ), we must have Ψ ⊂ ΦI , thus L ⊂ LI , thus there must be equality. Two Levi subgroups L, L of P are conjugate in P, since by (i) an element which conjugates a maximal torus T of L into L conjugates L to L . Modulo L, we can choose the conjugating element u in Ru (P). The unicity of u is equivalent to Ru (P) ∩ NP (L) = 1. Assume u ∈ Ru (P) ∩ NP (L); then for any l ∈ L we have [u, l] ∈ Ru (P) ∩ L = 1, thus u ∈ CP (L); but CP (L) ⊂ CG (T) = T thus u = 1.  Proposition 3.4.3 The G-conjugacy classes of Levi subgroups of parabolic subgroups of G are in bijection with the W-orbits of subsets of S, which are themselves in bijection with the W-conjugacy classes of parabolic subgroups of W. Proof Since all parabolic subgroups are conjugate to a PI , we may assume that we consider a Levi subgroup of some PI . Since by 3.4.2 such a Levi subgroup is G-conjugate to LI , the question becomes that of finding when LJ is G-conjugate to LI for two subsets I and J of S. If LJ = gLI for some g ∈ G, −1 then, since g T and T are two maximal tori of LI , there exists l ∈ LI such that −1 g T = lT and gl ∈ N (T) also conjugates L to L ; so the G-conjugacy classes G I J of LI are the same as the W (T)-conjugacy classes. Since LI = GΦI , the element w ∈ W conjugates LI to LJ if and only if wΦI = ΦJ . Since any two bases of ΦI are conjugate by an element of WI (see 2.2.6), we may assume that wI = J whence the first part of the statement. To see the second part it is enough to see  that if w ∈ NW (WI ) then wΦI = ΦI . This results from Lemma 3.3.7. The proof above shows that NG (LI )/LI is isomorphic to NW (WI )/WI . Proposition 3.4.4 Let L be a Levi subgroup of a parabolic subgroup P. Then R(P) = Ru (P)  R(L). Proof The quotient P/(R(L)Ru (P)) is isomorphic to L/R(L), so is semisimple. So R(P) ⊂ R(L)Ru (P). But R(L)Ru (P) is connected, solvable and normal in P as the inverse image of a normal subgroup of the quotient P/Ru (P)  L, whence the reverse inclusion.  We will now characterise parabolic subgroups in terms of roots. Proposition 3.4.5 A closed subgroup P of G containing T is parabolic if and only if P = GΨ for some parabolic subset Ψ.

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Proof We have seen that a parabolic subset Ψ is conjugate under W to Φ+ ∪ΦI ; thus GΨ is conjugate under W to PI . Conversely, assume that P is a parabolic subgroup. It contains a Borel subgroup containing T, thus up to conjugacy by  W it contains B (see Proposition 2.3.3), thus is of the form PI . We now give an important property of Levi subgroups. Proposition 3.4.6 Let L be a Levi subgroup of a parabolic subgroup of G; then L = CG (Z(L) 0 ). Proof We may assume that L = LI . Then by 2.3.4(i) the group Z(L) is the intersection of the kernels of the roots in ΦI . The group CG (Z(L) 0 ) is connected as it is the centraliser of a torus – Z(L) 0 is diagonalisable by 1.2.1(ii) and is a torus by 1.2.3(i) thus 1.3.3(iii) applies. It is normalised by T because it contains T, hence by 2.3.1(iv) it is generated by T and the Uα it con tains. If Uα ⊂ CG (Z(L) 0 ) then α is trivial on ( α ∈ΦI Ker α) 0 . This identity  component has finite index in α ∈ΦI Ker α, hence some multiple nα of α is  trivial on α ∈ΦI Ker α. With the notation of 1.2.12, this can be rewritten as ⊥ ⊥ ⊥ nα ∈ (ΦI ⊥ T )X(T) . But (ΦI T )X(T) /ΦI  is a torsion group (see 1.2.13). This implies that α ∈ ΦI  ⊗ Q, which in turn yields α ∈ ΦI by the definition of ΦI .  This proves that CG (Z(L) 0 ) ⊂ L. The reverse inclusion is obvious. The next proposition is a kind of converse. Proposition 3.4.7 For any torus S, the group CG (S) is a Levi subgroup of some parabolic subgroup of G. Proof Let T be a maximal torus containing S. As the group CG (S) is connected by 1.3.3(iii) and contains T, by 2.3.1(iv) we have CG (S) = T, Uα | Uα ⊂ CG (S). As S acts by α on Uα (see 2.3.1(i)), we have Uα ⊂ CG (S) ⇔ α|S = 0, where 0 is the trivial element of X(S). Let us choose a total order on X(S); that is, a structure of ordered Z-module. As X(S) is a quotient of X(T) (see 1.2.4) there exists a total order on X(T) compatible with the chosen order on X(S); that is, such that for x ∈ X(T) we have x ≥ 0 ⇒ x|S ≥ 0. This implies that the set Ψ = {α ∈ Φ | α > 0 or α|S = 0} is also equal to {α ∈ Φ | α|S ≥ 0}. This last definition implies that Ψ is closed, so (see 3.3.13 and 3.3.10) Ψ is also the set of α such that Uα ⊂ GΨ . Since Ψ is parabolic, it follows then from 3.4.5  that GΨ is a parabolic subgroup, of which CG (S) is a Levi complement. We now study the intersection of two parabolic subgroups. First note that by 3.1.4 the intersection of two parabolic subgroups always contains some maximal torus of G.

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Proposition 3.4.8 Let P and P be two parabolic subgroups of G and let L and L be respective Levi subgroups of P and P containing the same maximal torus T of G. Let U = Ru (P) and U = Ru (P ). Then (i) The group (P ∩ P ).U is a parabolic subgroup of G which has the same intersection as P with L, and it has L ∩ L as a Levi subgroup. (ii) The group P ∩ P is connected, as well as L ∩ L and we have the Levi decomposition P ∩ P = ((L ∩ U ).(L ∩ U).(U ∩ U ))  (L ∩ L ) where the right-hand side is a direct product of varieties – the decomposition of an element of P ∩ P as a product of four terms is unique. On the right-hand side the last factor is a Levi subgroup of P ∩ P and the first 3 factors form a decomposition of Ru (P ∩ P ). Proof Let Φ be the roots of G with respect to T and define subsets Ψ, Ψ ⊂ Φ by P = GΨ and P = GΨ . Let us show first that for any α ∈ Φ, either Uα or U−α is in the group (P ∩ P ) · U (it is a group since P normalises U). By the remarks before 3.3.11 and by 3.3.12 we have (P ∩ P ) 0 · U = GΨ∩Ψ · G∗Ψu , with the notation of 3.3.12. If neither α nor −α is in Ψu , they are both in Ψ in which case since one of them is in Ψ , one of them is in Ψ∩Ψ . Hence (Ψ∩Ψ )∪Ψu is a parabolic set; indeed, this set is closed as the sum of an element of Ψ and an element of Ψu which is a root is in Ψu – see the beginning of the proof of 3.3.13. Proposition 3.4.5 then shows that (P ∩ P ) 0 .U is a parabolic subgroup of G, equal to G(Ψ∩Ψ )∪Ψu . Then (P ∩ P )U, containing a parabolic subgroup is connected, hence equal to (P ∩ P ) 0 .U. Now (P ∩ P ).U = (P ∩ P ).G∗Ψu −Ψ since Ψ ∩ Ψu ⊂ Ψ ∩ Ψ . The set Ψu − Ψ is closed as the intersection of the closed subsets Ψu and the complement −Ψu of Ψ , hence the product (P ∩ P ).G∗Ψu −Ψ is a direct product of varieties as the intersection is a unipotent subgroup normalised by T containing no Uα , see 2.3.1(v). As the product is connected, each term is. Thus P ∩ P is connected equal to GΨ∩Ψ . The groups (P ∩ P ).U and P ∩ P have both (L ∩ L ) 0 as a Levi subgroup since ((Ψ ∩ Ψ ) ∪ Ψu )s = (Ψ ∩ Ψ )s = Ψs ∩ Ψs – indeed if α ∈ Ψu then −α  Ψ thus −α  (Ψ ∩ Ψ ) ∪ Ψu .    The decomposition Ψ∩Ψ = (Ψs ∩Ψs ) (Ψs ∩Ψu ) (Ψu ∩Ψs ) (Ψu ∩Ψu ) shows that P ∩ P = L ∩ L , L ∩ U , L ∩ U, U ∩ U . Using that U ∩ U is normal in P ∩ P , then that L ∩ L normalises L ∩ U and L ∩ U, we get P ∩ P = (L ∩ L ).L ∩ U , L ∩ U.(U ∩ U ).

3.4 Parabolic Subgroups and Levi Subgroups

55

Further, the commutator of an element of L ∩ U with an element of L ∩ U is in U ∩ U , thus P ∩ P = (L ∩ L ).(L ∩ U ).(L ∩ U).(U ∩ U ), Write now x = lu uv ∈ P ∩ P , where l ∈ L ∩ L , u ∈ L ∩ U , u ∈ L ∩ U, v ∈ U ∩ U . Then lu is the image of x by the projection P → L and l (resp. u) is the image of lu (resp. uv) by the morphism P → L . Thus the decomposition of x is unique, and the product map (L ∩ L ) × (L ∩ U ) × (L ∩ U) × (U ∩ U ) → P ∩ P is an isomorphism of varieties; the four terms are connected since the  product is. In particular L ∩ L is connected. Proposition 3.4.9 (i) Let P and P be two parabolic subgroups of G such that P ⊂ P, then Ru (P ) ⊃ Ru (P) and for any Levi subgroup L of P , there exists a unique Levi subgroup L of P such that L ⊃ L . (ii) Let L be a Levi subgroup of a parabolic subgroup P of G and L be a closed subgroup of L. Then the following are equivalent: (a) L is a Levi subgroup of a parabolic subgroup of L. (b) L is a Levi subgroup of a parabolic subgroup of G. Proof Let us prove (i); given a maximal torus T of L there is by 3.4.2(i) a unique Levi subgroup L of P containing T. Then by 3.4.8(ii) the group L ∩ L is a Levi subgroup of P = P ∩ P thus L ∩ L = L . Also Ru (P) is contained in all Borel subgroups of P, thus in P , whence Ru (P) ⊂ Ru (P ). Let us show (ii); if L is a Levi subgroup of the parabolic subgroup PL of L, then PL Ru (P) is a parabolic subgroup of G; indeed it is a group since L, thus PL , normalises Ru (P) and it clearly contains either Uα or U−α for any α ∈ Φ. Thus L is a Levi subgroup of PL Ru (P), since Ru (PL ).Ru (P) is unipotent normal in PL .Ru (P). We have shown that (a) implies (b). Conversely, let P be a parabolic subgroup of G with L as a Levi subgroup. By 3.4.8(ii) we have P ∩ P = L .(L ∩ U ).(U ∩ U ) thus (L ∩ U )  L is a Levi decomposition of L ∩ P , and this last group is a parabolic subgroup of L by 3.4.5.  When L is a Levi subgroup of some parabolic subgroup of G we will say (improperly) “L is a Levi subgroup of G” which is justified by statement (ii) of 3.4.9. Proposition 3.4.10 Let H be a closed connected reductive subgroup of G of maximal rank. Then:

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(i) The Borel subgroups of H are the B ∩ H where B is a Borel subgroup of G containing a maximal torus of H. (ii) The parabolic subgroups of H are the P ∩ H, where P is a parabolic subgroup of G containing a maximal torus of H. (iii) If P is a parabolic subgroup of G containing a maximal torus of H, the Levi subgroups of P ∩ H are the L ∩ H where L is a Levi subgroup of P containing a maximal torus of H. Proof Let T be a maximal torus of H; by assumption, it is also a maximal torus of G. Let B be a Borel subgroup of G containing T, and let B = U.T be the corresponding semi-direct product decomposition. The Borel subgroup B defines an order on the root system Φ (resp. ΦH ) of G (resp. H) with respect to T. The group U ∩ H is normalised by T, so is connected and equal to the product of the Uα it contains, that is those Uα such that α is positive and in ΦH , so (U ∩ H).T = B ∩ H is a Borel subgroup of H. This gives (i) since all Borel subgroups of H are conjugate under H. Let us prove (ii). If P is a parabolic subgroup of G containing T, it contains a Borel subgroup B containing T, so its intersection with H contains the Borel subgroup B ∩ H of H and thus is a parabolic subgroup. Conversely, let Q be a parabolic subgroup of H containing T, and let x be a vector of X(T)⊗Q defining Q as in 3.3.8(ii). Then x defines a parabolic subgroup P of G. It remains to show that P ∩ H = Q. The group P ∩ H is a parabolic subgroup of H by the first part. It is generated by T and the Uα it contains. But Uα ⊂ P ∩ H if and only if α ∈ ΦH and α, x ≥ 0; that is, if and only if Uα ⊂ Q by definition of x. Similarly, the Levi subgroup of Q containing T is the intersection of the Levi subgroup of P containing T with H, as it is generated by T and the Uα with α ∈ ΦH orthogonal to x, whence (iii). 

3.5 Centralisers of Semi-Simple Elements Proposition 3.5.1 Let s ∈ G be a semi-simple element, and let T be a maximal torus containing s and Φ be the set of roots of G relative to T; then (i) The identity component CG (s) 0 is generated by T and the Uα for α ∈ Φ such that α(s) = 1. It is a connected reductive subgroup of G of maximal rank. (ii) CG (s) is generated by CG (s) 0 and the elements n ∈ NG (T) such that ns = s. Proof (i) is an immediate consequence of 2.3.1(iv), and of the fact that the corresponding set of α is closed and symmetric – see 3.3.12 and 3.3.13.

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Let us prove (ii). Let B = U  T be the Levi decomposition of a Borel subgroup of G and let g ∈ CG (s); by Lemma 3.2.7 the element g has a unique decomposition g = unv with n ∈ NG (T), u ∈ U and v ∈ Uw where w is the image of n in W (T). As s normalises U, Uw and NG (T), this decomposition is invariant under conjugation by s, so each of u, n and v also centralises s.  Writing again a unique decomposition of the form u = α>0 uα we see that the α must satisfy α(s) = 1 so u ∈ CG (s) 0 , and the same argument applies to v. Thus we get (ii).  Remark 3.5.2 The Weyl group W 0 (s) of CG (s) 0 is thus the group generated by the reflections sα for which α(s) = 1. It is a normal subgroup of the Weyl group of CG (s) which is W (s) = {w ∈ W (T) | ws = s}. The quotient W (s)/W 0 (s) is isomorphic to the quotient CG (s)/CG (s) 0 . Proposition 3.5.3 If x = su is the Jordan decomposition of an element of G, where s is semi-simple and u unipotent, then x ∈ CG (s) 0 . Proof Let B = U  T be a Levi decomposition of a Borel subgroup containing  x, where T is a maximal torus of B containing s, and write u = uα (with uα ∈  Uα where U = Uα ). Then for any root α such that uα  1, we have α(s) = 1  which implies that Uα ⊂ CG (s) 0 , whence the result as s ∈ T ⊂ CG (s) 0 . Examples 3.5.4 (i) All centralisers in GLn are connected. Indeed, the centraliser in the variety of all matrices is an affine space, thus its intersection with GLn is an open subspace of an affine space, which is always connected. (ii) In the group SLn , centralisers of semi-simple elements are connected. Indeed such an element is conjugate to an element s = diag(t1 , . . . , tn ) of the torus T of diagonal matrices where we may assume, in addition, that equal ti are grouped in consecutive blocks, thereby defining a partition π of n. The elements of W (T) (permutation matrices) which centralise s are products of generating reflections sα which centralise s; that is, W (s) = W 0 (s) showing that the centraliser of s is connected. (iii) We finish with an example of a semi-simple element whose   centraliser is 1 0 not connected. Let s ∈ PGL2 be the image of ; in characteris0 −1 tic different from 2, CPGL2 (s) has two connected components, consisting   a 0 respectively of the images of the matrices of the form and of the 0 b   0 a form . b 0

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Notes A classic reference about (B, N)-pairs is Bourbaki (1968, Chapter IV). A detailed study of closed and quasi-closed subsets and reductive and parabolic subgroups is in Borel and Tits (1965). A detailed study of the centralisers of semi-simple elements can be found in, for example, Deriziotis (1984).

4 Rationality, the Frobenius Endomorphism, the Lang–Steinberg Theorem

4.1 k0 -Varieties, Frobenius Endomorphisms Let k be a (non-necessarily algebraically closed) field and let k0 be a subfield of k. A k0 -structure on a k-vector space V is a k0 -subspace V0 such that V = V0 ⊗k0 k. A k0 -structure on a finite type k-algebra A is a finite type k0 -algebra A0 such that A = A0 ⊗k0 k. A k0 -structure on an affine or projective k-variety V is the k0 -variety defined by A0 where the algebra A of V has a k0 -structure A0 . In general a k0 -structure on a variety is essentially given by a finite open affine covering where each open affine has a k0 -structure. In this book, all the varieties we need to consider are quasi-projective varieties. We assume all varieties are quasi-projective from now on. Definition 4.1.1 An algebraic variety V over k is said to be defined over k0 , if it has a k0 -structure V0 . In this case we write V = V0 ⊗k0 k. In the literature it is also sometimes said in the same circumstances that V is rational over k0 , or even when the context of k and k0 is unambiguous that V is rational. If the variety V has a k0 -structure, an element σ ∈ Aut(k/k0 ) acts on V by x ⊗ λ → x ⊗ σ(λ) on the corresponding coordinates for the variety. We assume now that k/k0 is a Galois extension; then one can see V0 as the fixed points of the action of Gal(k/k0 ). This results from the following proposition. Proposition 4.1.2 If V is a k-vector space (resp. a k-algebra) with a continuous action of Gal(k/k0 ) – as a profinite group, thus continuous means that V = ∪G V G where G runs over subgroups of finite index of 59

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Gal(k/k0 ) – (resp. compatible to the algebra structure), the fixed points of the action define a k0 -structure. Reference

See Springer (1998, 11.1.6).



Example 4.1.3 When k is an algebraic closure Fq of Fq , the Galois group Gal(Fq /Fq ) is the profinite completion Zˆ of Z; an element of Zˆ is defined by a sequence kn ∈ Z indexed by n ∈ N subject to the only condition kn ≡ km kn (mod m) if m divides n; this element acts on Fqn by x → xq . We have Zˆ   p prime Zp . Proposition 4.1.4 A subvariety (resp. subalgebra, vector subspace) is defined over k0 – equivalently has a k0 -structure which is a subvariety (resp. subalgebra, subspace) – if and only if it is stable under the action of Gal(k/k0 ). Proof We give the proof for vector spaces; the other cases follow. It is obvious that a vector subspace W ⊂ V which has a k0 -structure is stable by Γ = Gal(k/k0 ). Let us show the converse: assume that Γ(W) = W. Let W0 = W Γ . We need to show that W0 spans W over k. The space W0 is a k0 -subspace of V0 , the k0 -structure of V. Let W be a complementary k0 -subspace. Using that V0 is a k0 -structure for V, if W were not spanned by W0 , there would be a non zero element of W spanned by W . Let v = ni=1 ai vi be such an element with vi ∈ W , ai ∈ k× , and n minimal. We may assume a1 = 1 by dividing both  sides by a1 . For γ ∈ Γ we get γv − v = ni=2 (γai − ai )vi . The minimality of n implies that γv − v = 0, otherwise it would be a non-zero element of W giving rise to a smaller n. But this is true for all γ ∈ Γ, which implies v ∈ W0 , a contradiction.  Example 4.1.5 The affine line is A1 = Spec k[T]. The affine line on k0 , defined by the k0 -algebra k0 [T], is a k0 -structure since k[T] = k0 [T] ⊗k0 k. An element   σ ∈ Gal(k/k0 ) acts as i ai T i → i σ(ai )T i . A k-point of A1 is given by a ∈ k (or by the ideal which is the kernel of the morphism P → P(a) : k[T] → k); this point is defined over k0 if a ∈ k0 .

Frobenius Endomorphism We now take k = Fq , where q is a power of the prime p. Definition 4.1.6 Let V be an Fq -variety with an Fq -structure V0 . The associated geometric Frobenius endomorphism F : V → V is F0 ⊗Fq 1 where F0 is the endomorphism of V0 which raises the functions on V0 to the qth power. The endomorphism Φ of V induced by (λ → λ q ) ∈ Gal(Fq /Fq ) is called the arithmetic Frobenius endomorphism.

4.1 k0 -Varieties, Frobenius Endomorphisms

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On an affine variety Spec A the Fq -structure is of the form A = A0 ⊗Fq Fq and the geometric Frobenius endomorphism corresponds to a morphism F ∗ : a ⊗ λ → aq ⊗ λ in a corresponding coordinate system for the variety – the geometric Frobenius raises each coordinate to the qth power. The arithmetic Frobenius is given by Φ : a ⊗ λ → a ⊗ λ q . The composition F ∗ ◦ Φ raises each element of A to the qth power, which acts trivially on the Fq -points of Spec A. Example 4.1.7 On A1 , the geometric Frobenius is given by F ∗ : P(T) → P(T q ); thus F ∗ ◦ Φ maps P(T) to P(T) q . If a ∈ Fq is an Fq -point of A1 , the image of a by F ∗ ◦ Φ is defined by the kernel of P → P(a) q , which is the same as that of P → P(a). Note that the geometric Frobenius endomorphism is a morphism of Fq varieties, while the arithmetic Frobenius endomorphism is only a morphism of Fq -varieties. After this chapter, we will only consider the geometric Frobenius endomorphism and just call it “the Frobenius endomorphism”. Proposition 4.1.8 Let V be an affine or projective Fq -variety with algebra A. F∗

(i) A surjective morphism A −−→ A[q] , where A[q] = {xq | x ∈ A}, is the Frobenius endomorphism attached to an Fq -structure on V if and only if n for any x ∈ A there exists n such that F ∗n (x) = xq . (ii) If F is a Frobenius endomorphism on V attached to an Fq -structure, that Fq -structure is given by A0 = {x ∈ A | xq = F ∗ (x)}.  n Proof If A has an Fq -structure A = A0 ⊗Fq Fq  x = i xi ⊗ λ i then xq = n  qn n q q ∗n i xi ⊗ λ i thus x = F (x) when n is such that all λ i are in Fqn . n ∗ Conversely, if F is a surjective morphism as in the statement, since x → xq is injective then F ∗ must also be injective, thus bijective and we can define Φ by Φ(x) = F ∗−1 (xq ); then if we make the topological generator of Gal(Fq /Fq ) act by Φ, the assumptions of 4.1.2 are satisfied and Φ is the arithmetic Frobenius associated to F. The fixed points of Φ form the Fq -structure by 4.1.2 and are as described in the statement.  Exercise 4.1.9 Show that any Fq -structure on the affine line A1 is given by a × Frobenius endomorphism on Fq [T] of the form T → aT q with a ∈ Fq up to composition with an automorphism of A1 . Proposition 4.1.10

A Frobenius morphism is a finite morphism.

Proof Since being finite is a local property we need only to prove it for an affine variety Spec A with A = Fq [T1 , . . . , Tn ]/I for some ideal I. The image of q q q q A by F is A[q] = Fq [T1 , . . . , Tn ]/(I ∩ Fq [T1 , . . . , Tn ]). We have to show that the

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dimension of A over A[q] is finite. This property is true for Fq [T1 , . . . , Tn ] hence it is also true for A.  Proposition 4.1.11 Let V be an Fq -variety and F be the Frobenius endomorphism corresponding to an Fq -structure on V. (i) A subvariety of V is defined over Fq if and only if it is F-stable; the corresponding Frobenius endomorphism is the restriction of F. (ii) Let ϕ be an automorphism of V such that (ϕF) n = F n for some positive integer n; then ϕF is the Frobenius endomorphism attached to another Fq -structure on V. (iii) If F is a Frobenius endomorphism attached to another Fq -structure on V, there exists an integer n > 0 such that F n = F n . (iv) F n is the Frobenius endomorphism attached to an Fqn -structure on V. (v) Every closed subvariety of a variety defined over Fq is defined over a finite extension of Fq . Every morphism between varieties defined over Fq is defined over a finite extension of Fq . (vi) The orbits of F on the set of points of V are finite, as well as the set VF , also denoted by V(Fq ), which consists of the points of V defined over Fq . Proof (i) reflects 4.1.4. (ii) results from the fact that ϕF still satisfies 4.1.8(i). (iii) by considering an affine open covering it is sufficient to deal with the case V = Spec A. Then we use that A is of finite type, thus there exists n such n that F ∗n (x) = F ∗n (x) = xq for every generator x of A. (iv) results from 4.1.8(i). (v) has a proof similar to that of (iii): there exists n such that for any element a in a finite set of generators of the ideal I defining the subvariety (resp. n any coefficient a of an equation of the morphism) we have F ∗n a = aq , √ ∗n thus I ⊂ F (I). Let us show (vi). As in (iii) we may assume V = Spec A. Let A0 define the Fq -structure on A and let {a1 , . . . , am } be a generating set for A0 . A point x ∈ V is given by a morphism x : A → Fq . It is F ∗n -stable if and only if for any i we have x(ai ) ∈ Fqn , which happens for a sufficiently large n. It is F ∗ -stable if and only if x(ai ) ∈ Fq , or equivalently if we are given a morphism A(Fq ) → Fq ; there is a finite number of such morphisms.  Proposition 4.1.12 Fq -structure on V.

Let V  An as an Fq -variety. Then |VF | = qn for any

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Proof See Geck (2003, 4.2.4) for a (complicated) elementary proof in the case where V is a unipotent group. The proposition is an immediate consequence of the Lefschetz theorem in -adic cohomology, see 8.1.11(ii). 

4.2 The Lang–Steinberg Theorem; Galois Cohomology In the following, when we consider an algebraic group defined over Fq or endowed with a Frobenius endomorphism F, it always means that F is a group morphism. Example 4.2.1 Let us consider the group GLn over Fq . It is defined over Fq as its algebra is Fq [ti,j , det(ti,j ) −1 ], which is isomorphic to Fq [ti,j , det(ti,j ) −1 ] ⊗ Fq . Its points over Fq form the group GLn (Fq ). On the matrices representing GLn (Fq ), the endomorphism F raises the entries to the qth power. Notation 4.2.2 Let T be a torus with a surjective algebraic endomorphism F. We still denote by F the group endomorphism of X(T) given on α ∈ X(T) by α → α ◦ F. Proposition 4.2.3 If F is the Frobenius endomorphism on the torus T attached to an Fq -structure, there exists τ ∈ GL(X(T)) of finite order such that F = qτ on X(T). Proof If A = Fq [X(T)] is the algebra of T and Φ is the arithmetic Frobenius associated to F on A, and Φ0 is the arithmetic Frobenius on Fq [x, x−1 ] associated to the geometric Frobenius x → xq on Gm , using the invertibility of the arithmetic Frobenius as a semilinear morphism of algebras we can define a group automorphism τ on X(T) by the commutative diagram Φ0

Fq [x, x−1 ] −−→ Fq [x, x−1 ] ⏐ ⏐

τ (α)∗ ⏐

α∗ ⏐ Φ

A −→ A where α ∗ is the algebra morphism corresponding to α : T → Gm . We can extend this diagram to the following commutative diagram Φ0

F∗

Φ

F∗

Fq [x, x−1 ] −−→ Fq [x, x−1 ] −−→ ⏐ ⏐

τ (α)∗ ⏐

α∗ ⏐

Fq [x, x−1 ] ⏐

τ (α)∗ ⏐

A −→ A −−→ A where the downward map on the right is τ(α) ∗ since F ∗ ◦ Φ and F ◦ Φ0 are the algebra morphisms a → aq . The commutativity of the rightmost square diagram gives F = qτ. By 4.1.8(i), there is a power F a which is just multiplication  by qa on X(T), thus the order of τ is finite.

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A Frobenius endomorphism on a reductive group – having trivial kernel and being bijective on points – is an isogeny; but it is not an isomorphism, since it is not invertible as a morphism of varieties. Specifically, let G be a reductive group over Fq , let F be the Frobenius endomorphism attached to an Fq -structure and let T be an F-stable maximal torus – we will see below that such a torus always exists. As we saw in 4.2.3, F induces a morphism of the form qτ on X(T). By 2.4.8 the isogeny F maps a root subgroup Uα to Uτ (α) . Thus F is an isogeny associated to σ = τ and such that qα = q for any α; the p-morphism is qτ. Definition 4.2.4 (i) Let X be an algebraic variety over Fp and let F : X → X be an endomorphism such that some power is a Frobenius endomorphism. Then we call F a Frobenius root. (ii) If F is a Frobenius root and F m is the Frobenius endomorphism attached to an Fpn -structure we define a positive real number q attached to F by qm = pn . When X is a torus T, we have the following generalisation of 4.2.3, where F on X(T) is defined as in 4.2.2. Lemma 4.2.5 If F is a Frobenius root on T and q is the attached real number, then there exists a finite order τ ∈ GL(X(T) ⊗ R) such that F = qτ on X(T). Proof Define an element τ ∈ End(X(T) ⊗ R) by τ := q−1 F. By 4.1.8(i) there exists a power a of F such that F a = qa on X(T). It follows that τ is of finite order.  Definition 4.2.6 Let G be a reductive group over Fp ; we call Frobenius root on G a Frobenius root F on the variety G which is a group morphism. The group of fixed points GF is called a finite group of Lie type. Proposition 4.2.7 A Frobenius root on a reductive group is an isogeny. Proof A Frobenius root has trivial kernel since some power has trivial kernel, and is surjective since some power is surjective.  Lemma 4.2.8 Let G be an affine algebraic group over Fp and F be a Frobenius root. Then for g ∈ G the map ad g ◦ F is again a Frobenius root. Proof It is enough to check that some power of ad g ◦ F is equal to the same n−1 power of F. We have (ad g ◦ F) n = ad y ◦ F n with y = g Fg . . . F g; if n is such that g is F n -stable then y is also F n -stable and if y has order e then  (ad g ◦ F) ne = F ne .

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The fundamental theorem on connected algebraic groups over Fp – here connectedness is essential – follows next. Theorem 4.2.9 (Lang–Steinberg) Let G be a connected affine algebraic group over Fp , and F be a Frobenius root. Then the Lang map L : g → g−1 . Fg is a surjective endomorphism of G. Proof The morphism L has fibres isomorphic to GF , thus finite, thus dim Im L = dim G; as G is irreducible L is dominant (which means G is the closure of Im L), thus Im L contains a nonempty open subset of G. For a given x, the morphism g → g−1 .x. Fg has also finite fibres: indeed, a fibre has cardinality the number of solutions of g−1 x Fg = x, that is g = xFg and xF still has finitely many fixed points by Lemma 4.2.8. Thus the image of g → g−1 .x. Fg contains also a nonempty open subset of G, thus meets the image of L, using again the irreducibility of G. Thus there exist g and h such  that g−1 . Fg = h−1 .x. Fh, thus x = L(gh−1 ). Our statement is intermediate in strength between the original theorem of Lang, which gives the same statement assuming that F is a Frobenius endomorphism, and the stronger statement of Steinberg (1968), which uses only the assumption that F is a surjective morphism with GF finite. Some authors – see for instance Malle and Testerman (2011) – call Steinberg morphisms what we call Frobenius roots. A consequence of the Lang–Steinberg theorem is that for g ∈ G the group gF G is isomorphic to GF . Indeed, write g = h−1 . Fh then GgF = h−1 GF h.

Galois Cohomology Exact Sequence In this subsection we give a series of consequences of the Lang–Steinberg theorem, with corresponding elementary proofs. However these results can be understood in the more general context of Galois cohomology to which we first give an introduction. The impatient reader can skip this background information and use directly the elementary proofs we give. We follow Serre (1994, §5). If G is a profinite group acting continuously on a set E, we set H 0 (G, E) := EG . If E is a group on which G acts as group endomorphisms, then Serre defines a set H 1 (G, E) for which we do not give the ˆ If A ⊂ B is a group general definition, but rather give it below when G = Z. inclusion we have the “Galois cohomology exact sequence” p

i

− H 1 (G, A) → − H 1 (G, B). 1 → H 0 (G, A) → H 0 (G, B) → H 0 (G, B/A) → ˆ we will write H i (F, E) for When F is a topological generator of G = Z, 1 in this case H (F, E) is the set of F-conjugacy classes of E, equal

ˆ E); H i (Z,

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by definition to the E-conjugacy orbits in the coset E.F, or the classes of E under the twisted conjugacy e → e e. Fe −1 . In this book we will use the Galois cohomology exact sequence in the form shown in the following proposition. Proposition 4.2.10 Let A ⊂ B be an inclusion of algebraic groups, and F be a Frobenius root, then there is an exact sequence p

i

− H 1 (F, A) → − H 1 (F, B) 1 → AF → BF → (B/A) F → and when A is a normal subgroup of B, the sequence can be extended on the right to one more term H 1 (F, B/A). Reference and details For the last sentence see Serre (1994, §5, Proposition 38). The maps in 4.2.10 are the obvious ones except perhaps p which maps an F-stable coset bA to the F-conjugacy class of b−1 . Fb, an element of A since bA is F-stable. The first three terms form an exact sequence of groups. The exactness of the sequence at (B/A) F means that BF acts naturally on (B/A) F and that two elements of (B/A) F have the same image in H 1 (F, A) if and only if they are in the same BF -orbit. The exactness at H 1 (F, A) means that the image of p is the preimage by i of the F-conjugacy class of 1 and similarly for the exactitude at H 1 (F, B) when A is a normal subgroup of B: this is an “exact sequence of pointed sets”.  In the rest of this subsection F will be a Frobenius root. The Lang–Steinberg theorem can be rephrased as: Proposition 4.2.11 Let G be a connected algebraic group over Fp and F be a Frobenius root. Then: |H 1 (F, G)| = 1. Proposition 4.2.12 Let G be a connected algebraic group over Fp , let F be a Frobenius root and let V be a non-empty variety with an action of F on which G acts transitively and compatibly with F. Then VF  ∅. Proof Since the action is transitive, given v ∈ V, there exists g ∈ G such that = gv. Write g−1 = h−1 Fh, then F (hv) = Fhgv = hv. 

Fv

Lemma 4.2.13 Let G be a connected algebraic group over Fp and F be a Frobenius root. Let A ⊂ B be two closed and F-stable subgroups of G, with A connected; then: (i) We have (B/A) F = BF /AF . (ii) If, in addition, A is normal in B, the quotient map induces a bijection H 1 (F, B) → H 1 (F, B/A).

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Proof (i) and the injectivity in (ii) are 4.2.10 since |H 1 (F, A)| = 1 but let us give a naive proof. For (i), by 4.2.12, any F-stable coset bA contains an F-stable element, thus the natural map BF /AF → (B/A) F is surjective. It is injective since if x, y ∈ BF are in the same A-coset, then x−1 y ∈ AF . Let us show (ii). Surjectivity is clear. Conversely, if b, b ∈ B are F-conjugate modulo A, we have ab = xb Fx−1 , with x ∈ B and a ∈ A. We must see that ab is F-conjugate to b, that is there exists y ∈ B such that yab Fy−1 = b or equivalently a = y−1 bFy. This comes from 4.2.8 which shows that we may still apply the Lang–Steinberg theorem to ad b ◦ F.  Proposition 4.2.14 Let G be a connected algebraic group over Fp , let F be a Frobenius root and let V be a variety with an action of F on which G acts transitively and compatibly with F. Let x ∈ VF and g ∈ G. Then: (i) We have gx ∈ VF if and only if g−1 Fg ∈ CG (x). (ii) The map which sends the GF -orbit of gx ∈ VF to the F-conjugacy class of the image of g−1 Fg in CG (x)/CG (x) 0 is well-defined and bijective from the GF -orbits in O F , where O is the G-orbit of x, to H 1 (F, CG (x)) = H 1 (F, CG (x)/CG (x) 0 ). Proof The proposition translates 4.2.10 applied to the inclusion CG (x) ⊂ G, which gives 1 → CG (x) F → GF → VF → H 1 (F, CG (x)) → 1 since G is connected and since H 1 (F, CG (x)) = H 1 (F, CG (x)/CG (x) 0 ) by 4.2.13(ii). Again we will give a naive proof. (i) is an immediate computation. Let us show (ii). Let x ∈ VF , h, g ∈ G be such that hx, gx ∈ VF . Note that hx = gx if and only if h and g differ by an element of CG (x), and then h−1 Fh and g−1 Fg are F-conjugate in CG (x). We have thus a well-defined map from VF to the F-conjugacy classes of CG (x). On the other hand, if h ∈ GF , then gx and hgx have the same image g−1 Fg = (hg) −1 F (hg). Thus the map goes from the GF -orbits in VF to the F-conjugacy classes of CG (x). If g−1 Fg and h−1 Fh are F-conjugate by n ∈ CG (x), then gnh−1 ∈ GF and sends hx to gx. The map is thus injective. By the Lang–Steinberg theorem any element of CG (x) is of the form g−1 Fg with g ∈ G, which shows the surjectivity of the map. We finish the proof using 4.2.13(ii).  Corollary 4.2.15 Frobenius root.

Let G be a connected algebraic group over Fp and F be a

(i) F-stable Borel subgroups exist and form a single GF -conjugacy class. (ii) Let us say that two elements of GF are geometrically conjugate if they are conjugate under G, and define a geometric conjugacy class of GF as the

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intersection with GF of an F-stable G-conjugacy class. Then geometric conjugacy classes are non-empty, and if x is an element of such a class, the class splits into GF -conjugacy classes parametrised by H 1 (F, CG (x)/ CG (x) 0 ). (iii) In an F-stable Borel subgroup B of G, F-stable maximal tori exist and are BF -conjugate. Thus there exist F-stable pairs T ⊂ B of an F-stable maximal torus of G and an F-stable Borel subgroup of G containing it; any two such pairs are GF -conjugate. (iv) If G is reductive, any F-stable parabolic subgroup P has an F-stable Levi decomposition and two F-stable Levi subgroups of P are conjugate by an element of UF where U is the unipotent radical of P. Proof (i) comes from 4.2.14 applied with V the variety of Borel subgroups, using that for B a Borel subgroup NG (B) = B is connected. For (ii) we apply 4.2.14 with V the G-conjugacy class (and the action of G by conjugacy). For (iii) we first use the same argument as (i) using that for a torus T the group NB (T) = CB (T) (by 1.3.1(iii)) is connected by 1.3.3(iii) (if G is reductive it is equal to T by 2.3.1(iii)). Finally we use that all pairs T ⊂ B are G-conjugate and the normaliser of such a pair is CB (T) = T. For (iv) we use that by 3.4.2(ii) U acts transitively on the Levi decompositions, and the normaliser of a given Levi subgroup in the unipotent radical is trivial.  Example 4.2.16 As we saw in 3.5.4(i), all centralisers in GLn are connected, thus geometric conjugacy classes do not split. Example 4.2.17 Let G = PGL2 (Fq ) with q odd and let F be the “split” Frobenius   morphism which raises the matrix entries to the qth power. If s = 1 0 ∈ GL2 (Fq ) and s¯ is its image in G, then CG (¯s) is disconnected – see 0 −1   0 λ −1 q−1 = −1 the image m ¯ of m = 3.5.4(iii). For λ ∈ Fq2 with λ is in GF λ 0   1 1 F (since m = −m in GL2 ) and if x = then xsx−1 = m (mod Z(GL2 )) λ −λ   0 1 ¯ is and x−1 Fx = (mod Z(GL2 )) which is in CG (¯s) − CG (¯s) 0 thus m 1 0 geometrically conjugate but not GF -conjugate to s¯. Example 4.2.18 Two elements of SLn (Fq ) which are conjugate in GLn (Fq ) are conjugate in SLn (Fq ). Indeed, if x = yx y−1 , then we also have x = y x y −1

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with y = (det y) −1/n y ∈ SLn . But x, x ∈ SLFn which are conjugate under GLFn are not necessarily conjugate under SLFn , where F is the “split” Frobenius morphism which raises the matrix entries  to the qth power. Let us consider, 1 1 for instance the unipotent element u = of SLF2 . Its centraliser in GL2 0 1   × a b consists of all the matrices with a ∈ Fq and b ∈ Fq , so it is connected. 0 a But its intersection with SL2 is not connected if the characteristic is different from 2: it has two connected components corresponding respectively to a = 1 and to a = −1; so we have CG (u)/CG (u) 0  Z/2Z and the intersection of the geometric conjugacy class of u with SLF2 splits into two conjugacy classes. By 4.2.14(ii) and an easy computation, one sees that the class in SLF2 of the matrix   1 a with a ∈ F×q depends only on the image of a in F×q /(F×q ) 2 . 0 1 Definition 4.2.19 An F-stable maximal torus of the algebraic group G which is contained in an F-stable Borel subgroup of G is called a quasi-split torus. By 4.2.15(i) and (iii) quasi-split tori are GF -conjugate. Definition 4.2.20 A torus T defined over Fq is split if there is an isomorphism ∼ T− → Gnm defined over Fq . We will see in 7.1.3 that T is split if and only if the τ of 4.2.5 is the identity, and in 7.1.7 that a split torus is quasi-split, which is a kind of justification of the name “quasi-split”. Lemma 4.2.21 Let (X, Y, Φ, Φ∨ ) be the root datum of a connected reductive group G attached to a quasi-split torus T, and write F = qτ on X ⊗ R as in 4.2.5. Let Φ+ and Π be the positive roots and root basis defined by an F-stable Borel subgroup B containing T. Then τ permutes Φ+ and Π up to a positive scalar, and induces an automorphism of the Coxeter system (W, S) attached to the choice of B. Proof The p-morphism defined by F induces a permutation σ of the roots which differs from τ by positive scalars. By the definition of the system Φ+ attached to B – see 2.3.1(vi) – the permutation σ preserves Φ+ , thus τ does up to positive scalars. For any α ∈ Φ+ we have τ −1 (α) = b β for b ∈ R ≥0 and  β ∈ Φ+ , thus τ −1 (α) = γ ∈Π gγ γ with gγ ∈ R ≥0 . Applying τ to this equality, we find that every element of Φ+ is a positive linear combination of elements of τ(Π). From the minimality of Π, see 2.2.4, we deduce that τ preserves Π up to scalars. Since F, thus τ, permutes W it follows that τ permutes S. 

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Proposition 4.2.22 Let G be a connected reductive group over Fp , let F be a Frobenius root and let T be a quasi-split torus of G. Then the GF -conjugacy classes of F-stable maximal tori of G are parametrised by H 1 (F, WG (T)), by mapping an F-stable maximal torus gT where g ∈ G to the F-conjugacy class of w, the image in WG (T) of g−1 Fg ∈ NG (T). Proof We apply 4.2.14 with V the variety of maximal tori of G, on which G acts by conjugacy.  In the situation of 4.2.22 we call type of gT (with respect to T) the Fconjugacy class of w, or by abuse of notation w itself. Note that if gT is of type w, the pair ( gT, F) is sent by g−1 -conjugation to the pair (T, wF). Since quasisplit tori are GF -conjugate, and such a conjugation induces an F-equivariant isomorphism of Weyl groups, the set of types with respect to different quasisplit tori are in canonical bijection. Proposition 4.2.23 Let G be as in 4.2.9. Then every F-stable semi-simple element lies in some F-stable maximal torus of G. Proof If s ∈ G is semi-simple then s ∈ CG (s) 0 by 3.5.1(ii), and s being central in this group is in all its maximal tori, thus in particular in its F-stable maximal tori which are also maximal in G. 

4.3 Classification of Finite Groups of Lie Type Let G be a connected reductive group over Fp . Proposition 4.3.1 Let F be an isogeny, defined by the p-endomorphism f of the root datum (X(T), Y (T), Φ, Φ∨ ) of G; then F is the Frobenius endomorphism for an Fq -structure on G (where q is a power of p) if and only if F = qτ on X(T) where τ is an automorphism of finite order of (X(T), Φ). Proof We have seen that a Frobenius endomorphism is an isogeny of the above form in 4.2.3. Conversely, when τ = Id the fact that such an isogeny is a Frobenius endomorphism is a consequence, for instance of Springer (1998, Theorem 16.3.3). The case of an arbitrary τ follows then by 4.1.11(ii).  For a p-morphism as in 4.3.1 where τ = Id, we have σ = Id and qα = q for every α. The torus T is then split over Fq (by for example 7.1.3, since τ = Id). Definition 4.3.2 If F is as in 4.3.1 with τ = Id, that is F acts as q on X(T) for some maximal F-stable torus, we call F a split Frobenius endomorphism on G, or say that (G, F) is split.

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It follows from 4.3.1 that the converse of 4.2.5 holds, that is a Frobenius root is the same thing as an isogeny such that a power is a split Frobenius endomorphism. Example 4.3.3 An example of non-split Frobenius endomorphism is the Frobenius F which defines the unitary group GLFn , which is defined as F (x) = F( tx−1 ), where F is the split Frobenius endomorphism which maps the matrix entries to their qth power and tg denotes the transposed matrix of g. For the torus T of diagonal matrices, F acts by −q on X(T); the associated p-morphism has σ(α) = −α. The group GLFn is the group of unitary transformations of (Fq2 ) n . The group GLn with the Frobenius endomorphism F is called the unitary group, denoted by Un . This notation is somewhat inconsistent as it is not a different group but only a different rational form of GLn . In general if we want to classify isogenies which are Frobenius roots, note that qα and σ of the corresponding p-morphism determine completely the isogeny, thus the pair (G, F) and GF up to isomorphism, if and only if Φ spans X(T) ⊗ Q, that is if and only if G is semi-simple. We will classify now the finite groups of Lie type coming from semi-simple reductive groups, in particular those which correspond to simple finite groups. By 2.3.9 G is then the almost-direct product of quasi-simple groups; the endomorphism F permutes these quasi-simple components, and the orbits of F on the quasi-simple components give rise to F-stable normal subgroups of which G is the almost direct product. So we may assume there is only one orbit. We may assume n−1 n G quasi-simple, since if G = G1 × FG1 × · · · × F G1 with F (G1 ) = G1 , n we have GF  GF1 . The Coxeter diagram being then connected, we may assume up to multiplying F by an element of the Weyl group, that F preserves a Coxeter system corresponding to an F-stable Borel subgroup containing T – see 4.2.21. We first consider the case where F is a Frobenius. Then, on X(T) we have F = qτ; the σ and qα of the corresponding p-morphism are τ and q. The possibilities for τ correspond to the automorphisms of the Coxeter diagram which preserve the length of the roots. The possibilities for such a non-trivial automorphism τ on an irreducible root system are denoted by 2 An , (n ≥ 2), 2 D , 3 D and 2 E , where the exponent on the left is the order of τ. n 4 6 Remark 4.3.4 Note that by 2.4.8, any automorphism of the root datum of a reductive group gives rise to an isogeny which is an automorphism of the algebraic group, in particular the automorphisms defined above. We will use the same notation as for τ and say that the corresponding automorphism is of type 2An , 2Dn , 3D4 or 2E6 .

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Tits (1964) has shown, using the relative (B, N)-pair (see 4.4.8) that starting from a quasi-simple algebraic group G, to get a finite simple group we must take Der(GF ) divided by its centre, and apart from a few exceptions, Der(GF ) is generated by the unipotent elements of GF . According to Steinberg (1968, Theorem 12.4), if G is simply connected, then GF is generated by its unipotent elements. It follows that, apart from the exceptions, we get a simple finite group by taking GF /Z(GF ) where G is quasi-simple simply connected. We may compare this to (G/Z(G)) F using the following lemma. Lemma 4.3.5 We have 1 → Z(GF ) → GF → (G/Z(G)) F → Z(G)/(F − 1)Z(G) → 1. Proof This results from the Galois cohomology exact sequence 4.2.10 associated to the exact sequence 1 → Z(G) → G → G/Z(G) → 1, and from  Z(G) F = Z(GF ) (see 12.2.17) and H 1 (F, Z(G)) = Z(G)/(F − 1)Z(G). By 4.3.5, we see that GF /Z(GF )  (G/Z(G)) F if F − 1 is not surjective on Z(G). Classification 4.3.6 We now describe the groups corresponding to each case, and we list with each case the exceptions, which occur when the abelianised of GF is not formed of semi-simple elements, in which case the abelianised of GF /Z(GF ) is unipotent. • An−1 (n ≥ 2) – The simple algebraic group is G = PGLn  PSLn with split Frobenius endomorphism. The simple finite group is SLFn /Z(SLFn ); the centre Z(SLn ) identifies with the group μn (Fq ) of n-th roots of unity in Fq , on which F − 1 acts by raising to the power q − 1; thus SLFn /Z(SLFn )  PSLFn if q − 1 is not prime to n. The exceptions to simplicity are: – For q = 2, the group SLF2 /Z(SLF2 ) = PSLF2 = S3 . – For q = 3, the group SLF2 /Z(SLF2 ) = A4 . • 2An−1 (n ≥ 3) – The simple algebraic group is the special projective unitary group PSUn  PUn = Un /Z(Un ). A similar computation as in the split case applies to compute the simple finite group GF /Z(GF ) for G = SUn (the elements of Un of determinant 1), where F − 1 this time raises the elements of Z(G) to the power −q − 1. The exception to simplicity is: – For q = 2, the group SUF3 /Z(SUF3 )  (Z/3Z) 2  W (B2 ). • Cn (n ≥ 2) – The simple finite group is SpF2n /Z(Sp2n ) F where F is a split Frobenius endomorphism. It is equal to PSpF2n only in characteristic 2.

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73

The exception to simplicity is: – For q = 2, the group SpF4 /Z(SpF4 ) = PSpF4 = S6 . • Bn (n ≥ 2) – The simple finite group is SOF2n+1 /Z(SO2n+1 ) F where F is a split Frobenius endomorphism. It is equal to PSOF2n only in characteristic 2, in which characteristic Bn and Cn give isomorphic groups. • Dn (n ≥ 4) – In characteristic 2 the group SOF2n defined as (O02n ) F (F split) is simple. In odd characteristic, the simple algebraic group is PSO2n = SO2n /{±1}. To describe the finite simple group, we should start from the simply connected group Spin2n , but we will not describe it here. We will rather describe the simple finite group as Der(SOF2n )/Z(Der(SOF2n )) (F split) where Z(Der(SOF2n )) = {±1} ∩ SOF2n . The group Der(SOF2n ) is always of index 2 in SOF2n , but −1 is not always in Der(SOF2n ): it is in Der(SOF2n ) if either n is even or n is odd and q ≡ 1 (mod 4). • 2Dn – Similarly to the definition of Un , we will denote by O−2n the orthogonal group with a Frobenius endomorphism of the form εF composed of the split Frobenius endomorphism F on O2n with ad ε for some F-stable element ε ∈ O2n − SO2n . As two such endomorphisms differ by an element of SO2n , the Fq -structures they define are isomorphic. They are conjugate by an inverse image under the Lang map of the element by which the Frobenius endomorphisms differ. If q is odd we may take In−1  ε =  

0 1

1 0

  . 

In−1   This element normalises both the maximal torus and the Borel subgroup of respectively diagonal and upper triangular matrices. It induces the nontrivial automorphism of the root system of type Dn . Hence we get a group of type 2Dn . εF is the orthogonal group over F of a symmetric We show that O2n q bilinear form which is not equivalent to the form given by the matrix J of Example 1.5.5. Let h ∈ GL2n be such that h−1 . Fh = ε; then ad h maps εF into GLF . The the action of εF to that of F – in particular, it maps O2n 2n h element g preserves the form of matrix J if and only if g preserves the form of matrix J1 = th−1 Jh−1 . As J is F-stable and ε ∈ O2n , the matrix J1 is εF to the orthogonal group F-stable. Thus ad h is an isomorphism from O2n over Fq of J1 . Let us show that the bilinear forms given by J and J1 are not equivalent under GLF2n . If a ∈ GLF2n satisfies taJa = J1 , then ah is in the orthogonal group of J. But (ah) −1 . F (ah) = ε, so this would imply det(ε) = det(ah) q−1 =(±1) q−1 = 1, which is false, since det(ε) = −1, whence the result.

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εF The corresponding finite simple group is SO2n /{±1}. 3 • A similar analysis can be done for D4 , the triality group, and E6 , 2E6 , E7 , E8 , F4 and G2 . Using the root datum one sees that a semi-simple group of type E8 , F4 or G2 is both adjoint and simply connected. A semi-simple group of type E6 , 2E6 or E7 can only be either adjoint or simply connected. A semisimple root datum of type D4 has an automorphism of order 3 if and only if it is adjoint or simply connected (see Exercise 4.3.8), hence a semi-simple group of type 3D4 is either adjoint or simply connected. There is one exception to simplicity:

– For q = 2, the group GF where G is of type G2 whose derived subgroup, of index 2, is simple isomorphic to SUF3 for q = 3. In addition to the case where F is a Frobenius there are “exceptional” isogenies which on X(T) ⊗ R are of the form qτ where τ is an automorphism of finite order and q is a non-rational scalar. In each case τ induces an automorphism of the Coxeter diagram which does not extend to the root system because it does not preserve the length of the roots. Such automorphisms are of type 2B2 , 2F4 (resp. 2G2 ). To make √ τ map √ the root lattice into itself, we have to scale it by an odd power q of 2 (resp. 3). After scaling, we get a p-morphism defining an isogeny whose square is a Frobenius on the field Fq2 . Example 4.3.7 We assume G has a root system Φ of type B2 , with basis Π = {α = e1 − e2 , β = 2e2 }. If char k = 2 the formulae F(uα (x)) = u β (x2 ), F(uα+β (x)) = u2α+β (x2 ), F(u β (x)) = uα (x), F(u2α+β (x) ) = uα+β (x) define an isogeny. If t = diag(t1 , t2 , t2−1 , t1−1 ) ∈ T we have F(t) = diag(t1 t2 , t1 t2−1 , t1−1 t2 , t1−1 t2−1 ). The associated p-morphism is α → 2 β, β → α. We have F = qτ √ where q = 2 and τ 2 = 1. Then F 2 squares all coordinates, and is thus the split Frobenius over F2 . For any r the isogeny F 2r+1 has 22r+1 − 1 fixed points 2r+1 is the Suzuki group 2B2 (22r+1 ). on the torus; the group of fixed points GF We can construct similarly the Ree groups 2G2 (32r+1 ) and 2F4 (22r+1 ). In each case the simply connected algebraic group G has a trivial centre, thus the complications described above do not occur. The groups GF are simple apart from the exceptions: • 2B2 (2), which is the Frobenius group of order 20. • 2G2 (3), whose derived subgroup, of index 3, is SL2 (F8 ). • 2F4 (2) whose derived subgroup, of index 2, is Tits’ simple group. Adding to the above list the alternating groups, we have all the non-sporadic non-abelian finite simple groups.

4.4 The Relative (B, N)-Pair

75

Exercise 4.3.8 (i) Show that in a semi-simple root datum (X, Y, Φ, Φ∨ ), one has Φ ⊂ X ⊂ P where P := {x ∈ Φ ⊗ Q | α ∨ (x) ∈ Z for all α ∈ Φ}. (ii) With the notation of 2.2.15, show that if Φ is a root system of type D4 then {e1 , e1 + e2 , 12 (e1 + e2 + e3 − e4 ), 12 (e1 + e2 + e3 + e4 )} is a basis of P which is dual to the basis of simple coroots. (iii) Show that P/Φ  Z/2Z × Z/2Z. Deduce that a semi-simple root datum of type D4 can have an automorphism of order 3 only if X = P or X = Φ.

4.4 The Relative (B, N)-Pair Proposition 4.4.1 Let G be a connected reductive group over Fp and let F be a Frobenius root. (i) Let T be an F-stable maximal torus of G. Then W (T) F = NG (T) F /TF . Let T ⊂ B be an F-stable pair of a maximal torus and a Borel subgroup. Then:  (ii) GF = w∈W F BF wBF . (iii) If q ∈ R>0 is attached to F as in 4.2.4(ii), then  |GF | = ql(w0 ) |TF | ql(w) . w∈W F

(iv) Ru (B) F is a Sylow p-subgroup of GF . We will recognise (ii) as the Bruhat decomposition attached to a “relative” (B, N)-pair in 4.4.8 below. Proof (i) comes from 4.2.13(i). For (ii) we use the “unique Bruhat decomposition” 3.2.7 which implies that BwB is F-stable if and only if w ∈ W F and that an F-stable element of BwB is in BF nBF = BF nUFw where n ∈ NG (T) F is a representative of w.  Let us show (iii). By the proof of (ii) |GF /BF | = w∈W F |UFw |, and using |BF | = |TF ||UF | we get the stated formula if we show |UFw | = ql(w) , since U = Uw0 . If F is a Frobenius attached to an Fq -structure, this results from 4.1.12 and the fact that Uw is an affine space of dimension l(w). It is still true in general, as seen in Lemma 4.4.2. Lemma 4.4.2 Let F be a Frobenius root on G and B = U  T an F-stable Levi decomposition of a Borel subgroup of G. Then for any F-stable subgroup U1 ⊂ U normalised by T, we have |UF1 | = qdim U1 where q is associated to F as in 4.2.4(ii).

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 Proof We first show that |UF1 | = {α |Uα ⊂U1 } qα where the qα define the p-morphism corresponding to the isogeny F, see 2.4.8. By 2.3.1(v) we have   U1 = α ∈Ψ Uα and Der(U1 ) = α ∈Ψ Uα for some sets Ψ and Ψ of roots, in  particular these groups are connected. Thus we have U1 / Der(U1 )  α ∈Ψ−Ψ Uα and |UF1 | = |(U1 / Der(U1 )) F |.| Der(UF1 )|, since Der(U1 ) is connected. From 2.3.1(vii) we have Ψ  Ψ which allows us to argue by induction on Ψ. An ele ment of U1 / Der(U1 ) can be written u = α ∈Ψ−Ψ uα (xα ). Since F(uα (xα )) = qα uσ (α) (λ α xα ) for some non-zero constants λ α (see 2.4.8), the element u is q F-stable if and only if for each α ∈ Ψ − Ψ we have xσ (α) = λ α xαα . Solv ing these equations amounts, for each orbit o of σ in Ψ − Ψ , to solving an  q equation xα = λxαo for one chosen α ∈ o, where qo = β ∈o q β and λ is a product of the λ β to some powers. Such an equation has qo solutions,  whence |(U1 / Der(U1 )) F | = α ∈Ψ−Ψ qα . Since by induction | Der(U1 ) F | =  F {α ∈Φ+ \Ψ−Ψ } qα we get the formula for |U1 | claimed at the beginning of the proof. Let now o be a σ-orbit of roots in Ψ. To prove the lemma it suffices to show  that qo := α ∈o qα = q |o | . For α ∈ o we have F |o | (α) = qo α. Let m be such that F m is a split Frobenius; we have F m (α) = qm α. Hence F m|o | α = qm|o | α =  qm o α, whence the result.  We show (iv). As |GF /UF | = |TF |( w∈W F ql(w) ) is prime to p – since T is a  p -group and w∈W F ql(w) ≡ 1 (mod p) – we see that UF is a Sylow p-subgroup  of GF . Note that by an argument similar to the proof of the last statement of 3.4.1,  see Borel and Tits (1965, 5.19), we have NGF (UF ) = BF , thus w∈W F ql(w) is the number of Sylow p-subgroups of GF . The fact that the fixed points of the unipotent radical of a Borel subgroup form a Sylow p-subgroup extends to non-reductive (but connected) groups. Since Ru (G) is a p-group, it is in all unipotent radicals of Borel subgroups, and is connected, whence |GF | = |(G/Ru (G)) F ||Ru (G) F |. Corollary 4.4.3 Let G be a connected algebraic group over Fp and let F be a Frobenius root; any unipotent element of GF lies in an F-stable Borel subgroup. Proof Any F-stable unipotent element is in a Sylow p-subgroup by 1.1.5, so by 4.4.1(iv) (and the remarks above) is in the unipotent radical of an F-stable Borel subgroup.  Proposition 4.4.4 Let Γ be a group of automorphisms of the Coxeter system (W, S), that is of automorphisms of W preserving S. Let (S/Γ) 1. Notes General results about Fq -structures can be found for example in Deligne (1980). The groups GF of fixed points of an isogeny which has the property that GF is finite were first studied in Steinberg (1968); rationality properties over an arbitrary field are studied in Borel and Tits (1965).

5 Harish-Chandra Theory

A heuristic approach to building the irreducible representations of a finite group G is to find a family of subgroups of G of the same type as G which generate G, and then build representations of G by induction from these subgroups. A typical example is the construction of the irreducible representations of the n symmetric groups Sn as Z-linear combinations of IndS Sn1 ×···×Snk 1 with n1 + · · · + nk = n. In the case of a finite group of Lie type G = GF , a suitable family of subgroups consists of the groups LF where L runs over the F-stable Levi subgroups of F-stable parabolic subgroups of G. We could, as our introducF . But, tion suggests, build representations of GF from those of LF using IndG LF actually the representations thus obtained would not have the right properties; in particular, their decomposition into irreducible representations would be rather intractable. The right construction is to use the following Harish-Chandra induction, defined as a generalised induction associated to a bimodule.

5.1 Harish-Chandra Induction and Restriction Let A be a commutative ring with unit and let H and K be two finite groups. We will call “AH-module-AK” an A-module M endowed with a left H-action and a right K-action. We may also view it as an AH × AK opp -module where K opp denotes the opposed group to K. Definition 5.1.1

Given M, an AH-module-AK, we define two functors:

(i) RH K is a functor from AK-modules to AH-modules which maps the left AKmodule N to RH K (N) := M ⊗AK N = (M ⊗A N)K , which is the coinvariant space for the diagonal action of K given by m⊗n → mk−1 ⊗ kn. The action of H on M ⊗AK N comes from its action on M. A morphism f : N → N of AK-modules is mapped by RH K to the morphism IdM ⊗AK f : M⊗AK N → M ⊗AK N of AH-modules. 79

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(ii) ∗RH K is a functor from AH-modules to AK-modules which maps the AHmodule P to HomAH (M, P) = HomA (M, P) H , which is the invariant space for the action of H on HomA (M, P) given by ϕ → hϕ(h−1 ); here k ∈ K acts by ϕ → ϕ( k). A morphism f : P → P of AH-modules is mapped by ∗RH K to the morphism HomAH (M, P) → HomAH (M, P ) of AK-modules given by composition with f . Remark 5.1.2 If M is projective over AH then HomAH (M, ) is isomorphic to M ∨ ⊗AH  where ∨ means dual over AH – see Bourbaki (1971, II §4 no 2, H ∨ Corollaire); that is, ∗RH K is the functor RK associated with M , an AKmodule-AH. ∗ H Proposition 5.1.3 RH K is left-adjoint to RK .

Proof This is the classical “tensor-hom” adjunction, see Bourbaki (1971, II §4 no 1 Proposition 1 (a)).  Proposition 5.1.4 (Transitivity) Let G, H, and K be finite groups; let M be an AG-module-AH and N an AH-module-AK, then: G H (i) The functor RG H ◦ RK is equal to the functor RK defined by the AG-moduleAK given by M ⊗AH N. ∗ G ∗ G (ii) The functor ∗RH K ◦ RH is equal to the functor RK defined by the AGmodule-AK given by M ⊗AH N.

Proof The first item comes from the associativity of the tensor product. We get the second item from the first by adjunction.  Proposition 5.1.5 Assume |K| invertible in A. Let M be an AH-module-AK and N be an AK-module; then for h ∈ H we have  −1 N) = |K| Trace((h, k−1 ) | M) Trace(k | N). Trace(h | RH K k∈K

 |K| −1 k k−1

Proof The element eK := ⊗ k ∈ AK ⊗A AK is the projector on the K-invariants in M ⊗A N which are isomorphic to the K-coinvariants since the kernel of eK is the image of 1 − eK . Hence for h ∈ H we have Trace(h | M ⊗AK N) = Trace(he | M ⊗A N)  Trace((h, k−1 ) ⊗ k | M ⊗A N) = |K| −1 k∈K

= |K|

−1



k∈K

Trace((h, k−1 ) | M) Trace(k | N).



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81

Examples 5.1.6 (induction and restriction) We take for K a subgroup of H and for M the group algebra AH on which H acts by left multiplication and K by right multiplication. The above construction gives the induction functor AH ⊗AK  whose right-adjoint is HomAH (AH, ). Since for any AH-module P we have an isomorphism HomAH (AH, P)  P – where the action of K on the lefthand side becomes its action by restriction on P, we get that the right-adjoint of the induction is the restriction. We can also consider AH as an AK-module-AH where H acts by right multiplication and K by left multiplication. This defines a functor AH ⊗AH  whose right-adjoint is HomAK (AH, ). Since AH ⊗AH P  P, where the action of K becomes the restriction of the action of H, we see that AH ⊗AH  is the restriction functor. We leave as an exercise to show, using the fact that AH is free over AK, that HomAK (AH, ) is isomorphic to AH ⊗AK  and deduce that restriction and induction are biadjoint. The next example will play an important part in this book. It was first considered by Harish-Chandra (1970). Definition 5.1.7 Let G be a connected reductive group over Fp and F be a Frobenius root; let P be an F-stable parabolic subgroup of G and L an F-stable Levi subgroup of P, so that we have an F-stable Levi decomposition P = U  L. – is the functor ob(i) the Harish-Chandra induction – denoted by RG L⊂P tained by taking M in 5.1.1(i) to be the AGF -module-ALF given by A(GF /UF ) where GF acts by left translations and LF by right translations. This is possible since L normalises U. – is the fun(ii) the Harish-Chandra restriction – would be denoted by ∗ RG L⊂P F ctor obtained by taking M in 5.1.1(i) to be the AL -module-AGF given by A(UF \GF ) where GF acts by right translations and LF by left translations. The right-adjoint of RG is HomAGF (A(GF /UF ), ). The right-adjoint of L⊂P F F is HomALF (A(U \G ), ).

∗RG L⊂P

Proposition 5.1.8

Assume that the prime p is invertible in A. F

F

F

(i) RG = IndAG ◦ Inf AP (where the inflation Inf AP is the natural lifting L⊂P ALF ALF APF through the quotient PF → LF ). F (ii) ∗RG is ResAG followed with the taking of fixed points under UF . L⊂P APF G ∗ G (iii) RL⊂P and RL⊂P are exact and biadjoint. Proof We prove first (ii). Since p is invertible in A we have an A-module  isomorphism A(UF \GF )  eUF (AGF ) where eUF = |UF | −1 u∈UF u. For an F (X) = A(UF \GF ) ⊗AGF X  eUF X = X U as AGF -module X this gives ∗RG L⊂P ALF -modules whence (ii).

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Now for any AGF -module X we have HomAGF (A(GF /UF ), X)  X U as  HomAGF (A(GF /UF ), ). Hence ∗RG is rightALF -modules, hence ∗RG L⊂P L⊂P . adjoint to RG L⊂P is also left-adjoint to RG . Since A(GF /UF )  We now prove that ∗RG L⊂P L⊂P F F F AG eUF which is a direct factor of AG and AG is free over ALF we get = that A(GF /UF ) is a finitely-generated projective ALF -module. Thus RG L⊂P A(GF /UF ) ⊗ALF   HomALF (A(GF /UF ) ∨ , ) – see Bourbaki (1971, II §4 no 2, Corollaire) – where ∨ means dual over ALF , and this isomorphism is compatible with the GF -action. If we show that A(GF /UF ) ∨ is isomorphic to A(UF \GF ) as right AGF -modules, we will get the adjunction we want. We leave as an ex⎧ ⎪ 0 if gg  PF ercise to show that (UF g, g UF ) → ⎨ ⎪ l if gg ∈ lUF with l ∈ LF defines a ⎩ perfect ALF -duality compatible with the GF -actions. and ∗RG are exact since they are defined by tensor product Both RG L⊂P L⊂P with projective modules. Using adjunction and the fact that taking the fixed points under UF is adjoint  to lifting representations from ALF to APF we get (i). Remark 5.1.9 5.1.8(iii) shows that when p is invertible in A, we could have and ∗RG . used 5.1.1(ii) instead of 5.1.1(i) to define RG L L Remark 5.1.10 We have only defined here a functor RG when L is an L⊂P F-stable Levi subgroup of an F-stable parabolic subgroup P of G; later in 9.1.3 when A = Q we will define Deligne–Lusztig induction which generalises Harish-Chandra induction to the case of F-stable Levi subgroups of nonF-stable parabolic subgroups P. Proposition 5.1.11 (transitivity) Let P be an F-stable parabolic subgroup of G, and Q an F-stable parabolic subgroup contained in P. Let L be an F-stable Levi subgroup of P and M an F-stable Levi subgroup of Q contained in L. Then L G ∗ L ∗ G ∗ G RG L⊂P ◦ RM⊂L∩Q = RM⊂Q and RM⊂L∩Q ◦ RL⊂P = RM⊂Q .

Proof We prove the statement for Harish-Chandra induction; the statement for restriction follows by adjunction. Let U be the unipotent radical of P and V that of Q. We have U ⊂ V, so according to 3.4.8(ii) we have V = U(L ∩ V) and the uniqueness in that decomposition implies that VF = UF (LF ∩ VF ). Using 5.1.4, proving the proposition is equivalent to showing that ∼

A(GF /UF ) ⊗ALF A(LF /(L ∩ V) F ) − → A(GF /VF )

5.2 The Mackey Formula

83

as AGF -modules-AMF . But this clearly results from the isomorphism of “GF -sets-MF ” (with notation of 2.3.11) ∼

→ GF /VF GF /UF ×LF LF /(L ∩ V) F − which is induced by the map (gUF , l(L ∩ V) F ) → glVF from GF /UF × LF /(L ∩ V) F to GF /VF . This map is well defined since l normalises U and U ⊂ V. Using the decomposition VF = UF (L ∩ V) F , it is easy to see that it factors through the amalgamated product and defines an isomorphism.  We have the following character formula: Proposition 5.1.12 Assume the prime p invertible in A. If X is an AGF -module, for l ∈ LF we have  F −1 Trace(lu | X). Trace(l | ∗RG L⊂P (X)(l)) = |U | u∈UF

as the action of LF Proof This formula comes from the description of ∗RG L⊂P  F U F −1 on X using that eUF = |U | u∈UF u is the projector on that fixed point subspace. 

5.2 The Mackey Formula A fundamental property of Harish-Chandra induction and restriction is given in Theorem 5.2.1 below. This is an analogue of the usual Mackey formula for the composition of restriction and induction. A corollary will be that in characteristic 0 Harish-Chandra induction and restriction are independent of the parabolic subgroup used in their definition. The statement of the Mackey formula below remains valid for the Lusztig induction and restriction functors (dropping the assumption that the parabolic subgroups involved are F-stable) in the cases that we will be able to handle (see Chapter 9). This statement is due to Deligne but was first published in Lusztig and Spaltenstein (1985, 2.5). Theorem 5.2.1 (Mackey formula) Let P and Q be two F-stable parabolic subgroups of G, and L (resp. M) be an F-stable Levi subgroup of P (resp. Q). Then  xM ∗ G RL⊂P ◦ RG = RLL∩ xM⊂L∩ xQ ◦ ∗RL∩ xM⊂P∩ xM ◦ ad x, M⊂Q x

where ad x is the conjugation from AMF -modules to A xMF -modules, and x runs over a set of representatives of LF \S(L, M) F /MF , with S(L, M) = {x ∈ G | L ∩ xM contains a maximal torus of G}.

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In order to prove this theorem, we first study double coset representatives with respect to parabolic subgroups of G, using Proposition 3.2.3. Lemma 5.2.2 If S(L, M) is as in 5.2.1, then: (i) The natural map S(L, M) → P\G/Q induces an isomorphism ∼

L\S(L, M)/M − → P\G/Q. (ii) PF \GF /QF can be identified with (P\G/Q) F , and LF \S(L, M) F /MF with (L\S(L, M)/M) F . Proof

Let us first prove (ii): by 4.2.14, since P × Q, as well as the stabiliser (p,q)

isomorphic to P ∩ xQ of a point x ∈ G under the action x −−−−→ pxq of P × Q on G, are connected, the double cosets PF \GF /QF can be identified with (P\G/Q) F . Similarly, as L ∩ xM is connected when it contains a maximal torus (see 3.4.8(ii)), the double cosets LF \S(L, M) F /MF can be identified with (L\S(L, M)/M) F . Let us prove (i). We may conjugate Q by an element g ∈ G so that P and gQ contain a common Borel subgroup and that L and gM contain a common maximal torus T. Then for any x ∈ G we have PxQ = (Pxg−1 gQ)g; that is, the double cosets with respect to P and Q are translates of those with respect to P and gQ. Similarly S(L, M) = S(L, gM)g. Thus to show (i) we may replace Q by gQ; that is, assume that P and Q are standard and that L and M are standard Levi subgroups, that is, P = PI , Q = PJ , L = LI , and M = LJ . By Proposition 3.2.3(ii), there exist representatives in NG (T) of the double cosets in PI \G/PJ . Thus every double coset has a representative in S(L, M) and the map of (i) is surjective. Conversely, if x ∈ S(L, M), the group L ∩ xM contains a maximal torus. This torus is of the form lT for some l ∈ L and also of the form xmT for some m ∈ M, since all maximal tori in L or M are conjugate. But then l−1 xm ∈ NG (T) and is in the double coset LxM. Now let x and y be two elements of S(L, M) such that PxQ = PyQ. We just saw that there are elements of NG (T) in LxM and LyM. By Proposition 3.2.3(ii), these elements of NG (T) are in the same double coset with respect to WI and WJ , thus LxM = LyM.  Proof of Theorem 5.2.1 Let U (resp. V) denote the unipotent radical of P (resp. Q). By 5.1.4, the functor in the left-hand side of the Mackey formula corresponds to the ALF -module-AMF given by A(UF \GF ) ⊗AGF A(GF /VF ).

5.2 The Mackey Formula

85

This module is clearly isomorphic to A(UF \GF /VF ). Let us decompose GF into double cosets with respect to PF and QF . We get  UF \GF /VF = UF \PF xQF /VF . x∈LF \S(L,M) F /MF

We now use Lemma 5.2.3 to continue. Lemma 5.2.3

For any x ∈ S(L, M) F the map (l(L ∩ xV) F , ( xM ∩ U) F . xm) → UF lxmVF

LF /(L ∩ xV) F ×(L∩ xM)F ( xM ∩ U) F \ xMF → UF \PF xQF /VF , where the notation is as in 2.3.11, is an isomorphism. Proof It is easy to check that the map is well defined. The stabiliser – (u,v)

isomorphic to U ∩ xV – of a point x ∈ G under the action x −−−→ uxv of U × V is connected (see 2.3.1(v)). The stabiliser of a point of LF /(L ∩ xV) F × ( xM ∩ U) F \ xMF under the diagonal action of L ∩ xM is reduced to 1 and thus connected. Hence it is enough to show that the same map at the level of the algebraic varieties is an isomorphism, as it clearly commutes with F. The map is surjective since any element of PxQ has a decomposition of the form ulxmv (with each term in the group suggested by its name). We now prove its injectivity. If UlxmV = Ul xm V, then lxm = l uxvm for some u ∈ U and v ∈ V since l normalises U and m normalises V. Thus u−1 l −1 l = x (vm m−1 ) ∈ P ∩ xQ.

(5.2.1)

The component in L ∩ xM of the left-hand side of (5.2.1) is that of l −1 l, and the component in L ∩ xM of the right-hand side of (5.2.1) is that of x (m m−1 ) by the following lemma. Lemma 5.2.4 Let P = U  L and Q = V  M be respective Levi decompositions of two parabolic subgroups P and Q, and assume L ∩ M contains a maximal torus. If ul = vm ∈ P ∩ Q, then uM = mU , lV = vL and lM = mL , with obvious notation: u ∈ U, l ∈ L, v ∈ V, m ∈ M, uM is the component in U ∩ M of u ∈ U in the decomposition P ∩ Q = (U ∩ V)(U ∩ M)(L ∩ V)(L ∩ M) of 3.4.8(ii), . . . . Proof We have uV uM lV lM = vU vL mU mL = vU [vL , mU ]mU vL mL and the commutator which appears in this formula is in V ∩ U. The uniqueness of the decomposition in 3.4.8(ii) gives the result. 

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Thus (l(L ∩ xV), ( xM ∩ U) xm) and (l (L ∩ xV), ( xM ∩ U) xm ) are equal in the amalgamated product, whence the injectivity and proof of Lemma 5.2.3.  Taking then the union over all x, we get  ∼ UF \GF /VF − → LF /(L ∩ xV) F ×(L∩ xM)F ( xM ∩ U) F \ xMF , x∈LF \S(L,M) F /MF

and this bijection is compatible with the action of LF by left multiplication, and with that of MF by right multiplication on the left-hand side and by the composite of left multiplication and ad x on the x term of the right-hand side. We eventually get an isomorphism of ALF -modules-AMF : A(UF \GF /VF ) ∼

− →



A(LF /(L ∩ xV) F ) ⊗A((L∩ xM)F ) A(( xM ∩ U) F \ xMF ),

x∈LF \S(L,M) F /MF

where the action of MF on the right-hand side is as explained above. The ALF module-AMF which appears in the right-hand side is exactly the one which corresponds to the functor in the right-hand side of the Mackey formula, whence the theorem. 

5.3 Harish-Chandra Theory We explain now the concept of “cuspidal representations” which is due to Harish-Chandra. From now on A denotes a commutative ring in which p is invertible. In many proofs and statements we assume that A is a subfield K of C. In that case, for a finite group G, we denote by CK (G) (or C(G) when K is clear from the context) the space of K-valued class functions on G, which is endowed with the scalar  product f , f G = |G| −1 g∈G f (g)f (g). We will use 1G and regG respectively for the trivial and the regular characters of G (or just 1 and reg when G is clear from the context), and we will denote by Irr(G) the set of irreducible characters of G over K. and ∗RG are independent of the paraTheorem 5.3.1 The functors RG L⊂P L⊂P bolic subgroup used in their definition. Thus, from now on we will use for them and ∗ RG . the notations RG L L Proof The general result when p is invertible in A is due to Dipper and Du (1993) and Howlett and Lehrer (1994). We prove it only when A = K, thus we need only to prove the result for characters.

5.3 Harish-Chandra Theory

87

We will prove 5.3.1 for Harish-Chandra induction. The result for the restriction follows by adjunction. We proceed by induction on the semi-simple rank of G, which is by definition the rank of G/RG. Let us note that the semi-simple rank of a Levi subgroup L of a proper parabolic subgroup of G is strictly less than that of G since by 3.4.6 the radical Z(L) 0 of L contains strictly that of G. If L = G there is nothing to prove, so we may assume that the semi-simple rank of L is strictly less than that of G. By the Mackey formula applied to only one Levi subgroup L but with two different parabolic subgroups P and Q, we get for any character π of LF , using adjunction  xL x ∗ L G π, R π =  ∗RL∩ RG F xL⊂P∩ xL π, RL∩ xL⊂L∩ xQ πLF ∩ xLF . L⊂P L⊂Q G x∈LF \S(L,L) F /LF

By induction, the right-hand side does not depend on the parabolic subgroups P and Q, so the same is true for the left-hand side which means that, if we put π, we have f (P) = RG L⊂P f (P), f (P)GF = f (P), f (Q)GF = f (Q), f (Q)GF = f (Q), f (P)GF whence f (P) − f (Q), f (P) − f (Q)GF = 0 and so f (P) = f (Q).



Example 5.3.2 We construct a Hopf algebra whose commutativity relies on the fact that RG L⊂P is independent of P. Let CGL = n≥0 C(GLn (Fq )) where here C denotes the C-valued class functions. One can define on CGL a product by putting, for two class functions χ ∈ C(GLn (Fq )) and ψ ∈ C(GLm (Fq )), n+m χ ◦ ψ = RGL χ ⊗ ψ, GLn ×GLm

where GLn ×GLm is embedded in GLn+m as a group of block-diagonal matrices   0 GLn , 0 GLm using the parabolic subgroup Pn,m of upper block-triangular matrices with the same diagonal blocks (see Example 3.2.5). We thus get an algebra structure. Since the Levi subgroups GLn × GLm and GLm × GLn are conjugate in GLn+m (by 5.3.1) the product is commutative. Note that the parabolic subgroups Pn,m and Pm,n are not conjugate when n  m, thus the proof of commutativity relies in an essential way on 5.3.1. One can also define on CGL a cocommutative ∗ GLn coproduct CGL → CGL ⊗ CGL : χ ∈ C(GLn (Fq )) → i+j=n RGLi ×GLj χ, and the Mackey formula implies that these two laws define on CGL a Hopf algebra structure.

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The book Zhelevinski (1981) explains the theory of representations of general linear groups from the above viewpoint; this construction is originally due to Green (1955). (see 5.1.8 and 5.1.11), one Because of the transitivity and exactness of RG L can define a partial order on the set of pairs (L, M) consisting of an F-stable Levi subgroup of an F-stable parabolic subgroup of G, and of M a simple ALF -module, by putting (L , M ) ≤ (L, M) if L ⊂ L and there exists a surjective morphism RLL M  M. Proposition 5.3.3 The following properties are equivalent: (i) The pair (L, M) is minimal for the partial order defined above. (ii) For any F-stable Levi subgroup L of an F-stable parabolic subgroup of L, we have ∗RLL M = 0. If these conditions are fulfilled, the ALF -module M is said to be cuspidal and the pair (L, M) is said to be a cuspidal pair. Proof ∗RLL M  0 if and only if there exists an AL F -simple module M such that HomAL F (M , ∗RLL (M))  0. By adjunction this is equivalent to HomALF (RLL (M ), M)  0 which is equivalent to (L, M) not being minimal since, as M is simple, any non-zero morphism in this last set is surjective.  Notation 5.3.4 For a finite dimensional A-algebra H, we will denote by Simp(H) the set of simple H-modules up to isomorphism. If G is a finite group we will write Simp(G) for Simp(AG) if the base ring is clear from the context. We will denote by Irr(H) the set of irreducible characters of H, that is the characters of simple H-modules. Definition 5.3.5 The Harish-Chandra series associated to a cuspidal pair (L, Λ), denoted by Simp(GF | (L, Λ)) – or Simp(AGF | (L, Λ)) is the set of Λ. simple quotients of RG L Lemma 5.3.6 A simple AGF -module M is in the Harish-Chandra series (M). Simp(GF | (L, Λ)) if and only if Λ is a submodule of ∗RG L Proof Since RG and ∗RG are adjoint we have HomAGF (RG Λ, M)  HomALF L L L (Λ, ∗RG M), whence the result: since Λ (resp. M) is simple, any non-zero morL ∗ G G phism from Λ to RL M (resp. from RL Λ to M) is injective (resp. surjective).  Theorem 5.3.7 Assume that A is a field. Two Harish-Chandra series Simp(GF | (L, Λ)) and Simp(GF | (L , Λ )) have a non-empty intersection if and only if the cuspidal pairs (L, Λ) and (L , Λ ) are GF -conjugate, in which case the two series are equal.

5.3 Harish-Chandra Theory

89

Proof The last assertion comes the fact that for g ∈ GF we have RG Λ  L gΛ. Now if M is a quotient of RG Λ and if P and P are the = RG Λ M gL L respective projective covers of Λ and M (which exist since A is a field), we P  M using the exactness of RG . Hence PM must be a have a surjection RG L Λ L G Λ for some cuspidal pair P . If M is a quotient of R direct summand of RG Λ L L Λ )  0, whence HomAGF (RG P , RG Λ )  0. (L , Λ ), then HomAGF (PM , RG L L Λ L Using the Mackey formula and adjunction, we get  F x F HomLF ∩ xL F ( ∗RLLF ∩ xL F PΛ , ∗RLLF ∩ xL F xΛ )  0.

gRG Λ L

x

Λ

Since is cuspidal, the only non-zero terms in the above sum are for x such that xL ⊂ L. Thus there is an x ∈ GF such that xL ⊂ L. Exchanging the roles of L and L we see that there is a y ∈ GF such that L ⊂ yL . This implies that L and L have same dimension and we must have equalities. Hence there is an x ∈ GF such that L = xL and HomL (PΛ , xΛ )  0. Since Λ is simple there is a non-zero morphism from PΛ to xΛ if and only if Λ  xΛ , whence the result.  Corollary 5.3.8 If (L, Λ) and (L , Λ ) are cuspidal pairs, and A is a field large enough so that EndALF (Λ) = A, then G F x x F dim(HomAGF (RG L PΛ , RL Λ )) = |{x ∈ G | L = L and Λ  Λ }/L | G = dim(HomAGF (RG L Λ, RL Λ )),

where PΛ is the projective cover of Λ. Proof The proof of 5.3.7 gives the first equality. Now by the Mackey formula,  using the cuspidality of Λ and Λ we have HomAGF (RG Λ, RG Λ ) = x L L HomLF (Λ, xΛ ), where x runs over LF -representatives of {x ∈ GF | xL = L}, whence the second equality.  As a consequence of Theorem 5.3.7 we have a partition of Simp(GF ) into and ∗RG preserve the Harish-Chandra seHarish-Chandra series. Moreover RG L L ries in the following sense: Proposition 5.3.9 Let (L0 , Λ0 ) be a cuspidal pair with L0 a Levi subgroup (M) for M ∈ of a Levi subgroup L; then any simple quotient of RG L Simp(LF | (L0 , Λ0 )) is in Simp(GF | (L0 , Λ0 )). Similarly any simple submod(M) for M ∈ Simp(GF | (L0 , Λ0 )) is in Simp(LF | (L0 , Λ0 )). ule of ∗RG L Proof Both assertions are easy consequences of the transitivity and exactness of RG and ∗RG .  L L

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Remark 5.3.10 expressed as

When A = C – since PΛ = Λ – Corollary 5.3.8 can be

G F x x F RG L λ, RL λ GF = |{x ∈ G | L = L and λ = λ }/L |

where λ ∈ Irr(LF ) and λ ∈ Irr(L F ) are the characters of Λ and Λ . Remark 5.3.11 The union of the Harish-Chandra series Simp(GF | (L, Λ)) when Λ runs over the cuspidal representations of LF is called the Harish-Chandra series associated to L. When L is a quasi-split torus then all irreducible (thus degree one) representations of LF are cuspidal. The series associated to any such torus is called the principal series (all quasi-split tori are GF -conjugate). The set of cuspidal representations of GF is also the series associated to G, which is called the discrete series. So the first problem of Harish-Chandra theory is to study the discrete series, not only in GF but in any F-stable Levi subgroup of an F-stable parabolic subgroup of G. Notes The approach to generalised induction functors using bimodules was initiated by Broué around 1980. One of the first references to Harish-Chandra theory is Springer’s article “Cusp forms for finite groups” in Borel et al. (1970). In this chapter, as well as in the following one, we rely for the modular setting on lecture notes by Olivier Dudas, for which we thank him.

6 Iwahori–Hecke Algebras

In this chapter G is a connected reductive algebraic group over Fp , with a Frobenius root F; we consider a Harish-Chandra series Simp(GF | (L, Λ)) as in Chapter 5 and we assume that A is a field K of characteristic different from p which is large enough so that EndKLF (Λ) = K. We make no further assumption on K until 6.1.12.

6.1 Endomorphism Algebras We put NGF (L, Λ):= {n ∈ NGF (L) | nΛ  Λ} and WGF (L, Λ):= NGF (L, Λ)/LF . Λ) = |WGF (L, Λ)|. In this We have seen in 5.3.8 that we have dim EndKGF (RG L chapter we study the structure and representations of this algebra, which by the following theorem will lead to a parametrisation of Simp(GF | (L, Λ)). Theorem 6.1.1 Λ, M) induces a bijection from (i) The functor Θ : M → HomKGF (RG L F Λ)-modules up to isomorphism. Simp(G |(L, Λ)) to the simple EndKGF (RG L (ii) RG Λ is a semi-simple module if and only if EndKGF (RG Λ) is semi-simple. L L G F Λ)-module as In this case RL Λ decomposes as a KG × EndKGF (RG L Λ of a ⊕M ∈Simp(GF |(L,Λ)) M ⊗K Θ(M). In particular the multiplicity in RG L simple KGF -module M is equal to dim Θ(M) and the multiplicity of Θ(M) is equal to dim M. Proof We sketch the proof of (i), following Geck and Jacon (2011, 4.2.9). Lemma 6.1.2

RG Λ is a projective KGF /I-module where I is its annihilator. L

Proof Let PΛ be a projective cover of Λ; then P := RG P is projective and L Λ G G we have a surjection π : P  RL Λ, since RL is exact. We thus get an injective Λ, RG Λ) → HomKGF (P, RG Λ). Since these two spaces have map HomKGF (RG L L L 91

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Iwahori–Hecke Algebras

same dimension by 5.3.8, this last map is an isomorphism; any morphism ϕ : P → RG Λ factorises through π, hence Ker π ⊂ Ker ϕ. L Now we have IP ⊂ Ker π; we claim that Ker π = IP. Since P is finitely genΛ)  P∨ ⊗KGF RG Λ, see Bourbaki erated, projective, we have HomKGF (P, RG L L ∨ (1971, II §4 no 2, Corollaire), where P is the dual of P over KGF . This means Λ) is spanned by morphisms of the form ϕf,x : p → f (p)x with that Hom(P, RG L Λ. By the first part of the proof Ker π ⊂ ∩x,f Ker ϕf,x and f ∈ P∨ and x ∈ RG L this last intersection consists of the elements p ∈ P such that f (p) ∈ I for all f ∈ P∨ , that is it consists of IP, whence our claim. Λ is isomorphic to P/IP, hence is a projective We have proved that RG L  KGF /I-module. F Using that inflation is an equivalence of categories from KG /I-modules to KGF -modules annihilated by I we get that for such a module M we have Λ, M). Conversely any M in the head of RG Λ is annihiΘ(M) = HomKGF /I (RG L L lated by I. Hence to get (i) we are reduced to KGF /I-modules and properties of Λ, M). Property (i) is then proved the induced functor Θ : M → HomKGF /I (RG L G for Θ by noticing that, since RL Λ is projective, applying Θ to any surjective map RG Λ  M gives a surjective map EndKGF /I (RG Λ) = Θ(RG Λ)  Θ(M) L L L G which implies that Θ(M) is a simple EndKGF (RL Λ)-module and then it remains Λ)-simple modules are obtained in this way. For to prove that all EndKGF (RG L the details see Geck and Jacon (2011, 4.1.3). For (ii): the endomorphism algebra of a semi-simple module is semi-simple, see Bourbaki (1981, §5 no 4 Proposition 5 (a)). For the converse see Geck and Λ is semi-simple, it decomposes as in (ii), see Jacon (2011, 4.2.10). When RG L for example Bourbaki (1981, §4, no 7, Proposition 8).  Exercise 6.1.3 (i) Let G be a finite group and P be a subgroup. Show that if a KP-module Λ is given by KPe with e ∈ KP a primitive idempotent, then IndG P Λ = KGe and for any KG-module M one has HomKG (KGe, M)  eM. Deduce that H := EndKG (IndG P Λ)  eKGe, acting on eM by the restriction to H of the action of KG. Show that dimK (HomKG (KGe, M)) = Trace(e | M). (ii) Deduce that if KG is semi-simple, the bijection on characters induced by the functor Θ of 6.1.1 is given by the restriction from KG to H. Λ) and determine its algebra We will now produce a basis of EndKGF (RG L structure. Lemma 6.1.4 For n ∈ NGF (L, Λ) there exists a bijective linear map γn : Λ → Λ such that for l ∈ LF acting on Λ we have γn ◦ l = nl ◦ γn .

6.1 Endomorphism Algebras

93

Proof This results from the fact that the KLF -modules Λ and nΛ are isomorphic.  Since EndKLF (Λ) = K, for n, n ∈ NGF (L, Λ) there exist scalars λ(n, n ) ∈ K such that γnn = λ(n, n )γn γn , and λ is a 2-cocycle on NGF (L, Λ) × NGF (L, Λ) since it satisfies λ(n, n )λ(n, n n ) −1 λ(nn , n )λ(n , n ) −1 = 1, as can be seen by factorising nn n in two different ways. Lemma 6.1.5

We have dim(EndKGF (RG Λ)) = |WGF (L, Λ)| and L  HomKLF (Λ, nw Λ), EndKGF (RG L Λ)  w∈WGF (L,Λ)

where nw ∈ NGF (L, Λ) represents w. Proof This results from 5.3.8 and its proof.



Note that the vector space HomKLF (Λ, nw Λ) is equal to Kγnw . Our aim is to describe the algebra structure on the right-hand side of the isomorphism of 6.1.5. = K(GF /UF ) ⊗KLF  = (KGF )eUF ⊗KLF , where eUF is as in Recall that RG L the proof of 5.1.8. For nw ∈ NGF (L, Λ) we define a linear map G Bnw : RG L Λ → RL Λ

geUF ⊗KLF x → geUF n−1 w eUF ⊗KLF γnw (x). Using the equality in 6.1.4 and the fact that LF centralises eUF we get that Bnw is well defined. It clearly commutes with the action of GF . Lemma 6.1.6 By the isomorphism of 6.1.5, Bnw is mapped to γnw . In particuΛ). lar Bnw is invertible and {Bnw | w ∈ WGF (L, Λ)} is a basis of EndKGF (RG L Proof We have to follow Bnw through the isomorphism coming from the Λ, RG Λ)  HomKLF (Λ, ∗RG RG Λ) and then through the adjunction HomKGF (RG L L L L isomorphism coming from the Mackey formula. The adjunction isomorphism is the tensor-hom adjunction F G HomKGF ((KGF )eUF ⊗KLF Λ, RG L Λ)  HomKLF (Λ, HomKGF ((KG )eUF , RL Λ)),

under which Bnw corresponds to the map Λ → HomKGF ((KGF )eUF , RG L Λ) x → (geUF → geUF n−1 w eUF ⊗KLF γnw (x)); see Bourbaki (1971, II §4 no 1 Proposition 1 (a)). Under the identification HomKGF ((KGF )eUF , RG Λ)  eUF RG Λ as in the proof of 5.1.8, the above map L L −1 −1 becomes x → eUF nw eUF ⊗KLF γnw (x), whence the result since eUF nw eUF ⊗nKLF ∗ G G w γnw (x) is γnw (x) in the summand indexed by w in RL RL Λ = w∈WGF (L,Λ) Λ. 

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Λ) we will make a specific To describe the algebra structure of EndKGF (RG L choice of representatives of WGF (L, Λ). First we choose a quasi-split torus T of G contained in L and let W = NG (T)/T. Up to GF -conjugacy L is a standard Levi subgroup LI for some F-stable I ⊂ S. Lemma 6.1.7 For I ⊂ S we have NW (WI ) = WI  NI where NI = {w ∈ W | wI = I and w is reduced-I} is a subgroup of W. If I is F-stable, this decomposition is compatible with F so that NW (WI ) F = WIF  NIF . Proof First note that the elements of NI can be characterised equivalently by 3.2.2 as being I-reduced, since w ∈ NI has minimal length in its double coset WI wWI = wWI . We next show that NI is a group. By 3.2.1(iv) an element w ∈ NI is characterised by wI = I and N(w) ∩I = ∅. If v, w satisfy these conditions it is clear that ˙ −1 N(v)w: vw satisfies the first one; for the second one write N(vw) = N(w) +w −1 if N(v) does not meet I, since w normalises I the same is true of w N(v)w and if two sets do not meet I, neither does their symmetric difference. Using 3.2.1 we can write any w ∈ NW (WI ) as w = vw1 where v ∈ WI and w1 is I-reduced. Since w1 normalises WI , for s ∈ I we have w1 s = tw1 with t ∈ WI , and since w1 is I-reduced we must have l(t) = 1 thus w1 normalises I, whence NW (WI ) = WI  NI . The unicity of the decomposition vw1 gives the  decomposition of NW (WI ) F . By the remark following Proposition 3.4.3, we have NG (LI )/LI  NW (WI )/WI , thus if we set WGF (L):= NGF (L)/LF we have WGF (LI )  NIF , so that we can consider WGF (L, Λ) as a subgroup of W F . By Tits (1966) – using Borel and Tits (1965, Théorème 7.2) if (G, F) is not ˙ w˙ = (ww ) · split – W F has representatives w → w˙ ∈ NGF (T) such that w whenever l(w) + l(w ) = l(ww ). From now on we fix such a choice and for n ∈ NGF (L, Λ) we choose γn such that γlw˙ = lγw˙ on Λ. With this choice Bnw = Bw˙ for any representative nw of w. We will write Bw instead of Bw˙ . Lemma 6.1.8 If l(w) + l(w ) = l(ww ), then Bw Bw = λ(w, ˙ w˙ )Bww . Proof Bw Bw maps geUF ⊗ x to geUF w˙ −1 eUF w˙ −1 eUF ⊗ γw˙ γw˙ x. We claim that ˙ −1 eUF w˙ −1 eUF = eUF w˙ −1 w˙ −1 eUF . Indeed, using the unique Bruhat decomeUF w  ˙ F = UF wU ˙ Fw with Uw = {α |sα ∈N(w) } Uα . Now, position 3.2.7, we have UF wU since the lengths of w and w add, by 2.1.2(ii) and the remark following it we have N(w) ∩ N(w −1 ) = ∅, thus Uw w ⊂ w U, hence UFw w ⊂ w UF and ˙ F w˙ UF = UF w˙ w˙ UF , whence our claim, taking inverses. Using γw˙ γw˙ = UF wU ˙w ˙ = (ww ) · we get the lemma.  λ(w, ˙ w˙ )γw˙ w˙ and w

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95

Theorem 6.1.9 Assume K = C. Then the cocycle λ is trivial, thus there is a choice of the γw˙ in 6.1.4 ensuring that l(w) + l(w ) = l(ww ) implies Bw Bw = Bww . References This has been proven in Lusztig (1984a, 8.6) when Z(G) is connected, then in general in Geck (1993).  Λ) we To get a complete description of the algebra structure of EndKGF (RG L note the following lemma. Lemma 6.1.10 Let M be an F-stable Levi subgroup containing L of an Fstable parabolic subgroup such that |WMF (L, Λ)| = 2; then Bw satisfies a quadratic relation for the non-trivial w ∈ WMF (L, Λ) ⊂ WGF (L, Λ). M Proof For w ∈ WMF (L, Λ) we have an operator BM w ∈ EndKMF (RL Λ) and by G G definition, using the transitivity of the functor RL , we have Bw = RM (BM w ). If w M } is a basis of , B is the only non-trivial element of WMF (L, Λ), then, since {BM w 1 M ) 2 = αBM + βBM for some α, β ∈ K. Applying Λ), we have (B EndKMF (RG w w 1 L we get the result.  RG M

Note that β above is not 0 since Bw is invertible. To show that there exist M as in 6.1.10 and for computing α and β of 6.1.10 we need to know more on WG (L, Λ). The result is known in all cases only when K = C (or a large enough subfield of C). Assumption 6.1.11 subfield of C.

Until 6.1.19 we will assume that K is a large enough

Proposition 6.1.12 If (LI , Λ) is a cuspidal pair, then WGF (LI ) is normalised by wJ for any F-stable J ⊂ S containing I, where wJ is the longest element of the Coxeter group WJ ; it is a Coxeter group with generating simple reflections the elements wJ wI where J ⊃ I and J − I is a single orbit under F. Proof The first part is proved in Lusztig (1984a, 8.2) using a classification of cuspidal pairs. The second part follows from the next proposition applied to  (W F , S/F). Proposition 6.1.13 If (W, S) is a finite Coxeter system and I ⊂ S is such that WI is normalised by the wI∪{s} for s ∈ S − I and NI is as in 6.1.7, then (NI , {σs }s∈S−I ), with σs := wI∪{s} wI , is a Coxeter system. Further, if σs1 . . . σsk is a reduced expression of some w ∈ NI in the above Coxeter system, we have  l(w) = i=k i=1 l(σsi ), where l is the length on W with respect to S.

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Proof This proposition is proved for Weyl groups in Lusztig (1976a, §5) using root systems. We give a proof in the setting of the elementary theory of Coxeter groups. We first check that σs ∈ NI . Since wI∪{s} normalises WI and induces an automorphism of (WI∪{s} , I ∪ {s}) (see 2.1.5), it normalises I. Also by 2.1.5(ii) applied in WI∪{s} we have l(σs ) = l(wI∪{s} ) − l(wI ) which together with 3.2.1(i) implies that σs is shortest in the coset σs WI . We now follow the same outline as the proof of 4.4.4. Lemma 6.1.14 l(w) − l(σs ).

If w ∈ NI and l(ws) < l(w) where s ∈ S − I, then l(wσs ) =

Proof Since w normalises I it commutes with wI and since it is reduced-I we have l(wwI ) = l(w) + l(wI ). Thus N(wwI ) = N(wI w) = N(w) + w−1 N(wI )w = N(w) + N(wI ); since by assumption N(w)  s we have N(wI w)  s and N(wwI ) ⊃ I ∪ {s}. Thus by 3.2.1(i) and (iv) we have wwI = w wI∪{s} with l(w ) + l(wI∪{s} ) = l(wwI ), that is l(wσs ) = l(wwI ) − l(wI∪{s} ). Whence the  lemma since l(w) = l(wwI ) − l(wI ) and l(σs ) = l(wI∪{s} ) − l(wI ). Applying inductively the lemma, we get that any w ∈ NI can be written  σs1 . . . σsk with l(w) = i=k i=1 l(σsi ). This gives the final remark of 6.1.13 and proves in particular that the σs generate NI . We again use 2.2.9 to show that (NI , {σs }s∈S−I ) is a Coxeter system. For s ∈ S − I, let Ds = {w ∈ NI | s  N(w)}. We have clearly Ds  1 and Ds ∩ Ds σs = ∅, so it remains to show that if w ∈ Ds , s ∈ S − I are such that σs w  Ds , then σs w = wσs . Since w is in Ds , it is I ∪ {s}-reduced, hence N(w) ∩ Ref(WI∪{s} ) = ∅. On the other hand, we have s ∈ N(σs w) = −1 −1 w−1 N(σ ) +N(w) ˙ = ( w Ref(WI∪{s } ) − w Ref(WI )) +N(w), and s  N(w) and s ˙ −1 −1 −1 s  Ref(WI ) = w Ref(WI ), thus s ∈ w Ref(WI∪{s } ). Since I = w I ⊂ −1 −1 w−1 Ref(W w w Ref(WI∪{s } ), thus WI∪{s } ⊃ WI∪{s} . I∪{s } ), we have {s} ∪ I ⊂ As a non-trivial element v ∈ WI∪{s } ∩ NI satisfies l(vs ) < l(v), the lemma shows that WI∪{s } ∩ NI = {1, σs } and similarly WI∪{s} ∩ NI = {1, σs }, whence w−1 σ = σ .  s s For s, s ∈ SI , the length of the braid relation between σs and 2| Ref(WI∪{s,s } )| σs (the order of σs σs ) is equal to . | Ref(WI∪{s} )| + | Ref(WI∪{s } )|

Exercise 6.1.15

Proposition 6.1.16 Assume the centre of G connected. If (LI , Λ) is a cuspidal pair, the group WGF (LI , Λ) is a reflection subgroup of NIF with reflections Ref(NIF ) ∩ WGF (LI , Λ). For w ∈ WGF (LI , Λ) we define Tw := ql(w) Bw . We denote by ˜l the length in the Coxeter group WGF (LI , Λ). Then

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(i) Tw Tw = Tww if ˜l(w) + ˜l(w ) = ˜l(ww ). (ii) Let nσs be a reflection of WGF (LI , Λ) where n ∈ NIF and s ∈ S/F − I/F. Then T n2σs = (qcs −1)T nσs +qcs where cs is a positive integer such that qcs is −1

n Λ. the quotient of the degrees of the two irreducible constituents of RLLI∪s I

The “two irreducible constituents” in the last sentence refers to the fact that the condition of 6.1.10 holds: |WLI∪s (LI , Λ)| = 2. References

See Lusztig (1984a, 8.5.8, 8.5.13 and 8.6).



Remark 6.1.17 It can be shown that l(w) + l(w ) = l(ww ) implies ˜l(w) + ˜l(w ) = ˜l(ww ), hence relation 6.1.16(i) is compatible with 6.1.8. The converse implication is not true. Example 6.1.18 Consider the case where L = T is a maximal torus contained in an F-stable Borel subgroup B = TU. Any representation of TF is cuspidal in particular the trivial representation. In this example K can be any field (1) = K(GF /UF ) ⊗TF K = such that |BF | is invertible. The representation RG T F (1) is the permutation representation of GF on the F-stable K(GF /BF ) = IndG BF Borel subgroups. We have WGF (T, 1) = W F . The γn of 6.1.4 is the identity so that the cocycle is trivial. For w ∈ W F , the operator Bw is geBF → geBF w−1 eBF . The Levi subgroups M as in 6.1.10 are the LJ for J ∈ S/F, an orbit of F in S. We compute the quadratic relation satisfied by BwJ . We have B2wJ : geBF → geBF wJ eBF wJ eBF . Using the properties of the relative (B, N)-pair of 4.4.8, we have BF wJ BF wJ BF = BF wJ BF ∪ BF . The compuF ∩ wJ BF | F ∩ wJ BF | F ∩ wJ BF | )BwJ + |B |B B1 . We have |B |B = tation gives B2wJ = (1 − |B |B F| F| F| |UFwJ | −1 = q−l(wJ ) (see the proof of 4.4.1). If we define TwJ := ql(wJ ) BwJ we have Tw2 J = (ql(wJ ) − 1)TwJ + ql(wJ ) . We recover in this case 6.1.16 without needing the assumption that the centre of G is connected. Remark 6.1.19 It is shown in Lusztig (1978, 3.26) that for Λ a unipotent cuspidal module (see definition 11.3.4), the statements of 6.1.16 are true without any assumption on the centre of G. In this case one has WGF (L, Λ) = WGF (L).

6.2 Iwahori–Hecke Algebras We give now general results on algebras having the same kind of presentation (Λ)). as EndKGF (RG L Definition 6.2.1 Let A be a commutative ring, (W, S) be a Coxeter system, and (qs , qs )s∈S be pairs of elements of A depending only on the W-conjugacy

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class of s. The Iwahori–Hecke algebra H {qs,q s }s∈S (W, A) is the A-algebra with presentation  (Ts )s∈S | Ts Ts Ts . . . = Ts Ts Ts . . . for s, s ∈ S,    

order of ss

order of ss

 and (Ts − qs )(Ts − qs ) = 0 for s ∈ S .

In the particular case when qs = 1 and qs = −1, the Iwahori–Hecke algebra (Λ)), it would be enough is the group algebra AW. For the study of EndKGF (RG L to consider the case where qs = −1, but the consideration of the general case has some advantages: it is used when studying invariants of knots, and for rings A which have a Galois automorphism interchanging qs and qs , this exchange induces a duality on characters, see 6.2.3. Note that Ts is invertible if and only if qs and qs are invertible. Proposition 6.2.2 An equivalent presentation of H {qs,q s }s∈S (W, A) is given by 

(Tw )w∈W | for s ∈ S, ⎧ ⎪ Tsw Ts Tw = ⎨ ⎪ (q + q )T − q q T s s sw s w ⎩ s

if l(sw) = l(w) + 1  otherwise

.

Proof We first show that 6.2.2 implies 6.2.1. If s1 . . . sk ∈ W is a reduced expression of w ∈ W, iterating the first relation of 6.2.2 gives Tw = Ts1 . . . Tsk ; thus the Ts generate the algebra. The particular case of the second relation for w = s gives the second relation of 6.2.1. The first relation of 6.2.1 holds since both sides represent TΔs,s (see 2.1.1). We show now that 6.2.1 implies 6.2.2. If s1 . . . sk ∈ W is a reduced expression of w ∈ W, let Tw := Ts1 . . . Tsk ; this element does not depend on the reduced expression thanks to the first relation of 6.2.1 and 2.1.2(iv). Let s ∈ S and w ∈ W; if l(sw) = l(w) + 1 then we get a reduced expression of sw by prefixing by s a reduced expression of w, thus Tsw = Ts Tw ; if l(sw) = l(w) − 1 then we have Tw = Ts Tsw , hence using the second relation of 6.2.1 we have  Ts Tw = Ts2 Tsw = (qs + qs )Ts Tsw − qs qs Tsw = (qs + qs )Tw − qs qs Tsw . Theorem 6.2.3

The set {Tw |w∈W} is an A-basis of H {qs,q s }s∈S (W, A).

Proof By 6.2.2 the set {Tw |w∈W} spans the Iwahori–Hecke algebra. To prove the linear independence, for s ∈ S, we consider the operators Ls ∈ EndA (AW) defined by ⎧ ⎪ sw Ls (w) = ⎨ ⎪ (q + q )w − q q sw s s s ⎩ s

if l(sw) = l(w) + 1, otherwise.

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We get the theorem if we show that L : Ts → Ls defines a representation L : H {qs,q s }s∈S (W, A) → EndA (AW) and that the operators L(Tw ) are linearly independent. This will be a consequence of Lemma 6.2.4 Let λ ∈ EndA (AW) be in the subalgebra generated by the Ls . Then λ = 0 if and only if λ(1) = 0. Proof For t ∈ S define Rt ∈ EndA (AW) by ⎧ ⎪ wt Rt (w) = ⎨ ⎪ (q + q )w − q q wt t t t ⎩ t

if l(wt) = l(w) + 1, otherwise.

We claim that Ls and Rt commute. To simplify the computation we put Ls (w) = ls (w)sw + ls (w)w and Rt (w) = rt (w)wt + rt (w)w. We have Ls Rt (w) = ls (wt)rt (w)swt + ls (wt)rt (w)wt + ls (w)rt (w)sw + ls (w)rt (w)w and Rt Ls (w) = rt (sw)ls (w)swt + ls (w)rt (w)wt + ls (w)rt (sw)sw + ls (w)rt (w)w. The four basis elements involved are distinct unless sw = wt; in this case s and t are conjugate, hence ls (w) = rt (w), ls (wt) = rt (sw), lt (w) = rs (w) and ls (wt) = rt (sw), thus the two expressions are equal. Assume sw  wt. The coefficients of w are the same in both expressions. The coefficients of sw are the same unless l(wt) − l(w)  l(swt) − l(sw) which cannot happen since by the exchange property (see Exercise 2.1.4) it implies sw = wt. The same applies to the coefficients of wt. The coefficients of swt are different only if l(swt) − l(sw)  l(wt) − l(w) or l(swt) − l(wt)  l(sw) − l(w). Both cases imply sw = wt hence do not happen. We deduce the lemma: if λ(1) = 0, since λ commutes with Rs we have  λ(s) = λ(Rs (1)) = Rs (λ(1)) = 0 for all s ∈ S, hence λ = 0. The theorem follows: both Ls Ls . . . and Ls Ls . . . map 1 to Δs,s whence the  

 

order(ss )

order(ss )

equality of these two operators, and the quadratic relations are clear by definition, so we indeed get a representation of the Iwahori–Hecke algebra. We have L(w)(1) = w as can be seen by taking a reduced expression for w. If   a L(w) = 0 is a linear dependency relation we get 0 = w aw L(w)(1) = w w  w aw w so that aw = 0 for all w. Before going on, we need to recall the following theorem proved by Dyer (1990) and Deodhar (1989). Theorem 6.2.5 Let (W, S) be a finite Coxeter system and let W be a reflection subgroup of W – that is, W is generated by W ∩ Ref W.

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Let S(W ) = {r ∈ Ref(W) | N(r) ∩ W = {r}}. Then (W , S(W )) is a Coxeter system, called the canonical Coxeter system of W with respect to W. Proof We follow Dyer’s proof. Let W1 be the subgroup of W generated by S(W ). The first thing we prove is Any r ∈ Ref(W) ∩ W is W1 -conjugate to an element of S(W ).

(∗)

We shall prove (∗) by induction on l(r), where the induction assumption is that it holds for all r ∈ Ref(W) such that l(r ) < l(r) and all reflection subgroups of W. The start of the induction is when l(r) = 1 in which case r ∈ S(W ) and there is nothing to prove. For the general step of the induction, by 2.1.6(iv) there exists s ∈ S such that l(srs) = l(r) − 2. By the induction assumption, there exists r0 , . . . , rm ∈ S( sW ) such that srs = rm . . . r1 r0 . . . rm . There are now two cases: • If s ∈ W . Then sW = W and S( sW ) = S(W ) thus r is W1 -conjugate to r0 ∈ S(W ), as needed. • If s  W . We have r = srm . . . sr1 sr0 . . . srm where sri ∈ sS( sW ). It is thus sufficient to show that sS(W ) = S( sW ). Now let r ∈ Ref(W) ∩ W ; we have ˙ ∩ sW N( sr) ∩ sW = ({ srs}+˙ sN(r) +{s}) ˙ ˙ = s (({ rs}+N(r) +{s}) ∩ W ) = s (N(r) ∩ W ), where the first equality is by the defining relation of N, and the third one since s  W and rsW = rW . These equalities show that N( sr)∩ sW = { sr} is equivalent to N(r) ∩ W = {r}. We can now finish the proof of the theorem by applying criterion 2.1.2(ii) for being a Coxeter system, taking for N the function N on W defined by ˙ −1 N (x)y is an immediate N (w) := N(w) ∩ W ; the property N (xy) = N (y) +y consequence of the same property for N.  The theory of Iwahori–Hecke algebras applies in particular in the two following examples. Part (i) was proved by Iwahori, whence the terminology “Iwahori–Hecke” algebra. Proposition 6.2.6 (i) Let T be an F-stable maximal torus of an F-stable Borel subgroup B of G and let K be a field in which |BF | is invertible; then F EndKGF (RG T (1)) = H {ql(wJ ) ,−1}J∈S/F (W , K).

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(ii) Assume that the centre of G is connected; let (L, Λ) be a cuspidal pair over a large enough subfield K of C, then EndKGF (RG Λ) = H {qcs ,−1}s∈S(L,Λ) L (WGF (L, Λ), K), where cs is as in 6.1.16 and S(L, Λ) is the canonical set of Coxeter generators with respect to NIF of the reflection subgroup WGF (LI , Λ). Proof (i) is a consequence of what has been seen in Example 6.1.18. Λ) has a basis (Tw )w∈WGF (L,Λ) For (ii): by 6.1.16 and 6.1.6, EndKGF (RG L which satisfies the relations of 6.2.2 with parameters qcs and −1, whence the result.  Recall that if K is a field, a finite dimensional K-algebra H is absolutely semi-simple if H ⊗K K is semi-simple where K is an algebraic closure of K, in  which case H ⊗K K is isomorphic to a product of matrix algebras i Mni (K) and H ⊗K L is semi-simple for any extension L of K, see Bourbaki (1981, §13 no 3 Théorème 1). The multi-set {ni } is called the numerical invariant of H. A criterion for being absolutely semi-simple is given by the following Proposition 6.2.7

Let H be a finite dimensional algebra over a field K.

(i) If the bilinear form (a, b) → TraceH/K (ab) is non-degenerate then H is absolutely semi-simple. (ii) The converse of (i) is true when the characteristic of K is 0. Proof We recall that TraceH/K (a) is the matrix trace of the multiplication by a expressed in a K-basis of H. For (i): The form TraceH/K (ab) remains non-degenerate when extended to K. An element of the radical of H ⊗K K being nilpotent has trace 0. Thus for a in the radical we have TraceH ⊗K K/K (ab) = 0 for all b whence a = 0. For (ii): it is sufficient to prove the result for H = Mn (K). We can take the elementary matrices (Ei,j )i,j (see 1.5.3) as a K-basis for H. We get TraceH/K (Ei,j ) = nδi,j thus for h ∈ Mn (K) the matrix trace of h is equal to 1 n TraceH/K (h), since the characteristic is 0, hence the bilinear form is non degenerate since if the matrix trace of aEij is zero for all i, j then a = 0. Remark 6.2.8 If W is a finite group and K a field, then for w ∈ W we have TraceKW/K (w) = |W |δw,1 . Hence if the characteristic of K does not divide |W |, the form (a, b) → TraceKW/K (ab) is non-degenerate, and in particular KW is absolutely semi-simple. The next theorem, due to Tits, appears for the first time in Exercises 26 and 27 of Bourbaki (1968, IV, §2).

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Theorem 6.2.9 (Tits deformation theorem) Let H be a free finite dimensional A-algebra where A is an integral domain. Let K be the quotient field of A and let f : A → k be a ring morphism from A to some field k. (i) If the form (a, b) → TraceH ⊗A k (ab) is non-degenerate, then H ⊗A K is absolutely semi-simple. (ii) If H⊗k and H⊗K are absolutely semi-simple they have the same numerical invariant. (iii) If A is the integral closure of A in K then f can be extended to f : A → k; for any irreducible character χ of H ⊗A K, one has χ(H) ⊂ A and χ → (f ◦ χ) ⊗ 1 is a bijection from Irr(H ⊗A K) to Irr(H ⊗A k). Proof We write HK , HK , Hk , . . . , respectively for H ⊗A K H ⊗A K, H ⊗A k,. . . . An A-basis of H specialises to a k basis of Hk and is also a K-basis of HK . This shows that, under the assumption of (i), (a, b) → TraceHK /K (ab) is nondegenerate on HK , whence (i) by 6.2.7. For proving (ii), we can assume k and K algebraically closed. Then HK and Hk are products of matrix algebras. Let {ei | i = 1, . . . , r} be an A-basis of H and consider the algebra HK[x1, ...,xr ] where xi are indeterminates. We claim  that the characteristic polynomial of the multiplication by i xi ei is of the form s nj j=1 Pj ∈ K[x1 , . . . , xr ][t], where {n1 , . . . , ns } is the numerical invariant of HK and each Pj is irreducible of degree nj in t. It is sufficient to see it when HK is a matrix algebra Mn (K). Since the degree of the characteristic polynomial does not change by a (linear) base change, we can do the computation taking as basis of Mn (K) the elementary matrices Ei,j . The matrix of the multiplication by x := (xi,j ) is a block diagonal matrix with n blocks, all equal to x, thus the characteristic polynomial is Pn where P is the characteristic polynomial det(x − t In ) of the matrix x, an irreducible polynomial of degree n in t (since it is the “generic” characteristic polynomial of an n × n matrix), whence our claim. Now, the basis (ei ) of H specialises to a basis of Hk . The multiplication   by i xi ei specialises to the multiplication by xi ei in H ⊗A k[x1 , . . . , xn ], so  that the specialisation j f (Pj ) nj is the analogous polynomial over k[x1 , . . . , xn ]. Since this has to be the product of irreducible polynomials, each one raised to its degree and of the same degree r as P we must have f (Pj ) irreducible and {n1 , . . . , ns } must be the numerical invariant of Hk , whence (ii). We prove (iii). The fact that f can be extended to A is well known; see for example, Lang (2002, VII, Proposition 3.1). Each irreducible character χ of i=s M (K[x , . . . , x ]) is given by the matrix trace composed HK[x1, ...,xr ]  ⊕i=1 ni 1 r with the projection onto one of the Mni (K[x1 , . . . , xr ]). Its value χ(h) is thus the coefficient of degree ni − 1 of the irreducible factor Pi ∈ K[x1 , . . . , xr ], of  the characteristic polynomial P of the multiplication by h. For h = i xi ei ∈

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103

HK[x1, ...,xr ] the coefficients of P are in A[x1 , . . . , xr ], hence its roots are integral over this ring, so that the coefficients of its irreducible factors are in the integral closure A[x1 , . . . , xr ] of A[x1 , . . . , xr ] in K[x1 , . . . , xr ]. By Bourbaki (1975, V, §1 no 3, Proposition 13), we have A[x1 , . . . , xr ] = A[x1 , . . . , xr ]. Since f ( χ(h)) is  (−1) ni −1 times the coefficient of degree ni − 1 of f (Pi ) we get the result. Theorem 6.2.10 Let K ⊂ C be a splitting field forthe finite Coxeter group W √ and let qs , qs be indeterminates; the field K( qs , −qs )s∈S is a splitting field ±1 for H {qs,q s }s∈S (W, Q[q±1 s , qs ]s∈S ). Proof This can be deduced from Geck and Pfeiffer (2000, 9.3.1 and 9.3.5); see also Malle (1999, Corollary 4.8 and Proposition 5.1).  Corollary 6.2.11 Let (W, S) be a finite Coxeter system and K a field whose characteristic does not divide |W |. (i) If qs , qs are indeterminates, then H {qs,q s }s∈S (W, K(qs , qs )s∈S ) is absolutely semi-simple and has same numerical invariant as W. (ii) If as , as ∈ K are such that H {as,a s }s∈S (W, K) is absolutely semi-simple then H {as,a s }s∈S (W, K)  KW. (iii) If K is a splitting field for W, the characters of KW are specialised from the √ ±1  ±1 characters of H {qs,qs }s∈S (W, K[ qs , −qs ]s∈S ) for the specialisation √ qs → 1 and −qs → 1. Proof (i) is a consequence of Remark 6.2.8 and Theorem 6.2.9(i). (iii) is by 6.2.9(iii), taking in account 6.2.10. For (ii): by 6.2.9(ii) H {qs,q s }s∈S (W, K(qs , qs )s∈S ) has same numerical invariant as KW and as H {as,a s }s∈S (W, K). Hence these last two K-algebras are isomorphic.  ±1 Definition 6.2.12 The algebra H {qs,q s }s∈S (W, K[q±1 s , qs ]s∈S ), when qs , qs are indeterminates, is called the generic Iwahori–Hecke algebra of W over K.

Remark 6.2.13 Statement 6.2.11(i) is true for variations of the generic Iwahori–Hecke algebra like the algebra H {qcs ,−1}s∈S (W, K[q±1 ]) where q is an indeterminate and cs are positive integers. The same proof applies. Corollary 6.2.11 applies in particular in the two following cases: Corollary 6.2.14 (i) Let B be an F-stable Borel subgroup in a connected reductive group G; then for any algebraically closed field K in which |GF | is invertible, the F (1) are in bijection with the irreducible irreducible constituents of IndG BF characters of KW F . By this bijection the trivial representation of GF corresponds to the trivial representation of W F .

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(ii) We assume moreover that Z(G) is connected. Let (L, Λ) be a cuspidal pair over a large enough subfield K of C, then there is a bijective correspondence between Simp(KGF | (L, Λ)) and Simp(KWGF (L, Λ)). Proof If V is a representation of a finite group G over an algebraically closed field K whose characteristic does not divide |G| then EndG (V) is absolutely semi-simple. Thus in view of 6.2.6 (i) and (ii) we can apply 6.2.11(iii) to both cases of the Corollary, which, using 6.1.1(i), gives (ii) and the first assertion of (i). F (1) since this Now the trivial representation of GF is a constituent of IndG BF last representation is a permutation representation. We use the notation and F (1) = RG (1) where T is an F-stable maxiresults of 6.1.18. We have IndG T BF F

(1)) = H {ql(wJ ) ,−1}J∈S/F (W F , K). By 6.1.1(i) mal torus of B and EndKGF (IndG BF G the representation of EndKGF (RT (1)) corresponding to the trivial representa1, 1). A morphism in this last space maps all geBF tion of GF is HomKGF (RG T to the same element of K, and the action of TwJ is by composition with eBF → l(wJ ) . TwJ eBF = ql(wJ ) eBF w−1 J eBF , so that this action is just multiplication by q This one dimensional representation of the Iwahori–Hecke algebra is the specialisation of the representation of the generic algebra of 6.2.13 given by the same formula where q is an indeterminate. This last representation in turn specialises to the trivial representation of W F when the indeterminate q goes to 1, whence the last assertion of (i). 

6.3 Schur Elements and Generic Degrees We give now more properties of the characters of Iwahori–Hecke algebras. Proposition 6.3.1 elements of A.

Let A be an integral domain and let as , as be invertible

(i) Let τ be the linear map H {as,a s }s∈S (W, A) → A defined by τ(Tw ) = δw,1 . Then the bilinear form (a, b) → τ(ab) on H {as,a s }s∈S (W, A) is nondegenerate. (ii) The map Ts → as for s ∈ S (resp. Ts → as for s ∈ S) defines a linear character of H {as,a s }s∈S (W, A). If we denote by aw (resp. aw ) the value of −1 this character at Tw , then {(−1) l(w) a−1 w aw Tw−1 } is a basis dual to {Tw } for the bilinear form of (i). Proof We prove first that the maps of (ii) are characters. We need to check the compatibility with the relations; this is obvious for the quadratic relations. For an even length braid relation the number of si equal to a given s is the same on both sides of the relation thus the product of the asi is the same on both sides; if

6.3 Schur Elements and Generic Degrees

105

an odd length braid relation involves s and s , then s and s are conjugate, hence as = as , as = as , and the two products are again equal. We now prove by induction on l(w) that τ(Tw Tw ) = 0 unless w = w −1 and that τ(Tw Tw−1 ) = (−1) l(w) aw aw . This is true if w = 1. Otherwise write w = vs with l(v) = l(w) − 1 and s ∈ S. Then Tw Tw = Tv Ts Tw . If l(sw ) = l(w ) + 1, then Tw Tw = Tv Tsw . Since l(vs) > l(v) and l(w ) < l(sw ), we cannot have v = (sw ) −1 , thus by induction τ(Tw Tw ) = 0. If l(sw ) = l(w ) − 1 then w = sv for some s ∈ S by the exchange property, hence Tw Tw = Tv Ts2 Tv = (as +as )Tv Tw − as as Tv Tv . Since l(vs) > l(v) and l(sw ) < l(w ), we cannot have v = w −1 , hence we get by induction τ(Tw Tw ) = −as as τ(Tv Tv ) = (−1) l(v)+1 as as av av = (−1) l(w) aw aw if vv = 1 and 0 otherwise, which gives the result. This proves that we have dual bases, thus that the bilinear form is non-degenerate.  Note that when as = 1 and as = −1 for all s, the two characters of 6.3.1(ii) are the trivial and the sign character of AW and the dual basis to {w} is {w−1 }. Proposition 6.3.1 says in particular that H {as,a s }s∈S (W, A) is a symmetric algebra. A symmetric algebra over an integral domain is a finite dimensional free algebra on which there exists a linear form τ (the symmetrising form) such that (a, b) → τ(ab) is symmetric non-degenerate. We will use general properties of symmetric algebras which we reprove here for the convenience of the reader. Notation 6.3.2 Until 6.3.8 H will be a symmetric algebra over an integrally closed integral domain A with symmetrising form τ. We denote by K the quotient field of A and by HK the algebra H ⊗A K. Proposition 6.3.3  (i) The element I := i ei ⊗ ei ∈ H ⊗A H opp , where (ei ) and (ei ) are dual bases of H with respect to τ, does not depend on the choice of dual bases. (ii) If M and N are H-modules, then ϕ → Iϕ maps HomK (M, N) to HomH (M, N).  Proof For any h, h ∈ H we have (τ ⊗ τ)(I(h ⊗ h )) = i τ(ei h)τ(h ei ) =   i τ(τ(ei h)ei h ) = τ( i τ(ei h)ei h ) = τ(hh ). The last equality holds since, by  definition of dual bases, we have i τ(hei )ei = h. This shows the independence since τ ⊗ τ is non-degenerate. Since τ(hh ) = τ(h h), the above computation shows also I(h ⊗ h ) = I(h ⊗ h). In (ii) the action of H ⊗ H opp on ϕ is given by ((x ⊗ y)ϕ)(m) = xϕ(ym). We get the result from the equality I(1 ⊗ h) = I(h ⊗ 1) seen in the proof of (i).  Remark 6.3.4 If H ∗ is the dual of H, by the isomorphism H ⊗A H  HomA (H ∗ , H)  HomA (H, H) given by x ⊗ y →  (h → τ(xh)y) the element I goes

106

Iwahori–Hecke Algebras

 to h → i τ(ei h)ei = h hence to the identity, which gives another proof of the independence of I from the choice of basis. The symmetrising form τ induces an isomorphism f → fˇ : H ∗ → H, by f (h) = τ(fˇ h). By definition τˇ = 1. Note that f ∈ H ∗ is a trace on H, that is it satisfies f (hh ) = f (h h) for all elements h, h ∈ H, if and only if fˇ is in the  centre of H. For any pair of dual bases (ei ), (ei ) we have fˇ = i f (ei )ei . Definition 6.3.5 If χ is the character of an absolutely irreducible HK -module M, for h in the centre of HK , we denote by ω χ (h) ∈ K the scalar by which h ˇ ∈ A. acts on M and we call Schur element of χ the scalar S χ := ω χ ( χ) The name “Schur element” and the semi-simplicity criterion of 6.3.8 are due to Geck. Note that χˇ is in H since χ(H) ⊂ A, which is true since, A being integrally closed, the character of any finite dimensional HK -module takes values in A on H. Proposition 6.3.6 Assume moreover HK split semi-simple. (i) The central primitive idempotent corresponding to an irreducible characˇ ter χ is e χ = S1χ χ.  (ii) We have τ = χ ∈Irr HK S1χ χ. Proof For any h ∈ H we have χ(e χ h) = χ(h), hence τ( χe ˇ χ h) = τ( χh), ˇ ˇ Hence χˇ acts by 0 on any simple H-module with which implies χe ˇ χ = χ.  character different from χ. Since χˇ is central in H, we have χˇ = ϕ ∈Irr HK  ωϕ ( χ)e ˇ ϕ = S χ e χ , whence (i) since χˇ  0, whence S χ  0. From 1 = χ e χ ,  1  1 ˇ  we get thus 1 = χ S χ χˇ which is equivalent to τ = χ S χ χ since 1 = τ.  We define a scalar product on functions on HK by f , gHK := (f ⊗ g)(I) =  i f (ei )g(ei ). Proposition 6.3.7 we have

Assume HK split semi-simple; then for χ, ϕ ∈ Irr(HK )

⎧ ⎪ χ(1)S χ if χ = ϕ,  χ, ϕ = ⎨ ⎪0 if χ  ϕ. ⎩   Proof We have τ, χ = i τ(ei ) χ(ei ) = χ( i τ(ei )ei ) = χ(1). By 6.3.6,   the left hand side is equal to ϕ ∈Irr(HK ) S1ϕ ϕ, χ. But ϕ, χ = i ϕ(ei ) χ(ei ) =  ˇ i )ei ) = χ( ϕ). ˇ We have seen in the proof of 6.3.6 that ϕˇ = ϕe ˇ ϕ so χ( i τ( ϕe 1 that χ( ϕ) ˇ = 0 if ϕ  χ. We get then τ, χ = S χ  χ, χ = χ(1), whence the proposition. 

6.3 Schur Elements and Generic Degrees

107

The next result gives a criterion of semi-simplicity for a symmetric algebra. For a proof see Broué et al. (1999, 7.13) or Geck and Pfeiffer (2000, Theorem 7.4.7). Proposition 6.3.8 Assume HK split semi-simple and let A → k be a surjective morphism from A onto a field k. Assume Hk split, then Hk is semi-simple if and only if for all χ ∈ Irr(HK ) the Schur element S χ has a non-zero image in k. We now apply these results to Iwahori–Hecke algebras. As before (W, S) denotes a finite Coxeter system. Let A be an integrally closed integral domain with quotient field K. Definition 6.3.9 A representation of an Iwahori–Hecke algebra whose character vanishes on Tw for all w  1 is called trace-special. Corollary 6.3.10 Take H = H {as,a s }s∈S (W, A) and τ as in 6.3.1. Assume that HK is split semi-simple and that HK has a trace-special representation on a finite dimensional K-vector space V. Then the multiplicity in V of an irreducible V representation of HK with character χ is equal to dim Sχ . Proof This is a consequence of 6.3.6 since the character of V is (dim V)τ.



We can apply 6.3.10 to the two cases of 6.2.6. Proposition 6.3.11 (i) If B is an F-stable Borel subgroup in the reductive group G then the repF (1) over a field K whose characteristic does not divide resentation IndG BF F |B | is a trace-special representation of H {ql(wJ ) ,−1}J∈S/F (W F , K). (ii) With assumption and notation as in 6.2.6(ii), the representation of Λ is trace-special. H {qcs ,−1}s∈S(L,Λ) (WGF (L, Λ), K) on RG L Λ = K(GF /UF ) Proof Let us prove (ii). We can take as a basis of the space RG L ⊗KLF Λ the elements geUF ⊗ ei where g runs over a set of representatives of GF /LF UF and (ei ) is a basis of Λ. The operator Bw maps geUF ⊗ ei to F F F geUF n−1 w eUF ⊗ γnw (ei ). Since (U nw U ) ∩ U = ∅ unless w = 1 by 3.2.3(ii), the matrix of Bw has all diagonal entries 0, hence trace 0; whence (ii) since Tw is a  scalar multiple of Bw . The same argument applies to (i). ±1 Proposition 6.3.12 Let H = H {qs,q s }s∈S (W, K[q±1 s , qs ]s∈S ) be the generic Hecke algebra. Assume HL split semi-simple for some extension L of K(q±1 s , qs ±1 ). For any irreducible character χ of HL and any specialisation such that qs → 1 and qs → −1, the Schur element S χ specialises to |W |/ χ(1).

108

Proof

Iwahori–Hecke Algebras

By 6.3.7 and 6.3.1(ii), we have  −1 χ(Tw ) χ((−1) l(w) q−1 w qw Tw−1 ) = χ(1)S χ , w∈W

where qw is defined as aw in 6.3.1(ii). The left-hand side specialises to |W |, thus  S χ specialises to |W |/ χ(1). Definition 6.3.13 Let Hand L be as in 6.3.12; For χ ∈ Irr(HL ) the generic (−1) l(w) q q −1 degree of χ is d χ := w∈W S χ w w ∈ L.  Note that w∈W (−1) l(w) qw qw −1 is the Schur element of the linear character Ts → qs .   (−1) l(w) q q −1 Since d χ = w∈W S χ w w and w∈W (−1) l(w) qw qw −1 specialises to |W | one could say that the generic degree “specialises to χ(1)” although specialising from a field has no meaning. In the same vein we have

Proposition 6.3.14 Under the assumptions of 6.3.11(i), let {qJ , qJ | J ∈ S/F} F be indeterminates; the degree of the constituent γ χ ∈ Irr(GF ) of IndG (1) BF parametrised by χ ∈ Irr(W F ) is equal to the “specialisation” of the generic degree of the corresponding generic algebra for any specialisation such that qJ → ql(wJ ) and qJ → −1. Proof By 6.2.11(i) the generic algebra is absolutely semi-simple so that there exists a splitting field L as in 6.3.12. By 6.2.9(iii) Irr(LW F ) is in bijection with the irreducible characters of the generic algebra. By Benson and Curtis (1972, Theorem 2.6) except for Ree and Suzuki groups and by the tables in Lusztig (1984a, Appendix) for those last groups, we have d χ ∈ Q[qJ , qJ ]J ∈S/F for all χ ∈ Irr(W F ). The proposition is thus a consequence of 6.3.10 and 6.1.1(ii), F F F since dim(IndG F (1)) = |G /B |, which by 4.4.1(iii) is the specialisation of B  l(w) −1 qw qw where qw is defined as aw in 6.3.1(ii).  w∈W F (−1)

6.4 The Example of G2 We end this chapter with the example of the generic Hecke algebra of type G2 . Example 6.4.1 Let (W, {s, t}) be a Coxeter system of type G2 . Consider the Iwahori–Hecke algebra H := H(x,x ), (y,y ) (W, Z[x±1 , y±1 , x ±1 , y ±1 ]) where (x, x ) and (y, y ) are pairs of indeterminates corresponding to s and t – which are not conjugate. By 6.2.11(i) HQ(x,y,x ,y ) is absolutely semi-simple. We will compute the values of irreducible characters on the basis elements Tw and their generic

6.4 The Example of G2

109

√ degrees over a splitting field which we will show to be Q(x, y, x , y , xyx y ) (compare with 6.2.10). These irreducible characters are in bijection with Irr(W) thus there are four irreducible characters of degree 1 and two characters of degree 2. The representations of dimension 1 are determined by the possible eigenvalues for Ts which must be either x or x and the possible eigenvalues for Tt which must be y or y . These four possibilities give the four characters of degree 1. We now compute the irreducible representations of dimension 2. Since the values of the irreducible characters of W on s and t are 0, and are specialisations when x and y go to 1 and x , y go to −1 of the values of the corresponding character on Ts and Tt , the two eigenvalues must be distinct. Take a basis consisting of an eigenvector of Ts for the eigenvalue x and an eigenvector of Tt for the eigenvalue y , which are not colinear since the repre- x u sentation is irreducible. The matrices of Ts and Tt are then of the form 0 x   y 0 for some u and v which we can determine using the braid relation and v y √ between Ts and Tt . The computation gives uv = −(xx + yy ) + ε xyx y with √ ε = ±1. Thus a splitting field for H is Q(x, y, x , y , xyx y ), and the two repre√ √ sentations of dimension 2 are conjugate by xyx y → − xyx y , an element √ of Gal(Q(x, y, xyx y )/Q(x, y, x , y )). We can compute thus all character values, and then the Schur elements either by 6.3.7 or by solving equation 6.3.6(ii), and finally the generic degrees. We denote by 1, sgn, τ, and σ the four characters of degree 1 of W and by ρ and ρ the two characters of degree 2 and we use the same notation for the corresponding characters of H. The two characters ρ and ρ correspond to the two possible values of ε. In Table 6.1 we factorised generic degrees in terms of cyclotomic polynomials φi to save space, and we have only given the character values on elements Tw for one element w of minimal length in each W-conjugacy class. The values at other elements can be computed thanks to Theorem 6.4.2. Theorem 6.4.2

Let (W, S) be a finite Coxeter system. Then

(i) If w ∈ W is not of minimal length in its conjugacy class, there exists s ∈ S such that l(sws) = l(w) − 2; (ii) If w, w are of minimal length in the same conjugacy class, they are conjugate by a sequence of “cyclic” conjugations of the form w = xy, w = yx with l(w) = l(x) + l(y) = l(w ). Reference See Geck and Pfeiffer (2000, Theorem 3.2.9). Note that (i) for w ∈ Ref(W) follows from 2.1.6(iv). 

ρ

ρ

τ

σ

sgn

1

 xy 3 x y

 xφ1 (x/x )φ1 (y/y )φ3 ( xy/x y ) −  2x φ3 ( xy /x y)  xφ1 (x/x )φ2 (y/y )φ6 ( xy/x y ) −  2x φ6 ( xy /x y)

xφ (xy/x y ) − 3 x φ3 (yx /xy ) yφ (xy/x y ) − 3 y φ3 (xy /x y)

1 

Generic degree

2

2

1

1

1

1

T1

x + x

x + x

x

x

x

x

Ts

Table 6.1. Character table of H(x,x ), (y,y ) (W (G2 ), Z[x±1 , y±1 , x ±1 , y ±1 ])

y + y

−xx yy −xx yy

 − xx yy

(xy ) 2

(x y) 2

2

(xx yy ) 3 (xx yy ) 3

 

−2

(xy ) 3

(x y) 3

(x y ) 3

(xy) 3

(xy) 2 (x y ) 2

Tststst

Tstst

xx yy



y x

y y + y

x y

x y

xy

Tst

y

y

y

Tt

6.4 The Example of G2

111

Table 6.2. Character table of H(q,−1), (q,−1) (W (G2 ), BZ[q±1 ]) χ

Degree ofγ χ

T1

Ts

Tt

Tst

Tstst

Tststst

1

1

1

q

q

q2

q4

q6

sgn

q6

1

−1

−1

1

1

1

σ

q 3 φ3 (q)φ6 (q) q 3 φ3 (q)φ6 (q)

1

−1

q

−q

q2

−q3

1

q

−1

−q

q2

−q3

q 2 6 φ2 (q) φ3 (q) q 2 2 φ2 (q) φ6 (q)

2

q−1

q−1

q

−q2

−2q3

2

q−1

q−1

−q

−q2

2q3

τ ρ ρ

Since for elements as in (ii) we have Tw = Tx Ty and Ty Tx = Tw it follows that the Tw corresponding to elements of minimal length are conjugate in H, provided the parameters are invertible, thus have same character values. Assume now that w is not of minimal length in its conjugacy class, and let s and w := sws be as in (i) of the above theorem. Then Tw = Ts Tw Ts thus if χ ∈ Irr(H) then χ(Tw ) = χ(Ts Tw Ts ) = χ(Tw Ts2 ) = (qs + qs ) χ(Tw s ) − qs qs χ(Tw ), reducing the computation to character values on smaller length elements. F (1)) (with G of type G2 ) by By 6.3.14, when H is specialised to End(IndG BF √ x → q, y → q x → −1, y → −1 and xyx y → q, the generic degrees specialise to the dimensions of the elements of Simp(GF | (T, 1)), called the principal series unipotent representations. We give the character table of this specialised algebra in Table 6.2. By 6.2.14(i) γ1 is the trivial character and by 7.2.14 γsgn is the Steinberg character. Note that the four characters of degree 1 of the generic Iwahori–Hecke algebra are permuted by a Galois action, exchanging x and x and/or y and y . This Galois action disappears in the specialised algebra. The two characters of degree 2 of H are interchanged by the Galois action which changes the sign √ of xx yy . Exercise 6.4.3 shows that these properties generalise to all Hecke algebras of finite Coxeter groups. Exercise 6.4.3 (i) Show that for any field K, the map ϕ : Ts → qs qs Ts−1 defines an automorphism of the generic Iwahori–Hecke algebra H := H {qs,q s }s∈S (W, A) where ±1 A = K[q±1 s , qs ]s∈S . (ii) Denote by a → a the automorphism of A which maps qs and qs to their in  −1 verses. Show that j( w aw Tw ) = w aw (qw qw ) Tw is a semilinear automorphism of H.

112

Iwahori–Hecke Algebras

 √ (iii) Let K ⊂ C be a splitting field for W and let A = K[ qs ±1 , −qs ±1 ]s∈S . The automorphism ϕ extends naturally to H ⊗A A . Show that if χ1 denotes the specialisation to KW of a character χ of H⊗A A for some specialisation such that qs → 1 and qs → −1, then the character χϕ := χ ◦ ϕ specialises to χ1 ⊗ sgn. (iv) Let K and A be as in (iii). We denote again by j an extension of j to A . It can be shown, by similar arguments as in Geck and Pfeiffer (2000, 9.2.5), that for any irreducible character χ of H ⊗A A the value χ(Tw ) ∈ A is homogeneous of degree l(w) in qs , qs . Using this fact, show that if ρ is a representation of H over A with character χ then ρj (Tw ) := ρ(j(Tw )) is a representation of H whose character is χj (Tw ) = qw qw χ(Tw ) and the Galois automorphism σ of A specialises to χ1 ⊗ sgn. Deduce that  √ which for all s interchanges qs and −qs satisfies χϕ = σ ◦ χ for any character χ of H ⊗A A . We will see in 7.2.13 that the unipotent characters γ χ and γ χ ϕ are dual to each other. Thus the Galois automorphism of (iv) of the generic Hecke algebra corresponds to duality. Notes A good reference about Iwahori–Hecke algebras is Geck and Pfeiffer (2000). Λ), in particular Lemmas 6.1.8 and The structure of the algebra EndKGF (RG L 6.1.10, was first determined in Howlett and Lehrer (1980) in characteristic 0 and by Geck et al. (1996b) in general.

7 The Duality Functor and the Steinberg Character

This chapter is devoted to the exposition of the main properties of the “duality” functor for the characters of GF . This duality, introduced in 1979, has an elementary definition and allows substantial simplification in some areas of the representation theory of GF , for instance the theory of the Steinberg character.

7.1 F-rank First we attach signs to F-stable tori and reductive subgroups of a reductive group G with a Frobenius root F. Definition 7.1.1 If F is a Frobenius root on the torus T acting as qτ on E = X(T) ⊗ R – see 4.2.5 – we define the F-rank of T as dim Eτ , and define a sign ε T = (−1) F -rank(T) . Lemma 7.1.2

We have ε T = (−1) rank(T) det τ.

Proof Since τ is real and of finite order, its eigenvalues are roots of unity which are 1 with multiplicity dim Eτ , −1 with a certain multiplicity n−1 , and non-real roots of unity which come in pairs (λ, λ). Since λλ = 1, is follows that det τ = (−1) n−1 . We have dim Eτ + n−1 ≡ dim E (mod 2), which gives the result since dim E = rank(T).  When F is the Frobenius endomorphism attached to an Fq -structure, the F -rank is called also the Fq -rank, and Fq -rank(T) = rank(X(T)τ ) since τ is defined on X(T) in this case. Lemma 7.1.3 rank(T).

A torus T defined over Fq is split if and only if Fq -rank(T) =

113

114

Proof

The Duality Functor and the Steinberg Character Let {a1 , . . . , an } be a Z-basis of X(T). Using that ±1 T = Spec(Fq X(T)) = Spec(Fq [a±1 1 , . . . , an ]),

we see that it is equivalent that F acts by q on all ai or that the isomorphism ∼ ±1 → Gnm given by Ti → ai : Fq [T1±1 , . . . , Tn±1 ] → Fq [a±1 T− 1 , . . . , an ] be defined  over Fq . We have then the following alternative definition of Fq -rank: Proposition 7.1.4 A torus T defined over Fq contains a unique maximum split subtorus T1 and rank(T1 ) = Fq -rank(T). f

− S Proof The image of a split torus S by a surjective morphism of tori S → ∗ defined over Fq is split, as f : X(S ) → X(S) is injective. This implies that the product of two split subtori of T is again split, whence the existence of T1 . If we denote again by τ the transposed action of τ on Y (T), since rank (X(T)τ ) = rank(Y (T)τ ), to prove the second part of the statement it is sufficient to prove that rank(Y (T)τ ) = rank(Y (T1 )). The inclusion of T1 in T already proves that the dimension of the left-hand side is not smaller than the dimension of the right-hand side. On the other hand the sublattice Y (T)τ defines a split  subtorus, which must thus be T1 . Definition 7.1.5 Let F be a Frobenius root on the connected algebraic group G. We call F-rank of G the F-rank of a quasi-split torus of G, and define a sign ε G = (−1) F-rank(G) . Note that, since all quasi-split tori of G are GF -conjugate, this definition makes sense. We will also call Fq -rank the F-rank of G when F is a Frobenius endomorphism. Note that the F-rank of a group is equal to that of its reductive quotient, as tori map isomorphically in that quotient; that is, F-rank(G) = F-rank (G/Ru (G)). Lemma 7.1.6 Let T be an F-stable maximal torus of type w (see 4.2.22) of G. Then ε G ε T = (−1) l(w) . Proof By definition of the type, if T0 is quasi-split, then the pair (T, F) is conjugate to (T0 , wF). Thus if F = qτ on X(T0 ) ⊗ R, we have F = wqτ on X(T) ⊗ R and by 7.1.2 we have ε T ε G = det w. Now w is a product of l(w) reflections, of determinant −1, whence the result.  Proposition 7.1.7 An F-stable torus T of G is quasi-split if and only if F-rank(T) = F-rank(G), which is the maximum over all F-stable maximal tori T of G of F-rank(T). In particular, if F is a Frobenius endomorphism, an

7.1 F-rank

115

F-stable torus T is quasi-split if and only if maximally split; that is, T1 of 7.1.4 is a maximal split torus of G. Proof Taking the quotient by Ru (G) we can assume G reductive. We use again that any F-stable torus is conjugate to (T0 , wF) for some w, where T0 is quasisplit. Since F preserves a Borel subgroup B containing T0 , if Π is the basis of the roots corresponding to the order defined by B, then by 4.2.21 τ preserves the open cone C = {x ∈ E | α(x) > 0 for any α ∈ Π}. This implies that there is a τfixed point in C, thus lying outside all reflecting hyperplanes (indeed, any α ∈ Φ+ being a linear combination of elements of Π with nonnegative coefficients, we also have α(x) > 0 for x ∈ C). This means that τ has a regular eigenvector for the eigenvalue 1, see Springer (1974, 6.5), thus by Springer (1974, 6.2) Eτ has maximal dimension among all the Ewτ , and any other element such that dim Ewτ = dim Eτ satisfies wτ = vτv−1 ; that is, wτ is conjugate to τ, or in other terms w is F-conjugate to 1.  Example 7.1.8 Let us compute the Fq -rank of the unitary group Un , where the Frobenius F is as in 4.3.3. If B is the Borel subgroup of upper triangular matrices, the group F B is the Borel subgroup of lower triangular matrices, and is also equal to w0 B, where w0 is the longest element of W (T) where T is the torus of diagonal matrices (w0 is the permutation (1, 2, . . . , n) → (n, n − 1, . . . , 1)). Thus T has type w0 with respect to some quasi-split torus, so (T, w0 F ) is geometrically conjugate to (T0 , F ) where T0 is a quasi-split torus. The Fq -rank of Un is thus the dimension of (X(T) ⊗ R) w0 F /q ; that is, the number of eigenvalues q of w0 F , which is equal to n/2 . Thus (Un , F ) is not split and in particular not isomorphic to (GLn , F). The following lemma relates the Fq -rank of a group to that of its semi-simple quotient. Lemma 7.1.9

With the notation of 7.1.5(ii), we have F-rank(G) = F-rank(G/RG) + F-rank(RG).

Proof Taking the quotient by Ru (G) we can assume G reductive, then R(G) is a torus. Let T be an F-stable maximal torus of G. Applying to the exact sequence 1 → RG → T → T/RG → 1 the exact functor X, see 1.2.4, we get 0 ← X(RG) ← X(T) ← X(T/RG) ← 0.

116

The Duality Functor and the Steinberg Character

This sequence remains exact when tensoring with R, and it remains so when taking the fixed points under τ, as τ, being of finite order, is semi-simple. If we take for T a quasi-split torus of G, we get the result.  Definition 7.1.10 Let G be a connected algebraic group over Fq , with a Frobenius root F. We call semi-simple F-rank of G, denoted by r(G), the F-rank of G/RG. Lemma 7.1.11 Let T be an F-stable maximal torus of a connected reductive group G, and let E = X(T) ⊗ R; if Φ∨ is the set of coroots of G relative to T, we have F-rank(RG) = dim(Φ∨ ⊥ ∩ Eτ ). Proof We first use that (T, F) is GF -conjugate to (T0 , wF) for some quasisplit T0 and w ∈ WG (T0 ); as W acts trivially on Φ∨ ⊥ , the actions of τ and wτ on Φ∨ ⊥ are the same and we may assume w = 1. The lemma is a con∼ → Φ∨ ⊥ , which itself results from sequence of the isomorphism X(RG) ⊗ R − the second exact sequence of 7.1.9 tensored by R: indeed Φ is the root system of the semi-simple group G/RG relative to the quasi-split torus T/R(G), so it spans X(T/R(G)) ⊗ R (see 2.3.5(iii)).  Proposition 7.1.12 Let G be a connected reductive group with a Frobenius root F. In the context of the (B, N) pair for G defined by an F-stable pair T ⊂ B, let PI = BWI B be the F-stable parabolic subgroup of G defined by an F-stable subset I ⊂ S. Then r(PI ) = |I/F|. Proof Without changing the semi-simple rank, we may replace PI by its Levi subgroup LI . We may then also assume that LI = G and is semi-simple, so that I = S. We have then to show that dim(Eτ ) = |S/F|. As τ is of finite order and normalises W, it follows that the group τ, W is finite, thus there is a scalar product on E which is W and τ invariant. Thus τ is defined by the permutation it induces of the unitary vectors for this scalar product colinear to the elements of Π. The proposition then results from the fact that the dimension of the 1eigenspace of a linear map which permutes a basis is the number of orbits of the map on the basis. 

7.2 The Duality Functor Here C will denote class functions over C. In the rest of this chapter we assume that G is connected reductive. Definition 7.2.1 Let F be a Frobenius root on G over Fp and let B be an Fstable Borel subgroup of G. By duality we mean the operator DG on C(GF ) defined by

7.2 The Duality Functor DG =



117

∗ G (−1) r(P) RG L ◦ RL

P⊃B

where the summation is over the set of F-stable parabolic subgroups of G containing B and where L denotes an arbitrarily chosen F-stable Levi subgroup of P. Remark 7.2.2 properties:

This definition is consistent because of the two following

◦ ∗RG depends only on the parabolic subgroup P and not (i) The operator RG L L on the chosen Levi subgroup. (ii) DG does not depend on the F-stable Borel subgroup B. Property (i) comes from the fact that two F-stable Levi subgroups of P are ◦ ad x−1 = RG conjugate under PF and from the equality RG xL along with its L ∗ G ◦ ∗RG ( χ) = analogue for RL . We could also deduce it from the equality RG L L  GF F −1 F χ(pu) for p ∈ P , where U = R IndPF ( χ ) where χ (p) := |U | F u (P). u∈U Property (ii) comes from the fact that two F-stable Borel subgroups of G are GF -conjugate. Proposition 7.2.3 The functor DG is self-adjoint. Proof This is straightforward from the definition of DG and the fact that RG L and ∗RG are adjoint to each other (see 5.1.8(iii)).  L Theorem 7.2.4 (C. Curtis) For any F-stable Levi subgroup L of an F-stable parabolic subgroup of G we have G DG ◦ RG L = RL ◦ DL .

The proof we shall give is valid even when the parabolic subgroup is not Fwith the Lusztig functor which is stable, in which case we have to replace RG L – this functor will be extensively its generalisation and is again denoted by RG L studied later, starting from Chapter 9 – and the formula has then to be written G ε G DG ◦ RG L = ε L RL ◦ DL .

In the case of an F-stable parabolic subgroup this formula reduces to the previous one because L then contains a quasi-split torus of G, so that F-rank(G) = F-rank(L). that we shall use in the proof are transitivity, the The only properties of RG L G ◦ ad x (for x ∈ GF ) and the “Mackey formula”; these facts = R formula RG x L L hold for the Lusztig functor, with some restrictions for the Mackey formula (see Chapter 9). The reader will check that in the remainder of the book we shall use 7.2.4 for the Lusztig functor only in those cases where the Mackey formula has been proved.

118

The Duality Functor and the Steinberg Character

Proof By the Mackey formula, using the equality (MF \S(M, L) F /LF ) −1 = LF \S(L, M) F /MF we get DG ◦ RG L =



∗ G G (−1) r(Q) RG M RM RL

Q⊃B

=



Q⊃B

(−1) r(Q)



−1 M ∗ L RG M ad x R xM∩L R xM∩L . x

x∈LF \S(L,M) F /MF

G M G We then use the equalities RG ad x−1 = RG xM and R xM R xM∩L = R xM∩L = M RL to get RG L xM∩L x

  G  DG ◦ RG (−1) r(Q) RLxM∩L ∗RLxM∩L  . L = RL ◦  x∈LF \S(L,M) F /MF Q⊃B  We now transform the right-hand side of the above equality. Let BL be a fixed F-stable Borel subgroup of L. For each parabolic subgroup Q ⊃ B of G with Levi subgroup M we define a bijection P from LF \S(L, M) F /MF to the set of parabolic subgroups of G which are GF -conjugate to Q and contain BL . If x ∈ S(L, M) F , as xQ and L contain a common maximal torus, the group xQ∩L is a parabolic subgroup of L – see 3.4.8(i). As it is F-stable this parabolic subgroup contains an F-stable Borel subgroup of L and such a Borel subgroup is conjugate by an element l ∈ LF to BL . Then we have lx = x, where x denotes the image of x in LF \S(L, M) F /MF , and lxQ ⊃ BL . The map P : x → lxQ is well defined; that is, lxQ does not depend on the chosen l, since if l lxQ also contains BL then l lxQ ∩ L = lxQ ∩ L by 1.3.6(ii), thus l ∈ lxQ ∩ L. The map P is onto. Indeed, let xQ with x ∈ GF be a parabolic subgroup in GF containing BL , let T0 be a maximal torus of BL and let M be the unique Levi subgroup of xQ containing T0 (see 3.4.2). Then M and xM are conjugate under xQF ; that is, there exists q ∈ QF such that M = xqM. But then xqM∩L ⊃ T0 , so xq is in S(L, M) F and satisfies P (xq) = xqQ = xQ. We now prove the injectivity of P. If xQ = yQ with x and y in S(L, M) F then since NGF (Q) = QF there exists q ∈ QF such that y = xq. The groups L ∩ xM and L ∩ xqM are two F-stable Levi subgroups of the parabolic subgroup L ∩ xQ of L. Thus there exists l ∈ LF ∩ xQ such that L ∩ xM = l (L ∩ xqM) = L ∩ lxqM, so xM and lxqM are two Levi subgroups of xQ containing a common maximal torus (any maximal torus of L ∩ xM), so they are equal. Thus lxqx−1 ∈ NGF ( xM) ∩ xQ = xM (equality by 3.4.2), so y = xq ∈ l−1 xM ⊂ LxM which implies, using the Lang–Steinberg theorem in L ∩ xM, that y ∈ LF xMF , whence y = x.

7.2 The Duality Functor

119

We have noticed in 7.2.2(i) that the functor RLxM∩L ∗RLxM∩L depends only on xQ∩L. We denote by f ( xQ) this functor; we have f ( xQ) = f (P (x)) as P (x) and xQ are conjugate under LF . Since P is bijective and r(Q) = r( xQ) = r(P (x)), we get  G = R ◦ (−1) r(Q ) f (Q ) DG ◦ RG L L Q

where Q in the summation runs over all F-stable parabolic subgroups of G containing BL . The summation in the right-hand side can be written   (−1) r(Q )  RLM0 ◦ ∗RLM0 Q0 ⊃BL  {Q |Q ∩L=Q0 }  where Q0 runs over F-stable parabolic subgroups of L containing BL , where M0 denotes an F-stable Levi subgroup of Q0 , and Q runs over F-stable parabolic subgroups of G such that Q ∩ L = Q0 . Theorem 7.2.4 is thus a consequence of the following result. 

Proposition 7.2.5 Let H be an F-stable reductive subgroup of maximal rank of G, and let QH be an F-stable parabolic subgroup of H. Then we have  εG (−1) r(Q) = (−1) r(QH ) ε H {Q |Q∩H=QH }

where Q in the summation runs over F-stable parabolic subgroups of G such that Q ∩ H = QH . Proof Let T be an F-stable maximal torus of QH , and let E = X(T) ⊗ R. The coroots Φ∨ of G with respect to T define a hyperplane arrangement H = {α ⊥ | α ∈ Φ∨ } on E. By 4.2.5, we have F = qτ on E and τ induces a permutation of H since it differs by scalars from the permutation of the coroots attached to the p-morphism defined by F. Lemma 7.2.6 There is a bijection Q → FQ between parabolic subgroups of G containing T and facets of the hyperplane arrangement H . The parabolic subgroup Q is F-stable if and only if FQ ∩ Eτ  ∅, and then r(Q) = dim Eτ − dim(FQ ∩ Eτ ). Proof We call here facets intersection of hyperplanes and of open half-spaces defined by hyperplanes. Thus a facet can always be written {x ∈ E | x, α = 0 for α ∈ Φ∨1 , x, α > 0 for α ∈ Φ∨2 }, where Φ∨1 is some subset of Φ∨ and Φ∨2 is the half of Φ∨ − Φ∨1 for which the scalar products are positive. Then Φ∨1 ∪ Φ∨2 is clearly a parabolic subset thus defines a parabolic subgroup (see 3.4.5). Conversely, a parabolic subset Ψ∨ defines a facet where Φ∨1 = Ψ∨ ∩ − Ψ∨ and Φ∨2 = Ψ∨ − Φ∨1 .

120

The Duality Functor and the Steinberg Character

If Q is F-stable, then FQ is τ-stable, and the sum of the τ-orbit of x ∈ FQ defines an element of Eτ , non-zero since it has a positive scalar product with some coroot (unless FQ is the zero facet, which is in Eτ ). Conversely a facet which meets Eτ non-trivially must be τ-stable since τ permutes the facets. Finally, if Q is F-stable, it has same F-rank as G, thus r(Q) = F-rank(G) − F-rank(R(Q)) and if Q corresponds to a parabolic subset of the form Φ∨1 ∪ Φ∨2 as above, since Φ∨2 corresponds to the unipotent radical of Q, it follows from 7.1.11 that r(Q) = dim Eτ − dim(Φ∨1 ⊥ ∩ Eτ ) = dim Eτ − dim(FQ ∩ Eτ ).  In the same way we associate to QH a facet FQH of the set of hyperplanes HH = {α ⊥ | α ∈ Φ∨H } where Φ∨H is the set of coroots of H relative to T, and τ the same formula applies, thus, factoring out (−1) dim E , the formula we have to prove can now be written  τ τ (−1) dim( FQ ∩E ) = (−1) dim( FQH ∩E ) . {Q |Q∩H=QH }

Further using that a facet {x ∈ E | x, α = 0 for α ∈ Φ∨1 , x, α > 0 for α ∈ Φ∨2 } is in the facet of HH given by {x, α = 0 for α ∈ Φ∨1 ∩ Φ∨H , x, α > 0 for α ∈ Φ∨2 ∩ Φ∨H }, we see that Q ∩ H = QH ⇔ FQ ⊂ FQH , so we can rewrite the formula as  τ (−1) dim F = (−1) dim FQH ∩E . F ⊂ FQH ∩Eτ

But this last formula is well-known: it expresses the fact that the Euler characteristic of the convex open set FQH ∩ Eτ can be computed using the subdivision  into facets defined by HG ∩ Eτ .  Corollary 7.2.7 (of 7.2.4) Proof

∗RG L

◦ DG = DL ◦ ∗RG . L

This is just the adjoint formula to 7.2.4.



Corollary 7.2.8 DG is an isometry for the scalar product on C(GF ) and DG ◦ DG is the identity functor.

7.2 The Duality Functor

121

Proof The first assertion is a consequence of the second one using the fact that DG is self adjoint (7.2.3) which implies that it is self adjoint with respect to the scalar product on class functions. Let us prove the second assertion. Let T ⊂ B be an F-stable pair of a maximal torus and a Borel subgroup of G, and let (W, S) be the corresponding Coxeter system. The F-stable parabolic subgroups of G which contain B correspond one-to-one to the F-stable subsets of S; that is, to the subsets of the set S/F. If LI denotes a Levi subgroup of the parabolic subgroup corresponding to the F-stable subset of S of image I ⊂ S/F, then r(LI ) = |I| by 7.1.12.  ∗RG , whence Thus we can write DG = I ⊂S/F (−1) |I | RG LI LI   |I | G ∗ G ∗ G DG ◦ DG = (−1) RLI ◦ RLI ◦ DG = (−1) |I | RG LI ◦ DLI ◦ RLI I ⊂S/F

=



(−1)

I ⊂S/F

|I |

 K ⊂I

I ⊂S/F

(−1) |K | RG LI

◦

RLLIK

◦ ∗RLLIK ◦ ∗RG LI

 ∗ G (−1) |K |  (−1) |I |  RG LK ◦ RLK = Id; K ⊂S/F I ⊃K   the last equality uses the fact that I ⊃K (−1) |I | = 0 if K  S/F. =





The following result shows that the dual of γ ∈ Irr(GF ) is irreducible up to a sign determined by the Harish-Chandra series of γ. Corollary 7.2.9 Let (L, λ) be a cuspidal pair and let γ ∈ Irr(GF ) be such λGF  0. Then (−1) r(L) DG (γ) ∈ Simp(GF | (L, λ)). that γ, RG L Proof We have γ, γGF = DG γ, DG γGF (see 7.2.8), so DG (γ) is an irreducible character up to sign. As λ is cuspidal, all summands in the formula for DL (λ) are zero except one of them, thus DL (λ) = (−1) r(L) λ. So, by 7.2.3, 7.2.4 and this last fact, we have r(L) γ, DG RG (−1) r(L) DG γ, RG L λGF = (−1) L λGF G = (−1) r(L) γ, RG L DL λGF = γ, RL λGF > 0,



whence the result.

We want now to describe the action of DG on the principal series. The result will be given in 7.2.13. We start with some lemmas. Lemma 7.2.10 Let (W, S) be a Coxeter system. For any character χ ∈ Irr  W (CW) we have I ⊂S (−1) |I | IndW WI ResWI χ = χ ⊗ sgn. W W W Proof Since IndW WI ResWI χ = χ ⊗ (IndWI ResWI 1), the lemma results from the case χ = 1. Now for w ∈ W we have −1 x IndW WI 1(w) = |WI | |{x ∈ W | w ∈ WI }|.

122

The Duality Functor and the Steinberg Character

We first give the proof for Weyl groups, placing ourselves in the space E = X(T) ⊗ R, where T is a torus of a reductive group having W as Weyl group, and we apply the methods of the proof of 7.2.5. If Φ+ is the positive root system corresponding to S, to an x in the above set we attach the facet x(FI ) where FI = {e ∈ E | α, e = 0 for α ∈ Φ∨I , α, e > 0 for α ∈ Φ+∨ − φ∨I } is the facet attached to the standard parabolic subgroup subgroup PI . The condition that xWI  w is equivalent to x(FI ) ⊂ Ew (the fixed space of the parabolic subgroup xWI is a subspace of the fixed space of one of its elements w). Further the stabiliser in W of FI is WI , so the formula above becomes IndW WI 1(w) = |{F ∼ FI | F ⊂ Ew }| where ∼ means that F is a facet conjugate to FI , and the  total sum becomes F ⊂Ew (−1) codim F , since every facet is conjugate to some FI (every parabolic subgroup is conjugate to some PI ) and codim FI = |I|. By the argument at the end of 7.2.5 this sums to codim Ew , which is equal to (−1) l(w) by the argument in the proof of 7.1.2 applied to τ = w. For the general case where W is not a Weyl group, the above proof still applies verbatim using for E the space of the reflection representation of W, defined for example in Bourbaki (1968, V, §4, Proposition 3).  Lemma 7.2.11 Let LJ be an F-stable standard Levi subgroup of GF ; for F γ , λ  = ResW χ, ψW F , where ψ ∈ Irr WJF and χ ∈ Irr W F we have  ∗RG LJ χ ψ LF WF J

J

J

γ χ (resp. λψ ) are the characters of GF (resp. LFJ ) corresponding to χ (resp. ψ) by 6.2.14(i). LF

F

1) and HJ := EndCLF (IndBJF ∩L 1). Let χq Proof Let H := EndCGF (IndG BF J J (resp. ψq ) be the characters of H (resp. HJ ) corresponding to χ (resp. ψ). We have HJ = eBF ∩LJ CLFJ eBF ∩LJ . Note that if e ∈ HJ is a primitive idempotent such that HJ e is the representation with character ψq , then e is primitive in CLFJ and affords λψ . Indeed e is an idempotent which, being multiple of eBF ∩LJ , acts by 0 on irreducible modules not in the principal series and acts also by 0 on principal series modules other that the one with character ψq since the action of HJ on the principal series is by restriction from LFJ (see Exercise 6.1.3(ii)). Moreover λψ (e) = ψq (e) = 1, hence e is primitive in CLFJ . Thus, if UJ is the unipotent radical of the parabolic subgroup PJ , the idempotent eeUF affords the inflation of λψ to PFJ and CGF eeUF is a representation J

J

of GF with character RG λ (see Exercise 6.1.3(i)). Similarly He is a repreLJ ψ G sentation of H with character IndH HJ ψq . Thus λ χ (eeUF ) = λ χ , RLJ λψ GF (see J

Exercise 6.1.3(i)), and similarly χq (e) is the multiplicity of χq in IndH HJ ψq which is also the multiplicity of χ in IndW (ψ) as can be seen using the fact WJ that the scalar product on the generic algebra specialises to the scalar product , H and also that – by 6.3.7 and 6.3.12 – it specialises to |W F |, W F .

7.3 Restriction to Centralisers of Semi-Simple Elements

123

Since H = eBF CGF eBF and UJ ⊂ B we have He = HeeUF and χq (e) = χq (eeUF ) = λ χ (eeUF ), since χq is the restriction to H of λ χ . Thus the two multiplicities are equal, whence the result by adjunction.  Corollary 7.2.12 With notation as in 7.2.11, if T is a quasi-split torus of LJ , the map ψ → γψ extends to an isometry between the span of Irr(WJF ) and the span of the irreducible constituents of RLTJ (1) which, for any χ ∈ Irr W F , maps F χ to ∗RG (γ χ ). In particular ∗RG (1 ) = 1LF . ResW LJ LJ G F WF J

J

preserves HarishProof This follows from 7.2.11 and the fact that ∗RG L Chandra series.  Proposition 7.2.13 With same notation as in 7.2.11, for χ ∈ Irr W F , we have DG γ χ = γ χ ⊗sgn where sgn is the sign character of the Coxeter group W F .  ∗RG (see the proof of 7.2.8). Proof We have DG = I ⊂S/F (−1) |I | RG LI LI F By 7.2.12, for χ, χ ∈ Irr W we have ∗ G W W  ∗RG LJ γ χ , RLJ γ χ LF = ResW F χ, ResW F χ W F , J

F

F

J

J

J

whence by adjunction ∗ G W W RG LJ RLJ γ χ , γ χ GF = IndW F ResW F χ, χ W F .

We deduce DG γ χ , γ χ GF =

 I ⊂S/F

F

F

J

J

(−1) |I | IndW ResW χ, χ W F =  χ ⊗ sgn, χ W F , WF WF F

F

I

I

the last equality by 7.2.10. This gives the result using 7.2.9 and using again the F isometry between the spans of Irr(W F ) and of the constituents of IndG 1.  BF Corollary 7.2.14 Under the bijection of 6.2.14(i) the sign character of W F parametrises the Steinberg character of GF (see 7.4.1 for its definition). Proof This comes from 7.2.13 and the facts that the trivial character of W F parametrises the trivial character of GF by 6.2.14(i) and that DG StGF = 1GF . 

7.3 Restriction to Centralisers of Semi-Simple Elements Before stating the main result 7.3.7 of this section, we need to state some further properties of Harish-Chandra induction. Let P = U  L be an F-stable Levi decomposition of an F-stable parabolic subgroup of G. We begin with a finite group theoretic result.

124

The Duality Functor and the Steinberg Character

Proposition 7.3.1 Let s be the semi-simple part of an element l ∈ LF ; then the map UF × (CG (s) ∩ UF ) → lUF defined by (y, z) → y (lz) is onto and all its fibres have the same cardinality equal to |CG (s) ∩ UF |. Proof Let v be the unipotent part of l and let V = v, UF ; the group V is the unique Sylow p-subgroup of l, UF  because it is normal and the quotient is cyclic generated by a p -element (the image of s). For any u ∈ UF we must count the number of solutions (y, z) ∈ UF × (CG (s) ∩UF ) to the equation y (lz) = lu. Let s u be the Jordan decomposition of lu; since v normalises CG (s) ∩ UF , the element vz is unipotent in CG (s) and s.vz is the Jordan decomposition of lz, so the equation above is equivalent to the system of two equations ys = s and y (vz) = u . But s and s  are two p -complements of the normal subgroup V of l, UF  so, by the Schur–Zassenhaus theorem – see Gorenstein (1980, 6.2.1) – they are conjugate by an element of V. As s cannot be conjugate to one of its powers other than itself in the quotient group l, UF /V, there exists y ∈ V such that ys = s . The element y can be taken in UF as v centralises s. If we prove that the −1 element z defined by z = v−1 . y u is in CG (s) ∩ UF then we are done, as there are |CG (s) ∩ UF | solutions y to ys = s and the value of z is uniquely defined by that of y. −1 −1 But v and s commute, and y u commutes with y s = s, so z commutes −1 with s. It remains for us to show that z is in UF : we have lz = y (lu) ∈ −1 y (lUF ) = lUF (this last equality because y ∈ UF ), whence the result.  The following corollaries show the usefulness of 7.3.1. Definition 7.3.2 (i) We denote by C(H)p the set of class functions f on a finite group H such that f (x) = f (xp ) for any x ∈ H, where xp is the p -part of the element x. (ii) We denote by χp (or χG p to specify the group) the characteristic function F of Gu ; note that this function is in C(GF )p . Corollary 7.3.3 UF × LF .

If f ∈ C(GF )p , it satisfies f (lu) = f (l) for any (u, l) ∈

Proof By 7.3.1, if l = sv is the Jordan decomposition of l, there exists y ∈ UF and z ∈ CG (s) ∩ UF such that y (lz) = lu, so, in particular f (lz) = f (lu). As we have seen in the proof of 7.3.1 s.vz is the Jordan decomposition of lz, so f (lz) = f (s) = f (l), whence the result.  Corollary 7.3.4

f = ResG f and ∗RG ( χf ) = If f ∈ C(GF )p , then ∗RG L L LF

( ∗RG χ). ResG f for any χ ∈ C(GF ). L LF F

F

7.3 Restriction to Centralisers of Semi-Simple Elements



Proof This result follows from 7.3.3 and 5.1.12. Corollary 7.3.5 χ ∈ C(LF ).

125

GF

If f ∈ C(GF )p , then RG ( χ ResLF f ) = (RG χ).f for any L L 

Proof This is what we get by adjunction from 7.3.4.

The next property has been proved and used, under somewhat stronger assumptions, in Curtis (1980); we shall generalise it to Lusztig induction in Chapter 10. Corollary 7.3.6

Let s be the semi-simple part of l ∈ LF ; then

(ResLC

F L (s)

0F

∗ G RL χ)(l)

0

(s) G = ( ∗RCCG(s) 0 ResC L

F

G (s)

0F

χ)(l).

Proof First let us note that the above formula has a well-defined meaning because l ∈ CL (s) 0 by 3.5.3, and CL (s) 0 is a Levi subgroup of CG (s) 0 . Indeed, since CG (s) 0 is a connected reductive subgroup of maximal rank of G by 3.5.1(i), by 3.4.10(ii) and (iii) P ∩ CG (s) 0 is a parabolic subgroup of CG (s) 0 with Levi decomposition (U ∩ CG (s) 0 )  (L ∩ CG (s) 0 ). So L ∩ CG (s) 0 is connected and L ∩ CG (s) 0 = CL (s) 0 . We have by 5.1.12 and 7.3.1   F −1 χ(lu) = |CG (s) ∩ UF | −1 χ(lz). ( ∗RG L χ)(l) = |U | u∈UF

z∈CG (s)∩UF

We then get the result since CG (s) ∩ U = CG (s) 0 ∩ U, see 3.5.3 again. Proposition 7.3.7



If s is the semi-simple part of x ∈ GF , then

ε G (DG χ)(x) = ε CG (s)0 (DCG (s)0 ◦ ResG C

G (s)

0

χ)(x).

Proof If L is an F-stable Levi subgroup of an F-stable parabolic subgroup P F ◦ ∗RG χ = IndG ( χ ), if χ (p) = ∗RG ( χ)(p) where of G, we have by 7.2.2 RG L L L PF p is the image of p in L, whence   ∗ G F −1 g−1 x) = (RG ( ∗RG ( ∗RG L ◦ RL χ)(x) = |P | L χ)( L χ)(x) {g∈GF | gPx}

{P ∼ P |P x} GF

P

where in the last summation denotes an F-stable parabolic subgroup of G, where L is an F-stable Levi subgroup of P , and x denotes the image of x in L . Using this in the definition of DG gives    (−1) r(P) ( ∗RG χ)(x) = (−1) r(P ) ( ∗RG (DG χ)(x) = L L χ)(x). P⊃B

{P ∼ P |P x} GF

P x

(7.3.1)

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By 7.3.6 we then get    0 GF G (s) (DG χ)(x) = (−1) r(P) ∗RCL∩C Res χ (x) C (s) 0F (s) 0 G

Px

=

 P

G

   0 F  (−1) r(P)  ∗RCL G (s) ResG χ (x) CG (s) 0F 0 =P } {P |P∩C (s) G  

where P runs over the set of F-stable parabolic subgroups of CG (s) 0 and where L is any F-stable Levi subgroup of P . If we now apply 7.2.5 with H = CG (s) 0 and compare with the equality (7.3.1) applied in CG (s) 0 , we get the result. 

7.4 The Steinberg Character In this section we use duality to define and study the famous “Steinberg character” which was originally defined in Steinberg (1956, 1957). Its definition below as the dual of the trivial character is due to Curtis. Definition 7.4.1 The irreducible character StGF = DG (1GF ) where 1GF is the trivial character of GF is called the Steinberg character of GF . Note that StGF is a true character (and not the negative of a character) by 7.2.9 since – as we have seen in 6.2.14(i) – 1GF ∈ Simp(GF | (T, 1)) where T is a quasi-split torus of G, and r(T) = 0. Proposition 7.4.2 Proof

∗RG St L GF

= StLF .

As the duality and the Harish-Chandra restriction commute we get ∗ G RL StGF

∗ G = ∗RG L DG (1GF ) = DL RL (1GF ) = DL (1LF ) = StLF ,



the third equality by 7.2.12, whence the result. In the case of a torus we have the following more precise result.

Lemma 7.4.3 Let T be an F-stable maximal torus of an F-stable Borel subgroup B of G; then F

F

StGF = IndBTF 1. ResG BF Proof

Using the definitions of StGF and of DG we get  F (−1) |I | IndG 1 StGF = PF I ⊂S/F

I

7.4 The Steinberg Character

127

(the notation is the same as in the proof of 7.2.8). So we have  F F F ResG (StGF ) = (−1) |I | ResG IndG 1 BF BF PF I

I ⊂S/F



=



(−1) |I |

I ⊂S/F

w∈ZI

F

PF

IndBBF ∩ wPI ResBIF ∩ wP 1, I

the last equality following from the ordinary Mackey formula, where we have denoted by ZI the set of F-stable reduced-I elements of W = WG (T) where I ⊂ S is the preimage of I ⊂ S/F – the set ZI is a set of representatives for the F-stable double cosets B\G/PI and is also a set of representatives for the double cosets BF \GF /PFI ; see 3.2.3(ii), 4.4.8 and 5.2.2(ii). We have B ∩ wPI = B ∩ wB. Indeed, since for any v ∈ WI we have l(w) + l(v) = l(wv), for any v ∈ WI we have wBvB ⊂ BwvB; whence Bw ∩ wBvB ⊂ Bw ∩ BwvB, and this last intersection is empty unless v = 1. Hence we have  wBvB = Bw ∩ wB. Bw ∩ wPI = Bw ∩ wBWI B = Bw ∩  v∈WI  F

Using this result in the formula for ResG StGF and exchanging the summations BF give F

StGF = ResG BF

 F  (−1) |I |  IndBBF ∩ wB 1. w∈W  {I ⊂S/F |w∈ZI }  

For w ∈ W F let Sw := {O ∈ S/F | ∃s ∈ O, l(ws) > l(w)} = {O ∈ S/F | ∀s ∈ O, l(ws) > l(w)}, the last equality since w ∈ W F . We have ZI = {w ∈ W F | I ⊂ Sw }, so   (−1) |I | = (−1) |I | , {I ⊂S/F |w∈ZI }

I ⊂Sw

which is different from zero only if Sw = ∅, in which case w is the longest  element of W, see 2.1.5(i), and B ∩ wB = T, whence the result. Corollary 7.4.4

For x ∈ GF , we have

⎧ ⎪ ε G ε (CG (x)0 ) |CG (x) 0F |p StGF (x) = ⎨ ⎪0 ⎩

if x is semi-simple, otherwise.

Proof Let s be the semi-simple part of x. We have StGF (x) = DG (1)(x) = ε G ε CG (s)0 (DCG (s)0 (1))(x) = ε G ε CG (s)0 StCG (s)0F (x) by 7.3.7. So we may assume that s is central in G. But then there exists an F-stable Borel subgroup B which contains x. Indeed the unipotent part of x

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is contained in an F-stable Borel subgroup as is any F-stable unipotent element (see 4.4.3), and s, being central, is contained in all Borel subgroups. So F F StGF )(x) = (IndBTF 1)(x). Thus by Lemma 7.4.3 we have StGF (x) = (ResG BF StGF (x) = 0 if x has no conjugate in TF — that is, x is not semisimple; and if x = s we get StGF (x) = |BF |/|TF | = |GF |p , 

the second equality by 4.4.1(iv), whence the result. GF

is Corollary 7.4.5 (of 7.4.4) The dual of the regular character regGF of F F DG (regGF ) = γp , where γp denotes the element of C(G ) whose value is |G |p on unipotent elements and 0 elsewhere. Proof By 7.3.4 and 7.3.5 we have DG ( χ.f ) = DG ( χ).f for any χ ∈ C(GF ) and any f ∈ C(GF )p . So, since γp ∈ C(GF )p , we have DG (γp ) = DG (1.γp ) = DG (1)γp = StGF .γp = regGF , 

the last equality by 7.4.4. Corollary 7.4.6 (of 7.4.5) to (|GF |p ) 2 . Proof

The number of unipotent elements in GF is equal

This formula is easily deduced from 7.4.5 by writing regGF , regGF GF = γp , γp GF . 

Proposition 7.4.7 Let L be an F-stable Levi subgroup of an F-stable parabolic subgroup of G, and let χ ∈ Irr(LF ); then F

G StGF .RG L ( χ) = IndLF (StLF . χ).

Proof Let C(GFp ) be the space of class functions on GFp (the set of elements of order prime to p, which is also the set of semi-simple elements of GF ; see 1.1.5); we identify C(GFp ) with the subspace of C(GF ) consisting of functions which are zero outside GFp . For f ∈ C(GFp ), let us denote by Br f the element of C(GF ) defined by (Br f )(x) = f (s) where s is the semi-simple part of x (Br is similar to the “Brauer lifting” in block theory). Then we have Br f = DG (StGF .f ). Indeed Br f ∈ C(GF )p , so – as in the proof of 7.4.5 – we have DG (1. Br f ) = DG (1). Br f = StGF . Br f = StGF .f , whence the result as DG is an involution. So for any χ ∈ C(GF ) we have DG ( χ.StGF ) = Br( χ|GF ). But for any f ∈ C(GF )p , as Br f ∈ C(GF )p , we have, by 7.3.4, ∗ G RL (Br f )

F

GF

= ResG Br f = Br(ResLFp f ). LF p

p

7.4 The Steinberg Character

129

If we apply 7.2.4 and then the two previous formulae we find ∗ G ∗ G DL ( ∗RG L ( χ.StGF )) = RL (DG ( χ.StGF )) = RL (Br( χ|GF )) p

= Br( χ|LF ) = p

F DL (StLF . ResG LF

χ),

which gives after applying DL ∗ G RL ( χ.StGF )

F

= (ResG χ). StLF , LF

whence the proposition by adjunction since the multiplication by StGF is selfadjoint.  Notes The duality operator DG was introduced simultaneously by Curtis (1980), Alvis (1980), Kawanaka (1982), and Lusztig. Deligne and Lusztig studied it in a geometric way in Deligne and Lusztig (1982, 1983); see also Digne and Michel (1982) which we follow here, in particular for the proof of 7.2.4. In the case where F is a Frobenius endomorphism, the notion of F-rank and semi-simple F-rank (called Fq -rank and Fq -semi-simple rank) can be found in, for example, Borel and Tits (1965, 4.23 and 5.3.). Our exposition here of the properties of the Steinberg character, in particular in the proofs of 7.4.4, 7.4.5, 7.4.6 and 7.4.7, follows the ideas of Digne and Michel (1982), making a systematic use of the properties of the duality functor.

8 -Adic Cohomology

8.1 -Adic Cohomology Let X be an algebraic variety over Fp and  be a prime number different from p. Using the results of Grothendieck one can associate to X groups of -adic cohomology with compact support Hci (X, Q ) – see Deligne (1977, Rapport, 2.10) – which are Q -vector spaces of finite dimension. This cohomology theory is a powerful tool. It will be used in the next chapter to define Deligne–Lusztig induction. We recall in this chapter those properties of -adic cohomology that we need. Throughout this chapter X will stand for an algebraic variety over Fp . The reader who wants to keep within the framework of Chapter 4 may assume that all the varieties in this chapter are quasi-projective. Proposition 8.1.1 We have Hci (X, Q ) = 0 if i  { 0, . . . , 2 dim X }. If X is affine we have Hci (X, Q ) = 0 if i  { dim X, . . . , 2 dim X }. References See Grothendieck et al. (1972–1973, XVII, 5.2.8.1) for the first assertion and Grothendieck et al. (1972–1973, XIV, 3.2) together with Grothendieck et al. (1972–1973, XVIII, 3.2.6.2) for the affine case.  Henceforth we shall write Hci (X) for Hci (X, Q ), since we shall consider only cohomology with coefficients in Q . Proposition 8.1.2 Any proper morphism X → Y induces a linear morphism Hci (Y) → Hci (X) for any i, and this correspondence is functorial; a Frobenius root on X is a finite morphism and induces an automorphism of Hci (X). Proof The first assertion comes from Grothendieck et al. (1972–1973, XVII, 5.1.17). Now let F be a Frobenius root such that F n is a Frobenius endomorphism. By 4.1.10 F n is finite. It follows that F n is separated, which implies that, as a factor of a separated morphism, F is separated, see Hartshorne (1977, II, 4.6(e)). It also follows that F n is proper and writing F n = F n−1 ◦ F and 130

8.1 -Adic Cohomology

131

using that F n−1 is separated, we get that F is proper, see Hartshorne (1977, II, 4.8(e)). Finally, being proper and quasi-finite, F is finite, see Grothendieck (1967, 18.12.4). The assertion that F is an automorphism of the cohomology results from the same assertion for F n , for which see for example Deligne (1977, Rapport, 1.2).  Definition 8.1.3 If g ∈ Aut(X) has finite order, we call the Lefschetz number of g the number L(g, X) = Trace(g | Hc∗ (X)), where we put Hc∗ (X):=  i i i (−1) Hc (X). Hc∗ (X) is a virtual Q -vector space of finite dimension. The fundamental property of -adic cohomology is the “Lefschetz theorem”. Theorem 8.1.4 Let F be the Frobenius endomorphism associated to an Fq structure on X. We have |XF | = Trace(F | Hc∗ (X)). Reference



See Deligne (1977, Rapport, 3.2).

Corollary 8.1.5 Assume that X is defined over Fq , with Frobenius endomorphism F, and that g ∈ Aut(X) is an automorphism of finite order commuting with F; then we have L(g, X) = R(t)|t=∞ , where R(t) is the formal series  gF n |tn . − ∞ n=1 |X This result is an example of the power of -adic cohomology: it proves that R(t)|t=∞ is the value at g of some virtual character for any finite subgroup of Aut(X) containing g. No other proof of this fact is known. Proof As g and F commute, they can be reduced to a triangular form in the same basis of ⊕i Hci (X) (note that we do not assume that F is semi-simple). Let λ 1 , . . . , λ k be the eigenvalues of F and x1 , . . . , xk be those of g, and let ε i = ±1 be the sign of the cohomology space (in Hc∗ (X)) in which λ i and xi are eigenvalues. As gF n is also a Frobenius endomorphism on X for any n (see 4.1.11(ii)), we have by 8.1.4 R(t) = − whence R(t)|t=∞ =

k i=1

k ∞   n=1 i=1

ε i xi .

ε i λ ni xi tn =

k  i=1

ε i xi

−λ i t 1 − λit 

Corollary 8.1.6 For any g ∈ Aut(X) of finite order, the Lefschetz number L(g, X) is a rational integer independent of . Proof The independence of  is a consequence of 8.1.5 as any g ∈ Aut(X) is defined over some finite subfield of Fp , see 4.1.11(v). Moreover the proof of 8.1.5 shows that R(t) is in Q (t). As it is a formal series with integral coefficients, it has to be in Q(t). So we have L(g, X) ∈ Q. But a Lefschetz number is

-Adic Cohomology

132

 an algebraic integer since it is equal to ki=1 ε i xi where all xi are roots of unity (of order dividing that of g), whence the result.  In the following propositions we shall list properties of the Lefschetz numbers. When these properties can be proved directly using 8.1.5, we shall give that proof; but we shall also give, without proof, the corresponding result on -adic cohomology, if it exists. Proposition 8.1.7  (i) Let X = X1 X2 be a partition of X into two subvarieties with X1 open (thus X2 closed). We have a long exact sequence · · · → Hci (X1 ) → Hci (X) → Hci (X2 ) → Hci+1 (X1 ) → · · · , whence Hc∗ (X) = Hc∗ (X1 ) + Hc∗ (X2 ) as virtual vector spaces. (ii) If in (i) X1 is also closed, then the exact sequence of (i) splits, and for any i we have Hci (X)  Hci (X1 ) ⊕ Hci (X2 ).  (iii) Let X = j Xj be a finite partition of X into locally closed subvarieties; if g is an automorphism of finite order of X which stabilises (globally) this  partition, we have L(g, X) = {j | gXj =Xj } L(g, Xj ). Proof For (i) and (ii) see, for example, Grothendieck et al. (1972–1973, XVII, 5.1.16.3). For (iii), let F be a Frobenius endomorphism which commutes with g and is such that all Xj are F-stable, see 4.1.11(v). Assertion (iii) results from  n gF n  8.1.5 and the equality |XgF | = {j | gXj =Xj } |Xj |. Proposition 8.1.8 points.

Assume that the variety X is reduced to a finite number of

(i) We have Hci (X) = 0 if i  0 and Hc0 (X)  Q X. (ii) Any permutation g of the finite set X defines an automorphism of the variety X, and Hc∗ (X)  Q X is a permutation module under g. We have L(g, X) = |Xg |. Proof

These results are clear from 8.1.1 and 8.1.7(ii) and (iii).



Proposition 8.1.9 Let X and X be two varieties.  j i (i) We have Hck (X×X )  i+j=k Hc (X)⊗Q Hc (X ) (the “Künneth formula”). (ii) Let g ∈ Aut X (resp. g ∈ Aut X ) be automorphisms of finite order; then we have L(g × g , X × X ) = L(g, X)L(g , X ). Proof For (i) see Grothendieck et al. (1972–1973, XVII, 5.4.3). Let us prove  (ii). We write f ∗ h = i≥0 ai bi ti for the Hadamard product of two formal series   f = i≥0 ai ti and h = i≥0 bi ti . Let F be a Frobenius endomorphism of X

8.1 -Adic Cohomology

133

(resp. X ) which commutes with g (resp. g ). We have to show that the series   n n f = n≥1 |XgF |tn and h = n≥1 |X g F |tn satisfy the relation − ( f ∗ h)|t=∞ = −f |t=∞ × −h|t=∞ . But this result follows easily from the proof of 8.1.5, because these two series are linear combinations of series of the form t/(1 − λt) and such series satisfy the equality.  Proposition 8.1.10 Let H ⊂ Aut X be a finite subgroup such that the quotient variety X/H exists (which is always true if X is quasi-projective). Consider an automorphism g ∈ Aut X of finite order which commutes with all elements of H. Then: (i) The Q [g]-modules Hci (X) H and Hci (X/H) are isomorphic for any i,  (ii) L(g, X/H) = |H| −1 h∈H L(gh, X). Proof For (i) see, for example, Grothendieck et al. (1972–1973, XVII, 6.2.5). Let us prove (ii). Let us choose a Frobenius endomorphism F on X such that g and all elements of H commute with F. Then F is again a Frobenius morphism on X/H (for example, by 4.1.8(i)), and we have  n n |XghF |, |(X/H) gF | = |H| −1 h∈H

whence the result by 8.1.5. Proposition 8.1.11



Let X be an affine space of dimension n; then:

⎧ ⎪ 1 if i = 2n, (i) dim Hci (X) = ⎨ ⎪ 0 otherwise. ⎩ (ii) If F is the Frobenius endomorphism associated to some Fq -structure on X, we have |XF | = qn . (iii) For any finite order g ∈ Aut(X) we have L(g, X) = 1. Proof For (i) see, for example, Srinivasan (1979, 5.7). Note that (iii) is straightforward from (ii) by 8.1.5. Let us prove (ii). Let λ be the scalar by which F acts on the one-dimensional space Hc2n (X); in view of 8.1.4, we want to prove that λ = qn . We can invoke 8.1.12(iii), or reason as follows: for m any m > 0 we have |XF | = λ m so λ is a positive integer. We now show that, if A = A0 ⊗Fq Fq defines the Fq -structure on X, there exists n0 such that A0 ⊗Fq Fqn0  Fqn0 [T1 , . . . , Tn ]. Indeed we have A  Fq [T1 , . . . , Tn ], so if A0 is generated by t1 , . . . , tk and if n0 is such that all ti are in Fqn0 [T1 , . . . , Tn ] we have A0 ⊗Fq Fqn0 ⊂ Fqn0 [T1 , . . . , Tn ] and this inclusion has to be an equality because these two Fqn0 -spaces have equal tensor products with Fq . For any m multiple m  of n0 we have |XF | = qmn , so λ = qn , which proves (ii) by 8.1.4.

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-Adic Cohomology

Proposition 8.1.12 (i) Hc2 dim X (X, Q ) has a basis naturally indexed by the irreducible components of X of maximal dimension. (ii) An automorphism of X of finite order induces on Hc2 dim X (X, Q ) the permutation of the natural basis reflecting its action on the irreducible components of maximal dimension. (iii) If F is the Frobenius endomorphism attached to an Fq -structure on X, then F acts on Hc2 dim X (X, Q ) by qdim X times the permutation of the natural basis reflecting its action on the irreducible components of maximal dimension. Note that in (ii) and (iii), since the cohomology is contravariant the action induced on the cohomology is by the inverse of the permutation on the irreducible components. Proof Let Y be a smooth dense open subvariety of X. Since the complement of Y in X has lesser dimension than X, the long exact sequence 8.1.7(i) shows that Hc2 dim X (X, Q )  Hc2 dim X (Y, Q ). Further we may assume in (ii) (resp. (iii)) that Y is stable by the automorphism (resp. by F) since the union of its images by powers of the automorphism (resp. of F) has the same properties as Y. Thus we are reduced to the case where X is smooth. Then its irreducible components are connected components Xi , thus by 8.1.7(ii) we have canonically Hc2 dim X (X, Q ) = ⊕i Hc2 dim X (Xi , Q ) where the sum is over the connected components of maximal dimension. This reduces the question to the case X irreducible (in the case of F, one can replace F by the composed F0 = F ◦ σ where σ is the automorphism of X which just permutes the connected components in the inverse way of F, thus F0 is a Frobenius by 4.1.11(ii) which fixes the connected components). Finally, the case X irreducible is Milne (1980, Chapter VI, Lemma 11.3).  π

→ X be an epimorphism such that all fibres are Proposition 8.1.13 Let X − isomorphic to affine spaces of the same dimension n. Let g ∈ Aut X, g ∈ Aut X be finite order automorphisms such that g π = πg. (i) The Q [g]-module Hci (X) is isomorphic to Hci−2n (X )(−n), a “Tate twist” of the module Hci−2n (X ) (if g acts on this last module by the action of g ). (ii) We have L(g, X) = L(g , X ).

8.1 -Adic Cohomology

135

See Srinivasan (1979, III page 47) for the definition of a “Tate twist”; note j j that Hc (X )(−n) and Hc (X ) are isomorphic vector spaces and that for any Fq j structure on X the action of F on Hc (X )(−n) is qn times the action of F on j Hc (X ). Proof For (i) see Srinivasan (1979, 5.5, 5.7). Let us prove (ii). We choose Fq -structures on X and X such that π is defined over Fq . By 8.1.11(ii) we have  m m m |π −1 (y) gF | = |X g F |qmn , |XgF | = m

y∈X g F

whence the result by 8.1.5. Proposition 8.1.14 variety X.



Let G be a connected algebraic group acting on the

(i) Any element g ∈ G acts trivially on Hci (X) for any i. (ii) Let g ∈ G be such that the induced isomorphism has finite order; then L(g, X) = L(1, X). Proof The proof of (i) may be found in Deligne and Lusztig (1976, 6.4, 6.5). Let us prove (ii). By 4.1.11(v) there exist Frobenius endomorphisms F on G and X such that the action of G commutes with F, that is F (gx) = Fg Fx for all (g, x) ∈ G × X. If n is a positive integer, by the Lang–Steinberg theorem 4.2.9 n there exists h ∈ G such that h. F h−1 = g. But then x → h−1 x defines a bijection n n n n from XgF onto XF , so we have |XgF | = |XF |, whence the result by 8.1.5.  Proposition 8.1.15 Let g = su be the decomposition of the finite order automorphism g ∈ Aut X into the product of a p -element s and a p-element u. Then we have L(g, X) = L(u, Xs ). Proof There is no known proof of this result by means of a computation using only Lefschetz numbers. One has to use directly the definition of -adic cohomology, see Deligne and Lusztig (1976, 3.1).  Proposition 8.1.16 Let T be a torus acting on a variety X and let g ∈ Aut X be a finite order automorphism commuting with the action of T. Then for any t ∈ T we have L(g, X) = L(g, Xt ). Moreover, if X is affine, we have L(g, X) = L(g, XT ). Proof Let g = su be the decomposition of g into its p -part and its p-part; by 8.1.15 we have L(g, X) = L(u, Xs ). The action of T commutes with g, so commutes also with s, thus T acts on Xs . As this group is connected the action of any t ∈ T on Hc∗ (Xs ) is trivial by 8.1.14. Hence for any t ∈ T we have L(u, Xs ) = L(ut, Xs ) = L(u, (Xs ) t ) = L(u, (Xt ) s ) = L(g, Xt ),

136

-Adic Cohomology

the second and the last equalities coming from 8.1.15. If X is affine 1.2.7 shows that there exists t ∈ T such that XT = Xt .  Notes The theory of -adic cohomology is expounded in Grothendieck et al. (1972– 1973), Deligne (1977) and Grothendieck et al. (1977). For the properties of the Lefschetz numbers one can also refer to Lusztig (1978). A good survey, with some proofs, of properties needed in our case may be found in Srinivasan (1979, Chapter V).

9 Deligne–Lusztig Induction: The Mackey Formula

We are going now to extend the construction of the functor RG to the case L where L is an F-stable Levi subgroup of G which is not the Levi subgroup of any F-stable parabolic subgroup of G. Example We would like to generalise the construction of 5.3.2 to the case of the unitary groups (see 7.1.8), but the construction used there does not work, since   the Frobenius morphism on Un+m maps the parabolic subgroup Un ∗ Un 0 to the parabolic subgroup which is not conjugate to it 0 Um ∗ Um   Un 0 is not a if n  m. Actually, in this case the F-stable Levi subgroup 0 Um Levi subgroup of any F-stable parabolic subgroup. We are again in the setting of a connected reductive group G with a Frobenius root F.

9.1 Deligne–Lusztig Induction The idea that Deligne and Lusztig had was to associate to any parabolic subgroup P with Levi decomposition P = U  L, where L is F-stable, a GF -variety-LF , that is an algebraic variety XU over Fq endowed with commuting left action of GF and right action of LF , such that when P is F-stable as the functor associated we have Hc∗ (XU )  Q (GF /UF ), and to define RG L to the Q GF -module-Q LF afforded by Hc∗ (XU ). This construction of Deligne and Lusztig was first published in Lusztig (1976b), and is traditionally called the “Lusztig functor” (or the “Lusztig induction”). More precisely: Definition 9.1.1 Let P = U  L be a Levi decomposition of a parabolic subgroup of G, where L is F-stable. We define the variety 137

138

Deligne–Lusztig Induction: The Mackey Formula

XU := {gU ∈ G/U | gU ∩ F(gU)  ∅} = {gU ∈ G/U | g−1 Fg ∈ U · FU}  {g ∈ G | g−1 Fg ∈ FU}/(U ∩ FU). this variety when needed to specify the group involved. We will denote by XG U The first two formulae are clearly equivalent definitions of the same set; it is a variety, as well as the third set, since they are closed subvarieties of the quotient of G by a closed subgroup, see 1.1.4(i). To show that these varieties are isomorphic we use the following proposition. Proposition 9.1.2 Let f : X → Y be a morphism of Fp -varieties bijective on points. Assume that connected components of X are irreducible and that Y is normal. Then f is an isomorphism. Reference

See for example Digne et al. (2007, 2.2.2).



By the Lang–Steinberg theorem one sees that the map g(U ∩ FU) → gU is bijective on points from the second variety to the first and both are smooth (see below), so this map is an isomorphism. On each of the varieties, we have actions by translation of GF on the left and of LF on the right, using the fact that LF normalises U and FU. We record now some geometric properties of the varieties XU that we will not use later in this book. • They are smooth (for example by the same argument as in the proof of Digne et al., 2007, 2.3.5). • It is conjectured they are affine; this is known when q is greater than the Coxeter number of W (the order of the product of the elements of S, taken in any order) at least when F is a Frobenius endomorphism, see Lusztig (1990, 6.3(b)). • They are irreducible if and only if U is in no F-stable proper parabolic subgroup of G, see Bonnafé and Rouquier (2006). , where P = U  L is a Levi Definition 9.1.3 The Lusztig functor RG L⊂P decomposition of P, is the generalised induction functor associated to the Q GF -module-Q LF afforded by Hc∗ (XU ) with action of (g, l) ∈ GF × LF induced by the action x → g−1 xl−1 on XU (see 5.1.1). (and sometimes called the “Lusztig The adjoint functor is denoted by ∗ RG L⊂P restriction”). We have taken above the action of (g−1 , l−1 ) on XU to have an action of (g, l) on the cohomology, since the cohomology is contravariant. But note also Proposition 9.1.4.

9.1 Deligne–Lusztig Induction

139

Proposition 9.1.4 The Q LF -module-Q GF afforded by Hc∗ (XU ) with action of (g, l) ∈ GF × LF induced by the action x → gxl on XU is isomorphic in the Grothendieck group of Q LF -modules-Q GF to the module which . defines ∗RG L⊂P Proof By Remark 5.1.2 ∗RG is given by the LF -module-GF given by Hc∗ L⊂P ∨ (XU ) with transposed action. But the trace does not change by transposition. Moreover, since Lefschetz numbers are integers (see 8.1.6), (g, l) and (g−1 , l−1 ) have the same trace on the cohomology of XU . Thus the module defining ∗RG L⊂P and the module of the statement afford the same character. They are thus isomorphic, since group algebras over an algebraically closed field of characteristic 0 are semi-simple so an element of the Grothendieck group is defined by its character.  In general when a group G acts on a set X we write X opp for the same set where g ∈ G acts by g−1 . In particular we will write Hc∗ (XU ) opp when we consider the action 9.1.4 of GF × LF on the cohomology. Note that if P is F-stable then U is also, in which case if gU ∩ FgU  ∅ then gU = FgU and XU is the discrete variety (G/U) F = GF /UF whose cohomology is Harish-Chandra is Q (GF /UF ) placed in degree 0; thus in this case RG L⊂P induction. Note also Lemma 9.1.5. Lemma 9.1.5 We have an isomorphism of Q GF -modules-Q LF between Hc∗ (XU ) and Hc∗ (L −1 (U)). −1

Proof The quotient map L −1 (U) → {g ∈ G | g−1 Fg ∈ U}/( F U ∩ U) has all its fibres isomorphic to an affine space. By 8.1.13(i) we get an isomorphism of Q GF -modules-Q LF between the Hc∗ . Finally F maps the second variety to  XU and induces an isomorphism on the cohomology, see 8.1.2. With the above notation, for g ∈ GF we have  F −1 (RG χ)(g) = |L | Trace((g, l) | Hc∗ (XU )) χ(l−1 ) L⊂P

Proposition 9.1.6

l∈LF

and for l ∈ LF we have F −1 ( ∗RG L⊂P ψ)(l) = |G |



Trace((g, l) | Hc∗ (XU ))ψ(g−1 ).

g∈GF

Proof From 5.1.5, we immediately obtain the two formulae, using for the second formula Proposition 9.1.4 and the fact that the Lefschetz numbers are integers. 

140

Deligne–Lusztig Induction: The Mackey Formula

Proposition 9.1.7 Let π ∈ Irr(LF ) and let π be the contragredient represen(π) is RG (π). tation; then the contragredient of RG L⊂P L⊂P Proof The proposition results immediately from formula 9.1.6, using that the Lefschetz numbers Trace((g, l) | Hc∗ (XU )) are integers, and from the fact that the trace of an element x on a given representation is equal to the trace of x−1 on the contragredient representation.  Proposition 9.1.8 (Transitivity) Let Q ⊂ P be two parabolic subgroups of G, and let M ⊂ L be F-stable Levi subgroups of Q and P respectively. Then ◦ RLM⊂L∩Q = RG . RG L⊂P M⊂Q Proof Let P = U  L and Q = V  M be the Levi decompositions; then by 3.4.8(ii) L ∩ Q = (V ∩ L)  M is a Levi decomposition of a parabolic subgroup of L. Following 5.1.4 and 5.1.11 we have to show that there is an isomorphism of Q GF -modules-Q MF Hc∗ (XG U ) ⊗Q

L

F

Hc∗ (XLV∩L )  Hc∗ (XG V ).

This results from the isomorphism of GF -varieties-MF given by XG U ×

L LF XV∩L

 XG V

(gU, l(V ∩ L)) → glV, (where the notation is as in 2.3.11) and the properties 8.1.9 and 8.1.10 of the cohomology. 

9.2 Mackey Formula for Lusztig Functors The rest of this chapter is devoted to the study of the “Mackey formula” (5.2.1) for the Lusztig functors. At the present time, we do not know of a proof in all cases; we are going to give those reductions that we can make and deduce the formula when one of the Levi subgroups is a torus. It is conjectured that the formula always holds; when F is a Frobenius endomorphism, the best known result is that it holds unless q = 2 and G has a quasi-simple component of type 2E simply connected, E simply connected or E , see Bonnafé and Michel 6 7 8 (2011) and Taylor (2018). When F is a Frobenius root, we do not know of a proof for 2F4 . In what follows, we will work with L −1 (U) instead of XU , which is allowed by 9.1.5. Let us first consider the left-hand side of the Mackey for◦ RG where P = U  L and mula; that is, the composite functor ∗RG L⊂P L ⊂P P = U  L are two Levi decompositions where L and L are F-stable. Then, by 5.1.4, this composite functor is associated to the Q LF -module-Q L F

9.2 Mackey Formula for Lusztig Functors

141

given by Hc∗ (L −1 (U)) opp ⊗Q GF Hc∗ (L −1 (U )) where (g, l ) ∈ GF × L F acts on  L −1 (U ) by x → gx l and where Hc∗ (L −1 (U)) opp is as defined after 9.1.4. This tensor product of cohomology spaces is isomorphic, by the Künneth formula 8.1.9 and by 8.1.10 (see proof of 9.1.8 above), to the Q LF -module-Q L F given by Hc∗ (L −1 (U) ×GF L −1 (U )) – the quotient by the action of g ∈ GF on L −1 (U) × L −1 (U ) identifies (x, x ) with (gx, gx ). Lemma 9.2.1 Let Z = {(u, u , g) ∈ U × U × G | u. Fg = gu }. The map ϕ : (x, x ) → (L(x), L(x ), x−1 x ) is a bijective morphism of LF -varieties-L F from L −1 (U) ×GF L −1 (U ) to Z; it maps the action of (l, l ) ∈ LF × L F to the −1 action on Z given by (u, u , g) → ( lu, l u , lgl ). By 8.1.13 the map ϕ induces an isomorphism on cohomology groups. Proof Let us first consider the map on L −1 (U) ×L −1 (U ) defined by the same formula as ϕ. The image of this map is clearly in Z. On the other hand, an easy computation shows that (x, x ) and (y, y ) have the same image if and only if there is some γ ∈ GF such that (x, x ) = (γy, γy ), so the map factors through ϕ, and ϕ is injective. It remains to show that ϕ is surjective; let (u, u , g) ∈ Z and let x and x be any elements of G such that u = L(x) and u = L(x ). We also have u = L(γx) for any γ ∈ GF , so it is enough to show that there exists γ ∈ GF such that x−1 γ −1 x = g; but indeed xgx −1 ∈ GF because u. Fg = gu .  The formula for the LF × L F -action on Z is clear.  −1 −1 Since by 5.2.2 we have G = w∈L\S(L,L )/L F Pw F P , we have  Zw , Z= w∈L\S(L,L )/L

a union of the locally closed subvarieties −1

−1

Zw = {(u, u , g) ∈ U × U × F Pw F P | u. Fg = gu }. Introducing the new variables w = w. Fw−1 , F = w F and (u1 , u1 , g1 ) = (u, wu , gw−1 ), we have Zw  Zw , where −1

−1

Zw = {(u1 , u1 , g1 ) ∈ U × wU × F P F ( wP ) | u1 . Fg1 = g1 u1 w } −1

−1

and we have used the fact that wF P = F ( wP ). The action of LF × L F on Z induces an action of LF × ( wL ) F on Zw which is given in the new coordinates by the same formula as before. At this stage we may “forget w”; that is, express everything in terms of the new variables V = wU , Q = wP and M = wL . With this notation the variety −1

−1

Zw = {(u1 , v1 , g1 ) ∈ U × V × F P F Q | u1 . Fg1 = g1 v1 w }

142

Deligne–Lusztig Induction: The Mackey Formula

is endowed with an action of (l, m) ∈ LF × MF given by (u1 , v1 , g1 ) → −1 ( lu1 , m v1 , lg1 m). −1 −1 −1 −1 We are now going to use the decomposition F P F Q = F ULM F V. Lemma 9.2.2 The cohomology of the variety −1

−1

Zw = {(u, v, u , v , n) ∈ U × V × F U × F V × LM | u. Fn = u nv vw }

is isomorphic as an Q LF -module-Q MF to that of Zw ; the isomorphism of cohomology spaces is induced by the fibration

π : Zw → Zw : (u, v, u , v , n) → (u. Fu −1 , v. F v , u nv ),

and the isomorphism of Q LF -modules-Q MF is for the action induced by the action of (l, m) ∈ LF × MF on Zw given by −1

−1

(u, v, u , v , n) → ( lu, m v, lu , m v , lnm). Proof Once we get the isomorphism of cohomology spaces, everything else is a straightforward computation. To get that isomorphism we will use 8.1.13; for that we need to compute the fibres of π; that is, determine the quintuples (u, v, u , v , n) which are mapped by π to (u1 , v1 , g1 ). Since, once u1 and v1 are given, u and v are determined by u and v , it is equivalent to finding the triples (u , v , n) such that g1 = u nv , which is in turn equivalent to finding the pairs (u , v ) such that u −1 g1 v −1 ∈ LM. Let us choose some decomposition g1 = −1 −1 u1 lmv1 where u1 ∈ F U, v1 ∈ F V, l ∈ L and m ∈ M; the condition on (u , v ) −1 can be written as l l (u −1 u1 ) m (v1 v −1 )m ∈ LM; introducing the new variables −1 u = l (u −1 u1 ) and v = m (v1 v −1 ) the condition on the pair (u , v ) ∈ −1 F U × F −1 V becomes u v ∈ LM; that is, u −1 L ∩ v M  ∅. Using 5.2.4 −1 −1 for the parabolic subgroups F P and F Q it is easy to see that the solutions −1 −1 are all pairs of the form ( yx−1 a−1 , ay) where x ∈ F U ∩ M, y ∈ F V ∩ L −1 −1 and a ∈ F U ∩ F V. Thus all fibres of π are isomorphic to the affine space −1 −1 −1 −1  ( F U ∩ M) × ( F V ∩ L) × ( F U ∩ F V), whence the lemma. We do not know in general how to transform the left-hand side of the Mackey formula much further towards our goal; the idea that we will use now is to find an action on Zw of a subgroup of L × M whose identity component is a torus which commutes with LF × MF ; using 8.1.16 we may then replace Zw by its fixed points under that torus. This will work if we can find a large enough torus. Note that the image of some quintuple (u, v, u , v , n) ∈ Zw under (l, m) ∈ L × M F −1 −1 is in Zw if and only if n (l−1 . Fl) = w (m F m−1 ). Let Z(L) (resp. Z(M)) be the centre of L (resp. of M) and put Hw = {(l, m) ∈ Z(L) × Z(M) | l−1 . Fl =

w −1



(m F m−1 )}.

9.2 Mackey Formula for Lusztig Functors

143

Then the identity component H0w is a torus (a connected subgroup of Z(L) 0 × Z(M) 0 ), and the image of (u, v, u , v , n) ∈ Zw under (l, m) ∈ Hw is still in Zw . Indeed if we write n = λ μ with (λ, μ) ∈ L × M, the condition for the F −1 image of (u, v, u , v , n) under (l, m) to be in Zw can be written λ (l−1 Fl) = w −1 F μ (m F m−1 ) which holds when (l, m) ∈ H since l−1 F l ∈ Z(L) and m F m−1 w ∈ Z(M). The torus H0w is large enough for our purpose when one of the Levi subgroups is included in the other one, for example when M ⊂ L. From now on we will assume that hypothesis (beware that in our initial notation this means that wL ⊂ L for the given element w). To determine H0w in that case, we will use the “norm” on a torus. Notation 9.2.3 Let T be a torus with a Frobenius root F; given a nonzero integer n ∈ N, we define the morphism norm NFn /F :T → T by n−1 τ → τ. F τ . . . F τ. Lemma 9.2.4 (i) The first projection maps H0w surjectively to Z(L) 0 . (ii) We have H0w = {(NFn /F (τ), NF n /F ( w τ −1 )) | τ ∈ Z(L) 0 }, where n is such n n that F w = w (for such an n we have F n = F n and F w = w ) and n H0w ∩ (LF × MF ) consists of the pairs such that τ ∈ (Z(L) 0 ) F . (iii) H0w has a fixed point in Zw if and only if w has a representative in S(L, L ) F ; in that case we have H0w = {(l, l−1 ) | l ∈ Z(L) 0 } and −1

(Zw ) Hw  {(v, v , n) ∈ (V ∩ L) × ( F V ∩ L) × L | Fn = nv v}, 0

on which the action of (l, m) ∈ LF × MF is given by (v, v , n) → −1 −1 ( m v, m v , lnm).







Proof If l ∈ Z(L) 0 then w (l−1 Fl) ∈ w Z(L) 0 = w FZ(L) 0 ⊂ F Z(M) 0 = Z(M) 0 . Thus by the Lang–Steinberg theorem there is some m ∈ Z(M) 0 such that w (l−1 Fl) = m F m−1 , so the first projection maps Hw surjectively to Z(L) 0 . As Z(L) 0 is connected this projection is still surjective when restricted to H0w , whence (i). We now prove (ii). Let H1 stand for the group that we want H0w to be equal to. A straightforward computation shows that H1 ⊂ Hw (note that w τ ∈ w Z(L) 0 ⊂ Z(M) 0 ). As H1 is the image of the connected group Z(L) 0 , it is connected; thus H1 ⊂ H0w . On the other hand, the first projection maps H1 surjectively to Z(L) 0 n since by the Lang–Steinberg theorem any t ∈ Z(L) 0 can be written s. F s−1 with s ∈ Z(L) 0 . Thus we get t = NFn /F (τ) where τ = s Fs−1 , and t is the projection of (NFn /F (τ), NF n /F ( w τ −1 )). As two elements of Hw have the same

144

Deligne–Lusztig Induction: The Mackey Formula

first projection only if their second projections differ by an element of MF, the group H1 has finite index in H0w , thus is equal to it since H0w is connected. We prove (iii). Suppose there exists (u, u , v, v , n) ∈ Zw fixed by H0w . Then in particular for any (l, m) ∈ H0w we have lnm = n. Since n ∈ LM and l ∈ Z(L), m ∈ Z(M), we get lm = 1, whence the inclusion H0w ⊂ {(l, l−1 ) | l ∈ Z(L) 0 }. Since the first projection is surjective, this gives the second assertion of (iii). If the element (u, v, u , v , n) ∈ Zw is fixed by H0w then u, u , v and v centralise Z(L) 0 . But CG (Z(L) 0 ) = L (see 3.4.6), so this gives u ∈ U ∩ L = {1}, −1 −1 u ∈ F U ∩ L = {1}, v ∈ V ∩ L and v ∈ F V ∩ L. So the elements of 0 (Zw ) Hw are of the form (1, v, 1, v , n) with Fn = nv vw , which gives the last assertion of (iii) and proves that w−1 Fw = w ∈ L, since n ∈ L.M = L. Since w is determined up to left multiplication by an element of L, it has an F-stable representative. Conversely, if w is F-stable we have w = 1 and F = F; using the general form of the elements of H0w given in (ii) we see that the elements (1, v, 1, v , n)  are fixed by H0w . This completes the proof of (iii). We note for future reference that, by 8.1.14(i), a representation π ⊗ π ∈ Irr(LF × (MF ) opp ) can occur in some Hci (Zw ) only if its restriction to H0w ∩ (LF × MF ) is trivial. As stated above the virtual Q LF -modules-Q MF given by Hc∗ (Zw ) and 0 Hc∗ (Zw Hw ) are isomorphic. wL We now look at a term RLL∩ wL ⊂L∩ wP ◦ ∗RL∩ wL ⊂P∩ wL ◦ ad w of the righthand side of the Mackey formula indexed by some w ∈ S(L, L ) F (we make no particular assumption on L and L until the end of the current paragraph). As for the left-hand side, we first write the Q LF -module-Q ( wL ) F to which this functor is associated as w Hc∗ (LL−1 (L ∩ wU ) ×(L∩ wL )F L −1 wL (U ∩ L )).

As before, using V = wU , M = wL and Q = wP we may “forget w”; the module we study is thus the cohomology of the variety Sw = {(l, m) ∈ L × M | l−1 . Fl ∈ L ∩ V, m−1 . Fm ∈ U ∩ M}/(LF ∩ MF ) on which (λ, μ) ∈ LF × MF acts by (l, m) → (λl, μ−1 m). The next lemma shows that the terms corresponding to the same w on both sides of the Mackey formula are equal when wL (= M) ⊂ L. Lemma 9.2.5 Under the above hypothesis (M ⊂ L) the cohomology spaces 0 ⊕i Hci (Sw ) and ⊕i Hci (Zw Hw ) are isomorphic as Q LF -modules-Q MF .

9.2 Mackey Formula for Lusztig Functors

145

Proof We show first that the map ϕ from −1

(Zw ) Hw  {(v, v , n) ∈ (V ∩ L) × ( F V ∩ L) × L | Fn = nv v} 0

to LL−1 (V ∩ L) given by (v, v , n) → nv is surjective, with all its fibres isomorphic to the same affine space. Indeed, the image of ϕ is where we claim it is since nv ∈ L and L(nv ) = v −1 n−1 Fn Fv = v Fv ∈ V ∩ L; and if n1 ∈ LL−1 (V ∩ L), then F F −1 ϕ−1 (n1 ) = {(n−1 , v , n1 v −1 ) | v ∈ 1 n1 v

F −1

V ∩ L},

−1

which is isomorphic to the affine space F V ∩ L. The LF -action-MF on Zw is clearly mapped by ϕ to the natural LF -actionF M on LL−1 (L ∩ V) (remember that M ⊂ L), and thus for that action the cohomology spaces of Zw and LL−1 (L∩V) are isomorphic as Q LF -modules-Q MF . But M ⊂ L also gives M ∩ U = {1} whence Sw  {(l, m) ∈ L × MF | l−1 Fl ∈ L ∩ V}/MF  {l ∈ L | l−1 Fl ∈ L ∩ V}, which gives the result.



From Lemmas 9.2.1 to 9.2.5 we get in particular the following theorem. Theorem 9.2.6

The Mackey formula  wL ∗ G RL⊂P ◦ RG = RLL∩ wL ⊂L∩ wP ◦ ∗RL∩ wL ⊂P∩ wL ◦ ad w L ⊂P w∈LF \S(L,L ) F /L F

holds when either L or L is a maximal torus. Proof If L is a torus, then wL ⊂ L for any w ∈ S(L, L ) and all the preceding lemmas hold unconditionally. Since the adjoint of the Mackey formula is the same formula with L and L interchanged, the Mackey formula also holds when L is a maximal torus.  As we remarked in Chapter 6, when the Mackey formula holds, it implies does not depend on P. In what follows we will usually use the nothat RG L⊂P and ∗ RG ; we leave it to the reader to check that we do that in cases tations RG L L where either the choice of a parabolic subgroup is irrelevant or the Mackey formula holds. (θ) for θ ∈ Definition 9.2.7 When L = T is an F-stable maximal torus, RG T Irr(TF ) is called a Deligne–Lusztig character. These characters were introduced in Deligne and Lusztig (1976).

146

Deligne–Lusztig Induction: The Mackey Formula

9.3 Consequences: Scalar Products The Mackey formula gives the “scalar product formula for Deligne–Lusztig characters”. Corollary 9.3.1

Let T and T be two F-stable maximal tori of G. Then:

(i) For θ ∈ Irr(TF ) and θ ∈ Irr(T F ) we have G F −1 F n n RG T (θ), RT (θ )GF = |T | #{n ∈ G | T = T and θ = θ }.

(ii) The functor RG does not depend on the Borel subgroup B used in its T⊂B construction. (θ) where the pairs (T, θ) are taken up to GF -conjugacy form an (iii) The RG T orthogonal basis of the space they span. Proof (i) is just a way of writing the Mackey formula, and (ii) is clear from (θ) = RG (θ ) the remarks above 9.2.7. (iii) results from (i) and the fact that RG T T  when g (T, θ) = (T , θ ) for some g ∈ GF . In 11.1.3, in connection with Lusztig series, we will give a statement somewhat stronger than 9.3.1 which deals directly with the cohomology groups of L −1 (U), where B = U  T. The next corollary gives 9.3.1(i) in terms of the Weyl group when θ and θ are the trivial characters. Corollary 9.3.2 Let Tw , Tw be F-stable maximal tori of G of type w (resp. w ) (see 4.2.22); then wF ⎧ ⎪ G ⎨ |W | RG Tw (1), RTw (1)GF = ⎪ ⎩0

if w and w are F-conjugate in W, otherwise.

Proof The scalar product is 0 unless Tw and Tw are GF -conjugate; that is, unless w and w are F-conjugate (see 4.2.22). In this last case we may assume w = w and Tw = Tw . Then 9.3.1 gives G F −1 F RG Tw (1), RTw (1)GF = |Tw | |NGF (Tw )| = |W (Tw ) |.

As the action of F on the Weyl group of Tw can be identified with the action of wF on W, we get the result.  The next proposition shows that Deligne–Lusztig induction constructs many cuspidal representations, since it applies in particular for every θ ∈ Irr(Tw ) such that NGF (Tw , θ) = TFw .

9.3 Consequences: Scalar Products

147

Proposition 9.3.3 Let W be the Weyl group of a quasi-split torus T of G, and let Tw be an F-stable torus of type w with respect to T where no F-conjugate of w lies in an F-stable proper parabolic subgroup of W. Assume that θ ∈ (θ) is up to sign a true character. Then any irreducible Irr(TFw ) is such that RG Tw constituent of RG (θ) is cuspidal. Tw Proof It is sufficient to show that for any F-stable Levi subgroup L containing (RG (θ)) = 0 T of an F-stable proper parabolic subgroup of G, we have ∗RG L Tw G since, by the assumption that RTw (θ) is up to sign a true character, the same property will hold for the constituents of RG (θ). The Mackey formula implies Tw that this property holds if S(L, Tw ) F is empty; that is, if no GF -conjugate of Tw is conjugate to a torus of L. But this is a consequence of the assumption on w, since the type of any F-stable maximal torus of L is (up to F-conjugacy) in  WL , a proper F-stable parabolic subgroup of W. Notes The construction of RG is one of the fundamental ideas of Deligne and Lusztig T was first published in Lusztig (1976b) as a (1976). The construction of RG L natural extension of that construction. The Mackey formula when one of the Levi subgroups is a torus is given in Deligne and Lusztig (1983, Theorem 7), but the proof in that paper has an error. The proof we give here corrects this error, using an argument indicated to us by Lusztig. The case of two tori (9.3.1) was already in Deligne and Lusztig (1976).

10 The Character Formula and Other Results on Deligne–Lusztig Induction

In this chapter we give the character formulae for Deligne–Lusztig induction and restriction. We then use them to generalise the results of Chapter 5, as well as some results of Chapter 7, and we express the identity, Steinberg and regular characters, and the characteristic function of a semi-simple conjugacy class, as linear combinations of Deligne–Lusztig characters. We are in the context of a connected reductive group G with a Frobenius root F.

10.1 The Character Formula Definition 10.1.1 Given L, an F-stable Levi subgroup of G and a Levi decomposition P = U  L of a (possibly not F-stable) parabolic subgroup containing : GFu × LFu → Q is defined by L, the two-variable Green function QG L (u, v) → |LF | −1 Trace((u, v) | Hc∗ (XU )). We recall that Gu denotes the set of unipotent elements of G. The Green to take in account the parabolic function should be properly denoted by QG L⊂P we omit P from the notation with subgroup used in the definition, but as for RG L the same caveats. Proposition 10.1.2 (character formula for RG and ∗RG ) Let L be an F-stable L L F Levi subgroup of G and let ψ ∈ Irr(G ) and χ ∈ Irr(LF ). (i) If g = su is the Jordan decomposition of g ∈ GF (RG L χ)(g) = |LF | −1 |CG (s) 0F | −1



|C hL (s) 0F |

{h∈GF |s∈ hL}

 v∈C hL (s)u0F

148

0

QCCG (s) (u, v−1 ) h χ(sv). (s) 0 hL

10.1 The Character Formula (ii) If l = tv is the Jordan decomposition of l ∈ LF  0F 0F −1 ( ∗RG L ψ)(l) = |CL (t) ||CG (t) |

149

0

(t) −1 QCCG(t) )ψ(tu). 0 (u, v L

u∈CG (t)u0F

Proof The main step is the following lemma. Lemma 10.1.3

With the above notation, we have

Trace((g, l) | Hc∗ (XU )) = |CL (t) 0F ||CG (t) 0F | −1 = |CG (s) 0F | −1





0

{h∈GF | ht=s−1 }

−1 {h∈GF | h s=t−1 }

−1

(t) h QCCG(t) u, v) 0(

(∗)

L

0

|C hL (s) 0F |QCCG (s) (u, hv). (s) 0

(∗∗)

hL

Proof These two equalities are clearly equivalent; we shall prove the first one. We use again the variety L −1 (U). From 8.1.15 we get Trace((g, l) | Hc∗ (L −1 (U))) = Trace((u, v) | Hc∗ (L −1 (U)

(s,t)

)).

We first show that the morphism ϕ : {h ∈ GF | ht = s−1 } × {z ∈ CG (t) 0 | z−1 Fz ∈ U} → L −1 (U)

(s,t)

given by (h, z) → hz is surjective – it is easy to check that the image of ϕ (s,t) (s,t) since L −1 (U) = {x ∈ G | x−1 Fx ∈ U and sxt = x}. If is in L −1 (U) (s,t) −1 then sxt = x implies s Fxt = Fx = x(x−1 Fx) whence s Fxt = x ∈ L (U) −1 F sxt(x x); which can be written (x−1 Fx)t = t(x−1 Fx); that is, x−1 Fx ∈ CG (t). As x−1 Fx is unipotent, we even get x−1 Fx ∈ CG (t) 0 (see 3.5.3). Applying the Lang–Steinberg theorem in the group CG (t) 0 we may write x−1 Fx = z−1 Fz, where z ∈ CG (t) 0 . If we put then h = xz−1 we have h ∈ GF , ht = s−1 and ϕ(h, z) = x, whence the surjectivity of ϕ. The map ϕ is not injective, but ϕ(h, z) = ϕ(h , z ) if and only if h−1 h = zz −1 ∈ CG (t) 0F , thus ϕ induces an isomorphism ∼

→ L −1 (U) {h ∈ GF | ht = s−1 } × CG (t)0F {z ∈ CG (t) 0 | z−1 Fz ∈ U} − which may be written L −1 (U)

(s,t)





(s,t)

,

{z ∈ CG (t) 0 | z−1 Fz ∈ U}h

{h∈GF /CG (t) 0F | ht=s−1 }

=



{h∈GF /CG (t) 0F | ht=s−1 }

LC−1 (t)0 (U ∩ CG (t) 0 )h , G

where (u, v) ∈ CG (s)u0F × CL (t)u0F acts on the piece indexed by h by z →

h−1 uzv.

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More about Deligne–Lusztig Induction

We thus get Trace((g, l) | Hc∗ (L −1 (U))) = |CG (t) 0F | −1



−1

{h∈GF | ht=s−1 }

Trace(( h u, v)|Hc∗ (LC−1 (t)0 (U ∩ CG (t) 0 ))), G



whence the lemma. We now prove Proposition 10.1.2. From Proposition 9.1.6 we have  F −1 (RG χ)(g) = |L | Trace((g, l) | Hc∗ (L −1 (U))) χ(l−1 ) L l∈LF

and F −1 ( ∗RG L ψ)(l) = |G |



Trace((g, l) | Hc∗ (L −1 (U)))ψ(g−1 ).

g∈GF

Applying Lemma 10.1.3 in both sums, and interchanging the sums, we get from the first formula and (∗∗) (RG L χ)(g) = |LF | −1 |CG (s) 0F | −1

 {h∈GF | h

−1



|C hL (s) 0F | s∈LF }

v∈CL ( h

−1

C (s) 0 h h−1 −1 sv ) 0 (u, v) χ( hL (s)

QCG s)u0F

and from the second formula and (∗) ( ∗RG L ψ)(l) =

 |CL (t) 0F | F 0F |G ||CG (t) | F



0

−1

(t) h QCCG(t) u, v)ψ( htu−1 ). 0(

h∈G u∈CG ( ht)u0F

L

We then get the result by changing the variable over which we sum to hv−1 −1 in the first formula and to h u−1 in the second.  Corollary 10.1.4 If as in 10.1.2 g = su is a Jordan decomposition, and M is an F-stable connected reductive subgroup of G such that M ⊃ CG (s) 0 , we have  |( hL ∩ M) F | 1 RM χ)(g) = ( h χ)(g). (RG h L |CMF (s)||LF | |classMF (s) ∩ hL| L∩M F h {h∈G |s∈ L}

When M = CG (s) 0 we may rewrite this as  (RG L χ)(g) =

h RM hL∩M ( χ)(g)

h∈MF \S(M,L) F /LF

where S(M, L) = {h ∈ G | M ∩ hL contains a maximal torus of G}.

10.1 The Character Formula

151

Proof Note that any maximal torus containing s lies in M, so that M is a subgroup of maximal rank; thus by 3.4.10(iii) hL ∩ M is a Levi subgroup of a parabolic subgroup of M so that RM hL∩M makes sense. Let us show the first 0 statement when M = CG (s) . In the case where s ∈ Z(G), Lemma 10.1.3 (∗) reduces to |LF | −1 Trace((g, l) | Hc∗ (L −1 (U)))

 −1 ⎧ |GF | −1 h∈GF QG ( h u, v) = QG (u, v) ⎪ L L ⎨ =⎪ ⎩0

if s = t−1 , otherwise,

where the last equality in the first line comes from the fact that QG is the reL striction to unipotent elements of a central function on GF × LF . So in that case 9.1.6 gives  −1 χ)(g) = QG (∗) (RG L L (u, v ) χ(sv). v∈LFu 0

h Applying this formula to RCCG (s) 0 ( χ)(g) in the right-hand side of the equality hL (s) we want to prove, we see that it is equivalent to 10.1.2(i) thus we have proved the first formula of the statement when M = CG (s) 0 . For a more general M, we use the formula just obtained replacing G by M and L by hL ∩ M to express each term on the right-hand side of the sought formula as: h RM hL∩M ( χ)(g)



= |CG (s) 0F | −1 |( hL ∩ M) F | −1

{h ∈MF |s∈ h hL}



0

|(CG (s) 0 ∩ h hL) F |RCG (s) 0



CG (s) ∩ h hL

h h

χ(g).

Hence the sum over h in the right-hand side of the formula to prove becomes 



|(CG (s) 0 ∩ h hL) F | |classMF (s) ∩

hL||C (s) 0 F | G



0

hh RCCG (s) χ(g), (s) 0 ∩ h hL G



where the sum is over {(h, h ) ∈ GF × MF | s ∈ hL ∩ h hL}. Since |classMF (s) ∩ hL| = |classMF (s) ∩ h hL| we can sum over h = h h with the condition s ∈ h L, each term occurring with cardinality the number of (h, h ) ∈ GF × MF such that s ∈ hL and h h = h , or equivalently the number of h such that h s ∈ h L. For each h s in classMF (s) ∩ h L there are |CMF ( h s)| = |CMF (s)| such h . Putting things together, the right-hand side of the equality we want to prove becomes its particular case for M = CG (s) 0 , whence the result. We finally show the last statement of the corollary. Any h such that s ∈ hL is in S(M, L) since C (s) 0 contains a maximal torus of G which is in hL

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CG (s) 0 = M. Conversely, s is in all maximal tori of M so for any element of S(M, L) we have hL  s. We conclude by observing that the coefficients in the first statement count the number of h ∈ GF which have a given image in  MF \S(M, L) F /LF . We now use the character formula to extend 7.3.6 to the Deligne–Lusztig induction. Proposition 10.1.5 Let L be an F-stable Levi subgroup of G, let ψ ∈ Irr(GF ) and let l = su be the Jordan decomposition of some element l ∈ LF . Then ((ResLC

0

F L (s)

0F

∗ CG (s) G ◦ ∗RG L )ψ)(l) = (( RC (s) 0 ◦ ResC

F

G (s)

L

0F

)ψ)(l).

Proof This results from the fact that in formula 10.1.2(ii) the right-hand side  does not change if we replace G by CG (s) 0 and L by CL (s) 0 . The next proposition and its corollary extend 7.3.4 and 7.3.5. Proposition 10.1.6 Let f ∈ C(GF )p ; then, for any F-stable Levi subgroup L of G and any λ ∈ C(LF ) (resp. γ ∈ C(GF )), we have F

G G RG L (λ. ResLF f ) = (RL λ).f , GF ( ∗RG L γ). ResLF

Proof

f =

(i)

∗ G RL (γ.f ).

(ii)

Proposition 10.1.2 gives F

G RG L (λ. ResLF f )(g)

= |LF | −1 |CG (s) 0F | −1

 {h∈GF |s∈ hL}

|C hL (s) 0F |

 v∈C hL (s)u0F

0

QCCG (s) (u, v−1 ) h λ(sv) hf (sv), (s) 0 hL

which gives (i) using hf (sv) = f (sv) = f (s) = f (g); equality (ii) is proved similarly (it can also be obtained from (i) by adjunction).  Corollary 10.1.7

f = ResG f. For f ∈ C(GF )p , we have ∗RG L LF F

Proof This results from the special case of 10.1.6(ii) where γ = 1, and the (1 ) = 1LF . Let us prove this last fact: by definition ∗RG (1 ) remark that ∗RG L GF L GF F F ∗ −1 G is the character afforded by the Q L -module Hc (L (U)) ; by 8.1.10 this module is isomorphic to Hc∗ (L −1 (U)/GF ), and the Lang map induces an  isomorphism from L −1 (U)/GF to U, whence the result by 8.1.11.

10.2 Uniform Functions

153

10.2 Uniform Functions As discussed in the proof of 7.2.4 the validity of the Mackey formula when one of the Levi subgroups is a torus (see 9.2.6) allows us to state the following analogue of 7.2.4, using the fact that DT = Id for a torus T. Theorem 10.2.1

For any F-stable maximal torus T of G we have G ε G DG ◦ RG T = ε T RT .

(1) given in the proof of 10.1.7 As in 7.4.2 we deduce, using the value of ∗RG T instead of 7.2.12 that ∗ G RT StGF

= ε G ε T StTF = ε G ε T 1TF .

(θ). We can now give the dimension of the (virtual) characters RG T Proposition 10.2.2 For any F-stable maximal torus T of G and any θ ∈ (θ) = ε G ε T |GF |p |TF | −1 . Irr(TF ), we have dim RG T St F = ε G ε T 1TF , Proof Taking the scalar product with θ of the equality ∗RG  G T G (θ), St we get RG F GF = ε T ε G δ 1,θ , whence  θ R (θ), StGF GF = ε T ε G . G T T  (θ) is the character afforded by the module Hc∗ (L −1 (U)), where U But θ RG T is the unipotent radical of some Borel subgroup containing T. By 8.1.15 this character vanishes on all non-unipotent elements, as a non-trivial semi-simple element has no fixed points on L −1 (U). Since StGF vanishes outside semisimple elements, the scalar product above reduces to |GF | −1 StGF (1)

 θ

dim(RG T (θ)).

But by 10.1.2 dim(RG (θ)) does not depend on θ. This gives the result after T  replacing StGF (1) by its value |GF |p . To get further properties of the Lusztig functor, in particular the dimension ( χ) and the analogue of 7.4.7, we first need to prove that the identity and of RG L the regular representations are both linear combinations of Deligne–Lusztig characters. We will use the following terminology. Definition 10.2.3 We call uniform functions the class functions on GF that are linear combinations of Deligne–Lusztig characters. (θ) where (T, θ) runs over GF -conjugacy classes Note that by 9.3.1 the RG T of pairs of an F-stable maximal torus T and of θ ∈ Irr(TF ) form an orthogonal (but not orthonormal) basis of the uniform functions. In the formulae up to the end of this chapter, we will let T denote the set of all F-stable maximal tori of G, and [T /GF ] denote a set of representatives

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More about Deligne–Lusztig Induction

of GF -conjugacy classes of F-stable maximal tori. We fix a quasi-split T1 ∈ T and we put W = W (T1 ). We recall that T /GF is in one-to-one correspondence with the F-classes of W (see 4.2.22). Finally for each w ∈ W we choose some F-stable maximal torus Tw of type w. Proposition 10.2.4 Denote by p (or pG if needed to specify the group) the orthogonal projection of class functions onto the subspace of uniform functions. Then:   ∗ G ∗ G RG |W (T) F | −1 RG p = |W | −1 Tw ◦ RTw = T ◦ RT T∈[T /GF ]

w∈W

= |GF | −1



∗ G |TF |RG T ◦ RT .

T∈ T

Proof The equality of the three expressions for p results from a straightforward computation. Let us check that the middle one is a projector on uniform functions. Since p( χ) is clearly uniform for any χ ∈ Irr(GF ), it is enough to (θ)GF = check that for any T ∈ [T /GF ] and any θ ∈ Irr(TF ), we have  χ, RG T G p( χ), RT (θ)GF . We have  ∗ G G p( χ), RG |W (T ) F | −1 RG T (θ)GF =  T RT χ, RT (θ)GF T ∈[T /GF ]

=



∗ G G |W (T ) F | −1 ∗RG T χ, RT RT (θ)GF

T ∈[T /GF ]

but, by 9.3.1 we have: ∗ G G RT RT (θ)

 ⎧ ⎪ w∈W (T)F wθ =⎨ ⎪0 ⎩

if T = T if T and T are not GF -conjugate

so ∗ G F −1 p( χ), RG T (θ)GF =  RT ( χ), |W (T) |



w

θGF =  χ, RG T (θ)GF

w∈W (T) F

the rightmost equality since for any θ we have RG ( wθ) = RG (θ). T T Proposition 10.2.5 (i) 1GF is a uniform function; we have  RG 1GF = |W | −1 Tw (1) = w∈W



|W (T) F | −1 RG T (1)

T∈[T /GF ]

= |GF | −1



T∈ T

|TF |RG T (1),



10.2 Uniform Functions

(ii) StGF is a uniform function; we have  StGF = |W | −1 ε G ε Tw RG Tw (1) = w∈W

 T∈[T

155

|W (T) F | −1 ε G ε T RG T (1)

/GF ]

= |GF | −1



|TF |ε G ε T RG T (1).

T∈ T

Proof By 10.2.1, (ii) is equivalent to (i). We prove (i). Since by 10.1.7 we (1 ) = 1TF , the three expressions in (i) above all represent p(1GF ). have ∗RG T GF It is enough to check that 1GF has the same scalar product with one of these expressions as with itself. But indeed we have   −1 RG  ∗RG 1, |W | −1 Tw (1)GF = |W | Tw (1), 1TFw = 1 w∈W

w∈W

(1 ) = 1TFw . using again that ∗RG Tw G F



Corollary 10.2.6 The character regGF of the regular representation of GF is a uniform function; we have  G dim(RG regGF = |W | −1 Tw (1))RTw (regTFw ) w∈W

=

|GF |p−1



F −1 ε G ε T RG T (regTF ) = |G |p

T∈ T



ε G ε T RG T (θ).

T∈ T θ ∈Irr(TF )

Proof Again, the equality of the three expressions is straightforward. Let us get the first one. As seen in the proof of 7.4.5 regGF = StGF γp . Using the first (1)γp = dim fomula for StGF in 10.2.5(ii), is enough to see that ε G ε Tw RG Tw F

RG (1)RG (regTFw ). This comes from the equality RG (1)γp = RG (ResG (γp )) Tw Tw Tw Tw TF w

F

given by 10.1.6, from the fact that ResG (γp ) has value |GF |p at 1 and 0 elseTF w

where, so is equal to |GF |p |TFw | −1 regTFw , and from 10.2.2.



The next corollary is in Steinberg (1968, 14.14) with a completely different proof. We follow here the arguments of Digne and Michel (1987). Corollary 10.2.7 The number of F-stable maximal tori of G is equal to |GF |p2 . F

Proof For any F-stable maximal torus T, we have IndG (regTF ) = regGF = TF  GF −1 |T | T∈ T IndTF regTF . We now use Lemma 10.2.8. Lemma 10.2.8

For any F-stable maximal torus T we have F

IndG (regTF ) = ε T ε G RG T (regTF )StGF . TF

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More about Deligne–Lusztig Induction

Proof This is just the special case of 10.2.10 below for RG . We can give this T lemma because the proof of 10.2.10 in this special case needs only 10.2.2 above and not 10.2.9.   (regTF ) and Applying this lemma, we get regGF = |T | −1 StGF T∈ T ε T ε G RG T the right-hand side of this equality is equal to |T | −1 StGF |GF |p regGF by 10.2.6.  Taking the value at 1 of both sides, we get the result, as StGF (1) = |GF |p . We will now give, using 10.2.2, the analogue for RG of 10.2.2 and 10.2.8. L These properties could be proved directly, using the same arguments as for 10.2.2 and 10.2.8, if we knew the validity of the Mackey formula in general, and thus the truth of 7.2.4 for a general Lusztig functor. Proposition 10.2.9 Let L be a Levi subgroup of G, and let ϕ ∈ Irr(LF ); then F F dim(RG L ϕ) = ε G ε L |G /L |p dim(ϕ).

We have ϕ(1) = ϕ, regLF LF and similarly  G G F −1 (RG ε G ε T RG L ϕ)(1) = RL ϕ, regGF GF = RL ϕ, |G |p T (regTF )GF ,

Proof

T∈ T

the last equality by 10.2.6. We now use adjunction to transform the last term above, then apply to it the Mackey formula 9.2.6, and then again take adjoints; we get  F −1 G ε T ε G  ∗RG (RG L ϕ)(1) = |G |p T ◦ RL ϕ, regTF TF T∈ T

=

|GF |p−1

=

|GF |p−1



T∈ T



T∈ T



εT εG 

ad x−1 ◦ ∗RLxT ϕ, regTF TF

LF \{x∈GF | xT⊂L}



εT εG

ϕ, RLxT x regTF LF .

LF \{x∈GF | xT⊂L}

In the last expression we may take as a new variable xT which is equivalent to summing over all F-stable maximal tori of L, if we multiply the expression by  |GF |/|LF |. We get |GF |p |LF | −1 ε G T⊂L ε T ϕ, RLT regTF LF , which is equal by  10.2.6 to ε G ε L |GF |p /|LF |p ϕ, regLF LF , whence the result. Corollary 10.2.10

For any ϕ ∈ Irr(LF ) (resp. ψ ∈ Irr(GF )) we have F

(i) (RG ϕ).ε G StGF = IndG (ϕ.ε L StLF ). L LF (ii) ∗RG (ψ.ε G StGF ) = ε L StLF . ResG ψ. L LF F

10.3 The Characteristic Function of a Semi-Simple Class

157

Proof It is enough to prove (ii) as (i) is its adjoint. Applying 10.1.2, and using the value of StLF , the truth of (ii) is equivalent to the fact that for any ψ ∈ Irr(GF ) and any semi-simple element s ∈ LF we have 0

(s) 0F −1 ε CG (s)0 |CG (s) 0F |p−1 |ψ(s)QCCG(s) ) |CL (s) 0 (1, v L

⎧ ⎪0 =⎨ ⎪ε 0F ⎩ CL (s)0 |CL (s) |p ψ(s)

if v  1, if v = 1

(∗)

(we have used the fact that the Steinberg character vanishes outside semi-simple elements). On the other hand, 10.2.9 gives that for any ϕ ∈ Irr(LF ) we have F F RG L ϕ, regGF GF = ε G ε L |G /L |p ϕ, regLF LF

whence ∗ G RL regGF

= ε G ε L |GF /LF |p regLF .

If in the equality above we apply the character formula 10.1.2 to ∗RG regGF , L and compute both sides at some given unipotent element v of LF , we get ⎧ ⎪ −1 ⎨0 QG (1, v ) = L ⎪ ε ε |GF /LF | p ⎩ G L

if v  1, if v = 1.

(∗∗)

The equality (∗) we want to prove results immediately from (∗∗) applied with  G replaced by CG (s) 0 and L replaced by CL (s) 0 .

10.3 The Characteristic Function of a Semi-Simple Class We will show in 13.3.4 that the characteristic function of a geometric conjugacy class is uniform. We end this chapter with a characterisation of uniform functions which proves in particular that the characteristic function of a semi-simple class is uniform. Definition 10.3.1 For s ∈ GF semi-simple, we define a map dsG : C(GF ) → C((CG (s) 0 ) F ) by ⎧ ⎪ f (sx) (dsG f )(x) = ⎨ ⎪0 ⎩ Proposition 10.3.2

We have 0

(s) G (i) dsL ◦ ∗RG = ∗RCCG(s) 0 ◦ ds , L L

(ii) dsG ◦ RG = RG ◦ dsL for s ∈ Z(GF ). L L

if x is unipotent, otherwise.

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More about Deligne–Lusztig Induction 0

0

F

Proof We have dsG (f ) = ts ( χCp G (s) · ResG f ), where χCp G (s) is as in 7.3.2 CG (s) 0F and ts denotes the translation by the central element s. By the character for(s) 0 mula 10.1.2(ii) ∗RCCG(s) 0 commutes with ts . By 10.1.6(ii), it commutes with the L

0

multiplication by χCp G (s) . We then get (i) by 10.1.5. G For s ∈ Z(GF ) we have dsG (f ) = ts ( χG p f ). Since ts commutes with RL by G the character formula 10.1.2(i) and the multiplication by χp also commutes by 10.1.6(i), we get (ii).  with RG L Proposition 10.3.3 For s ∈ GF semi-simple we have dsG ◦ pG = pCG (s)0 ◦ dsG . Proof If x ∈ (CG (s) 0 ) F is not unipotent we have (dsG ◦ pG f )(x) = (pCG (s)0 ◦ dsG f )(x) = 0. For u ∈ (CG (s) 0 )uF we have  |TF | ∗ G (RG T RT f )(su) F| |G T∈ T   0 |TF | = RChTG (s) ( h∗RG T f )(su) F 0 F T∈ T |CG (s) ||G | h∈GF | hTs  |TF ||N F (T)|  0 G = RCT G (s) ( ∗RG T f )(su) F 0 F T∈ T |CG (s) ||G | T s  0 |TF | RCG (s) ( ∗RG = T f )(su) F T 0 T∈ T ,Ts |CG (s) |

(dsG ◦ pG f )(u) = pG f (su) =

where the first line is by the definitions, the second line is using the first formula in 10.1.4 for M = CG (s) 0 , the third line is taking T = hT as a variable and the last line results from counting tori. Now we have 0

0

0

CG (s) RCTG (s) ( ∗RG (RCTG (s) ( ∗RG T f )(su) = ds T f ))(u) 0

= RCTG (s) (dsT ∗RG T f )(u) 0

0

= RCTG (s) ( ∗RCTG (s) dsG f )(u) where the second equality is 10.3.2(ii) and the third equality is 10.3.2(i). Applying this transformation to the terms above, one recognises (pCG (s)0 ◦  dsG f )(u). From the above proposition we immediately get the following corollary. Corollary 10.3.4 f ∈ C(GF ) is uniform if and only if ds f is uniform for all s ∈ GF semi-simple. It results that if f is the characteristic function of a GF semi-simple class, it is uniform since dsG f is 0 or a multiple of regCG (s)0F , which is uniform. We can

10.3 The Characteristic Function of a Semi-Simple Class

159

get actually a nice formula for the “normalised” characteristic function of the class of s. Notation 10.3.5 Given a finite group H and some x ∈ H, we denote by πxH the function whose value is |CH (x)| on the H-conjugacy class of x and 0 on other elements of H. With this notation, we can write the following proposition. Let s be a semi-simple element of GF ; then  0 TFw = |W 0 (s)| −1 dim(RCTGw (s) (1))RG Tw (πs )

Proposition 10.3.6 πsG

F

w∈W 0 (s)

= ε CG (s)0 |CG (s) 0F |p−1



F

T ε T RG T (πs ),

T∈ T Ts

where in the first sum W 0 (s) is as in 3.5.2, and Tw has type w with respect to some quasi-split torus of CG (s) 0 . Note that if T0 is a quasi-split torus of CG (s) 0 , of type v with respect to a quasi-split torus T1 of G, then (Tw , F) is CG (s) 0 -conjugate to (T0 , wF), and there is g ∈ G such that g (T0 , wF) = (T1 , v gwF), thus Tw is of type v gw in G. F

F

Proof By 7.4.4 we have StGF πsG = ε G ε CG (s)0 |CG (s) 0F |p πsG , thus it is enough F to give a formula as a uniform function for StGF πsG , and for that, since we F know that πsG is uniform, to compute F

 |TF | GF RG ∗RG T (StGF πs ) F| T |G T∈ T  |TF | GF GF = ε G ε T F RG T ResTF πs , |G | T∈ T

pG (StGF πsG ) =

where the second equality is 10.2.10(ii) and the fact that StTF = 1TF . For t ∈ ⎧ ⎪ |CGF (t)| if t is conjugate to s F F which is equal TF we have ResG π G (t) = ⎨ TF s ⎪0 otherwise ⎩  F F F πG = to |{g ∈ GF | gs = t}| = |TF | −1 {g∈GF | gs∈T} π Tgs (t), hence ResG TF s  F |TF | −1 {g∈GF | gs∈T} π Tgs . Plugging everything back together we get   ε CG (s)0 F TF pG (πsG ) = ε RG T T π gs F 0 F |CG (s) |p ||G | T∈ T {g∈GF | gs∈T} −1

which, taking g T as a new variable and counting tori, gives the second formula of the statement. The first one results from a straightforward computation. 

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More about Deligne–Lusztig Induction

Remark 10.3.7 uniform.

Another corollary of 10.3.4 is that any function in C(GF )p is

Exercise 10.3.8 Let x = su be the Jordan decomposition of some element x ∈ GF , and let L be an F-stable Levi subgroup of G containing CG (s) 0 ; prove that F

F

L G RG L (πx ) = πx .

Hint

ψ)(x), for any ψ ∈ Irr GF . Use 10.1.5 to compute ( ∗RG L



Notes is in Deligne and Lusztig The character formula 10.1.2 for the functor RG T G (1976, 4.2). The generalisation to RL was first published in Digne and Michel (1983); we have since learned that it was already known by Deligne at the time of Deligne and Lusztig (1976).

11 Geometric Conjugacy and the Lusztig Series

We now introduce Lusztig’s classification of irreducible characters of GF . In this classification, a character is parametrised by a pair: an F ∗ -stable semisimple class (s) of the dual group G∗ of G, and a “unipotent” irreducible character χ of the centraliser CG∗F∗ (s).

11.1 Geometric Conjugacy In this chapter, we have all the tools to deal with the ingredient (s) of the classification; see Chapter 14 for the ingredient χ. We start with an immediate corollary of 10.2.6. Proposition 11.1.1 For any χ ∈ Irr(GF ), there exists an F-stable maximal (θ)GF  0. torus T and θ ∈ Irr(TF ) such that  χ, RG T We note also that 9.3.1(i) implies that RG (θ) is irreducible up to sign if and T only if no non-trivial element of W (T) F stabilises θ. Furthermore, by 10.2.2, in (θ) is a true character. that case ε T ε G RG T We now give a condition for two Deligne–Lusztig characters to have a common irreducible constituent, using the norm NFn /F on an F-stable torus T, see n 9.2.3. Note that NFn /F maps TF to TF . By 9.3.1(i), if the pairs (T, θ) and (θ) and RG (θ ) are orthogonal to each (T , θ ) are not GF -conjugate, then RG T T other; but they may have a common constituent as they are virtual characters. Definition 11.1.2 Let T and T be two F-stable maximal tori, and let θ and θ be characters respectively of TF and T F . We say that the pairs (T, θ) and (T , θ ) are geometrically conjugate by g ∈ G if T = gT and for any n such n n that g ∈ GF the two characters of T F given by θ ◦ NFn /F ◦ ad g and by θ ◦ NFn /F coincide.

161

162

Geometric Conjugacy and the Lusztig Series

Proposition 11.1.3 Let T (resp. T ) be an F-stable maximal torus, and let U (resp. U ) be the unipotent radical of some Borel subgroup containing T (resp. T ). Let θ ∈ Irr(TF ) (resp. θ ∈ Irr(T F )). Assume that there exist i and j j such that the Q GF -modules Hci (L −1 (U)) ⊗Q TF θ and Hc (L −1 (U )) ⊗Q T F θ   have a common irreducible constituent. Then the pairs (T, θ) and (T , θ ) are geometrically conjugate. Proof Let χ be the common irreducible constituent of the statement. We remark first that, by 8.1.6, χ occurring in Hci (L −1 (U)) ⊗Q TF θ is equivalent

to χ occurring in Hci (L −1 (U)) ⊗Q θ

opp

⊗Q

T

F



F T

θ, which implies that χopp occurs in

Hci (L −1 (U)) opp . As χ occurs in Hc (L −1 (U )) ⊗Q

resentation θ ⊗ θ j Hc (L −1 (U )),

opp

j

F T

θ , the rep-

of TF × T F occurs in the module Hci (L −1 (U)) opp ⊗Q i+j

G

F

which with the notation of 9.2.1 is a submodule of Hc (Z). However, we have the following lemma.

Lemma 11.1.4 If the Q TF -module-Q w (T F ) given by θ ⊗ wθ opp occurs in some cohomology group of Zw for some w ∈ S(T, T ) (see 9.2.2) and if n > 0 n is such that F w = w, then θ ◦ NFn /F = θ ◦ NFn /F ◦ ad Fw−1 . Proof Using the remark which follows the proof of 9.2.4, we get the result from 9.2.4(ii) applied with L = T and M = wT , using the fact that F −1  NF n /F ( w τ −1 ) = wNFn /F ( w τ −1 ). Under the hypothesis of 11.1.4 the element Fw is the required element g of 11.1.2, and the proposition is proved. As the cohomology of the TF -varietywT F given by Z is isomorphic to that of the TF -variety-T F given by Z (via w w ad w), the following lemma thus completes the proof of the proposition. Lemma 11.1.5 If the character θ ⊗ θ opp does not occur in Hck (Zw ) for any k and w, then θ ⊗ θ opp does not occur in Hci (Z) for any i. Proof

We use the following lemma.

Lemma 11.1.6 (i) With the notation of 9.2.1, if L = T and L = T are tori, and if we let B and −1 −1 B denote the Borel subgroups F P and F P , then for any v ∈ W (T) the  union v ≤v Zv w1 is closed in Z where w1 ∈ S(T, T ) is such that w1 B = B and where v ≤ v is the Bruhat–Chevalley order of 3.2.8.  (ii) The connected components of the union l(v)=n Zvw1 are the Zvw1 . Proof Property (i) results from the fact that in the present case Zvw1 is the inverse image in Z by the third projection of the subset Bvw1 B = BvBw1 of G,

11.1 Geometric Conjugacy

163

and from 3.2.8. Furthermore, (i) shows that the closure of Zvw1 in Z does not  meet Zv w1 if l(v) = l(v ) and v  v , whence (ii). This lemma will allow us to apply repeatedly the long exact sequence 8.1.7(i)  to Z = v Zvw1 . We now prove 11.1.5 by showing by induction on n that the  hypothesis of 11.1.5 implies that Hci ( l(v) ≤n Zvw1 )θ ⊗θ opp = 0, where the subscript θ ⊗ θ opp denotes the subspace of the cohomology where TF × T F acts through θ ⊗ θ opp . This is true for n = 0 by hypothesis. Suppose it for n − 1. We have Zvw1 = ( Zvw1 ) ( Zvw1 ), l(v) ≤n

l(v) ≤n−1

l(v)=n

and this last union is closed since by 11.1.6(i) it is a finite union of closed subsets. There is thus a cohomology long exact sequence relating these three unions; this sequence remains exact restricted to the subspaces where TF × T F acts through θ ⊗ θ opp . But 11.1.6(ii)implies, according to 8.1.7(ii), that  k for any k we have Hck ( l(v)=n Zvw1 ) = l(v)=n Hc (Zvw1 ), so by assumption  k opp Hc ( l(v)=n Zvw1 )θ ⊗θ = 0. Since, by the induction hypothesis, for any k we  have Hck ( l(v) ≤n−1 Zvw1 )θ ⊗θ opp = 0, at each step two out of three terms of the  long exact sequence are 0, so the third one Hck ( l(v) ≤n Zvw1 )θ ⊗θ opp is also 0.   We will now introduce the dual of G, which will allow us to give a nice interpretation of geometric conjugacy (and a justification of this term), using the duality between the groups X(T) and Y (T) for a torus T. × ∼ → We shall assume we have chosen once and for all an isomorphism Fp − × × (Q/Z)p and an embedding Fp → Q . Proposition 11.1.7 Let T be a torus with a Frobenius root F; as in 4.2.2 the action of F on X := X(T) (resp. Y := Y (T)) is α → α ◦ F (resp. β → F ◦ β). Then: (i) The sequence F−1

0 → X −−−→ X → Irr(TF ) → 1 is exact, where the right map is the restriction to TF of characters (for this to make sense we identify Hom(TF , Gm ) with Irr(TF ), using the chosen × × embedding Fp → Q ). (ii) The sequence F−1

π

0 → Y −−−→ Y − → TF → 1

164

Geometric Conjugacy and the Lusztig Series is exact, where π is defined by y → NFn /F (πn (y)) for any n such that F n is a split Frobenius on T, where πn (y) = y(1/(qn − 1)). Here q is the real number associated with F, and we have used the chosen isomorphism to × identify 1/(qn − 1) ∈ (Q/Z)p with an element of Fp .

Proof We have seen (i) in the proof of 4.4.9. Let us prove (ii). Let n be the smallest integer such that F n is a split Frobenius on T – with the notation of 4.2.5, this is the lcm of the order of τ and of the smallest m such that qm is an integer, and the set of m such that F m is a split Frobenius is {kn | k ∈ N − {0}}. For any positive integer k, we have NFn /F (πn (y)) = NFn /F (NFkn /Fn (πkn (y))) = NFkn /F (πkn (y)), so the map π is well-defined. To show the exactness we use the commutative diagram 0

→

0

→ Y

F n −1

Y −−−−→ ⏐

NFn /F ⏐ F−1

−−−→

πn

Y 

−−→

Y

− →

π

n

TF → ⏐

NFn /F ⏐

1

→

1

TF

where NFn /F := Id +F + · · · + F n−1 on Y. We show first the exactness of the top sequence: the injectivity of F n − 1 is clear as it is the multiplication by qn − 1, since T is split over Fqn ; the surjectivity of πn is straightforward by 1.2.10 applied with k = Fqn . Moreover if y is in the kernel of πn then it is trivial on all n (qn − 1)-th roots of unity, so is constant on all fibres of the map x → xq −1 from Gm to itself; so it factors through that endomorphism of Gm ; that is, there exists y1 ∈ Y such that y = (qn − 1)y1 ; that is, y is in the image of F n − 1. The exactness of the bottom sequence is deduced from that of the top sequence: the n surjectivity of π follows by the surjectivity of NFn /F : TF → TF . The injectivity of F − 1 is clear as F is the transpose of the endomorphism qτ of X(T), see 4.2.5, so has no eigenvalue 1. The image of F − 1 is obviously in the kernel of π; Take now y in the kernel of π; that is, such that NFn /F (y(1/(qn − 1))) = 1. This means that NFn /F (y) is in the kernel of πn , so NFn /F (y) ∈ (F n − 1)Y = NFn /F ◦ (F − 1)(Y). As NFn /F : Y → Y is injective, since the composite  (F − 1) ◦ NFn /F = F n − 1 is injective, we get y ∈ (F − 1)Y. We note that by 11.1.7(ii) any character of TF gives a character of Y (T) with values in the p -roots of unity in Q ; using our chosen isomorphism, we will consider it as an element of Hom(Y (T), (Q/Z)p ). We may now reinterpret the notion of geometric conjugacy. Proposition 11.1.8 Let T and T be two F-stable maximal tori of G and let θ ∈ Irr(TF ) and θ ∈ Irr(T F ); then (T, θ) and (T , θ ) are geometrically conjugate by g ∈ G if and only if T = gT and g conjugates θ , considered

11.1 Geometric Conjugacy

165

as an element of Hom(Y (T ), (Q/Z)p ) via 11.1.7(ii), to θ, considered as an element of Hom(Y (T), (Q/Z)p ). Proof The characters θ and θ ◦ NFn /F are identified with the same element of Hom(Y (T), (Q/Z)p ) by the construction of 11.1.7(ii), as seen in the proof of that proposition. This gives the result.  Corollary 11.1.9 Let us fix an F-stable maximal torus T; then geometric conjugacy classes of pairs (T , θ ) (where θ ∈ Irr(T F )) are in one-to-one correspondence with F-stable W (T)-orbits in X(T) ⊗ (Q/Z)p . Proof Since all tori are G-conjugate we may assume that T = T, so gT = T becomes g ∈ W (T). The result then comes from the remark that Hom(Y (T),  (Q/Z)p ) is isomorphic to X(T) ⊗ (Q/Z)p . We now introduce the dual of a reductive group. Definition 11.1.10 Two connected reductive algebraic groups G and G∗ are said to be dual to each other if there exists maximal tori T of G and T∗ of ∼ → G∗ such that their root data are dual; that is, there is an isomorphism X(T) − Y (T∗ ) inducing an isomorphism ∼

(X(T), Y (T), Φ, Φ∨ ) − → (Y (T∗ ), X(T∗ ), Φ∗∨ , Φ∗ ). In particular, two tori T and T∗ are said to be dual to each other if we have ∼ → Y (T∗ ). When we talk of groups dual to been given an isomorphism X(T) − each other, we will always assume that we have chosen corresponding dual tori; we will say that (G, T) is dual to (G∗ , T∗ ) when we need to specify the tori used. Note that by 2.4.2 every connected reductive group has a dual and, given G, the isomorphism class of G∗ is well defined. Using the exact pairing ∼ → Y (T∗ ) is equivalent to between X(T) and Y (T), being given a duality X(T) − ∼ → Y (T). Given ϕ ∈ Hom(X(T), X(T )) we will being given a duality X(T∗ ) − ∗ ∗ denote by ϕ ∈ Hom(X(T ), X(T∗ )) the transposed of the image of ϕ by the duality isomorphism. Proposition 11.1.11 ϕ

f

− G inducing the p-morphism Given an isogeny G → f∗

→ X(T), then ϕ∗ is a p-morphism induced by an isogeny G ∗ −→ G∗ , X(T ) − that we call the dual isogeny to f . Proof We just need to check that the dual of a p-morphism is a p-morphism. This results from the definition and from the fact that a map ϕ between lattices has a finite cokernel K if and only if its transposed is injective. Indeed, ϕ → X → K → 0 by the left exactness of Hom(, Z) the exact sequence X −

166

Geometric Conjugacy and the Lusztig Series ϕ∨

gives the exact sequence 0 → Hom(K, Z) → Hom(X , Z) −−→ Hom(X, Z) and Hom(K, Z) = 0 if and only if K is finite.  Definition 11.1.12 If F is a Frobenius root on G which stabilises T and F ∗ is the dual isogeny, we say then that the pair (G, F) is dual to the pair (G∗ , F ∗ ). Note that F ∗ is a Frobenius root by the sentence just after 4.3.2. Note that w → w∗ is an anti-isomorphism W (T) → W (T∗ ), and that through that anti-isomorphism the action of F on W (T) is identified with the inverse of the action of F ∗ on W (T∗ ). Examples 11.1.13 The group GLn is its own dual, and the same is true for the unitary group which is GLn with the Frobenius endomorphism which sends q a matrix (ai,j ) to t (ai,j ) −1 , see 4.3.3. The groups SLn and PGLn are dual to each other. The group SO2l is its own dual, with either the split Frobenius endomorphism or the Frobenius endomorphism of the non-split group. The symplectic group Sp2n is dual to SO2n+1 . It follows from the definitions that the dual of a semi-simple group is semisimple, and that the dual of an adjoint group is simply connected, and conversely. We get as an immediate corollary of 11.1.7: Proposition 11.1.14

∗

If (T∗ , F ∗ ) is dual to (T, F) then Irr(TF )  T∗F .

The next proposition interprets geometric conjugacy using G∗ . Proposition 11.1.15 Assume that (G, F) and (G∗ , F ∗ ) are dual to each other with corresponding dual tori T and T∗ . Geometric conjugacy classes of pairs (T , θ ) in G are in one-to-one correspondence with F ∗ -stable conjugacy classes of semi-simple elements of G∗ . F ∗ -stable conjugacy classes of semi-simple elements of G∗ are in bijection ∗ with their intersection with G∗F , the semi-simple geometric conjugacy classes ∗ ∗F of G . Proof We apply 11.1.9. Using the fact that the isomorphism from X(T) to Y (T∗ ), is compatible with the action of the Weyl group, we get a bijection between geometric conjugacy classes of pairs (T , θ ) and F ∗ -stable W (T∗ )orbits in Y (T∗ ) ⊗ (Q/Z)p . But by 1.2.10 there is an isomorphism from Y (T∗ ) ⊗ × (Q/Z)p to T∗ (which depends on the fixed isomorphism Fq  (Q/Z)p ), so geometric conjugacy classes of pairs (T , θ ) are in one-to-one correspondence with F ∗ -stable W (T∗ )-orbits in T∗ . As any semi-simple element is in a maximal torus and all maximal tori are conjugate, any semi-simple class meets T∗ . Furthermore – by 1.3.6(iv) – W (T∗ )-orbits in T∗ are the intersections of

11.2 More on Centralisers of Semi-Simple Elements

167

G∗ -conjugacy classes with T∗ , hence F ∗ -stable W (T∗ )-orbits are the intersec tions of F ∗ -stable conjugacy classes with T∗ , whence the result. After 9.3.1 we remarked that the RG (θ) are parametrised by GF -conjugacy T classes of pairs (T, θ). Using the dual group, we may give another parametrisation Proposition 11.1.16 The GF -conjugacy classes of pairs (T, θ) where T is an F-stable maximal torus of G and θ ∈ Irr(TF ) are in one-to-one correspon∗ dence with the G∗F -conjugacy classes of pairs (T∗ , s) where s is a semi-simple ∗ element of G∗F and T∗ is an F ∗ -stable maximal torus containing s. Proof Let T be a fixed F-stable maximal torus. By the remark after 4.2.22 the GF -conjugacy classes of pairs (T , θ ) correspond one-to-one to the conjugacy classes under W (T) of pairs (wF, θ) with θ ∈ Irr(TwF ). If (G∗ , T∗ ) is dual to (G, T), these pairs are by 11.1.14 in one-to-one correspondence with the ∗ ∗ conjugacy classes under W (T∗ ) of pairs (F ∗ w∗ , s) with s ∈ TF w which are in ∗ turn, by the remark after 4.2.22, in one-to-one correspondence with the G∗F  conjugacy classes of F ∗ -stable pairs (T ∗ , s ). Using this proposition, we will, in what follows, sometimes use the notation (s) for RG (θ). RG T∗ T

11.2 More on Centralisers of Semi-Simple Elements Lusztig’s classification of characters is considerably simpler when centralisers of semi-simple elements in G∗ are connected. We will sometimes need this hypothesis in this chapter and the next. We now show that it holds if the centre of G is connected, and give some other consequences of the connectedness of Z(G) we will need in the next chapter. Lemma 11.2.1 Let T be a maximal torus of G and Φ be the set of roots of G relative to T; then: (i) The group Irr(Z(G)/Z(G) 0 ) is canonically isomorphic to the torsion group ⊥ of X(T)/(Φ⊥ T )X(T) . (ii) If the centre of G is connected, then the centre of any Levi subgroup of G is also connected. (iii) If (G∗ , T∗ ) is dual to (G, T), then for any s ∈ T∗ the group W (s)/W 0 (s) (isomorphic to CG∗ (s)/CG∗ (s) 0 , see 3.5.2) is isomorphic to a subgroup of Irr(Z(G)/Z(G) 0 ).

168

Geometric Conjugacy and the Lusztig Series

Proof By 1.2.3(i) and 1.2.4 the torsion of X(D) is isomorphic to Irr(D/D0 ) for any diagonalisable group D. As we have Z(G) = Φ⊥ T , it follows using ⊥ , whence (i). ) 1.2.4 that X(Z(G)) = X(T)/(Φ⊥ T X(T) ⊥ It follows that Z(G) is connected if and only if X(T)/(Φ⊥ T )X(T) has no torsion, which is equivalent to X(T)/Φ having no p -torsion by 1.2.13; but the subgroup ΦL  of Φ spanned by the roots of a Levi subgroup L containing T is a direct summand of Φ, so if X(T)/Φ has no p -torsion then X(T)/ΦL  has none either, so Z(L) is connected, whence (ii). By 1.2.10 an element s ∈ T∗ can be identified with an element of Y (T∗ ) ⊗ (Q/Z)p ; that is, of X(T) ⊗ (Q/Z)p . Let us write such an element x/m with x ∈ X(T) and m ∈ Z relatively prime to p. We have W (s) = {w ∈ W | w(x) − x ∈ mX(T)}. Consider the map W (s) → X(T)/Φ w(x) − x (mod Φ); w → m it is a group morphism because, as W acts trivially on X(T)/Φ (this is true for a reflection, whence also for an arbitrary product of reflections), we have ww (x) − w(x) w (x) − x ≡ m m

(mod Φ)

which is equivalent to ww (x) − x w (x) − x w(x) − x ≡ + m m m

(mod Φ).

The kernel of this morphism is {w ∈ W | w(x) − x ∈ mΦ} which is a reflection group by Bourbaki (1968, VI, Ex. 1 of §2), so is equal to W 0 (s) by 3.5.2. Thus we get an embedding of W (s)/W 0 (s) into X(T)/Φ. The image is a p -torsion group since for any w we have w(x) − x ∈ Φ, so m((w(x) − x)/m) ∈ Φ, which means that the exponent of the image divides m. We then get (iii) by (i) and 1.2.13.  Remark 11.2.2 (i) Note that the above proof, using 3.5.2, shows that, for any semi-simple element s of a connected algebraic group G, the exponent of CG (s)/CG (s) 0 divides the order of s. (ii) A consequence of 11.2.1(iii) is that, if the centre of G is connected, the centraliser of any semi-simple element of G∗ is connected.

11.2 More on Centralisers of Semi-Simple Elements

169

We will now give further information on CG (s)/CG (s) 0 . We first show that we can reduce to adjoint groups. Proposition 11.2.3 Let s ∈ G be semi-simple and let s¯ be the image of s in G := G/Z(G); then CG (s)/CG (s) 0 identifies with a subgroup of CG (¯s)/CG (¯s) 0 . Proof Let T be a maximal torus containing s and let T be its image in G; the roots of G with respect to T identify with those of G with respect to T and the Uα are isomorphic. By 3.5.1 CG (s) 0 = T, Uα | α(s) = 1 and CG (¯s) 0 = T, Uα | α(¯s) = 1 but α(s) = α(¯s), hence CG (¯s) 0 = CG (s) 0 /Z(G). The proposition follows.  Now, if G is adjoint it is a direct product of quasi-simple adjoint groups, so we may reduce the study of CG (s)/CG (s) 0 to the quasi-simple and adjoint case. Then G∗ is quasi-simple simply connected. We will describe Z(G∗ ) in this case, from which the description of the groups CG (s)/CG (s) 0 follows by 11.2.1(iii). We first need: Proposition 11.2.4 Let Φ be an irreducible crystallographic root system, and let Φ+ be a positive subsystem and Π the corresponding basis of Φ. Then there is a unique root (in Φ+ ), called the highest root, such that the sum of its coefficients on Π is maximal. Proof Denote by λ the linear form which takes value 1 on Π. Let α be a root with maximal value of λ. Note that α, β ≥ 0 for any β ∈ Φ+ otherwise by  3.3.3(i) α+ β would be a root with greater value of λ. Now write α = γ ∈Π nγ γ. We show that no nγ is 0; otherwise let I = {γ ∈ Π | nγ = 0} and let J = Π − I.  Then for δ ∈ I we have 0 ≤ α, δ =  γ ∈J nγ γ, δ ≤ 0, the left inequality by what we just showed and the right one by 2.2.5. It follows that we must have γ, δ = 0 for any γ ∈ J, δ ∈ I which contradicts the irreducibility of Φ if I  ∅. We deduce the uniqueness of α: if β is another root with maximal value of λ,  in 0 ≤ α, β =  γ ∈Π nγ γ, β we must have some term γ, β  0 otherwise β = 0. It follows that α, β > 0 thus by 3.3.3(i) both α − β and β − α are roots. One of them must be positive, which contradicts the maximality of λ(α) or that of λ( β).  Definition 11.2.5 Let Φ be an irreducible crystallographic root system, let Φ+ be a positive subsystem and Π be the corresponding basis. We call extended ˜ Π ∪ {α0 } where α0 is the negative of the highest root. basis of Φ the set Π=

170

Geometric Conjugacy and the Lusztig Series

Proposition 11.2.6 Let G be a quasi-simple simply connected reductive group, ˜ be an extended root basis of the root system of G. Then Z(G) is and let Π ˜ induced by the elements of W isomorphic to the group Ω of permutations of Π ˜ which stabilise Π. Reference

See for example Bonnafé (2006, Proposition 5.3).



Table 11.1 illustrates such groups Ω. The extended root basis gives rise to an extended Coxeter diagram which records the scalar products between el˜ (or the order of st, where s and t are the corresponding reflecements of Π tions). The extended Coxeter diagram is also the Coxeter diagram describing a Coxeter group, the affine Weyl group, which is the affine reflection group in Y (T) ⊗ R generated by W and the reflection with respect to the affine hyperplane of equation α0 , y = 1, see for example Bourbaki (1968, Chap. VI, §2.3). We describe the extended diagrams and generators of the group Ω, described by their cycle decomposition, in each case. In Table 11.1, we just write 0, 1, . . . , n for the elements α0 , α1 , . . . , αn of an extended root basis, where Π = {α1 , . . . , αn }. We give another application of the extended basis of an irreducible root system Φ, due to Borel and De Siebenthal: ˜ − {α}, for α ∈ Π, ˜ are bases of Proposition 11.2.7 The sets of the form Π closed root subsystems of Φ. Every maximal proper closed root subsystem of Φ has a basis conjugate to one of these sets. Reference

See for example Bourbaki (1968, VI, Ex. 4 of §4).



11.3 The Lusztig Series The next definition is a first step towards the classification of characters of GF . Definition 11.3.1 The Lusztig series E (GF , (s)) associated to a semi-simple ∗ geometric conjugacy class (s) ⊂ G∗F is the set of irreducible characters of (s), for some F ∗ -stable maximal torus T∗ of G∗ GF which occur in some RG T∗ containing an element of (s). The series E (GF , (s)) are sometimes called “geometric Lusztig series”; we ∗ ∗ will also (see 12.4.4) consider “G∗F -Lusztig series” where the G∗F -class of s is fixed. The two notions coincide, by 4.2.15(ii), when CG∗ (s) is connected.  Proposition 11.3.2 We have Irr(GF ) = (s) E (GF , (s)) where (s) runs over ∗ the semi-simple geometric conjugacy classes of G∗F .

11.3 The Lusztig Series

171

Table 11.1. Groups Ω for extended Coxeter diagrams

 0

A˜n : 



1

Ω = (0, 1, . . . , n)

 n

2

0 ˜n :  B

· · ·  

1

2

C˜n :   0

· · ·  

1

2

0



Ω = (0, n)(1, n − 1)(2, n − 2) · · ·

n

n−1



 ˜n : D

Ω = (0, 1)

n

n− 1

n−1





n−2

2

 n

1

Ω = (1, n, 0, n − 1)(2, n − 2)(3, n − 3) · · · n odd Ω = (0, 1)(n, n − 1), (0, n)(1, n − 1)(2, n − 2) · · · n even 0 2 ˜6 :  E 1





3



4



5

Ω = (1, 0, 6)(3, 2, 5)

6

2 ˜7 :  E 0





1

3









Ω = (0, 7)(1, 6)(3, 5)











4

5

6

7

2 ˜8 :  E





F˜4 : 



 

˜2 :  G

 

1

0

0

3

4

1

1

2

5

3

6

7

8

Ω = {1}

0



Ω = {1}

4

Ω = {1}

2

Proof By 11.1.3 and 11.1.15 two Deligne–Lusztig characters in different series have no common constituent, and by 11.1.1 any irreducible character is in some series.  Remark 11.3.3 If U  T is a Levi decomposition of a Borel subgroup of G, by 11.1.3 any γ ∈ Irr(GF ) occurring in some Hci (L −1 (U)) ⊗Q TF θ is in the series 

172

Geometric Conjugacy and the Lusztig Series

E (GF , (s)), where (s) is the semi-simple conjugacy class of G∗ corresponding to the geometric class of (T, θ). A particularly important series, which is a kind of “prototype” for the other ones, is the series associated to the identity element of G∗ . Definition 11.3.4 The elements of E (GF , 1) (that is, the irreducible constituents (1)) are called unipotent characters. of the RG T We introduce a variety whose cohomology gives directly the unipotent characters. Definition 11.3.5 If B is a Borel subgroup containing an F-stable maximal torus, we define the variety XB := {gB ∈ G/B | gB ∩ F(gB)  ∅} = {gB ∈ G/B | g−1 Fg ∈ B · FB}  {g ∈ G | g−1 Fg ∈ FB}/(B ∩ FB). The group GF acts by left translations on XB . The same arguments as after 9.1.1 show that the three formulas define isomorphic varieties. They are smooth by, for example, the proof of Digne et al. (2007, 2.3.5). Proposition 11.3.6 If B = U  T is a Levi decomposition of a Borel subgroup where T is F-stable then XU /TF  XB as GF -varieties. Proof We work with the models XU = {g ∈ G | g−1 Fg ∈ FU}/(U ∩ FU) and XB = {g ∈ G | g−1 Fg ∈ FB}/(B ∩ FB). By 3.4.8(ii), we have B ∩ FB = (U ∩ FU)T. The morphism g(U ∩ FU) → g(B ∩ FB) maps XU to XB . If g(B ∩ F B) ∈ X , we have g−1 F g = tu with t ∈ T and u ∈ F U. By the Lang–Steinberg B theorem in T, we write t = τ. F τ −1 so that gτ(U∩ FU) is in XU and is a preimage of g(B∩ FB). Moreover τ is well defined up to multiplication by TF . This shows that the morphism g(U ∩ FU) → g(B ∩ FB) is surjective and that it defines a bijective morphism from XU /TF to XB . The variety XB being smooth, it is an  isomorphism by 9.1.2, which is compatible with the GF -actions. Corollary 11.3.7 (i) With notation as in 11.3.6, the character RG (1) is the character of the T virtual GF -module Hc∗ (XB ). (ii) The unipotent characters of GF are the irreducible constituents of the Hci (XB ) when B runs over Borel subgroups containing an F-stable maximal torus. Proof Both statements are consequences of the proposition and of 8.1.10(i), using Remark 11.3.3 for (ii). 

11.3 The Lusztig Series

173

The next statement shows that the set of unipotent characters depends only on the isomorphism type of the root system of G. Proposition 11.3.8 Let (G, F) and (G1 , F1 ) be two connected reductive groups with Frobenius roots, and let f : G → G1 be a morphism of algebraic groups with a central kernel, such that f ◦ F = F1 ◦ f and such that f (G) contains Der G1 ; then the unipotent characters of GF are the χ ◦ f , where χ runs over the unipotent characters of GF1 1 . Proof We first remark that in the situation of the proposition we have G1 = Z(G1 ) 0 .f (G) by 2.3.8. The Borel subgroups, the maximal tori and the parabolic subgroups of f (G) are thus the intersections of those of G1 with f (G). On the other hand, f induces a bijection from the Borel subgroups, the maximal tori and the parabolic subgroups of G to those of f (G), so there is a bijection from the sets of Borel subgroups, maximal tori and parabolic subgroups of G to the corresponding sets for G1 . The unipotent characters of GF are those which occur in some cohomology space of some variety XB where B runs over Borel subgroups of G containing an F-stable maximal torus. Let B1 = f (B)Z(G1 ) 0 be the Borel subgroup of G1 corresponding to B by the above bijection. The morphism gB → f (gB) : XB → XB1 is compatible with the actions of GF and GF1 1 . Since Ker f ⊂ B this morphism is injective. Let us show that it is an isomorphism, using the second model of the varieties. Let x1 ∈ G1 be such that x1−1 F1 x1 ∈ B1 F1 B1 ; multiplying by an element of Z(G1 ) 0 does not change x1 B1 , hence we can assume x1 ∈ f (G). A preimage x of x1 in G satisfies x−1 Fx ∈ FB whence the surjectivity. We conclude using that XB1 is smooth and 9.1.2. 1 (1) = RG (1) ◦ f , where T is an F-stable maxiWe have thus proved that RG T T1 0 mal torus of B, and T1 = f (T)Z(G1 ) is the corresponding F1 -stable maximal torus of G1 . Moreover as GF -classes of F-stable maximal tori are in one-toone correspondence with GF1 1 -classes (they are parametrised by the F-classes of the Weyl group), the “restriction through f ” defines a bijection from the set (1) of GF onto the similar set for GF1 1 . It reof Deligne–Lusztig characters RG T mains to see that the restriction through f maps the irreducible constituents of one to those of the other. We have GF1 1 = TF1 1 . f (GF ). Indeed, any y ∈ G1 can be written y = zf (x) with z ∈ Z(G1 ) 0 and x ∈ G, and y is F1 -stable if and only if f (x Fx−1 ) = z−1 F1 z. This element is in Z(G1 ) 0 ∩ f (G) ⊂ Z(f (G)) and this last group is in f (T) which is a maximal torus of the reductive group f (G) (see the beginning of the present proof). So x Fx−1 ∈ T, since the kernel of f is in T, and by the

174

Geometric Conjugacy and the Lusztig Series

Lang–Steinberg theorem applied in the group T we can find t ∈ T such that tx ∈ GF . So y = (zf (t−1 ))f (tx) ∈ TF1 1 . f (GF ). It follows that GF1 1 /f (GF ) is a quotient of TF1 1 thus commutative, so we can use the following result from Clifford’s theory, see Curtis and Reiner (1981, 11.4, 11.5). Lemma 11.3.9 Let G be a finite group and H be a normal subgroup of G such that G/H is abelian. Then for χ, χ ∈ Irr(G), we have G ResG H χ, ResH χ H = #{ζ ∈ Irr(G/H) | χζ = χ }.

From this lemma we see that the restriction of a unipotent character χ of GF1 1 to f (GF ) is irreducible if and only if for any irreducible non-trivial character ζ of GF1 1 /f (GF ) we have χ  χζ. But χζ is not unipotent if ζ is not trivial, be1 1 (1) then χζ is a constituent of RG (Res cause if χ is a constituent of RG T1 T1

F

G1 1 F T1 1

ζ)

and the restriction of ζ to TF1 1 is not trivial, since GF1 1 = TF1 1 .f (GF ), so cannot be geometrically conjugate to the trivial character, whence the proposition.  There is a similar – though weaker – result for a general Lusztig functor. Proposition 11.3.10 We assume the same hypotheses as in 11.3.8, and assume, in addition, that the kernel of f is connected. Then, if L1 is an F1 -stable Levi subgroup of some parabolic subgroup P1 of G1 and if λ 1 ∈ Irr(LF1 1 ), γ1 ∈ Irr(GF1 1 ), we have 1 (λ 1 ) ◦ f = RG RG L1 ⊂P1 f −1 (L

1 ) ⊂f

−1 (P

1)

(λ 1 ◦ f )

and ∗ G1 RL1 ⊂P1 (γ1 )

◦ f = ∗RG f −1 (L

1 ) ⊂f

−1 (P

1)

(γ1 ◦ f ).

Proof By the remarks we made at the beginning of the proof of 11.3.8, the group P := f −1 (P1 ) = f −1 (P1 ∩ f (G)) is a parabolic subgroup of G which has L := f −1 (L1 ) = f −1 (L1 ∩ f (G)) as Levi subgroup, so the statement makes sense. If P = U  L is the Levi decomposition of P, and P1 = U1  L1 that of P1 , then we have U1 = f (U) since U1 ⊂ Der G1 ⊂ f (G). To prove the proposition, we will use the following. Lemma 11.3.11 −1 (U)/(Ker f ) F  L −1 (U ) and (i) If f is surjective, it induces isomorphisms LG 1 G1

−1 (U)) (Ker f )  H ∗ (L −1 (U )). Hc∗ (LG 1 c G1 (ii) If f is an embedding of G as a closed subgroup of G1 containing Der G1 , −1 (U) =  −1 (U) =  g · LG · we have U = U1 and LG F F g ∈ G1 1 /GF l∈L1 1 /LF 1 −1 LG (U) · l. F

11.4 Lusztig’s Jordan Decomposition of Characters

175

Proof For (i) we first prove the surjectivity of the morphism induced by f . If LG1 (g1 ) = u1 ∈ U1 then any g ∈ f −1 (g1 ) satisfies LG (g) = uz with z ∈ Ker f and u ∈ U. Since Ker f is connected we may find z ∈ Ker f such that LG (z ) = −1 (U). z−1 . Then LG (gz ) = u (using that z is central), thus gz ∈ LG −1 We now show that the fibres of f on L (U) are isomorphic to (Ker z) F . If g and gz are in L −1 (U), where z ∈ Ker f , then LG (gz) = LG (z)LG (g) ∈ LG (z)U, and LG (z)U is disjoint from U unless LG (z) = 1, that is z ∈ (Ker f ) F . Using the smoothness of L −1 (U) and 9.1.2, we get the first assertion of (i). The second assertion of (i) follows from 8.1.10(i). −1 (U) ⊂ L −1 (U) and since the For the first equality of (ii) it is clear that LG G1 right variety is stable by left translation by GF1 1 , it is sufficient to prove that it −1 (U). If g ∈ L −1 (U) then by the Lang– is covered by GF1 1 -translates of LG 1 G1 −1 (U) such that L (g) = L (g ), which Steinberg theorem there exists g ∈ LG G G1 1 F1 −1 implies that g1 g ∈ G1 , whence the result. We deduce the last equality using  GF1 1 /GF  LF1 1 /LF . The restriction from LF1 1 to LF through f is Q LF1 1 ⊗Q

F1  L1

. Decomposing L

F1

f into a surjective map followed by an embedding, we thus get RG ◦ ResL1F = L −1 (U)) ⊗ Hc∗ (LG Q

F

Q LF1 1 ⊗Q

F

. Since both cases of the lemma imply that

1  L1 F1 ∗ −1 ∗ −1 (U )) as GF -modules-LF1 , the action of Hc (LG (U)) ⊗Q LF Q L1  Hc (LG 1 1 1  GF on the right-hand side being by restriction through f we get the first equality L

of the proposition. The proof of the second equality is similar, using 9.1.4, interchanging G and L and simultaneously G1 and L1 and using that both cases of the lemma F1 −1 (U)) opp ⊗ ∗ −1 opp as LF -modules-GF1 , imply Hc∗ (LG Q GF Q G1  Hc (LG1 (U1 )) 1  the action of LF on the right-hand side being by restriction through f .

11.4 Lusztig’s Jordan Decomposition of Characters: The Levi Case We explain now Lusztig’s “Jordan decomposition of characters” which gives a parametrisation of E (GF , (s)) by the unipotent characters of CG∗ (s) F . We first explain how to get explicitly such a bijection when CG∗ (s) is a Levi subgroup of a parabolic subgroup of G∗ ; this is in a sense the general case as “most” semi-simple elements have this property: we will show (see just after 12.1.12) that, modulo the centre, there is only a finite number of semi-simple conjugacy classes whose centraliser is not in a proper Levi subgroup of G. We first explain the correspondence between F-stable Levi subgroups of G and F ∗ -stable Levi subgroups of G∗ . We first classify GF -conjugacy classes of F-stable Levi subgroups.

176

Geometric Conjugacy and the Lusztig Series

Proposition 11.4.1 Let G be a connected reductive group over Fp with a Frobenius root F. Let T ⊂ B be a pair of an F-stable maximal torus and an F-stable Borel subgroup and let (W (T), S) be the corresponding Coxeter system. Then the GF -classes of F-stable Levi subgroups of G are parametrised by F-conjugacy classes of cosets WI w where I ⊂ S and w are such that wFWI = WI . Proof According to 4.2.14 the GF -classes of Levi subgroups conjugate under G to L are parametrised by H 1 (F, NG (L)/L). We may conjugate L to some LI by conjugating a maximal torus of L to T but then, by a similar argument to the remark after 4.2.22, the action of F on L becomes that of v˙ F on LI where v˙ ∈ NG (T) is a representative of some v ∈ W (T); the condition that L is Fstable becomes v˙ FLI = LI , which is equivalent to vFWI = WI as seen in the proof of 3.4.3. So the GF -classes of Levi subgroups conjugate under G to L are parametrised by H 1 (˙vF, NG (LI )/LI )  H 1 (vF, NW (WI )/WI ), which is the same as the F-classes under NW (WI ) of cosets WI zv where z ∈ NW (WI ). This last condition is equivalent to zvFWI = WI . We thus get the proposition by putting w = zv; the F-classes under NW (WI ) are replaced by F-classes under W since, if the F-class of WI w and that of WJ w are equal, then WI is F-conjugate  under W to WJ . Note that, as the parametrisation given by 11.4.1 is the same in G and G∗ , the GF -classes of F-stable Levi subgroups of G correspond one-to-one to the similar classes in G∗ . If the class of L corresponds to the F-class under W (T) of WI w (that is, to the W (T)-class of WI wF), we make L correspond to a Levi subgroup L∗ of G∗ whose class is parametrised by F ∗ w∗ WI∗ . As seen in the proof of 11.4.1, we may find in L a torus Tw of type w; that is, (Tw , F) is geometrically conjugate to (T, wF); similarly we may find in L∗ a torus T∗w such that (T∗w , F ∗ ) is geometrically conjugate to (T∗ , F ∗ w∗ ). With these choices ∼ → X(T∗w ) is (L, Tw , F) is dual to (L∗ , T∗w , F ∗ ), where the isomorphism Y (Tw ) − ∼ → X(T∗ ) by the chosen geometric “transported” from the isomorphism Y (T) − conjugations. Given the duality between (G, T) and (G∗ , T∗ ), the duality thus defined between L and L∗ is defined up to the automorphisms of L induced by G; that is, by NG (L). It is easy to check that this duality is compatible with 11.1.15 and 11.1.16; that is, if (T, θ) where T ⊂ L corresponds to (T∗ , s) where T∗ ⊂ L∗ for this duality, then it also corresponds to (T∗ , s) for the duality in G. Definition 11.4.2 A semi-simple element s of a reductive group G is quasiisolated if CG (s) is not contained in any proper Levi subgroup of G. We may then state the following result, which reduces the problem of parametrising E (GF , (s)) to the case of quasi-isolated elements. In particular the case of elements such that CG∗ (s) is a Levi subgroup is reduced to the case of central elements.

11.4 Lusztig’s Jordan Decomposition of Characters

177

∗

Theorem 11.4.3 Let s be a semi-simple element of G∗F and let L∗ be an F ∗ -stable Levi subgroup of G∗ which contains CG∗ (s). Let L be an F-stable ∗ Levi subgroup of G whose GF -class corresponds to the G∗F -class of L∗ as explained above, and let U  L be the Levi decomposition of some parabolic subgroup of G containing L; then: (i) For any λ ∈ E (LF , (s)) there exists an integer i(λ) such that the space Hci (XU ) ⊗Q LF λ is zero for i  i(λ) and affords an irreducible representa tion of GF for i = i(λ); we thus have (−1) i(λ) RG (λ) ∈ Irr(GF ). When XU L is known to be affine then i(λ) = dim XU . induces a bijection from E (LF , (s)) to E (GF , (s)). (ii) The functor ε G ε L RG L Proof We begin with a lemma which translates for the group G the condition CG∗ (s) ⊂ L∗ . Lemma 11.4.4

In the situation of 11.4.3

(i) Let T and T be two F-stable maximal tori of L and let θ ∈ Irr(TF ) and θ ∈ Irr(T F ) be such that (T, θ) and (T , θ ) are geometrically conjugate ∗ in L and their geometric class corresponds to (s) ⊂ L∗F by the duality between L and L∗ ; then any g ∈ G which geometrically conjugates (T, θ) to (T , θ ) is in L. (ii) Let T, an F-stable maximal torus of G and θ ∈ Irr(TF ) be such that the geometric conjugacy class of (T, θ) corresponds to the geometric class of ∗ s in G∗F . Then T is GF -conjugate to some torus of L. Proof To prove (i) it is clearly enough to consider the case where T = T and θ = θ . By 11.1.8 g is then an element of NG (T) which fixes θ seen as an element of Hom(Y, (Q/Z)p ) as in 11.1.7(ii). The assumption does not change if we replace F by F n . Thus we can assume that T is split and g ∈ GF . Then the Weyl group W of G is NG (T)/T and the image w of g in W fixes the element of Hom(Y, (Q/Z)p ) defined by θ by 11.1.7(ii); that is, by the proof of 11.1.15, fixes the corresponding element of Y (T∗ ) ⊗ (Q/Z)p which is s by definition. Hence w ∈ W (s) ⊂ WL so that g is in L. We now prove (ii). Let T0 ⊂ L and θ 0 ∈ Irr(TF0 ) be such that the geometric class in L of the pair (T0 , θ 0 ) corresponds to the class in L∗ of s. Then (T0 , θ 0 ) also corresponds to (s) for the duality between G and G∗ , so (T, θ) and (T0 , θ 0 ) are geometrically conjugate in G. Let g ∈ G be an element which geometrically conjugates (T, θ) to (T0 , θ 0 ). Then Fg also geometrically conjugates (T, θ) to (T0 , θ 0 ), so g. Fg−1 geometrically conjugates (T0 , θ 0 ) to itself. Thus by (i) we have g. Fg−1 ∈ L. Using the Lang–Steinberg theorem to write g. Fg−1 = l−1 . Fl with l ∈ L, we get an element lg which is F-stable and conjugates T to some torus of L. 

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Geometric Conjugacy and the Lusztig Series

We now place ourselves in the setting of 9.2.1, where we take L = L and = P = LU and we work with the variety L −1 (U) instead of XU as is allowed by 9.1.5. The key step for 11.4.3 is Proposition 11.4.5.

P

Proposition 11.4.5 Under the same assumptions as 11.4.3, let λ and λ be two irreducible representations in E (LF , (s)), then λ opp ⊗Q

L

F

Hci (Z, Q ) ⊗Q

L

λ

F

is 0 except for i = 2d, where d = dim U + dim(U ∩ FU), where Z is as in 9.2.1 and λ opp is as in the proof of 11.1.3. In addition, we have dim(λ opp ⊗Q

L

F

Hc2d (Z, Q ) ⊗Q

L

F

λ ) = λ, λ LF .

Proof We may choose two F-stable maximal tori T and T in L, characters θ ∈ Irr(TF ) and θ ∈ Irr(T F ) and TV (resp. T V ) (the Levi decomposition of) a Borel subgroup of L containing T (resp. T ) such that there exists j j (resp. k) for which λ (resp. λ ) is a constituent of Hc (L −1 (V)) ⊗Q TF θ (resp.  of Hck (L −1 (V )) ⊗Q T F θ ); let Zw be as defined after 9.2.1, then for all i, λ opp ⊗Q





θ opp ⊗Q

LF

Hci (Zw , Q ) ⊗Q

L

F

λ is a subspace of

Hc (L −1 (V)) opp ⊗Q j

T

F

L

F

Hci (Zw , Q ) ⊗Q

L

F

Hck (L −1 (V )) ⊗Q

F T

θ .

By similar arguments to those used in the proof of 9.1.8, the Q TF -moduleQ T F given by Hc (L −1 (V)) opp ⊗Q j

L

F

Hci (Zw , Q ) ⊗Q

L

F

Hck (L −1 (V ))

i+j+k  is isomorphic to a submodule of Hc ( w1 Z1w1 , Q ), where Z1w1 is the variety analogous to Zw , relative to T and T , and where w1 runs over elements of T\S(T, T )/T having the same image in L\S(L, L)/L as w. By an argument similar to the proofs of 11.1.4 and 11.1.5, the character θ ⊗ θ opp does not occur in this module if none of the w1 geometrically conjugate θ to θ . By 11.4.4(i), this implies that w is in L. Thus λ opp ⊗Q LF Hci (Zw , Q ) ⊗Q LF λ is zero if w is   not in L. If w is in L, that is w = 1 in L\S(L, L)/L, the variety Zw = Z1 may be simplified. We have −1

Z1 = {(u, u , g) ∈ U × U × F P | u Fg = gu }, −1

so (u, u , g) ∈ Z1 implies u Fgu −1 = g ∈ P ∩ F P. Using the decomposition −1 −1 3.4.8(ii) P ∩ F P = L.(U ∩ F U), and taking projections to L, we see that −1 g ∈ LF .(U ∩ F U); so the variety projects to LF with all fibres isomorphic to U × (U ∩ FU). Thus, up to a shift of 2d, the cohomology is that of the discrete variety LF , thus

11.4 Lusztig’s Jordan Decomposition of Characters

λ opp ⊗Q

L

F

Hci (Zw , Q ) ⊗Q

L

F

λ =

⎧ λ opp ⊗Q LF Q LF ⊗Q LF λ ⎪   ⎨ ⎪0 ⎩ and the dimension of λ opp ⊗Q

L

F

179

Q LF ⊗Q

L

F

if w ∈ L and i = 2d, otherwise,

λ is λ, λ LF . Using the argu-

ments of 11.1.5 we get the required result for λ opp ⊗Q

L

F

Hci (Z, Q )⊗Q

L

F

λ .



λ, RG λ GF is equal (using 8.1.6) to the alternating As the scalar product RG L L sum of the dimensions of λ opp ⊗Q

L

F

Hc∗ (L −1 (U)) opp ⊗Q

G

F

Hc∗ (L −1 (U)) ⊗Q

L

F

λ ,

the above proposition thus shows, using 9.2.1, that G RG L λ, RL λ GF = λ, λ LF .

Remark 11.4.6 We note that under the assumption of 11.4.3, even if we do not assume P = P , the preceding proof shows that w ∈ L and this together with Lemma 9.2.5 shows that the following statement (equivalent to the Mackey formula when λ or λ is cuspidal) holds: G RG L⊂P λ, RL⊂P λ G = λ, λ LF .

As seen in the proof of 5.3.1, this implies that RG λ is independent of P in L⊂P that case. We now prove theorem 11.4.3. From 11.4.5 the dimension of ⊕i+j=2d λ opp ⊗Q

L

F

Hci (L −1 (U)) opp ⊗Q

Hc (L −1 (U)) ⊗Q j

G

F

L

 λ opp ⊗Q

F

L

F

λ Hc2d (Z) ⊗Q

L

F

λ

is equal to 1, so all the summands have dimension 0 except one, say λ opp ⊗Q

L

F

Hci0 (L −1 (U)) opp ⊗Q

G

F

Hc2d−i0 (L −1 (U)) ⊗Q

L

F

λ

which has dimension 1. Suppose now that the Q GF -module given by Hc (L −1 (U)) ⊗Q j

L

F

λ

is not 0. Let χ be one of its irreducible constituents. Then χ is in E (GF , (s)), so χ is in E (GF , (s−1 )): indeed in the exact sequence of 11.1.7(i), if θ ∈ Irr(TF ) is ∗ the image of x ∈ X then θ = θ −1 is the image of −x and if s ∈ T∗F is the image of x in the exact sequence analog to 11.1.7(ii) for T∗ then s−1 is the image of (θ) where T is a maximal torus of L −x. By 11.4.4(ii), χ is a constituent of RG T

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Geometric Conjugacy and the Lusztig Series

and θ is given by the geometric class of s. So χ is a constituent of RG RL (θ) and L T (λ ) with λ ∈ E (LF , (s)). Then χopp occurs in particular appears in some RG L in some λ opp ⊗Q LF Hck (L −1 (U)) opp and thus 

E := λ opp ⊗Q

Hc (L −1 (U)) opp ⊗Q j

L

F

But E is a subspace of λ opp ⊗Q

G

F

Hck (L −1 (U)) ⊗Q

j+k

L

F

Hc (Z, Q ) ⊗Q

L

F

L

F

λ  0.

λ , so this last space is

not 0, which proves by 11.4.5 that j + k = 2d and λ = λ thus by definition of i0 we have j = i0 . Since in that case E has dimension 1, we see that χ must be, in addition, the only irreducible constituent of Hci0 (L −1 (U)) ⊗Q LF λ. Whence  (i) of the theorem with i(λ) = i0 − 2 dim(U ∩ FU). When XU is affine, its cohomology is zero in degree less than dim XU , so that of L −1 (U) is zero in degree less than dim XU + 2 dim(U ∩ FU) = d, hence i0 ≥ d and 2d − i0 ≥ d, thus i0 = d, whence i(λ) = d − 2 dim(U ∩ FU) = dim XU . maps up to sign E (LF , (s)) to E (GF , (s)). Let us show (ii). (i) implies that RG L That the sign is ε G ε L follows immediately from 10.2.9. Let us show that is surjective. Let χ ∈ E (GF , (s)); by definition χ is a constituent ε G ε L RG L G of RT (θ ) for some torus T and some character θ which is in the class defined by s. By 11.4.4 (ii), the torus T has a GF -conjugate in L. We may thus assume (θ ) = RG ◦ RLT (θ ), so χ is indeed the RG of a that T is in L. In that case RG T L L F  character of E (L , (s)). Exercise 11.4.7 (i) With the notation as in 11.4.3, let BL be an F-stable Borel subgroup of L and let w ∈ W be such that the pair (BL U, F) is G-conjugate to (B0 , wF) where B0 is an F-stable Borel subgroup of G. Show that dim XU = l(w). (ii) Show that a quasi-split torus of L contained in BL is of type w in G. (iii) If XU is known to be affine, under the assumption of Theorem 11.4.3, show maps a true using (i) of the theorem without using 10.2.9 that ε G ε L RG L l(w) character to a true character (hint: we have ε G ε L = (−1) ). The special case of Theorem 11.4.3 where L∗ = CG∗ (s) shows that RG is L then up to sign a bijection from E (CG∗ (s) ∗F , (s)) to E (GF , (s)). Moreover, Proposition 11.4.8 Let s be an F ∗ -stable central element of G∗ . Then: (i) There exists a linear character sˆ ∈ Irr(GF ) such that, for any F-stable maximal torus T of G, the pair (T, sˆ |TF ) is in the geometric conjugacy class defined by s. (ii) Taking the tensor product with sˆ defines a bijection ∼

E (GF , 1) − → E (GF , (s)).

11.4 Lusztig’s Jordan Decomposition of Characters

181

Proof We follow Deligne and Lusztig (1976, 5.11). ∗ As s ∈ Z(G∗F ), its geometric class defines a character of T F for any Fstable maximal torus T ⊂ G since, if (G, T) is dual to (G∗ , T∗ ), the element s ∗ ˜ T) ˜ be a dual pair to (G∗ /Z(G∗ ), T∗ /Z(G∗ )). is in T∗wF for any w. Let (G, Proposition 11.4.9 The pair (G∗ /Z(G∗ ) 0 , T∗ /Z(G∗ ) 0 ) is dual to the pair ˜ → Der G (Der G, T ∩ Der G). Let π0 be the inclusion Der G → G and f : G be the dual isogeny to f ∗ : G∗ /Z(G∗ ) 0 → G∗ /Z(G∗ ); then the composed map π := π0 ◦ f is an F-equivariant map with central kernel which induces an ˜ ⊂ T. isomorphism on the Uα and such that π(T) Proof The groups G and Der G have same root systems (Φ, Φ∨ ) with respect to T and T ∩ Der G respectively and the groups G∗ and G∗ /Z(G∗ ) 0 have same root systems (Φ∨ , Φ) with respect to T∗ and T∗ /Z(G∗ ) 0 respectively. We have X(T∗ /Z(G∗ ) 0 ) = ((Φ∨ ) ⊥ ) ⊥ = Y (T ∩ Der G), the first equality by 2.3.10(i) and the second one by 2.3.10(ii). This shows that (G∗ /Z(G∗ ) 0 , T∗ /Z(G∗ ) 0 ) is dual to (Der G, T ∩ Der G). The map f is F-equivariant since f ∗ is F ∗ -equivariant. The other statements are obvious since they are true for an isogeny and for the inclusion Der G ⊂ G.  ˜ = Der G, and π(T) ˜ = T ∩ Der G. We thus get π(Z(G)) ˜ = We have π(G) Z(Der G) since the centre consists of the elements in a maximal torus which act trivially on the Uα . Since the kernel of π is central, we deduce that π induces ˜ ˜  (T ∩ Der G)/Z(Der G). an isomorphism T/Z( G) Lemma 11.4.10

˜ F ) = π(T˜ F ). ˜ = π(T) ˜ and TF ∩ π(G T ∩ π(G)

˜ Proof We have seen the first statement. The second comes from π −1 (T) = T, F F ˜ ˜ since Ker π ⊂ T, thus an element of T ∩ π(G ) is the image of an element in ˜ F ∩ π −1 (T) = T˜ F .  G Lemma 11.4.11 ˜ F ). GF /π(G

The inclusion T ⊂ G induces an isomorphism TF /π(T˜ F ) 

˜ the first equality by 2.3.8. For g ∈ Proof We have G = T Der G = Tπ(G), F ˜ Since g is F-stable we G , let us write g = tπ(g˜ ) with t ∈ T and g˜ ∈ G. −1 F F −1 ˜ = π(T), ˜ the last equality by the previous have t t = π(g˜ g˜ ) ∈ T ∩ π(G) ˜ By the lemma. Since the kernel of π is contained in T˜ this implies g˜ Fg˜ −1 ∈ T. F −1 −1 F ˜ ˜ ˜ ˜ Lang–Steinberg theorem we can write g˜ g˜ = t1 t1 with t1 ∈ T. Then ˜t1 g˜ ˜ F and tπ(˜t−1 ) = gπ(˜t1 g˜ ) −1 is in TF . Thus we have g = tπ(˜t−1 )π(˜t1 g˜ ) ∈ is in G 1 1 F ˜ F ), hence GF = TF π(G ˜ F ). Using the second assertion of the previous T π(G ˜ F )  TF /π(T˜ F ).  lemma we get GF /π(G

182

Geometric Conjugacy and the Lusztig Series ∗

Proposition 11.4.12 There is a natural bijection between Z(G∗ ) F and the ˜ F ). characters of the abelian group GF /π(G ˜ = X(T∗ /Z(G∗ )) = ((Φ∨ ) ⊥∗ ) ⊥ ∗ where (Φ, Φ∨ ) is the Proof We have Y (T) T X(T ) root system of G. A character θ ∈ Irr(TF ) vanishes on π(T˜ F ) that is on the ∗ images of the coroots if and only if the corresponding element s ∈ G∗F is in ∗ (Φ∨ )T⊥∗ that is s ∈ Z(G∗ ) F . Conversely for any F-stable T, an element s ∈ ∗ Z(G∗ ) F defines a character of TF which vanishes on π(T˜ F ) hence a character ˜ F ). This character is independent of the torus T since the characters of GF /π(G obtained in various tori are geometrically conjugate, and Lemma 11.4.13 Geometrically conjugate characters of F-stable maximal ˜ F ). tori induce the same character of GF /π(G Proof Let T and T be two F-stable maximal tori of G and let n be such that n n there exists x ∈ GF such that T = xT. It is sufficient to show that for t ∈ TF ˜ F ), where the elements NFn /F (t) and NFn /F ( xt) have same image in GF /π(G n−1 F F 0 ˜ we may y. Since G = Z(G) π(G) NFn /F : G → G is the map y → y y . . . n F ˜ and take n large enough so that x˜ ∈ G ˜ . assume x = π(˜x) ∈ π(G) Let φG : T → G be the map t → NFn /F (t) −1 NFn /F ( xt), and let φG˜ be the ˜ defined using x˜ . analogous map T˜ → G n F ˜ such that π(˜t ) = t; then ˜t−1 Fn ˜t ∈ Take now t ∈ T . Lift it to ˜t ∈ G ˜ Using that for any y in an F-stable torus we have FNFn /F (y) = Ker π ⊂ Z(G). n n n −1 F yNFn /F (y), we get F φG˜ (˜t ) = (˜t−1 F ˜t ) −1 φG˜ (˜t ) x˜ (˜t−1 F ˜t ) = φG˜ (˜t ), the last y n ˜ F ).  equality since ˜t−1 F ˜t is central. Thus φG (t) = π(φG˜ (˜t )) is in π(G  We have seen in the proof of 11.4.12 that the character of TF corresponding ∗ to s ∈ Z(G∗ ) F is in the geometric class defined by s, whence 11.4.8(i). Let us now prove 11.4.8(ii). Taking the tensor product with sˆ permutes the (θ)ˆs = RG (θˆs) for any T. irreducible characters, and by 10.1.2(i) we have RG T T ∗ w∗ wF ∗F is compatible with multiplication, if As the isomorphism Irr(T )  T ∗ the geometric class of (T, θ) corresponds to that of s1 ∈ G∗F , then the class of  (T, θˆs) corresponds to that of s1 s, whence 11.4.8(ii). ˜ F ) is the semi-simple quotient of the abelian Remark 11.4.14 Actually GF /π(G F quotient of G (for quasi-simple groups, the abelian quotient of GF has no unipotent elements except for the cases mentioned in 4.3.6). Indeed, the theo˜ F is generated by its unipotent rem of Steinberg recalled above 4.3.6 says that G F ˜ F ) is generated ˜ elements, thus the same holds for π(G ); thus, by 11.4.11, π(G F by the unipotent elements of G . The result then follows from the theorem of Tits recalled above 4.3.6.

11.5 Lusztig’s Jordan Decomposition: The General Case

183

Proposition 11.4.8 together with 11.4.3(ii) gives the special case of Theorem 11.5.1 below where CG∗ (s) is a Levi subgroup, if we show that the unipotent characters of a group and of its dual are in one-to-one correspondence; this results from 11.3.8 when the root systems of G and G∗ have same type; that is, unless G has a quasi-simple component of type Bn or Cn ; it was proved in this last case in Lusztig (1977).

11.5 Lusztig’s Jordan Decomposition of Characters: The General Case We now give a statement of Lusztig’s “Jordan decomposition of characters”. The statement is more complicated when Z(G) is not connected; to give a general statement, we first generalise the definition of Deligne–Lusztig characters F 0 (θ) = IndG0 F (RG (θ)), where T0 is an to a disconnected group G by setting RG T0 T0 G

(θ) is a sum of some F-stable maximal torus of G0 – we will not use this, but RG T0 G characters RT (θ ) which may be defined using a “quasi-torus” T = NG (T0 , B) where B is a Borel subgroup which contains T0 , see Digne and Michel (1994). This definition allows us to extend to the case of disconnected G the definition of unipotent characters, keeping the same definition; that is, E (GF , 1) is (1). With these definidefined as the set of irreducible constituents of the RG T0 ∗ tions, and using the definition 12.4.3 of G∗F -Lusztig series E (GF , (s)G∗F∗ ), we may state the following theorem. Theorem 11.5.1 Let G be a connected reductive group with a Frobenius root ∗ F; for any semi-simple element s ∈ G∗F , there is a bijection from E (GF , (s)G∗F∗ ) ∗ to E (CG∗ (s) F , 1). This bijection may be chosen such that, extended by lin(s) (see remark after 11.1.16 ) to earity to virtual characters, it sends ε G RG T∗ CG∗ (s) ∗ ε CG∗ (s)0 RT∗ (1) for any F -stable maximal torus T∗ of CG∗ (s). Comments and Proof. When the centre of G is connected, the group CG∗ (s) is connected for any semi-simple element s ∈ G∗ – see 11.2.2(ii) – and the remarks above about disconnected groups are not needed. When F is a Frobenius endomorphism and G has a connected centre Theorem 11.5.1 is the main topic of Lusztig (1984a). The description of the bijection in Lusztig (1984a) leaves some ambiguity, but this ambiguity may be lifted – see Digne and Michel (1990, Theorem 7.1). The bijection was extended in Lusztig (1988) to the disconnected centre case, but it is not explicit. We will first in 11.5.2 and 11.5.3 state the theorem of Lusztig (1988) and explain how to deduce our statement 11.5.1 when F is a Frobenius morphism.

184

Geometric Conjugacy and the Lusztig Series

An easy computation shows that conjugation by g ∈ G preserves GF if and only if LF (g) ∈ CG (GF ); the induced automorphisms of GF obtained this way are called diagonal automorphisms of GF . Now CG (GF ) = Z(G) – see 12.2.17 – and the condition LF (g) ∈ Z(G) is equivalent to the image g¯ ∈ G/Z(G) being F-stable. So we have an action of (G/Z(G)) F on GF by diagonal automorphisms. Further, we have the Galois cohomology exact sequence 4.3.5 1 → Z(G) F → GF → (G/Z(G)) F → H 1 (F, Z(G)) → 1 thus the image of the diagonal automorphisms in the outer automorphisms of GF factors through H 1 (F, Z(G)). We thus get an action of the group H 1 (F, ∗ Z(G)) (see 12.2.13) on Irr(GF ). To simplify notation, let H = CG∗ (s) F and ∗ H 0 = CG∗ (s) 0F . We denote by CH ( ρ) the inertia group in H of ρ ∈ Irr(H 0 ). Lusztig (1988, 5.1) can then be stated as follows. Proposition 11.5.2 Assume that F is a Frobenius endomorphism. There is a bijection between H 1 (F, Z(G))-orbits on E (GF , (s)G∗F∗ ) for the action through diagonal automorphisms and H-orbits on E (H 0 , 1). If the orbits O ⊂ E (GF , (s)G∗F∗ ) and O ⊂ E (H 0 , 1) correspond, then |O| = |CH ( ρ )/H 0 | for any ρ ∈ O . Further, for any ρ ∈ O and any F ∗ -stable torus T∗ of CG∗ (s) 0 we have  C ∗ (s) 0 ε G ρ, RG (s) = ε  ρ , RTG∗ (1)H 0 . F ∗ ∗ C (s) G G T ρ ∈ O

By 11.4.3(ii) it is enough to prove 11.5.1 when s is quasi-isolated. In this case the statement, when F is a Frobenius endomorphism, follows from Lusztig’s statement and from the following. Proposition 11.5.3 Assume that F is a Frobenius endomorphism. Let s ∈ ∗ ∗ ∗ G∗F be quasi-isolated, and let H = CG∗ (s) F and H 0 = CG∗ (s) 0F . The  restriction of any ρ˜ ∈ E (H, 1) to H 0 is of the form ρ ∈ O ρ where O is an H-orbit on E (H 0 , 1). There are |CH ( ρ )/H 0 | elements of E (H, 1) with a given restriction to H 0 . Proof By Clifford’s theory – see for example Curtis and Reiner (1981, §11) – all statements result from the first one, and the first one is equivalent to the claim that any ρ ∈ E (H 0 , 1) can be extended to its inertia group CH ( ρ ). The following lemma reduces the proof of 11.5.3 to the case where Z(G∗ ) = {1}. Lemma 11.5.4 Let (H, F) and (H1 , F1 ) be two reductive groups with Frobenius roots and let f : H → H1 be a morphism such that the morphism f0 : H0 → H01 induced on the identity components satisfies the assumptions of 11.3.8; that is, Ker f0 ⊂ Z(H0 ), and f0 ◦ F = F1 ◦ f0 , and f0 (H0 ) contains Der H01 . Assume

11.5 Lusztig’s Jordan Decomposition: The General Case

185

1 moreover that all unipotent characters of H0F 1 can be extended to their inertia F1 group in H1 . Then all unipotent characters of H0F can be extended to their inertia group in HF .

Proof By 11.3.8 the unipotent characters of H0F are of the form χ ◦ f0 where 1 χ is a unipotent character of H0F 1 . We have f (CHF ( χ ◦ f0 )) ⊂ CHF1 1 ( χ). Hence, if χ˜ is an extension of χ to CHF1 ( χ), then χ˜ ◦ f is an extension of χ ◦ f0 to 1 CHF ( χ ◦ f0 ).  ∗

Let s¯ be the image of s in G := G∗ /Z(G∗ ). Note that s¯ is still quasi-isolated. ∗ If we prove that all unipotent characters of CG∗ (¯s) 0F can be extended to their ∗ inertia group in CG∗ (¯s) F , we will get the extension property, thus 11.5.3, in G∗ by applying 11.5.4 with H = CG∗ (s) and H1 = CG∗ (¯s). We thus assume now Z(G∗ ) = {1}. We next reduce to the case where G∗  (thus G also) is simple. Since Z(G∗ ) = 1 we have G∗ = ki=1 G∗i , a direct product of simple groups; we can write s = s1 s2 . . . sk with si ∈ G∗i , and the centraliser of s is the direct product of the centralisers of si in G∗i . Splitting  ∗ ∗k the product into orbits under F ∗ , we get CG∗ (s) F  i∈I CG∗i (si ) F i , with I ⊂ {1 . . . k} such that {G∗i | i ∈ I} is a set of representatives of the orbits of F ∗ and where ki is the size of the corresponding orbit. Thus it is sufficient to prove the result for one of these simple factors. We assume now G∗ simple. The first statement of 11.5.3 holds when H/H 0 is cyclic, since it is always possible to extend characters to their inertia group in this case; see Curtis and Reiner (1981, 11.47). By 11.2.1(iii) the group H/H 0 is isomorphic to a subgroup of Irr(Z(G)) (we have Z(G) 0 = {1}); by 11.2.6 and table 11.1, this group is cyclic unless G∗ is adjoint of type D2n , in which case it is isomorphic to (Z/2Z) 2 . Thus the result is proved unless s is a quasiisolated element of G = PSO4n such that CG (s)/CG (s) 0  (Z/2Z) 2 . These ˜ be elements are completely described by Bonnafé (2005b, Theorem 5.1): let Π an extended basis of the root system of type D2n . By Bonnafé’s theorem quasiisolated elements such that CG (s)/CG (s) 0  (Z/2Z) 2 correspond to orbits of Ω ˜ − Ω. We thus ˜ and the group CG (s) 0 is generated by T and Uα for α ∈ Π on Π, 0 find two cases; in the first case CG (s) is of type A2n−3 ; CG (s) is generated, in addition, by two elements of W which act respectively by an automorphism of type 2A2n−3 and an automorphism of the radical. We can reduce to the adjoint quotient of CG (s) 0 by applying Lemma 11.5.4 with H = CG (s) and H1 = CG (s)/Z(CG (s) 0 ). Then H1 is the direct product of Z/2Z by a reductive group H2 such that H2 /H01 is cyclic, and the extendibility of characters follows. In the second case CG (s) 0 is of type Dn−i × Dn−i × A2i−1 and CG (s) is generated, in addition, by the representatives of two elements of W:

186

Geometric Conjugacy and the Lusztig Series

• An element σ which acts by an automorphism of type 2Dn−i on both components of type D. • An element τ which interchanges these two components and acts by an automorphism of type 2A2i−1 on the other component. By Adams and He (2017, Theorem A), we can lift the Weyl group to NG (T), in particular we can assume that σ and τ are commuting involutions. We can as in the first case reduce to the adjoint quotient and we get H01 = PSO2n−2i × PSO2n−2i ×PGL2i . A character τ and σ-invariant is of the form ρ⊗ ρ⊗ ρ where ρ is σ-invariant and ρ is τ-invariant. We can extend it first to σ as ρ˜ ⊗ ρ˜ ⊗ ρ , taking care to take the same extension of both components ρ. Such a character can then be extended to τ by extending ρ and making τ act on ρ˜ ⊗ ρ˜ by the interchange of the two factors of the tensor product.  Note that Proposition 11.5.3 just determines the existence of an (arbitrary) bijection between O and O in 11.5.2; the existence of a “canonical” bijection is an open problem. We finally indicate how to proceed to prove 11.5.1 when F is a Frobenius root which is not a Frobenius morphism. Using our arguments, we can reduce to the case where G is simple and using 11.4.3 and 11.4.8 we can reduce to the case where s is a non-central quasi-isolated element. Such elements do not exist in 2B2 and 2G2 ; in 2F4 there is a single conjugacy class of such elements where s is of order 3 and CG∗ (s) is connected of type A2 × A2 where F ∗ interchanges the two components. The corresponding characters were computed by Malle, see Malle (1990) and Geck et al. (1996a).  We still call uniform functions in a disconnected group the class functions which are linear combination of Deligne–Lusztig characters. Note that, since the Deligne–Lusztig characters of a disconnected group G are defined as induced from the Deligne–Lusztig characters of G0F , the induced function of a uniform function on G0F is uniform; in particular – by 10.2.6 – regGF is uniform. We generalise 9.3.1 to disconnected groups in Lemma 11.5.5. Lemma 11.5.5 (i) Let T and T be two F-stable maximal tori of the (possibly disconnected) reductive group G; for θ ∈ Irr(TF ) and θ ∈ Irr(T F ) we have G F −1 F n n RG T (θ), RT (θ )GF = |T | #{n ∈ G | T = T and θ = θ }.

(θ) where the pairs (T, θ) are taken up to GF -conjugacy form an (ii) The RG T orthogonal basis of the space of uniform functions.

11.5 Lusztig’s Jordan Decomposition: The General Case

187

Proof We have RG (θ), RG (θ )GF = ResG RG (θ), RG (θ )GoF . By the orT T T GoF T dinary Mackey formula this is equal to  0 x G0 RG xT ( θ), RT (θ )GoF . F

0

0

x∈GF /G0F

By 9.3.1(i) the summand indexed by x is equal to |TF | −1 |{g ∈ G0F | (T , θ )}|, whence (i). (ii) is an immediate consequence of (i).

gx (T, θ)

= 

If ψs is any bijection as in 11.5.1, then for any χ ∈

Proposition 11.5.6 E (GF , (s)) we have

χ(1) =

|GF |p ψs ( χ)(1). |CG∗ (s) F∗ |p

Proof Since the characteristic function of the identity is uniform, see 10.2.6, χ has the same dimension as its projection p( χ) to the space of uniform functions. By 11.5.5(ii) this projection is equal to p( χ) =

 (T,θ)

 χ, RG (θ)GF T RG (θ), RG (θ)GF T T

RG T (θ),

(11.5.1)

where (T, θ) is taken up to GF -conjugacy. Since χ ∈ E (GF , (s)) the only nonzero terms in this sum are those such that the pair (T, θ) corresponds to (T∗ , s) under the bijection of 11.1.16. Hence p( χ) =



 χ, RG (s)GF T∗

(T∗ )

RG (s), RG (s)GF T∗ T∗

RG T∗ (s)

F∗

where T∗ runs over CG∗ (s) -classes of F ∗ -stable maximal tori of CG∗ (s). By (s), RG (s)GF = 9.3.1 applied in G and 11.5.5 applied in CG∗ (s), we have RG T∗ T∗ CG∗ (s) CG∗ (s) RT∗ (1), RT∗ (1)CG∗ (s)F∗ , whence p( χ) =

 (T∗ )

 χ, RG (s)GF T∗ C

RTG∗

∗ (s)

C

(1), RTG∗

∗ (s)

(1)CG∗ (s)F∗

RG T∗ (s).

We have RG T∗ (s)(1) = ε G ε CG∗ (s) 0

|GF |p C ∗ (s) R G∗ (1)(1) |CG∗ (s) F∗ |p T ∗

∗

by 10.2.2 in both sides, using that by 11.2.1(iii) CG∗ (s) F /CG∗ (s) 0F is a p -group. By 11.5.1 we have C

G  χ, RG T∗ (s)GF = ε G ε CG∗ (s) 0 ψs ( χ), RT∗

∗ (s)

(1)CG∗ (s)F∗ .

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Geometric Conjugacy and the Lusztig Series

We deduce χ(1) =

|GF |p |CG∗ (s) F∗ |p 

C

ψs ( χ), RTG∗

∗ (s)

(1)CG∗ (s)F CG∗ (s) CG∗ (s) (1), RT∗ (1)CG∗ (s)F∗ (T∗ ) RT∗ ∗

C

RTG∗

∗ (s)

(1)(1).

By (11.5.1) applied in CG∗ (s), which is valid in a disconnected group by ∗ 11.5.5(ii), and using that by definition of unipotent characters of CG∗ (s) F , the CG∗ (s) character ψs ( χ) is orthogonal to RT∗ (θ) for θ  1, the right-hand side is equal to the dimension of the uniform projection of ψs ( χ) that is to ψs ( χ)(1) since the characteristic function of the identity is uniform (see the remark above 11.5.5). We thus get χ(1) =

|GF |p ψ ( χ)(1). ∗ |CG∗ (s) F |p s



11.6 More about Unipotent Characters Theorem 11.5.1 reduces the problem of decomposing RG (θ) to the case of T unipotent characters. We give here a construction which is a first step for de(1). This construction solves the problem for general linear and composing RG T unitary groups. Let W be the Weyl group of a quasi-split torus. We consider the coset WF ˜ := W  F, where F is the group generated by the automorphism of W in W of finite order induced by F on W. Definition 11.6.1 Let C(WF) be the space of functions WF → C invariant by W-conjugacy. Then for f ∈ C(WF) we define   f (wF)RG |CW (wF)| −1 RG Rf := |W | −1 Tw (1) = Tw (1). w∈W

w∈H 1 (F,W)

In particular, if πwF is the normalised characteristic function of the class of (1). wF, equal to |CW (wF)| on that class and 0 outside it, we have RπwF = RG Tw Lemma 11.6.2

If we define a scalar product on C(WF) by  f , f WF = |W | −1 f (wF)f (wF), w∈W

then the map f → Rf is an isometry between C(WF) and unipotent uniform functions on GF . Proof

This is an immediate consequence of 9.3.2.



˜ By abuse of notation, we will still write Rf when f is a class function on W, where we interpret f as its restriction to the coset WF.

11.6 More about Unipotent Characters

189

Proposition 11.6.3 ˜

˜ vanishes on WF unless ResW χ ∈ Irr(W). (i) χ ∈ Irr(W) W ˜ such that ResW˜ χ and ResW˜ ψ are in Irr(W). Then (ii) Let χ, ψ ∈ Irr(W) W

⎧ ⎪0  χ, ψWF = ⎨ ⎪1 ⎩

W

˜

˜

W unless ResW W χ = ResW ψ,

if χ = ψ.

˜ does not have an irreducible restricProof Let us prove (i). If χ ∈ Irr(W) ˜ tion to W then by 11.3.9 there exists a non-trivial character ζ of W/W = F such that ζ χ = χ. Thus λ = ζ (F) is a non-trivial root of unity; let o be its  i order. Then for w ∈ W we have o χ(wF) = o−1 i=0 ζ χ(wF) = (1 + λ + · · · + o−1 λ ) χ(wF) = 0 which proves (i). Now let δ = |F|, let ζ be a character of F such that λ = ζ (F) is ˜ set p( χ) = χ + a primitive root of unity of order δ and for χ ∈ Irr(W) λ −1 ζ χ + · · · + λ 1−δ ζ δ−1 χ. Then an easy computation shows that for w ∈ W ⎧ ⎪ δ χ(wF) if i = 1 ˜ we have . Thus if χ, ψ ∈ Irr(W) we have p( χ)(wF i ) = ⎨ ⎪0 otherwise ⎩ p( χ), p(ψ)W˜ = δ χ, ψWF . Now, if χ and ψ have irreducible restrictions to W, using again 11.3.9 we get ⎧ ⎪δ p( χ), p(ψ)W˜ = ⎨ ⎪0 ⎩

˜

˜

W if ResW W χ = ResW ψ,

otherwise, 

whence the result. It follows from 11.6.2 and 11.6.3 that:

˜ of each Corollary 11.6.4 Let E be a set consisting of one extension to W F character in Irr(W) . Then {R χ˜ } χ˜ ∈ E is an orthonormal basis of the space of unipotent uniform functions on GF . Some R χ˜ are irreducible characters, for instance 10.2.5(i) and 10.2.5(ii) together with 7.1.6 can be interpreted respectively as R1 = 1GF and Rsgn = StGF , where sgn is the sign character w → (−1) l(w) of W, extended trivially to WF. We will see that all R χ˜ are irreducible characters for the general linear groups. In general they are only Q-linear combinations of irreducible characters; but to (1) it is enough to decompose the R χ˜ since: decompose the RG Tw Lemma 11.6.5

(1) = RG Tw



χ˜ ∈ E

χ(wF)R ˜ χ˜ .

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Geometric Conjugacy and the Lusztig Series

Proof The formula is a consequence of the “second orthogonality relation for C(WF)”  χ ∈E

⎧ ⎪ |CW (wF)| χ(wF) χ(w F) = ⎨ ⎪0 ⎩

if wF and w F are W-conjugate otherwise,

˜ which is itself a consequence of the second orthogonality relation for W:  ˜ χ ∈Irr(W)

⎧ ⎪ |CW˜ (wF)| χ(wF) χ(w F) = ⎨ ⎪0 ⎩

if wF and w F are conjugate otherwise,

since: ˜ if and only if they are W-conjugate. • wF and w F are conjugate in W • In the formula only the characters with an irreducible restriction to W contribute and the contribution of two characters with the same restriction is the same. • There are |F| characters with the same restriction and |CW˜ (wF)| = |F||CW (wF)|.  Let W be a standard parabolic subgroup of W which is w F-stable for some WF ∈ W. By restricting IndW w F to cosets we define an induction between cosets given, for f ∈ C(W w F), by the formula  −1 (IndWF f (hwFh−1 ). W w F f )(wF) = |W |

w

{h∈W |hwFh−1 ∈W w F }

With this definition we can make the following proposition. Proposition 11.6.6 Let L be an F-stable Levi subgroup of a parabolic subgroup of G, which as in 11.4.1 is parametrised by a coset W w where W is a (Rf ) = RIndWF (f ) . w F-stable standard parabolic subgroup of W. Then RG L W wF

W , where Proof It is enough to prove the proposition for the functions πww F the exponent shows the considered group. It is straightforward using the above W W definition that IndWF W w F (πww F ) = πww F ; the corresponding functions on the L (1), thus the proposition is a consereductive group side are RT (1) and RG Tww ww quence of transitivity of the Lusztig induction: RG (RLT (1)) = RG (1).  T L ww

ww

11.7 The Irreducible Characters of GLFn and UFn

191

11.7 The Irreducible Characters of GLFn and UFn We will express all irreducible characters of the general linear and unitary groups as explicit combinations of Deligne–Lusztig characters, following the method of Lusztig and Srinivasan (1977). Note that this will prove in particular that all class functions are uniform in these groups. We will denote by G the group GLn and by F a Frobenius endomorphism which may be either the split one or that of the unitary group (called F in 7.1.8). The torus of diagonal matrices is F-stable in both cases; it is split for general linear groups, but in the unitary case the Borel subgroup B of upper triangular matrices is sent by F to the Borel subgroup w0 B (see example 7.1.8), thus the torus of diagonal matrices is of type w0 , which implies that the Frobenius acts by conjugation by w0 on its Weyl group W = Sn . Thus in both cases the set E of 11.6.4 consists in one extension of each irreducible character of W, and for χ ∈ Irr(W), we set χ(wF) ˜ = χ(w) in the case of GLn and χ(wF) ˜ = χ(ww0 ) in the case of Un . In both cases we write R χ for R χ˜ . Lemma 11.6.5 becomes: Lemma 11.7.1 RG Tw (1)

 ⎧ ⎪ χ ∈Irr(W) χ(w)R χ ⎨ = ⎪ ⎩ χ ∈Irr(W) χ(ww0 )R χ

for GLn , for Un .

We may now prove: Theorem 11.7.2 In the general linear and unitary groups, the set E (GF , 1) is equal (up to signs in the unitary case) to {R χ } χ ∈Irr(W) . Proof If we prove that the R χ are virtual characters, they will be up to sign distinct irreducible characters (since they form an orthonormal set) and by 11.7.1 they will span the unipotent characters. We know that R1 = 1 is irreducible. It is also well known that in the symmetric group any irreducible character is a linear combination with integral coefficients of the character induced from the trivial character of various parabolic subgroups; that is, any χ ∈ Irr(W) is a linear combination with integral coefficients of IndW WL (1) where L runs over diagonal Levi subgroups of G. In the unitary case this correspond to cosets WI w0 where WI runs over the standard parabolic subgroups of W. Applying the isometry 11.6.2 and proposition (1), 11.6.6, we get an expression of R χ as an integral linear combination of RG L which proves indeed that it is a virtual character. To prove that the R χ are actual characters for general linear groups, we note that, in that case, as the split torus T1 of diagonal matrices is contained in the

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Geometric Conjugacy and the Lusztig Series F

F-stable Borel subgroup B of upper triangular matrices, RG (1) = IndG (1) is T1 BF  G an actual character; since RT1 (1) = χ χ(1)R χ , and χ(1) > 0, this proves the result.  In the unitary case, it is not difficult to obtain a formula for the dimension of R χ and deduce from it, for each χ, which of R χ or −R χ is an irreducible  = χ χ(w0 )R χ is an character. The same argument as in GLn , using that RG T1 actual character, shows that χ(w0 )R χ is an actual character. So, when χ(w0 )  0, the sign we have to give to R χ to make it an actual character is that of χ(w0 ). For the sign in general, we will see in 13.1.14(ii) that R χ (1) is a polynomial in q, and the sign is (−1) B χ where B χ is the degree of that polynomial. Once we know the unipotent characters of GF , we can easily get all characters. Theorem 11.7.3 The irreducible characters of the general linear and unitary groups are (up to sign) the  G χ(ww ˜ R χ (s) := |WI | −1 1 )RTww (s), w∈WI

1

where (s) runs over F-stable semi-simple conjugacy classes of G; then CG (s) is a Levi subgroup parametrised by a coset WI w1 as in 11.4.1. The character χ runs over w1 -stable irreducible characters of WI and χ˜ stands for a real extension to WI .w1  of χ. Proof We can take (G∗ , F ∗ ) to be (G, F); see examples 11.1.13. Moreover, by the description of parabolic subgroups in GLn – see 3.2.5 – and that of centralisers of semi-simple elements – see 3.5.4(ii) which applies also to GLn – the centraliser of a semi-simple element is a Levi subgroup conjugate to a group of block-diagonal matrices. If s ∈ GF is semi-simple, by 11.4.1 the action of F on CG (s) permutes blocks of equal size. An orbit of blocks is of the form Gk1 where F cyclically permutes the copies of G1 and F k still acts on G1 as a split or unitary type Frobenius endomorphism (on a bigger field). If W1 is the Weyl group of G1 , an F-invariant character of W1k is of the form χ ⊗ · · · ⊗ χ and it has a natural real extension which on (w1 , . . . , wk )F takes the value χ(w1 · · · wk F k ). With this extension Theorem 11.7.2 can be extended easily to such groups. Then, using 11.4.8 and the fact that by 11.4.3 the functor F F RG CG (s) is an isometry from the series E (CG (s) , (s)) to E (G , (s)), we get the theorem.  Example 11.7.4 The character table of GL2 (Fq ). (See Table 11.2). The elements of GL2 (Fq ) are the invertible 2×2 matrices over Fq ; the Frobenius F raises the matrix entries to their qth power. The diagonal matrices T form

11.7 The Irreducible Characters of GLFn and UFn

193

a split F-stable maximal torus. The Weyl group W = WG (T) can be identified as the two permutation matrices, which we denote by 1 and s; there are thus two GF -conjugacy classes of F-stable maximal tori, that of T and of Ts such that (Ts , F) is G-conjugate to (T, sF). We have TF  F×q ×F×q , thus an element of Irr(TF ) is given by a pair (α, β) of characters of F×q , and s interchanges α and β. When α = β the character may be written α ◦ det thus extends to a linear character of GF . By 9.3.1, the Deligne– (α, β), which are true characters since RG is a HarishLusztig characters RG T T Chandra induction in this case (see remark before 9.1.5), are distinct irreducible characters when {α, β} runs over (non-ordered) pairs of distinct characters of (α◦det), where α runs over characters of F×q , are pairwise orthogonal F×q ; the RG T and are each one sum of two distinct irreducible constituents. q We have TFs  TsF = {diag(t1 , t1 ) | t1 ∈ F×q2 }  F×q2 . The element s acts on

Irr(F×q2 ) by ω → ωq . Thus by 9.3.1 the Deligne–Lusztig characters RG (ω) are Ts pairwise orthogonal when ω runs over representatives of Irr(F×q2 ) mod ω ≡ ωq ; these characters are the negative (by 10.2.9) of irreducible characters unless ω = ωq ; when they are not irreducible they have two irreducible constituents. We now decompose the non-irreducible Deligne–Lusztig characters. Note that ω ∈ Irr(F×q2 ) such that ω = ωq is equal to α ◦ NFq2 /Fq for some α ∈ Irr(F×q ), and ω is the restriction to TFs of α ◦ det. Thus by 10.1.6 we have (α ◦ det) = RG (1) · α ◦ det and RG (ω) = RG (1) · α ◦ det and 11.7.1 gives RG T T Ts Ts G G RT (1) = R1 + Rε and RTs (1) = R1 − Rε , which using the remarks after 11.6.4 becomes RG (1) = 1GF + StGF and RG (1) = 1GF − StGF . T Ts The list of irreducible characters of GL2 (Fq ) is thus: • RG (α, β) where {α, β} ⊂ Irr(F×q ) is of size 2. T

• −RG (ω) attached to {ω,ωq } ⊂ Irr(F×q2 ) of size 2. Ts • 1.(α ◦ det) and StGF .(α ◦ det) where α ∈ Irr(F×q ). We now give the list of conjugacy classes of GL2 (Fq ). In general, to list the conjugacy classes of a group GF , we may first describe the geometric semisimple classes and their centralisers, and get the GF -classes by 4.2.15(ii). We are then reduced for each centraliser CG (s) to describe the unipotent classes of CG (s), which are actually in CG (s) 0 by 3.5.3. By 3.5.4(i), geometric classes coincide with the GF -classes for GLFn . In GL2 , the only non-semi-simple classes unipotent element; the are of the form zu with z ∈ Z(GL2 ) and u a non-trivial   1 1 only non-trivial Jordan form in GL2 is u := ; thus we get q − 1 classes 0 1 for z ∈ Z(GF ) and q − 1 classes for zu.

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Table 11.2. Character Table of GL2 (Fq ) (note that |GL2 (Fq )| = q(q − 1) 2 (q + 1)) Classes



 a 0 0 a a ∈ F× q

 a 0 0 b a, b ∈ F× q ab





 x 0 F 0 x x ∈ F×2 q



 a 1 0 a a ∈ F× q

x  Fx

Number of classes of this type

q−1

(q − 1)(q − 2) 2

q(q − 1) 2

q−1

Cardinal of the class

1

q(q + 1)

q(q − 1)

q2 − 1

(α, β) RG T α, β ∈ Irr(F× q) α β

(q + 1)α(a) β(a)

α(a) β(b)+ α(b) β(a)

0

α(a) β(a)

−RG (ω) Ts ω ∈ Irr(F×2 ) q ω  ωq

(q − 1)ω(a)

0

−ω(x) − ω( Fx)

−ω(a)

α ◦ det α ∈ Irr(F× q)

α(a2 )

α(ab)

α(x. Fx)

α(a2 )

StGF .(α ◦ det) α ∈ Irr(F× q)

qα(a2 )

α(ab)

−α(x. Fx)

0

In GLFn the semi-simple elements of TFw  TwF are conjugate in GLn (Fq ) q to diag(t1 , . . . , tn ) where tw(i) = ti . Two such elements are conjugate in GLn if and only if they are conjugate under W, see 1.3.6(iii) and (iv). The semi-simple classes in GL2 with distinct eigenvalues are: • In TF , the classes diag(a, b), determined by {a, b} ∈ F×q of size 2. • In TFs , the classes which over Fq diagonalise to diag(x, Fx) where {x, Fx} ⊂   0 1 × F Fq2 is of size 2. Conjugates in G are . −x. Fx x + Fx All the entries in the character table of GL2 (Fq ) (Table 11.2) may be obtained by applying the character formula 10.1.2(i), once we know the value of (1) and RG (1) on non-trivial unipotent elements. This is 1, since RG T Ts G RT (1) = 1GF + StGF and RG (1) = 1GF − StGF , and StGF vanishes on non-trivial Ts unipotents.

11.7 The Irreducible Characters of GLFn and UFn

195

Notes The statements in this chapter about geometric conjugacy of (T, θ) and its interpretation in the dual group are in Deligne and Lusztig (1976). Langlands had introduced before a dual group which has the same root system as G∗ but is over the complex field. Most of 11.4.3 is already in Lusztig (1976b). Subsequent works Lusztig (1976a), Lusztig (1977), Lusztig (1984a) first gave the definition of series, and then parametrised, initially unipotent characters, then all characters, and gave the decomposition of Deligne–Lusztig characters into irreducible constituents. A good survey on the representation theory of reductive groups with disconnected centre is Bonnafé (2006).

12 Regular Elements; Gelfand–Graev Representations; Regular and Semi-Simple Characters

The aim of this chapter is to expound the main results about Gelfand–Graev representations. As the properties of these representations involve the notion of regular elements, we begin with their definition and some of their properties, in particular of regular unipotent elements.

12.1 Regular Elements Definition 12.1.1 An element x of an algebraic group G is said to be regular if the dimension of its centraliser is minimal. If G is reductive, this minimal dimension is rank(G). Proposition 12.1.2 Let G be a connected linear algebraic group. (i) For any element x ∈ G we have dim CG (x) ≥ rank(G). (ii) Assume G reductive; then in any torus T of G there exists an element t such that CG (t) = T. Proof Let B be a Borel subgroup of G containing x and let U be its unipotent radical. The conjugacy class of x under B is contained in xU, so its dimension is at most dim(U) = dim(B) − rank(G). As this dimension is equal to dim(B) − dim CB (x), we get dim CB (x) ≥ rank(G), whence dim CG (x) ≥ rank(G). We now prove (ii). If we apply 1.2.7 to the action of T on G by conjugation, we get an element t ∈ T such that CG (t) = CG (T). But by 2.3.2(iii) this means that  CG (t) = T, whence the result.

196

12.1 Regular Elements

197

Examples 12.1.3 Take G = GLn ; its rank is n (see 1.5.1). Any diagonal matrix with diagonal entries all distinct is regular, as its centraliser is the torus which consists of all diagonal matrices. The unipotent matrix 1 1   1   

.. ..

. .

   1 1

is regular, as its centraliser is the group of all upper triangular matrices with constant diagonals. From now on G will denote a connected reductive group over an algebraically closed field. The following result gives a characterisation of regular elements. Theorem 12.1.4 Let x = su be the Jordan decomposition of an element of G. The element x is regular if and only if u is regular in CG (s) 0 . Proof We have CG (su) = CG (s)∩CG (u), so dim CG (su) = dim CCG (s)0 (u). But su is regular if and only if this dimension is equal to rank(G) = rank(CG (s) 0 ). By 12.1.2 applied in CG (s) 0 this is equivalent to u being reg ular in CG (s) 0 . By 12.1.2 regular semi-simple elements exist. The existence of regular unipotent elements is much more difficult, and will be proved later. Once this existence is known, Theorem 12.1.4 shows that there are regular elements with arbitrary semi-simple parts. Exercise 12.1.5 Show that in GLn or SLn a matrix is regular if and only if its characteristic polynomial is equal to its minimal polynomial. We first study semi-simple regular elements. Proposition 12.1.6 (i) Let T be a maximal torus of G; an element s of T is regular if and only if no root of G relative to T is trivial on s. (ii) A semi-simple element is regular if and only if it is contained in only one maximal torus. (iii) A semi-simple element s is regular if and only if CG (s) 0 is a torus. Proof Assertions (i) and (iii) are straightforward from Proposition 3.5.1. Assertion (ii) is a direct consequence of assertion (iii). 

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Corollary 12.1.7 Let B be a Borel subgroup of G. A semi-simple element of B is regular in B if and only if it is regular in G. Regular semi-simple elements are dense in B. Proof Let B = U  T be a Levi decomposition of B. The first assertion results from 12.1.6(i) and (iii) which are valid in B as well as in G: if Φ is the root system of G with respect to T, and Φ+ is the positive subsystem defined by B, the condition for t ∈ T that α(t)  1 for any α ∈ Φ is equivalent to the same condition for α ∈ Φ+ ; that is, for Uα ⊂ B. Now consider the map f : (u, t) → ut : U × Treg → B where Treg is the set of regular elements of T. The image of f is the set of regular semi-simple elements of B since all maximal tori of B are conjugate under U to T. By 12.1.6(i) Treg is an open (hence dense) subset of T; further the fibres of f are reduced to a point since an equality ut = u t implies u −1 u t (u −1 u) −1 ∈ T, thus u −1 u = t (u −1 u) which implies u = u since t is regular. Thus the image of f has same dimension as B, hence f is dominant since B is irreducible, so the image of f is dense.  Remark 12.1.8 The method of the above proof shows that if t ∈ Treg , the image of the map u → u−1 tu : U → U is dense in U; actually it can be proved, using the method of the proof of 4.2.9, that this map is surjective. Corollary 12.1.9

Regular semi-simple elements are dense in G.

Proof This is clear from 12.1.7 as G is the union of all Borel subgroups by 1.3.3(ii).  We now consider regular unipotent elements. The proof of their existence (due to Steinberg) is not easy. We shall deduce it from the fact that there is only a finite number of unipotent classes in G. This result was proved by Kostant and Dynkin in characteristic 0 and by Richardson in good characteristic – see definition below 12.2.5 – by classifying the unipotent classes. It can be shown that to prove it in characteristic p it is enough to prove it over Fp . The general finiteness property was first proved in Lusztig (1976b) whose proof we follow; it is a good illustration of the use of the theory of representations of finite groups of Lie type. As the quotient morphism G → G/Z(G) induces a bijection on unipotent elements and also on unipotent classes it is enough to show the finiteness when G has a connected centre. The idea of the proof is to use the representation theory of GF , where F is a Frobenius endomorphism associated to the integer q, to separate the unipotent classes of GF by a finite set of class functions whose cardinality is bounded independently of q, which  n gives the result as G = n∈N∗ GF . Now G will be a connected reductive group over Fq , with a Frobenius root F, and (G∗ , T∗ , F ∗ ) are dual to (G, T, F). Note that an F-stable semi-simple

12.1 Regular Elements

199

element is quasi-isolated if and only if CG (s) is not contained in any proper F-stable Levi subgroup. Indeed, if s is not quasi-isolated – thus CG (s) ⊂ L for i some proper Levi subgroup L – then ∩i F L is an F-stable Levi subgroup coni taining CG (s) by 3.4.8(i) since all F L contain a given F-stable maximal torus of CG (s). We define E (GF ) to be the linear span of the Lusztig series E (GF , (s)) when ∗ s runs over the set S(G∗ ) of semi-simple elements of G∗F such that there exists an F ∗ -stable Levi subgroup L∗ of G∗ such that s ∈ Der L∗ and such that s is quasi-isolated in L∗ . We denote by E (GF )u the space of restrictions of E (GF ) to unipotent elements. The main lemma is the following. Lemma 12.1.10 Let G be a connected reductive group over Fq with a connected centre and a Frobenius root F; the restriction of any class function f on GF to unipotent elements is in E (GF )u . Proof The proof is by induction on the dimension of G. If Z(G)  {1}, the π → G/Z(G) induces a bijection on F-stable unipotent elements and their map G − conjugacy classes, so the restriction of f to unipotent elements is equal to f ◦ π for a class function f on unipotent elements of (G/Z(G)) F ; this last group is equal to GF /Z(G) F since Z(G) is connected. By the induction hypothesis we (θ) ◦ π = RG (θ ◦ π) have f ∈ E (GF /Z(G) F )u . By 11.3.10 we have RG/Z(G) T/Z(G) T ∗ and the dual group of G/Z(G) is isomorphic to Der(G ) – see 11.4.9 – thus the semi-simple element defining the series of (T, θ ◦ π) is in Der(G∗ ) whence f ∈ E (GF )u in this case. We now assume that Z(G) = 1, so G and G∗ are semi-simple. Let ρ be an irreducible character of GF ; it is in some Lusztig series E (GF , (s)). Assume first that the centraliser of s is contained in some proper F ∗ -stable Levi subgroup (λ), where λ is, up to sign, an L∗ of G∗ . Then, by 11.4.3(ii), we have ρ = RG L F element of E (L , (s)). By the induction hypothesis applied to L, the restriction commutes with the of λ to the unipotent elements of L is in E (LF )u . As RG L restriction to the unipotent elements (see 10.1.6 and 7.3.2(ii)) and maps E (LF ) (λ) to the unipotent elements is in E (GF )u , into E (GF ), the restriction of RG L whence the result in this case. Assume now that the centraliser of s is not contained in a proper F-stable Levi subgroup. We have s ∈ Der(G∗ ) = G∗ , so  E (GF , (s)) ⊂ E (GF ) and we get the result in this last case. Theorem 12.1.11 In a connected reductive algebraic group G over Fq , the number of unipotent classes is finite. Proof We will show that the dimension of E (GF ) is bounded independently of q, which proves the theorem, using 12.1.10. For that, we first show that

200

Regular Elements; Gelfand–Graev Representations

S(G∗ ) is a finite union of conjugacy classes, in number bounded independently of q. By 11.2.1(iii), in the dual of a group with connected centre the semi-simple elements have connected centralisers. So the condition that the s is quasi-isolated in L∗ can be seen on the root system of CG∗ (s) (with re∗ spect to some torus containing s). Explicitly, an element s ∈ G∗F is in S(G∗ ) if and only if there exists a Levi subgroup L∗ such that s ∈ Der L∗ and that the root system of CL∗ (s) is not contained in a proper parabolic subsystem of the root system of L∗ . But in a given root system there is only a finite number of root subsystems, whence only a finite number of G∗ -conjugacy classes of possible groups CG∗ (s), depending only on the type of the root system of G. Any one of these groups is the centraliser of a finite number of elements – bounded independently of q – of the corresponding Der L∗ by the following lemma applied to H = CG∗ (s) ∩ Der L∗ and G = Der L∗ and by 11.2.6. Lemma 12.1.12 Let G be a connected reductive group and let H be a connected subgroup of maximal rank of G. Assume that the root system of H (relative to some maximal torus T) is not contained in any parabolic subsystem of the root system of G relative to T; then Z(H) 0 = Z(G) 0 . In particular, if G is semi-simple, then so is H. Proof The centraliser in G of Z(H) 0 is a Levi subgroup of G containing H (see 3.4.7), so is equal to G and we have Z(H) 0 ⊂ Z(G) 0 . As H has maximal  rank it contains Z(G) 0 , whence the result. We have thus proved that E (GF ) is the union of a finite number of series E (GF , (s)) in number bounded independently of q. We shall be done if we bound independently of q the cardinality of E (GF , (s)); we claim |W | 2 is such a bound: each Deligne–Lusztig character has at most |W | irreducible constituents by 9.3.1, and the number of Deligne–Lusztig characters whose irreducible constituents lie in a Lusztig series E (GF , (s)) is the number of classes of F-stable maximal tori in the centraliser of s, thus is bounded by  |H 1 (F, W (s))| ≤ |W |. We now recall a well-known result from algebraic geometry which will be used in the proofs of 12.2.2 and of 12.2.9. Lemma 12.1.13 The dimension of any fibre of a surjective morphism from an irreducible variety X to another Y is at least dim(X) − dim(Y); moreover this inequality is an equality for an open dense subset of Y. Proof For the proof of the inequality see Hartshorne (1977, II, Exercise 3.22). The second statement is proved in Steinberg (1974, Appendix to 2.11, Prop. 2). 

12.2 Regular Unipotent Elements

201

12.2 Regular Unipotent Elements We can now state the properties of regular unipotent elements. We first introduce a definition. Definition 12.2.1 For x ∈ G, we define the variety Bx (called the Springer fibre of x) as the variety of Borel subgroups containing x; equivalently if we fix a Borel subgroup B of G it identifies with the fixed points of x for left translation in G/B. The Springer fibres are projective varieties as closed subvarieties of a projective variety. Proposition 12.2.2 Let u be a unipotent element of G; the following properties are equivalent: (i) (ii) (iii) (iv)

u is regular. Bu is finite. Bu consists of only one Borel subgroup. There exists a Borel subgroup B containing u and a maximal torus T of B  such that in the decomposition u = α ∈Φ+ uα (xα ), where Φ+ is the set of positive roots of G relative to T ⊂ B (see 2.3.1(vi)), none of the xα for α simple is equal to 0. (v) Property (iv) is true for any Borel subgroup B containing u and any maximal torus of B.

Proof It is obvious that (iii) implies (ii) and that (v) implies (iv). We prove that (ii) implies (v). Let B = U  T be a Levi decomposition of a  Borel subgroup containing u and let α ∈Φ+ uα (xα ) be the decomposition of u. If xα = 0 for a simple α then for any v ∈ Uα the element vu has also a trivial component in Uα by the commutation formulae 2.3.1(vii). Let n ∈ NG (T) be a representative of the simple reflection sα of W (T) corresponding to α; then nvu is in B since the only root subgroup of B which is mapped onto a negative root subgroup under the action of n is Uα , thus u is in vsα B for any v ∈ Uα . The Borel subgroups vsα B are all distinct since an equality vsα B = v sα B is equivalent to vsα B = v sα B and by the unique decomposition 3.2.7 this is possible only if v = v . Thus we have got an infinite number (at least an affine line) of Borel subgroups containing u. We show now that (iv) implies (iii). Let B = U  T, be as in (iv), let B1 be another Borel subgroup containing u and let T1 be a maximal torus contained in B ∩ B1 (see 3.1.4); then T = vT1 with v ∈ U, so that vB1 is a Borel subgroup containing T and so equals wB for some w ∈ W (T). Thus we see that wB contains vu. By the commutation formulae 2.3.1(vii), the element vu still −1 satisfies (iv), but if α is a simple root such that w α is negative, then the root

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Regular Elements; Gelfand–Graev Representations

subgroup Uα has a trivial intersection with wB, so the component of vu in Uα is necessarily trivial. This is a contradiction if w is not 1; that is, if B1  B. So far we have proved the equivalence of (ii), (iii), (iv) and (v). We shall show now that (i) is equivalent to these four properties. First we prove that (i) implies (iv). Let u be a regular unipotent element and B = U  T  be a Borel subgroup containing u, and let u = α ∈Φ+ uα (xα ) be the corresponding decomposition. Assume that for some simple α we have xα = 0. The element u is in the parabolic subgroup P = B ∪ Bsα B where sα is the reflection with respect to α. In that parabolic subgroup the conjugacy class of u contains only elements of U whose component in Uα is 1, so has dimension at most dim(U) − 1. Hence the dimension of the centraliser of u in P is at least dim(P) − (dim(U) − 1). As dim(P) = dim(U) + dim(T) + 1, we get CP (u) ≥ rank(G) + 2, which is a contradiction by 12.1.2. We will now prove that some element satisfying (iv) is regular. By 12.1.11, the variety Gu of unipotent elements is a finite union of conjugacy classes, so has the same dimension as one of them. We know codim(Gu ) ≥ rank(G), as the codimension of any conjugacy class is at least rank(G) by 12.1.2 (i). Consider the morphism G × U → G given by (g, u) → gu. Its image is Gu . Let u be as in (iii). Then (g, v) is in the fibre of u if and only if g ∈ NG (B) = B and −1 v = g u; thus the fibre of u is isomorphic to B. By 12.1.13 we have dim(B) ≥ dim(G) + dim(U) − dim(Gu ). As dim(B) − dim(U) = dim(T) = rank(G), we get codim(Gu ) ≤ rank(G). Thus codim(Gu ) = rank(G), so a class having same dimension as Gu is regular (and this class satisfies (iv)). Let us prove now that the elements satisfying (iv) form a single conjugacy class under B which will prove that every element satisfying (iv) is regular. Let u satisfy (iv) and be regular, and let v satisfy (iv); replacing v by a T-conjugate, we may assume that the components of v and u in Uα for α simple are equal  (see 2.3.1(i) and 1.2.11), so that vu−1 is in U∗ = α ∈Φ+ −Π Uα . As the orbits of a unipotent group acting on a variety are closed, see Steinberg (1974, 2.5, Proposition), the set [x, U] := {[x, u] | x ∈ U} is a closed subvariety of U∗ of dimension dim(U) − dim(CU (u)). If we prove dim([x, U]) = dim(U∗ ), we shall get U∗ = [x, U], thus the class (u)U of u under U is equal to uU∗ , thus v ∈ uU∗ is U-conjugate to u. Thus we have to prove dim(CU (u)) = dim(U) − dim(U∗ ) = rank(G) − dim(Z(G)). Since u is regular, this follows from the following lemma. Lemma 12.2.3

If u satisfies (iv) then CG (u) = Z(G).CU (u). −1

Proof If g centralises u then u is in g B, thus since u satisfies (iii) we get g ∈ NG (B) = B, thus g ∈ CB (u). Let g = tv be the Jordan decomposition of g

12.2 Regular Unipotent Elements

203

with t semi-simple and v unipotent, both in CB (u); the fact that u is of the form (iv) implies that t ∈ Z(G), whence the lemma.   We have seen in the proof of 12.2.2 Corollary 12.2.4 gacy class of G.

Regular unipotent elements exist and form a single conju-

We have also proved that any element of U∗ is a commutator of a regular element and another element of U, so in particular the following proposition is valid.  Proposition 12.2.5 We have Der U = α ∈Φ+ −Π Uα . To state the next result we need to define good and bad characteristics. Definition 12.2.6 The characteristic is good for G if it does not divide the coefficients of the highest root of the positive root system associated to a Borel subgroup of G. Bad (that is, not good) characteristics are p = 2 for a root system of type Bn , Cn or Dn , p = 2, 3 for types G2 , F4 , E6 , E7 and p = 2, 3, 5 for type E8 , see Bourbaki (1968, VI §4). Proposition 12.2.7 Let u be a regular unipotent element of G; if the char0 (u) is a acteristic is good for G then CU (u) is connected, otherwise CU (u)/CU non-trivial cyclic group generated by the image of u. References See Springer (1966b, 4.11, 4.12) and Lou (1968). Most of the proof is by a case-by-case check.  From the above results on unipotent and semi-simple regular elements, we can deduce results for all regular elements. Corollary 12.2.8 An element x is regular if and only if Bx is finite, and then |Bx | = |W/W 0 (s)| where s is the semi-simple part of x. Proof By 12.1.4 if su is the Jordan decomposition of x, then x is regular if and only if u is regular in CG (s) 0 , which is equivalent to u being contained in a finite number of Borel subgroups (in fact one) of CG (s) 0 by 12.2.2(iii). But a Borel subgroup of CG (s) 0 is of the form B∩CG (s) 0 where B is a Borel subgroup of G containing a maximal torus of CG (s) 0 (see 3.4.10(i)). As the number of Borel subgroups containing a given maximal torus in a connected reductive group is finite, we get the first assertion. This number is equal to the cardinality of the Weyl group, so is |W | in G and |W 0 (s)| in CG (s) 0 ; moreover any two Borel

204

Regular Elements; Gelfand–Graev Representations

subgroups of CG (s) 0 are contained in the same number of Borel subgroups of G since they are conjugate, whence the second assertion.  We now give a general result about centralisers. This result is due to Springer (1966a). We follow here the proof given in Steinberg (1974, 3.5, remark following Proposition 1). Proposition 12.2.9 The centraliser of any element of G contains an abelian algebraic subgroup of dimension at least rank(G). Proof The result is clear for semi-simple elements. The idea of the proof is to use the fact that semi-simple elements are dense in G (see 12.1.9). Consider the variety Sn = {(x1 , x2 , . . . , xn ) ∈ Gn | all xi belong to the same torus}; then the projection (x1 , x2 , . . . , xn ) → (x1 , x2 , . . . , xn−1 ) is surjective from Sn to Sn−1 . Moreover, if T is a maximal torus of G, as Sn is the image of G × Tn by the map (g, t1 , . . . , tn ) → ( gt1 , . . . , gtn ), it is irreducible, so its closure Sn is also irreducible. It follows that the projection is also surjective from Sn to Sn−1 . Note also that the components of an element of Sn commute with each other, so the same property is true for Sn . The density of semi-simple elements means that S1 = G. Now let x be an arbitrary element of G and choose n and (x1 , x2 , . . . , xn−1 ) ∈ Gn−1 such that (x, x1 , . . . , xn−1 ) ∈ Sn and that CG (x, x1 , . . . , xn−1 ) is minimal among all such sequences when n and xi vary, which is possible since G is a Noetherian topological space. Then any z ∈ G such that (x, x1 , . . . , xn−1 , z) ∈ Sn+1 , satisfies CG (x, x1 , . . . , xn−1 , z) = CG (x, x1 , . . . , xn−1 ), so is in the centre of CG (x, x1 , . . . , xn−1 ). The set of such elements z is thus an abelian subgroup of CG (x), so we shall be done if we prove that the fibres of the projection Sn+1 → Sn have dimension at least rank(G). By 12.1.13, which we can use since the morphism is surjective and Sn+1 is irreducible, the dimension of any fibre is at least dim(Sn+1 ) − dim(Sn ). But the fibre of an element of Sn contains at least a maximal torus, so has dimension at least rank(G) whence  dim(Sn+1 ) − dim(Sn ) ≥ rank(G), which gives the result. Corollary 12.2.10

If x is regular then CG (x) 0 is abelian.

Proof We know that the dimension of CG (x) 0 is equal to rank(G), but by 12.2.9, CG (x) 0 contains an abelian subgroup of that dimension, whence the result.  We now study GF -classes of regular unipotent elements, where F is a Frobenius root on G.

12.2 Regular Unipotent Elements

205

Remark 12.2.11 Note that all the GF -classes of regular unipotent elements have the same cardinality: indeed by 12.2.3, 12.2.7 and 12.2.10 the group CG (u) is abelian, and we have the following lemma. Lemma 12.2.12 If g ∈ GF is such that CG (g) is abelian, then all GF -classes in the geometric conjugacy class of g have same cardinality. Proof By 4.2.15(ii) an element of GF geometrically conjugate to g is of the form xg where x ∈ G is such that x−1 F(x) ∈ CG (g). But, since CG (g) is abelian, for any h ∈ CGF (g) we have xh ∈ GF , thus ad x is a bijection from CGF (g) to  CGF ( xg). Remark 12.2.13 Note that if H is a finite abelian group with an automorphism F we have H 1 (F, H)  H/L(H); as in this case L is a group morphism with kernel H F , we get |H 1 (F, H)| = |H F |. Thus for an algebraic group H with a Frobenius root F and such that H/H0 is abelian we get |H 1 (F, H)| = |H 1 (F, H/H0 )| = |(H/H0 ) F | = |HF |/|H0F |. We use this remark in the proof of the following proposition. Proposition 12.2.14 The number of regular unipotent elements in GF is |GF |/(|Z(G) 0F |ql ) where l is the semi-simple rank of G. Proof A regular unipotent element lies in only one Borel subgroup, so the number of regular unipotent elements in GF is equal to the product of the number of F-stable Borel subgroups – which is |GF /BF | for any F-stable Borel subgroup B – with the number of such elements in a given Borel subgroup. By 12.2.2(v) and 12.2.5, the number of regular unipotent elements in BF , where B = TU, is |(Der U) F | times the number of elements of UF /(Der U) F whose  every component is non-trivial in the decomposition U/ Der U  Uα ; we shall say that these elements are regular in U/ Der U. By 2.3.1(i) and 1.2.11 any two regular elements of U/ Der U are conjugate by an element of T, but the centraliser of a regular element in T is Z(G), so by 4.2.14 the number of TF -orbits of regular elements in UF /(Der U) F is |H 1 (F, Z(G))|. In each such orbit there are |TF /Z(G) F | elements, so the number of regular elements in GF is |GF |/(|TF ||UF |)|(Der U) F ||H 1 (F, Z(G))||TF |/|Z(G) F |. Now UF /(Der U) F has cardinality ql (by 4.4.2 applied to UF and (Der U) F ) and |H 1 (F, Z(G))| =  |Z(G) F /Z(G) 0F | since Z(G)/Z(G) 0 is abelian. The result follows. Note that by 4.2.14(i) and the above proof LT−1 (Z(G)) permutes transitively the regular elements of UF /(Der U) F . Proposition 12.2.15 If the characteristic is good for G, the GF -conjugacy classes of regular unipotent elements are parametrised by the F-conjugacy classes H 1 (F, Z(G)/Z(G) 0 ).

206

Regular Elements; Gelfand–Graev Representations

Proof If u is an F-stable regular unipotent element, by 4.2.15(ii) the GF classes inside its G-class are parametrised by H 1 (F, CG (u)/CG (u) 0 ). If the characteristic is good, by 12.2.7 we have CG (u)/CG (u) 0 = Z(G)/Z(G) 0 , whence the result.  The following corollary is clear. Corollary 12.2.16 If the characteristic is good for G and Z(G) is connected, there is a single class of regular unipotent elements in GF . We can also use regular unipotent elements to prove the following proposition. Proposition 12.2.17

CG (GF ) = Z(G).

Proof Let u be a regular unipotent element of GF . Let B be the unique Borel subgroup of G containing u, which is necessarily F-stable, and let B = U  T be an F-stable Levi decomposition of B. We have CG (GF ) ⊂ CG (u) = Z(G) · CU (u), where the right equality is by 12.2.3. Let now w0 ∈ WG (T) be the longest element which, being unique, is F-stable and thus has an F-stable  representative w˙ 0 ∈ NG (T). Since U ∩ w0 U = {α ∈Φ+ |w0 (α) ∈Φ+ } Uα = {1}, the Borel subgroup w0 B satisfies w0 B ∩ B = T – such a Borel subgroup is said to be opposed to B. Since w˙ 0 u is also regular, it follows that CG (GF ) ⊂  CG ( w˙ 0 u) ∩ CG (u) = Z(G), whence the result. Note that the parametrisation of 12.2.15 is well-defined only once we have chosen a particular regular unipotent element u1 ∈ GF which corresponds to the trivial element of H 1 (F, Z(G)/Z(G) 0 ). From now on we fix a regular unipotent u1 ∈ GF . In arbitrary characteristic, by 12.2.3 the GF -classes of regular unipotent elements are parametrised by 0 (u1 )); H 1 (F, Z(G)/Z(G) 0 × CU (u1 )/CU

the first projection maps these F-classes onto the F-classes of Z(G)/Z(G) 0 . We shall denote by Uz the set of regular unipotent elements of GF in a class parametrised by the fibre of z ∈ H 1 (F, Z(G)/Z(G) 0 ); by construction we have u1 ∈ U1 . If B = U  T is an F-stable Levi decomposition of some Borel subgroup, the sets Uz are characterised by the following property. Proposition 12.2.18 With the above notation, two regular elements u and u of UF are in the same set Uz if and only if there exists t ∈ TF such that u−1 tu is in Der(U). Proof There is an F-stable conjugate of u1 in U, which we denote again by u1 . Any element which conjugates u1 to u is in B by 12.2.2(iii). So we have

12.3 Gelfand–Graev Representations

207

u = svu1 with s ∈ T and v ∈ U. The class of u is parametrised by the F-class of v−1 s−1 Fs Fv in H 1 (F, Z(G)/Z(G) 0 × CG (u1 )/CG (u1 ) 0 ) – we know by 12.2.3 that s−1 Fs is in Z(G). The element u is in the same Uz as u if and only if it can be written s v u1 with s−1 Fs and s −1 Fs F-conjugate by some z1 ∈ Z(G). This implies that we have ts v u1 = sv u1 with t = sz1 s −1 ∈ TF . So u and tu are conjugate under U, which implies by 2.3.1(vii) that they have the same image in U/ Der U. Conversely, if svu1 and ts v u1 have the same image in U/ Der U, the element s−1 ts must be central by 12.2.2(v) and 2.3.1(i), so s−1 Fs and s −1 Fs are in the same F-class of Z(G), whence the result. 

12.3 Gelfand–Graev Representations In the following we fix an F-stable Levi decomposition B = U  T of an Fstable Borel subgroup B. We denote by Π the corresponding basis of the root system of G relative to T. Then F defines a permutation σ of Π (see 4.2.5 and  2.4.8). For any orbit O of σ in Π, we denote by U O the image of α ∈ O Uα in U/ Der U. It is a commutative group isomorphic to Ga| O | . The action of F maps Uα on U σ α for each α ∈ Π – and is an isomorphism of abstract groups. The action of F | O | stabilises each Uα for α ∈ O, so the group UFO is isomorphic to |O|

UFα for any α ∈ O. We choose such an α; by the Lang–Steinberg theorem we uα |O| can choose the isomorphism Ga −−→ Uα such that uα (1) ∈ UFα ; using 4.1.9 we then get UFO  F+q|O| – the additive group. In the following we shall assume that we have made such choices for each orbit O.

Definition 12.3.1 A linear character φ ∈ Irr(UF ) is regular if its restriction to Der(U) F is trivial and its restriction to UFO is not trivial for any orbit O of σ in Π. Note that (Der U) F contains Der(UF ) – actually it is almost always equal to that derived group, the only exceptions for quasi-simple groups being groups of type Bl , Cl , G2 or F4 over F2 and groups of type G2 over F3 ; see Howlett (1974, Lemma 7) where the proof is given in the case of split groups, for non-split groups the proof is similar. Note also that the quotient map is an isomorphism  ∼ → U/ Der U compatible with the actions of T and that of varieties α ∈Π Uα − this isomorphism induces a bijection on F-stable elements. We thus have an action of TF on regular characters. Proposition 12.3.2 The TF -orbits of regular characters of UF are in oneto-one correspondence with H 1 (F, Z(G)). Proof By the remark after 12.2.14 LT−1 (Z(G)) acts transitively on regular elements of UF /(Der U) F . By the remarks before 12.3.1, a regular character of

208

Regular Elements; Gelfand–Graev Representations 

+ O ∈Π/σ {Irr(Fq|O| )

− {1}} and a regular el ement of is identified with an element of O ∈Π/σ {Fq|O| − {0}}. So we see that LT−1 (Z(G)) acts transitively on the set of regular characters of UF . The stabiliser in LT−1 (Z(G)) of a regular character of UF is Z(G) and LT−1 (Z(G)) F = TF . So, by 4.2.14(ii) applied with G replaced by LT−1 (Z(G)) we get the result (in 4.2.14(ii) G is assumed to be connected but the only property actually needed is that the stabiliser of an element is contained in the image  of the Lang map, so we can apply 4.2.14(ii) to LT−1 (Z(G)) in our case. UF is identified with an element of UF /(Der U) F

Note that, by the above proof, the regular characters of UF are parametrised by LT−1 (Z(G))/Z(G). As in the case of regular unipotent elements, this parametrisation is well-defined once we have chosen a regular character. From now on we shall fix such a character which will be denoted by ψ1 . We make the choice in the following way: we fix a non-trivial additive character χ0 of F+p ; by composing with the trace, for any N we get a character χN of F+qN . We take for ψ1 , parametrised by 1 ∈ H 1 (F, Z(G)), the product of the characters χ | O | of the additive groups F+q|O| . The orbit of regular characters parametrised by

z ∈ H 1 (F, Z(G)) will be denoted by Ψz . If z˙ is the image in LT−1 (Z(G))/Z(G) of an element t ∈ LT−1 (Z(G)) such that the F-conjugacy class of L(t) is equal to z, then the character ψz˙ := tψ1 is in Ψz . Definition 12.3.3 For z ∈ H 1 (F, Z(G)) we define the Gelfand–Graev repreF (ψz˙ ). sentation Γz (or ΓzG ) by Γz = IndG UF Note that Γz = t Γ1 if z˙ is the image of t as above. The following result is due to Steinberg: Theorem 12.3.4

The Gelfand–Graev representations are multiplicity free.

Proof We shall not give the proof here. It may be found, for example, in Carter (1985, 8.1.3); the proof given there is written under the assumption that Z(G) is connected and F is a Frobenius, but it applies without any change to our case.  We now study the effect of Harish-Chandra restriction and of duality on the Gelfand–Graev representations. For this we need some notation and preliminary results. If L is an F-stable Levi subgroup containing T of an F-stable parabolic subgroup of G, for parametrising the regular characters of UF ∩ LF by H 1 (F, Z(L)) we have to fix a particular regular character: we choose F ψ , which corresponds for L to the same definition we gave of ψ1 for ResU UF ∩LF 1 G. The next lemma will allow us to compare the parametrisations in G and in L.

12.3 Gelfand–Graev Representations

Lemma 12.3.5

209

The inclusion Z(G) ⊂ Z(L) induces a surjective map hL : H 1 (F, Z(G)) → H 1 (F, Z(L)).

Proof The inclusion Z(G) ⊂ Z(L) gives rise to the Galois cohomology exact sequence, see 4.2.10 and the remark following for normal subgroups 1 → Z(G) F → Z(L) F → (Z(L)/Z(G)) F → hL

→ H 1 (F, Z(G)) −−→ H 1 (F, Z(L)) → H 1 (F, Z(L)/Z(G)). Thus it is enough to show that the quotient Z(L)/Z(G) is connected, which results from 11.2.1(ii) as – by 2.3.4(ii) – Z(L)/Z(G) is the centre of the Levi subgroup L/Z(G) of the group G/Z(G).  Proposition 12.3.6

With the above notation we have ∗ G G RL (Γz )

= ΓhLL (z) .

Proof We have to show that, for any class function f on LF , we have G L RG L (f ), Γz GF = f , ΓhL (z) LF .

Up to GF -conjugacy, we may assume that L is the standard Levi subgroup LI , and we may compute the Harish-Chandra induction using the parabolic subgroup P having L as a Levi subgroup and containing n0 B, where n0 is a F f˜ where (f ) = IndG representative of the longest element of W. We have RG L PF F the value of f˜ = Inf P F f is given by f˜ (lu) = f (l) for u ∈ V, where V = Ru (P). L

So the left-hand side is equal to F F f˜ , IndG ψ F= IndG PF UF z˙ G



f˜ , gψz˙ PF ∩ gUF ,

g∈PF \GF /UF

this equality is by the ordinary Mackey formula. Now – by 5.2.2(ii) – ∼ → PF \GF /BF . Since L\S(L, T)/T = WI \W is represented (L\S(L, T)/T) F − by the I-reduced element, we get – using the unicity of an I-reduced element in its coset – that the F-stable I-reduced elements in W F form a set of representatives of PF \GF /BF . So we can choose as a cross section of PF \GF /UF a set of representatives in NG (T) F of I-reduced elements of W F . If n is such an element – a representative of w ∈ W F – we have PF ∩ nUF = (LF ∩ nUF ).(VF ∩ nUF ), F n F F and the restriction of f˜ to PF ∩ nUF is equal to Inf PLF∩∩ nUUF ◦ ResLLF ∩ nUF (f ) in that decomposition. So we have F

f˜ , nψz˙ PF ∩ nUF = ResLLF ∩ nUF f , nψz˙ LF ∩ nUF 1, ψz˙  n−1 VF ∩UF .

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Regular Elements; Gelfand–Graev Representations

But, by the definition of ψz˙ , the scalar product 1, ψz˙  n−1 VF ∩UF is not zero (and −1 is equal to 1) if and only if n V ∩ U contains no Uα for α ∈ Π. However, as w is I-reduced, using 2.3.1(v) we get n−1

n−1 n0

V∩U=

U ∩ U.

This group contains no Uα with α ∈ Π if and only if w = 1; that is, n ∈ TF . We then have L ∩ nU = L ∩ U, so F

L n RG L (f ), Γz GF = ResLF ∩UF f , ψz˙ LF ∩UF .

So we shall have proved the proposition if we show that the TF -orbit of F z˙ |LF ∩UF , that is of ψz˙ |LF ∩UF , is the T -orbit ΨhL (z) of regular characters of F F t L ∩ U . But we have ψz˙ = ψ1 for any t ∈ LT−1 (Z(G)) such that z˙ = tZ(G) F F and – by the definition of hL – the character ResU ψ = t (ResU ψ ) is UF ∩LF z˙ UF ∩LF 1  in ΨhL (z) . nψ

Remark 12.3.7 Proposition 12.3.6 can be extended to Lusztig induction when the characteristic p is good for G and q is large enough, in the form ∗RG (ΓG ) = L z L ε G ε L ΓhL (z) , see Bonnafé (2005a, 15.2). We now give two results on the dual of a Gelfand–Graev representation. Proposition 12.3.8 (i) The dual of any Gelfand–Graev representation is zero outside regular unipotent elements. (ii) We have Γz , (−1) S/F DG Γz GF = |Z(G) F |δz,z . Proof

By 12.3.6 and 7.1.12 we have  DG Γz = (−1) |J | RG LJ (ΓhLJ (z) ). J ⊂S/F

F

By the definition of RG , and using ResU (ψz˙ ) ∈ ΨhLJ (z) , which we saw at LJ UF ∩LF J

the end of the proof of 12.3.6, we have F

PF

LF

F

G U J J RG LJ (ΓhLJ (z) ) = IndPF ◦ Inf LF ◦ IndUF ∩LF ◦ ResUF ∩LF (ψz˙ ). J

If v ∈ Ru (PJ PF

)F

and l ∈

LFJ ,

for any f ∈

J

f )(vl) = |UF | −1 (IndUJF ◦ Inf U UF ∩LF F

J

J

J

C(UF

∩ 

LFJ )

we have

{p∈PFJ | p (vl) ∈UF }

F

(Inf U f )( p (vl)), UF ∩LF J

where the inflation in the left-hand side is defined using UF = Ru (PJ ) F  (UF ∩ LFJ ). The condition p (vl) ∈ UF is equivalent to l l ∈ UF where l F F is the component in LJ of p in the decomposition PJ = Ru (PJ ) F  LFJ and

12.3 Gelfand–Graev Representations

PF



F

211

(Inf U f )( p (vl)) = f ( l l) when the condition is satisfied. Thus we get Inf LJF ◦ UF ∩LF J

LF

PF

J

F

. By transitivity of induction we get IndUJF ∩LF = IndUJF ◦ Inf U UF ∩LF J

J

F

F

F

G U U RG LJ (ΓhLJ (z) ) = IndUF ◦ Inf UF ∩LF ◦ ResUF ∩LF (ψz˙ ). J

J

Thus  F  F F  DG (Γz ) = IndG (−1) |J | Inf U ◦ ResU ψ  . F U UF ∩LFJ UF ∩LFJ z˙ J ⊂S/F  To prove assertion (i) of the proposition it is sufficient to show that  F F (−1) |J | Inf U ◦ ResU ψ UF ∩LF UF ∩LF z˙ J

J ⊂S/F

J

is zero outside regular unipotent elements of UF . We use the notation of 12.3.1.   Let u be in UF and let O ∈Π/σ u O denote the projection of u on O UFO ; if we  F ψ , we have ψz˙ (u) = O ∈Π/σ ψ O (u O ) whence put ψ O = ResU UF z˙ 

J ⊂S/F

O

(−1) |J | (Inf U ◦ ResU ψ )(u) = UF ∩LF UF ∩LF z˙ F

F

J

J



(1 − ψ O (u O ))

O ∈Π/σ

which is zero outside regular unipotent elements, as u is regular if and only if u O  1 for all O. F ϕ, We now prove (ii). By the above computations we have DG Γz = IndG UF where  F F (−1) |J | Inf U ◦ ResU ψ, ϕ= UF ∩LF UF ∩LF z˙ J

J ⊂S/F

J

which is zero outside regular unipotent elements. By the ordinary Mackey formula, using the fact that NG (T) F is a cross section for UF \GF /UF , we have  ϕ, nψz˙ UF ∩ nUF . DG Γz , Γz GF = n∈NG (T) F

But UF ∩ nUF does not contain any regular unipotent element if n is not in T F F ◦ ResU ψ , ψ  F is equal to zero if J  S/F. by 12.2.2(iii), and Inf U UF ∩LF UF ∩LF z˙ z˙ U So we have

J

J

DG Γz , Γz GF = (−1) S/F



ψz˙ , tψz˙ UF .

t∈TF

The characters ψz˙ and tψz˙ can be equal only if z = z . There are then |Z(G) F |  values of t such that ψz˙ = tψz˙ , whence the result. Using the Gelfand–Graev representations, we can give the value of any character on a regular unipotent element at least when the characteristic is good.

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Regular Elements; Gelfand–Graev Representations

Definition 12.3.9 We define σz := fixed after 12.2.17.



ψ ∈Ψz−1

ψ(u1 ), where u1 is the element we

Theorem 12.3.10 (i) For any χ ∈ Irr(GF ) we have   |Uz | −1 χ(u) = z ∈H 1 (F,Z(G))

u∈Uz

(ii) We have 

σzz −1 (−1) |S/F | DG ( χ), Γz GF .





Γz ,

z∈H 1 (F,Z(G))

Γz GF = |H 1 (F, Z(G))||Z(G) F |ql ,

z∈H 1 (F,Z(G))

where l is the semi-simple rank of G. Proof In the following we shall denote by γz the class function on GF whose value is zero outside Uz and |GF |/|Uz | on Uz . We have seen in the proof of F ϕ, where ϕ is a function on UF /(Der U) F which is 12.3.8 that DG Γz = IndG UF zero outside the set of regular elements. So by 12.2.18 DG Γz is constant on  Uz , and there exist coefficients cz,z such that DG Γz = z ∈H 1 (F,Z(G)) cz,z γz . By 12.3.8 (ii) the matrix (cz,z )z,z is invertible and its inverse is ((−1) |S/F | |Z(G) F | −1 Γz , γz GF )z,z . Let t ∈ LT−1 (Z(G)) be such that the F-class of L(t) is equal to z; we have  Γz , γz GF = |Uz | −1 Γz (u) = Γz ( tu1 ), u∈Uz

the last equality because Γ is constant on Uz , as it is induced from ψz˙ which factorises through (U/ Der U) F . So z

Γz , γz GF = Γz ( tu1 ) = Γz z−1 (u1 ). We now apply the following lemma. Lemma 12.3.11 Proof

We have σz = |Z(G) F | −1 Γz−1 (u1 ).

By definition Γz (u1 ) = |UF | −1



ψz˙ ( gu1 ).

{g∈GF | gu1 ∈UF }

As u1 is regular, we must have g ∈ BF in the above summation and, if g = tv with t ∈ TF and v ∈ UF , then ψz˙ ( gu1 ) = ψz˙ ( tu1 ), whence, as Z(G) F is the kernel of the TF -action on Ψz , we get  |Z(G) F | −1 Γz (u1 ) = ψ(u1 ) = σz−1 . ψ ∈Ψz



12.3 Gelfand–Graev Representations

213

So we get |Z(G) F | −1 Γz , γz GF = σzz −1 . So (−1) |S/F | (σzz −1 )z,z is the inverse matrix of (cz,z )z,z ; that is,  γz = σzz −1 (−1) |S/F | DG Γz , z ∈H 1 (F,Z(G))

whence 12.3.10(i) by taking the scalar product of both sides with χ.     Let us prove (ii). We have  z Γz , z Γz GF =  z DG Γz , z DG Γz GF and, with the above notation, it follows that      DG Γz , DG Γz GF =  cz,z γz , cz,z γz GF z

z

z,z

z

But



z,z

   =  cz,z γz , cz,z γz GF . z

z

cz,z = DG Γz , 1GF = Γz , StGF GF = |GF | −1 Γz (1)StGF (1)

z

= |GF | −1 (|GF |/UF |)|GF |p = 1 (see 4.4.1 (iv)), whence



= 1 as cz,z = c1,z−1 z . Thus     DG Γz GF = γz , γz GF = |GF |/|Uz |.  DG Γz , z cz,z

z

z

z

z

All the sets Uz have the same cardinality as they are geometrically conjugate, so their cardinality is |H 1 (F, Z(G))| −1 times the number of regular unipotent elements. Hence the above sum is equal to |Z(G) 0F |ql |H 1 (F, Z(G))| 2 by 12.2.14. As Z(G)/Z(G) 0 is abelian we have |H 1 (F, Z(G))| = |H 1 (F, Z(G)/Z(G) 0 )| = |Z(G) F /Z(G) 0F |, whence the result.



Corollary 12.3.12 If the centre of G is connected and Γ denotes the unique Gelfand–Graev representation, then DG Γ is equal to |Z(G) F |ql on regular unipotent elements and to zero elsewhere. If moreover the characteristic is good F we have DG Γ = πuG , where u ∈ GF is regular unipotent. Proof In that case there is only one cz,z whose value has to be 1 by the proof of 12.3.10(ii). If moreover the characteristic is good the set of regular unipotent  elements is a single conjugacy class of GF by 12.2.16.

214

Regular Elements; Gelfand–Graev Representations

Corollary 12.3.13 If the characteristic is good for G, for any χ ∈ Irr(GF ) and any u ∈ Uz we have  χ(u) = σzz −1 (−1) |S/F | DG ( χ), Γz GF . z ∈H 1 (F,Z(G))

Proof This is 12.3.10(i) since when the characteristic is good the set Uz is a single conjugacy class by 12.2.7.  Note that, since for bad characteristic the distinct conjugacy classes into which Uz splits have all the same cardinality (see 12.2.11), the left-hand side of 12.3.10(i) is the mean of the values of χ at these classes. Finally, note that there are convenient formulas to compute σz in terms of Gauss sums, see Digne et al. (1997, 2.4) and the computation at the end of Section 12.5.

12.4 Regular and Semi-Simple Characters The final part of this chapter is devoted to the study of irreducible constituents of Gelfand–Graev representations and of their dual. We shall prove in particular that when the centre of G is connected these representations are uniform. Definition 12.4.1 An irreducible character of GF is regular if it is a constituent of some Gelfand–Graev character. An irreducible character whose dual is (up to sign) a regular irreducible character will be called semi-simple. F

StGF = regUF (see 7.4.4 For example the character StGF is regular as ResG UF and 4.4.1(iv)), so by Frobenius reciprocity StGF is a constituent of any Gelfand– Graev representation. The character 1GF is thus semi-simple. The following definition will allow us to describe the regular and semi-simple characters (see 12.4.10 and 12.4.12 below). ∗

Definition 12.4.2 If s is a semi-simple element of G∗F and if T∗1 is quasi-split maximal torus of CG∗ (s) 0 , we define a class function χ (s) on GF by  χ (s) = |W 0 (s)| −1 ε G ε T∗w RG T∗w (s), w∈W 0 (s)

W 0 (s)

where is as in 3.5.2 and (see 4.2.22).

T∗w

is a maximal torus of CG∗ (s) 0 of type w

Note that ε G ε T∗w = ε T∗1 (−1) l(w) , see remark after 10.3.6. ∗

Definition 12.4.3 A G∗F -Lusztig series of characters, denoted by (s) where the E (GF , (s)G∗F∗ ), is the set of irreducible constituents of the RG T∗ ∗ semi-simple class (s)G∗F∗ of s in G∗F is fixed.

12.4 Regular and Semi-Simple Characters

215

Thus E (GF , (s)G∗F∗ ) is a subset of E (GF , (s)). These series, in the case ∗ where F ∗ is a Frobenius endomorphism and thus the G∗F -classes are the “rational classes”, are also called in the literature “rational Lusztig series”. We see that χ (s) is in E (GF , (s)G∗F∗ ). Note that by 11.2.2(ii) and 4.2.14(ii), if ∗ the centre of G is connected, the G∗F -Lusztig series and the geometric series are the same. This is not true in general, as can be seen from the following proposition. Proposition 12.4.4 Irr(GF ).

∗

The G∗F -Lusztig series of characters form a partition of

This result will be proved after 12.4.12. We shall also use the next result. ∗

Proposition 12.4.5 The number of geometric semi-simple classes of G∗F is |Z(G) 0F |ql , where l is the semi-simple rank of G. Proof Let T∗ be a quasi-split maximal torus of G∗ . Any semi-simple class of G∗ has a representative in T∗ and two elements of T∗ are conjugate in G∗ if and only if they are conjugate by the Weyl group W of G∗ with respect to T∗ , see 1.3.6(iv). So semi-simple conjugacy classes of G∗ are in one-to-one correspondence with T∗ /W and the F ∗ -stable ones – which are in bijection with ∗ ∗ the geometric semi-simple classes of G∗F – are parametrised by (T∗ /W) F . Now t ∈ T∗ has an F-stable image in T∗ /W if and only if t ∈ T1 := {t ∈ T∗ | ∃w ∈ W, wFt = t}. Thus if O(t) denotes the W-orbit of t ∈ T1 we get |(T∗ /W) F | = |T1 /W | =

 t∈T1

1 |O(t)|

1  1  |{w ∈ W | wt = t}| = |{w ∈ W | = |W | t∈T |W | t∈T 1

wF

t = t}|,

1

the last equality since for t ∈ T1 there exists v ∈ W such that vFt = t thus wt = t is equivalent to wvFt = t. Thus we get 1  1  ∗F |(T∗ /W) F | = |{t ∈ T1 | wFt = t}| = |T |. |W | w∈W |W | w∈W w Thus, using 4.4.9 and using that (−1) l(w) = det w = det w−1 and that det τ = det τ −1 since it is a sign, we get 1  |(T∗ /W) F | = det(q − τ −1 w | X(T∗ ) ⊗ R). |W | w∈W If T is a maximal torus of G in duality with T∗ , we can replace X(T∗ ) by Y (T). Further, since Y (T) = Y (Z(G) 0 ) ⊕ Y (T ∩ Der G) and W acts trivially on the first factor, we have

216

Regular Elements; Gelfand–Graev Representations

det(q − τ −1 w | Y (T) ⊗ R) = det(q − τ −1 | Y (Z(G) 0 ) ⊗ R) det(q − τ −1 w | Y (T ∩ Der G) ⊗ R) = |Z(G) 0F | det(q − τ −1 w | Y (T ∩ Der G) ⊗ R), the last equality by 4.4.9 applied to the group Z(G) 0 . So we are reduced to  prove qdim V = |W1 | W det(q − τ −1 w | V) where V = Y (T) ⊗ R, when G is semi-simple. Now it is well-known that det(q − τ −1 w | V) = and since

1 |W |

 w∈W

dim V

(−1) i qdim V−i Trace(τ −1 w | V ∧i ),

i=0

w is the projector onto the fixed points of W, we get

dim V 1  det(q − τ −1 w | V) = (−1) i qdim V−i Trace(τ −1 | (V ∧i ) W ). |W | W i=0

But it is a result of Steinberg (1968, 14.1) that if V W = 0 (which is the case when G is semi-simple), then we also have (V ∧i ) W = 0 for i > 1. Thus we only have the term for i = 0 in the above sum, whence the result.  ∗

Proposition 12.4.6 We have  χ (s) , χ (s) GF = |(W (s)/W 0 (s)) F |. Proof

By definition

 χ (s) , χ (s) GF = |W 0 (s)| −2

 w,w ∈W 0 (s)

= |W 0 (s)| −2



G ε T∗w ε T∗w RG T∗w (s), RT∗ (s)GF w

∗

|W (s) wF ||W 0 (s) ∩ {F ∗ -class of w in W (s)}|

w∈W 0 (s)

= |W 0 (s)| −2



∗

|{v ∈ W (s) | vw F v−1 ∈ W 0 (s)}|,

w∈W 0 (s)

the second equality by 9.3.1 – since the proof of 11.1.16 shows that the sta∗ biliser of θ in W (T) F is isomorphic to that of s in W (T∗ ) F for corresponding pairs (T, θ) and (T∗ , s). But, as w is in W 0 (s), the F ∗ -conjugate by v ∈ W (s) of w is in W 0 (s) if and only if the image of v in W (s)/W 0 (s) is F ∗ -stable, so we get  ∗ |(W (s)/W 0 (s)) F ||W 0 (s)|,  χ (s) , χ (s) GF = |W 0 (s)| −2 w∈W 0 (s)



whence the result. Proposition 12.4.7  χ (s) , Γz GF = 1. Proof

∗F ∗

For any s ∈ G

and z ∈ H 1 (F, Z(G)) we have

We shall use the following property of Green functions.

12.4 Regular and Semi-Simple Characters

217

Lemma 12.4.8 The value of any Deligne–Lusztig character at a regular unipotent element is 1. Sketch of the proof. The proof is in Deligne and Lusztig (1976, 9.16). Its main ingredient is the use of a compactification of the variety L −1 (U)/TF . The trace of u on Hc∗ (L −1 (U)/TF ) is then computed, using the fact that u, being regular unipotent, has only one fixed point a on that compactification, and by taking local coordinates at a. The proof in Deligne and Lusztig (1976) is given for a Frobenius endomorphism, but applies verbatim for a Frobenius root.  By definition we have  χ (s) , Γz GF = |W 0 (s)| −1



ε G ε T∗w RG T∗w (s), Γz GF .

w∈W 0 (s)

But G ε G ε T∗w RG T∗w (s) = DG (RT∗w (s))

by 10.2.1 and the proof of 11.1.16, so the above scalar product is equal to G DG (RG T∗w (s)), Γz GF = RT∗w (s), DG (Γz )GF .

If we express DG (Γz ) as a linear combination of the γz as in the proof of  (s), γz GF ; but – by 12.4.8 – RG (s), γz GF is 12.3.10, we get z cz,z RG T∗w T∗w  equal to 1 and by the proof of 12.3.10, z cz,z is equal to 1, whence the result.  We can now prove the following proposition. Proposition 12.4.9

We have  ∗ −1 |(W (s)/W 0 (s)) F | χ (s) = |Z(G) F |/|Z(G) 0F | (s)



Γz ,

z∈H 1 (F,Z(G))

where in the sum of the left-hand side (s) runs over semi-simple conjugacy ∗ classes of G∗F . Proof To prove the result we shall show that the scalar product of the two sides is equal to the scalar product of each side with itself. We first compute the scalar product of the left-hand side with itself. By 12.4.4 this is equal to  ∗ −2 |(W (s)/W 0 (s)) F |  χ (s) , χ (s) GF , |Z(G) F | 2 /|Z(G) 0F | 2 (s)

which, by 12.4.6 equals  ∗ −2 ∗ |Z(G) F | 2 /|Z(G) 0F | 2 |(W (s)/W 0 (s)) F | |(W (s)/W 0 (s)) F | (s)

= |Z(G) F | 2 /|Z(G) 0F | 2

 (s)

∗

−1

|(W (s)/W 0 (s)) F | .

218

Regular Elements; Gelfand–Graev Representations

By 4.2.15(ii), 3.5.2 and 12.2.13 – which applies since W (s)/W 0 (s) is abelian, ∗ ∗ see 11.2.1(iii) – |(W (s)/W 0 (s)) F | is the number of G∗F -classes in the geometric conjugacy class of s, the last sum is equal to ∗

|Z(G) F | 2 /|Z(G) 0F | 2 |{ geometric semi-simple classes of G∗F }|. By 12.4.5, this equals |Z(G) F ||Z(G) F /Z(G) 0F |ql , which, by 12.2.13, can be written |Z(G) F ||H 1 (F, Z(G))|ql . The scalar product of the right-hand side with itself was computed in 12.3.10 and has the same value. The scalar product of one side with the other one is easily computed from 12.4.7: it is the product of |Z(G) F /Z(G) 0F |, the number of geometric conjugacy classes of semi-simple elements of G∗ and the number  of z, which gives again |Z(G) F |ql |H 1 (F, Z(G))|. Corollary 12.4.10

Assume that the centre of G is connected. Then:

(i) χ (s) is an irreducible character of GF for any (s). (ii) The (unique) Gelfand–Graev representation of GF is the sum of all the χ (s) (which are all the regular characters). F (iii) The dual of the Gelfand–Graev representation (which is πuG if the characteristic is good; see 12.3.12) is the sum of all the semi-simple characters of GF up to signs. Proof By hypothesis, from 11.2.2(ii) and 3.5.2, for all s we have ∗ |(W (s)/W 0 (s)) F | = 1, thus  χ (s) , χ (s) GF = 1 by 12.4.6. By the disjunction ∗ of the geometric series, which are in the present case equal to the G∗F -series, χ (s) which is the projection of the unique Gelfand–Graev character on the series E (GF , (s)) is a true character. The corollary is then straightforward (note that in the present case we do not actually have to use 12.4.4).  We now want to get a result similar to 12.4.10 for groups with a nonconnected centre. First we shall embed G in a group with connected centre and the same derived group. The construction is as follows: take any torus S ˜ = G ×Z(G) S, where Z(G) acts by translation on containing Z(G) and put G ˜ is isomorphic to S and clearly Der G ˜ = Der G. G and on S. The centre of G If T is a maximal torus of G then it is contained in a unique maximal torus ˜ and T = T˜ ∩ G. We shall choose, in addition, S T˜ = T ×Z(G) S (see 1.2.3) of G ˜ F-stable and extend F to G in the obvious way. We then have a bijection T → T˜ ˜ If θ is a character of from the set of F-stable maximal tori of G onto that of G.

12.4 Regular and Semi-Simple Characters

219

T˜ F (where T is an F-stable maximal torus of G), then by 11.3.10 the restriction ˜F ˜ (θ) to GF is RG (ResTTF θ). of RG T T˜ We now consider dual groups. If T˜ ∗ is a torus dual to T˜ and T∗ a torus dual ˜ so Y (T∗ ) is a quotient of Y (T˜ ∗ ), so T∗ is to T, then X(T) is a quotient of X(T), ∗ ˜ ∗ be a group dual to G ˜ (the torus T˜ ∗ ˜ by 1.2.10. Let G naturally a quotient of T ˜ ∗ being dual of the torus T ˜ of G); ˜ then the kernel of T˜ ∗ → T∗ is a central of G ∗ ∗ ˜ ∗ by this central torus is clearly dual to G ˜ torus of G and the quotient G of G ∗ ˜ ∗ has a Frobenius (the torus T being dual of T). We shall assume also that G ∗ ∗ ˜∗ root F dual to F, so the same is true for G . Note that the root systems of G (relative to T˜ ∗ ) and of G∗ (relative to T∗ ) are the same. Proposition 12.4.11 proper character.

∗

For any s ∈ G∗F the class function χ (s) on GF is a

˜ ∗F∗ be a semi-simple element whose image is s ∈ G∗F∗ (˜s Proof Let s˜ ∈ G ˜ ∗ → G∗ is connected); we shall show that χ (s) is the exists as the kernel of G ˜ F , which implies the result by 12.4.10(i). restriction of the character χ (˜s) of G ˜ is connected, the centralisers of semi-simple elements in Since the centre of G ∗ ˜ G are connected, hence by 3.5.2 W (˜s) is the group generated by the reflections ˜ ∗ with respect to the roots which are trivial on s˜ so is the same as W 0 (s). of G (s) is the Since s˜ maps to s, for any maximal torus T∗  s the character RG T∗ ˜ G F restriction to G of the character RT˜ (˜s), and, from the definitions of χ (s) and  χ (˜s) the result follows. Once we know that χ (s) is a proper character, we can give the decomposition of the Gelfand–Graev representation in general. Theorem 12.4.12 For any z ∈ H 1 (F, Z(G)) and any semi-simple conjugacy ∗ class (s) of G∗F , there is exactly one irreducible common constituent χs,z of χ (s) and Γz ; it has multiplicity 1 in both χ (s) and Γz and we have  Γz = χs,z , (s) ∗

where (s) runs over the set of semi-simple classes of G∗F . Proof The result is straightforward by 12.4.7 and 12.4.9.



Note that we have not given a parametrisation of the regular characters: indeed it is possible to have χs,z = χs,z . For instance, when s = 1 we have χ (s) = StGF as observed in the proof of 10.2.6. It can be shown that χs,z = χs,z if and only if z−1 z is in a certain subgroup of H 1 (F, Z(G)) which can be related to CG∗ (s)/CG∗ (s) 0 via duality, see Digne et al. (1992, Proposition 3.12(ii)).

220

Regular Elements; Gelfand–Graev Representations ∗

Proof of 12.4.4 Let s be an element of G∗F and s˜ be a preimage of s in ˜ ∗F∗ . The series E (GF , (s)G∗F∗ ) consists, by 11.3.10, of the irreducible conG ˜ F , (˜s)). Assume that two sestituents of the restrictions of elements in E (G F F ∗ ∗ ries E (G , (s)G∗F ) and E (G , (s )G∗F ) have a non-empty intersection. Let s˜ ˜ ∗F∗ of s ; then there exist two irreducible characters χ ∈ be a preimage in G ˜ F , (˜s )) whose restrictions to GF have a common ˜ F , (˜s)) and χ ∈ E (G E (G constituent. By Clifford’s theory (see 11.3.9) we see that the restrictions of two ˜ F to GF are disjoint or equal and that they irreducible characters χ and χ of G ˜ F /GF . are equal if and only if χ = χ.ζ, where ζ is a linear character of G ∗ F F ∗F ˜ ), since ˜ /G is of the form zˆ with z ∈ Z(G By 11.4.12 a character of G ˜ F . Since multiplication by zˆ maps RG˜ (˜s) to such a character is trivial on Der(G) T˜ ˜ ˜ F , (˜s )) and E (G ˜ F , (˜sz)) have to be equal as they have χˆz (˜ s z), the series E ( G RG T˜ ˜ F , (˜s)) as a common element. But zˆ is trivial on GF , so we see that the series E (G F F ˜ , (˜sz)) have the same restrictions, and thus E (G , (s)G∗F∗ ) = E (GF , and E (G (s )G∗F∗ ).



∗

Theorem 12.4.13 Let s and s be two semi-simple elements of G∗F . Then two (s) and RG ∗ (s ) have a common constituent if Deligne–Lusztig characters RG T∗ T ∗ and only if s and s are G∗F -conjugate. Proof The “only if” part is 12.4.4. A similar computation to that of 12.4.6 (s)GF . As shows that for any w ∈ W 0 (s) we have  χ (s) , χ (s) GF =  χ (s) , RG T∗w G F ˜ RT∗ (s) is invariant by conjugation under G , all the irreducible constituents w (s), so they must have multiplicity 1. of χ (s) have the same multiplicity in RG T∗ w

∗

Thus we see that, when s and s are conjugate in G∗F , all irreducible con (s) and RG  stituents of χ (s) (= χ (s ) ) occur in both RG (s ). T∗ T∗

12.5 The Character Table of SL2 (Fq ) The greater part of the character table of SL2 (Fq ) (Table 12.1) will be obtained by restriction from that of GL2 (Fq ) and by Clifford theory, using that GL2 (Fq )/SL2 (Fq )  F×q . As in 11.7.4, F will denote the Frobenius endomorphism raising the matrix entries to their qth power, T the diagonal torus, and {1, s} the elements of W (T). 0 −1 This time s can be represented by the matrix . 1 0 The group TF , formed of the matrices diag(x, x−1 ) where x ∈ F×q is isomorphic to F×q . An element of Irr(TF ) is given by α ∈ Irr(F×q ) and s acts by changing α into α −1 .

12.5 The Character Table of SL2 (Fq )

221

By 9.3.1, the Deligne–Lusztig characters RG (α) are pairwise orthogonal for T distinct subsets {α, α −1 } ⊂ Irr(F×q ); they are irreducible when α 2  1, and otherwise have two distinct irreducible constituents. For the other GF -conjugacy class of maximal tori, we have q

q+1

TFs  TsF = {diag(t1 , t1 ) | t1

= 1}  μq+1 .

The element s acts by ω → ωq = ω−1 on Irr(μq+1 ). (ω) are pairwise orthogonal By 9.3.1, the Deligne–Lusztig characters RG Ts −1 for distinct subsets {ω,ω } ⊂ Irr(μq+1 ); they are the negative (by 7.1.6) of irreducible characters when ω2  1 and when they are not irreducible they have (α, β) two irreducible constituents. When they are irreducible the characters RG T (ω) are all distinct since they are in different Lusztig series. and −RG Ts We now decompose the non-irreducible RG (θ). By 11.3.8, the unipotent T characters of SL2 (Fq ) are the restriction of those of GL2 (Fq ), and we still have (1) = 1GF + StGF and RG (1) = 1GF − StGF . In characteristic 2, any character RG T Ts of order 2 is trivial and thus there are no other non-irreducible Deligne–Lusztig characters. Otherwise, let α0 (resp. ω0 ) be the non-trivial character of order 2 of F×q (resp. μq+1 ). By 11.3.10, for any F-stable maximal torus T˜ of GL2 and any θ ∈ Irr(T˜ F ) we have ˜F

SL2 RT∩SL (ResTT˜ F ∩SL θ) = Res ˜ 2

2

GLF2 SLF2

2 (RGL (θ)). T˜

Applying this to an extension of α0 , being the restriction of an irreducible char(α0 ) is the sum of two distinct irreducible characters χ+α0 and χ−α0 ; acter, RG T + + χ − ; the characters χ ± must be distinct from (ω0 ) = χω similarly −RG ω0 α0 0 Ts ± G χω0 , since RTs (ω0 ) and RG (α ) are orthogonal. Actually RG (α0 ) = χ (s1 ) and 0 T T G RTs (ω0 ) = χ (s2 ) where s1 is the element of order 2 of the split torus of PGL2 and s2 is the element of order 2 of the non-split torus of PGL2 , and, since s1 and s2 are not rationally conjugate, they are in distinct rational series (see 12.4.2, 12.4.4). The list of irreducible characters of SL2 (Fq ) is thus • • • • •

RG (α) attached to {α, α −1 } ⊂ Irr(F×q ) of size 2. T (ω) attached to {ω,ω−1 } ⊂ Irr(μq+1 ) of size 2. −RG Ts The characters 1GF and StGF . (α0 ). If q is odd, the two irreducible constituents χ+α0 and χ−α0 of RG T + and χ − of −RG (ω ). If q is odd, the two irreducible constituents χω 0 ω0 0 Ts

We now give the list of conjugacy classes of SL2 (Fq ), by describing how the classes of GL2 (Fq ) which are in SL2 (Fq ) split. The semi-simple classes do not split since they have a connected centraliser since the dual group PGL2

222

Regular Elements; Gelfand–Graev Representations 

 1 1 has a connected centre, see 11.2.2(ii). But the class of which has been 0 1 analysed in 4.2.18(ii) splits in odd characteristic. The list of semi-simple classes we get in SL2 (Fq ) is: • The centre, formed of diag(1, 1) and in odd characteristic diag(−1, −1). • In TF the classes diag(a, a−1 ) for {a, a−1 } ⊂ F×q of size 2. • In TFs , the classes conjugate over Fq to diag(x, x−1 ) for {x, x−1 } ⊂ μq+1 of size 2. In addition, we have the non-semi-simple classes zu where z is central and u  1 unipotent. From the above descriptions there are 4 such classes in odd characteristic, and only one in characteristic 2 – these unipotent elements are regular so we could also take the analysis from 12.2.15. We get thus in odd  a b characteristic the four classes with a ∈ {−1, 1} and b ∈ (F×q ) 2 or b ∈ 0 a F×q − (F×q ) 2 . Since the classes do not split, in characteristic 2 we get the character table of SL2 (Fq ) by restriction form that of GL2 (Fq ). Thus from now on and in Table 12.1. we assume q odd. Most of the table of SL2 (Fq ) (q odd) is obtained by restriction from GL2 (Fq ), using 11.3.10; the only values which are not given by restriction are those of ± . Since χ + + χ − is the restriction of an irreducible characχ±α0 and χω α0 α0 0 ter of GL2 (Fq ), the characters χ+α0 and χ−α0 are conjugate under GL2 (Fq ) – because SL2 (Fq ) is a normal subgroup of GL2 (Fq ). They have thus the same dimension, and more generally the same value on all classes of SL2 (Fq ) that are invariant under the action of GL2 (Fq ); in particular they also have the same central character α0 . These remarks show  that  to get all their values it is enough 1 b + to compute their values on the classes . The same analysis holds for χω 0 0 1 − and χω0 . These classes are regular unipotent; if 1 and z are the two elements of H 1 (F,Z(SL2 )) and we choose for u1 – see  definition after 12.2.17 – the 1 x 1 1 class of with x ∈ F×q − (F×q ) 2 . For any , then uz is the class of 0 1 0 1 χ ∈ Irr(SL2 (Fq )), by 12.3.10(i) we have − χ(u1 ) = σ1 DG ( χ), Γ1 GF + σz DG ( χ), Γz GF and − χ(uz ) = σ1 DG ( χ), Γz GF + σz DG ( χ), Γ1 GF .

⎫ ⎪ ⎬ ⎪ ⎭

(12.5.1)

+ and As we have seen χ+α0 and χ−α0 are the two constituents of χ (s1 ) and χω 0 + + are the two constituents of χ (s2 ) . Let us choose for χ α0 (resp. χω0 ) the common constituent of χ (s1 ) and Γ1 (resp. of χ (s2 ) and Γ1 ) – see 12.4.12. − χω 0

12.5 The Character Table of SL2 (Fq )

223

Table 12.1. Character Table of SL2 (Fq ) for q odd (note that |SL2 (Fq )| = q(q − 1)(q + 1)) 

 a 0 0 a a ∈ {1, −1}



 a 0 0 a−1 a ∈ F× q a  {1, −1}



 x 0 0 Fx x. Fx = 1 x  Fx



Number of classes of this type

2

(q − 3)/2

(q − 1)/2

4

Cardinal of the class

1

q(q + 1)

q(q − 1)

(q2 − 1)/2

RG (α) T α ∈ Irr(F× q) α2  1

(q + 1)α(a) α(a) + α

χ εα 0 ε ∈ {1, −1}

q+1 α0 (a) α0 (a) 2

0

−RG (ω) Ts ω ∈ Irr(μq+1 ) ω2  1

(q − 1)ω(a) 0

−ω(x) − ω( Fx) −ω(a)

q−1 ω0 (a) 0 2

−ω0 (x)

1GF

1

1

1

1

StGF

q

1

−1

0

Classes

ε χω 0 ε ∈ {1, −1}

  1 0 a

 a b 0 a a ∈ {1, −1}, b ∈ {1, x} with × 2 x ∈ F× q − (Fq )

α(a)

α0 (a)  1+ε 2   α0 (ab) α0 (−1)q

ω0 (a)  −1 + ε 2   α0 (ab) α0 (−1)q

+ and χ − are cuspidal, we have D ( χ + ) = − χ + and Since by 9.3.3, χω G ω0 ω0 ω0 0 − − ∗RG χ + − χ + . Now, by Frobenius DG ( χω0 ) = − χω0 . We have DG ( χ+α0 ) = RG α0 T T α0 ( χ+α0 ), hence ∗RG χ+ = α0 since ∗RG reciprocity α0 has multiplicity 1 in ∗RG T T α0 T preserves Harish-Chandra series –see 5.3.9– and α0 is the only character of TF in its series. So we get DG ( χ+α0 ) = χ−α0 and DG ( χ−α0 ) = χ+α0 . Thus (12.5.1) gives + − (u1 ) = − χω (uz ) = −σ1 χ+α0 (uz ) = χ−α0 (u1 ) = − χω 0 0

and + − (uz ) = − χω (u1 ) = −σz . χ+α0 (u1 ) = χ−α0 (uz ) = − χω 0 0

224

Regular Elements; Gelfand–Graev Representations

To compute σ1 and σz , we notice that if χ1 ∈ Irr(F+q ) is a character as defined above 12.3.3, then by 12.3.11 we have  χ1 (x), σ1 = |Z(GF )| −1 Γ1 (u1 ) = x∈(F×q ) 2

and σz = |Z(GF )| −1 Γz (u1 ) = |Z(GF )| −1 Γ1 (uz ) =



χ1 (x),

x∈Fq \(Fq ) 2

the second equality since Γz = t Γ1 and uz = tu1 , where t is as in the remark after 12.3.3. To compute them notice that σ1 + σz = −1 and that σ1 − σz is the Gauss sum  α0 (x) χ1 (x), x∈Fq

so (σ1 − σz ) 2 = α0 (−1)q – seefor example Digne et al. (1997, above 2.8). In Table 12.1 we denote by α0 (−1)q the square root of α0 (−1)q which is equal to σ1 − σz . Notes Many results about regular elements can be found in Steinberg (1974). The Gelfand–Graev representation was first introduced by Gelfand and Graev (1962), who proved that it is multiplicity free in the case of SLn ; this result was proved in general by Steinberg. All results about semi-simple characters and the decomposition of the Gelfand–Graev representation in the case of a group with connected centre are due to Deligne and Lusztig (1976). The Gelfand– Graev representations for groups with non-connected centre were studied in Digne et al. (1992), using earlier work of Lehrer (1978). For an interesting account of the theory of representations of finite reductive groups in the setting of SL2 , see Bonnafé (2010).

13 Green Functions

In this chapter we explain the Lusztig–Shoji algorithm for computing Green functions. First, we must relate the values R χ˜ (1), see 11.6.1 and 11.6.4, to the invariant theory of W.

13.1 Invariants Let V be a vector space of dimension n over a field k of characteristic 0, and let W ⊂ GL(V) be a finite group. Then the action of W extends naturally to the symmetric algebra SV which, choosing a basis x1 , . . . , xn of V, is isomorphic to the polynomial algebra k[x1 , . . . , xn ]. Thus SV is graded by polynomial degree with (SV)1 = V and (SV)0 = k. A pseudo-reflection is an element s ∈ GL(V) such that Ker(s − 1) is a hyperplane; if k ⊂ R, a pseudo-reflection of finite order is a reflection. If W ⊂ GL(V) is a finite group, we will denote by Ref(W) the set of pseudo-reflections in W. Theorem 13.1.1 Assume W ⊂ GL(V) is a finite subgroup generated by Ref(W). Then the subalgebra SV W ⊂ SV of W-invariants is a polynomial algebra k[f1 , . . . , fn ] where f1 , . . . , fn ∈ SV are algebraically independent homogeneous polynomials. References This theorem has been proved case-by-case by Shephard and Todd (1954), by classifying all finite groups generated by pseudo-reflections, and then in a case-free way by Chevalley (1955).  In the situation of the theorem the fi are called fundamental invariants for W. They are not unique but their degrees are – a consequence of Lemma 13.1.4 below; these are called the reflection degrees of W. The converse of the above theorem is easier and we will prove it here. 225

226

Green Functions

Theorem 13.1.2 Assume W ⊂ GL(V) is a finite subgroup and SV W  k[f1 , . . . , fn ] where f1 , . . . , fn ∈ SV are algebraically independent homogeneous polynomials of degree d1 , . . . , dn . Then: (i) W is generated by Ref(W). (ii) |W | = d1 . . . dn .  (iii) | Ref(W)| = ni=1 (di − 1). Proof The tool for proving this theorem is to use Hilbert–Poincaré series. If X= i≥0 Xi is a graded k-vector space such that all Xi are finite dimensional, and g ∈ End(X) is a graded endomorphism (that is, it preserves the Xi ), we de  fine PX (g):= i≥0 Trace(g | Xi )ti ∈ k[[t]], and PX := PX (IX ) = i≥0 dim(Xi )ti . Lemma 13.1.3

(Molien)

(i) If w ∈ End(V) is semi-simple, then PSV (w) = det(IV −wt | V ⊗ k[t]) −1 . (ii) For W ⊂ GL(V) a finite group, we have  PSV W = |W | −1 det(IV −wt | V ⊗ k[t]) −1 . w∈W

Proof For (i) we first notice that if V = V1 ⊕ V2 , then SV = SV1 ⊗ SV2 , from which it follows that if w respects the decomposition V1 ⊕ V2 we have PSV (w) = PSV1 (w)PSV2 (w). Since both sides of formula (i) do not change if we extend k, we may assume k algebraically closed and diagonalise w; let x1 , . . . , xn be a basis of V formed of eigenvectors of w for the eigenvalues λ 1 , . . . , λ n . Then PSV (w) = Pk[x1 ] (λ 1 ) . . . Pk[xn ] (λ n ) and it is enough to observe that Pk[x] (λ) =  1 i i i≥0 λ t = 1−λt .  (ii) is an immediate consequence of (i) since |W | −1 w∈W w is a projector on the W-invariants for any kW-module. Thus the graded dimension of SV W is given by the graded trace of this projector, which is formula (ii).  Using 13.1.3(ii) we see that for any finite group W, the formal series in 1 t given by PSV W has a Laurent expansion at 1 whose first terms are |W |   1 | Ref(W)| + + · · · . Here the first term corresponds to 1 ∈ W and (1 − t) n 2(1 − t) n−1 the second to Ref(W), taking in account that each s ∈ Ref(W) of non-trivial 1 −1 we have terms eigenvalue λ contributes a term (1−λ)(1−t) n−1 , and that if s  s 1 1−λ

+

1 1−λ −1

= 1 and if s = s−1 then λ = −1 thus

1 1−λ

=

1 2

whence the factor 12 .

13.1 Invariants

227

We now compare with the Laurent expansion at 1 of the series Pk[f1, ...,fn ] = 1 1 1 ··· . The first terms are + Pk[f1 ] . . . Pk[fn ] = nd . . . d d1 dn (1 − t) 1 − t 1 − t 1 n n i=1 (di − 1) + · · · . Here we obtain the first term by using that (1 − td ) = (1 − t) n−1 2d1 . . . dn (1 − t)(1 + t + · · · + td−1 ), and to get the second one we multiply the series by (t − 1) n , derive with respect to t and set t = 1. Comparing both developments, it follows that if for a finite group W we  have SW  k[f1 , . . . , fn ] then |W | = d1 . . . dn and | Ref(W)| = ni=1 (di − 1). We have proved (ii) and (iii) of 13.1.2. We now prove (i). Lemma 13.1.4 Let k[f1 , . . . , fn ] ⊂ k[f1 , . . . , fn ] be two polynomial subalgebras of SV where the fi and fi are homogeneous, the degrees of the fi are d1 ≤ · · · ≤ dn , and the degrees of the fi are d1 ≤ · · · ≤ dn . Then for all i we have di ≥ di . Proof There are polynomials p1 , . . . , pn such that fi = pi (f1 , . . . , fn ). It follows that di ≥ di otherwise for an i such that di < di then p1 , . . . , pi can only which is impossible since f , . . . , f are algebraically indeinvolve f1 , . . . , fi−1 1 i pendent.  Let W be the subgroup of W generated by Ref(W). By 13.1.1, we have = k[f1 , . . . , fn ] for some homogeneous fi of degrees d1 , . . . , dn and by the above lemma di ≥ di if we order them in ascending order. Now we have   | Ref(W)| = i (di − 1) ≥ i (di − 1) = | Ref(W )| and as the extreme terms are equal we must have di = di thus |W | = d1 . . . dn = |W | and W = W is generated by Ref(W).  SV W

Exercise 13.1.5 Show that (i) of Molien’s Lemma 13.1.3 remains true without assuming w semi-simple. We will need the following definition. Definition 13.1.6 Let I be the ideal of SV generated by the W-invariants of positive degree. Then SVW := SV/I is called the coinvariant algebra of W. Since I is a graded ideal, the coinvariant algebra inherits a grading from SV. This leads to the following theorem. Theorem 13.1.7 As a kW-module, SVW is isomorphic to the regular representation of W, and there is a graded W-equivariant isomorphism of SV W -modules SVW ⊗ SV W  SV. References

See Chevalley (1955) or Hiller (1982, II §3).  di −1 Corollary 13.1.8 PSVW = ni=1 t t−1 and for w ∈ W we have n di i=1 (1 − t ) PSVW (w) = . det(1 − wt | V ⊗ k[t])



228

Green Functions

Proof By the previous theorem we have PSV (w) = PSVW (w)PSV W (w) and 1  PSV W (w) = PSV W = n (1−t di ) , see the proof of 13.1.2. i=1

Definition 13.1.9 For χ ∈ Irr(W), we define the fake degree Feg χ of χ to be the graded multiplicity of χ in SVW . For χ ∈ Irr(W) we have n (1 − tdi )  χ(w) . Feg χ =  χ, PSVW W = i=1 −1 t | V ⊗ k[t]) |W | det(1 − w w∈W

Corollary 13.1.10

Proof By definition, χ(1) Feg χ is the graded dimension of the χ-isotypic  part of SVW , that is i  χ, (SVW )i W ti =  χ, PSVW W and we obtain the result  using the formula for PSVW (w) in 13.1.8. In order to apply this theory to a reductive group with a Frobenius root, we need to generalise it to reflection cosets. Proposition 13.1.11 Let τ ∈ NGL(V) (W) be an element of finite order. Assume k large enough so that it contains roots of unity of the same order as τ. Then it is possible to find fundamental invariants for W which are τ-eigenvectors. The corresponding pairs of degrees and eigenvalues (d1 , ε 1 ), . . . , (dn , ε n ) depend only on the coset Wτ and are called the generalised reflection degrees of Wτ. Proof We show the proposition by induction on the degree. Assume we have chosen invariants which are eigenvectors of τ for degrees less than d. Since τ normalises W it preserves the space (SV W )d of homogeneous invariants of degree d. The subspace V1 of (SV W )d spanned by monomials in the invariants of degree less than d is τ-stable by induction. Since τ is of finite order V1 has a τ-stable complement V2 by Maschke’s theorem. A basis of V2 formed of τeigenvectors will give fundamental invariants of degree d.  We assume now that k is a subfield of C. Let χ˜ be as in 11.6.4 an extension to W  τ of χ ∈ Irr(W)τ . We then generalise the definition of the fake degree to functions like w → χ(wτ) ˜ by using the scalar product 11.6.2 on C(Wτ). Definition 13.1.12 If we denote by PSVW ( τ) the function w → PSVW (wτ), we define Feg χ˜ :=  χ, ˜ PSVW ( τ)Wτ . We note that by 11.6.3 Feg χ˜ is a polynomial which coefficients algebraic integers – it is a polynomial since SVW is finite dimensional, see 13.1.7. Proposition 13.1.13 If (d1 , ε 1 ), . . . , (dn , ε n ) are the generalised reflection degrees of Wτ, we have n (1 − ε i tdi )  χ(wτ) ˜ . Feg χ˜ = i=1 |W | det(1 − (wτ) −1 t | V ⊗ k[t]) w∈W

13.1 Invariants

229

Proof The proof is the same as for 13.1.10, except that instead of PSV W we   use PSV W (τ) = ( ni=1 (1 − ε i tdi )) −1 . In view of the above formula and the remark before 13.1.13, if χ˜ takes integral values and τ is real, Feg χ˜ is a polynomial with integer coefficients. Formula (i) below was given in Steinberg (1968, 11.16), with a different proof. Theorem 13.1.14 Let G be a connected reductive group over Fq , and let F be a Frobenius root acting as qτ on X(T) where T is a quasi-split torus of G. Let (d1 , ε 1 ), . . . , (dn , ε n ) be the generalised reflection degrees of Wτ on X(T) ⊗ C. Then n  (i) |GF | = q i=1 (di −1) ni=1 (qdi − ε i ). ˜ (ii) R χ˜ (1) = Feg χ(q).

Proof We first compute R χ˜ (1). We have  G R χ˜ (1) = |W | −1 χ(wτ)R ˜ Tw (1)(1) w∈W

= |W | −1



w∈W

χ(wτ)ε ˜ G ε Tw

|GF |p |TFw |

=

|GF |p  χ(wτ) ˜ |W | w∈W det τ det(wqτ − 1 | X(T))

=

|GF |p Feg χ(q) ˜ n det τ i=1 (ε i qdi − 1)

where the first equality is the definition 11.6.1, the second is by 10.2.2, the third by 7.1.6 and 4.4.9, and the last is by 13.1.13 taking into account that wτ and (wτ) −1 being real and of finite order have the same set of eigenvalues, thus det(wqτ − 1) = (−1) n det(1 − (wτ) −1 q). We now apply the above computation to the function χ˜ defined by χ(wτ) ˜ = 1 for all w ∈ W, an extension of the trivial character of W. Then by 10.2.5 we have R χ˜ (1) = 1 and by definition, using that the only component of SVW where W acts trivially is the component of degree 0 where τ acts trivially too, we have  Feg χ˜ = 1. We thus get (ii) of the theorem and |GF |p = det τ ni=1 (ε i qdi − 1). This last equality is equivalent to (i) of the theorem: indeed we have |GF |p = ql(w0 ) = q | Ref(W) | by 4.4.1(iii) and, by the proof of 13.1.11, the ε i attached to a given degree are globally invariant by complex conjugacy, thus by inverse, thus    det τ ni=1 (ε i qdi − 1) = (det τ i ε i ) ni=1 (qdi − ε i ), so that to get the positive  integer |GF | we must have det τ = ni=1 ε −1 i . One could also deduce this last

230

Green Functions

formula from the fact that τ has a regular vector for the eigenvalue 1 – see the proof of 7.1.7 and Springer (1974, 6.4 (v)).  Remark 13.1.15 The reflection degrees of W on the orthogonal of Φ∨ are all equal to 1. One often gives a formula for 13.1.14(i) which only involves the non-trivial degrees dr+1 , . . . dn where r = rank(Z(G) 0 ): n

|GF | = |Z(G) 0F |q

i=r+1 (di −1)

n 

(qdi − ε i ).

i=r+1

This is obtained by using formula 13.1.14(i) for the group G/Z(G) 0 . Remark 13.1.16 A proof for |GF |p in the spirit of this book would be to use the Lefschetz formula for computing |(G/B) F |, and to use that Hc∗ (G/B) is isomorphic to the coinvariant algebra. However this last fact is beyond the scope of this book. Examples 13.1.17 For applying Theorem 13.1.14, one needs a list of generalised degrees. Here they are for quasi-simple groups (we give just the degrees when the group is split): • • • • • • • • • • • • • • •

An : 2, . . . , n + 1. Bn , Cn : 2, 4, . . . , 2n. Dn : 2, 4, . . . , 2n − 2 and n. 2A : (2, 1), (3, −1), . . . , (n + 1, (−1) n+1 ). n 2D : (2, 1), (4, 1), . . . , (2n − 2, 1) and (n, −1). n 3D : (2, 1), (4, ζ ), (6, 1), (4, ζ 2 ). 4 3 3 E6 : 2, 5, 6, 8, 9, 12. 2E : (2, 1), (5, −1), (6, 1), (8, 1), (9, −1), (12, 1). 6 E7 : 2, 6, 8, 10, 12, 14, 18. E8 : 2, 8, 12, 14, 18, 20, 24, 30. F4 : 2, 6, 8, 12. G2 : 2, 6. 2B : (2, 1), (4, −1). 2 2F : (2, 1), (6, −1), (8, 1), (12, −1). 4 2G : (2, 1), (6, −1). 2

Exercise 13.1.18

This exercise recovers four lines of the above list.

(i) Show that the group G(de, e, n) of monomial n × n complex matrices whose entries are in μde and whose product of entries is in μd is generated by pseudo-reflections in its action on Cn . (ii) Show that the elementary symmetric functions f1 , . . . , fn where fi is the coefficient of tn−i in the polynomial (t −x1 ) . . . (t −xn ) is a set of fundamental invariants of the symmetric group Sn  G(1, 1, n).

13.2 Green Functions and the Springer Correspondence

231

(iii) Show that the fundamental invariants of G(de, e, n) are {fj (x1de , . . . , xnde ) | j = 1, . . . , n−1} and fn (x1de , . . . , xnde ). Hint: compute |G(de, e, n)| by writing G(de, e, n) as the semi-direct product of a group of diagonal matrices by Sn . (iv) Show that W (An−1 )  G(1, 1, n), W (Bn )  G(2, 1, n), W (Dn )  G(2, 2, n), W (G2 ) = G(6, 6, 2).

13.2 Green Functions and the Springer Correspondence A consequence of the results of this chapter is that one can compute in principle (θ) on all elements of GF , at least in good characthe character values of RG T teristic and when F is a Frobenius morphism. By the character formula 10.1.2 the problem is in principle reduced to the computation of the Green functions (u, 1). By the character formula, applied to g = u (thus s = 1) we have QG T (θ)(u) = QG (u, 1) for any θ ∈ Irr(TF ); it follows that these Green functions RG T T take their values in Z. Our goal now is to explain the Lusztig–Shoji algorithm to by 0 on non-unipotent elements compute these Green functions. We extend QG T by the following definition: Definition 13.2.1 Let G be a connected reductive group with a Frobenius root F. If T is an F-stable maximal torus of G we define the (one-variable) Green as RG (1) · χp where χp is as in 7.3.2(ii) the characteristic function function QG T T F of the set Gu of unipotent elements. Proposition 13.2.2

For F-stable maximal tori T, T of G we have

| ⎧ if T and T are GF -conjugate, ⎪ |W|T(T) F| =⎨ ⎪0 otherwise. ⎩  G Proof Since regTF = θ ∈Irr(TF ) θ, we have for u unipotent QT (u) = (regTF )(u). This equality holds everywhere since RG (regTF ) = |TF | −1 RG T T G T G (reg RT ( χp · regTF ) = RG F ) · χ p where the second equality is by 10.1.6 T T (regTF ), RG (regT F )GF = which holds since χp ∈ C(GF )p . Now RG T T ∗ G G regTF , RT ◦ RT (regT F )TF and by the Mackey formula 9.2.6 we have, using that regTF is invariant by conjugation by W (T) F , F

G QG T , QT GF

∗ G RT

F ⎧ ⎪ ⎨ |W (T) | regTF ◦ RG T (regT F ) = ⎪ ⎩0

which gives the result.

if T and T are GF -conjugate, otherwise, 

232

Green Functions

The computation of Green functions relies on the above scalar product formula and an approach due to Springer relating Green functions to the cohomology of Springer fibres (see 12.2.1). Theorem 13.2.3 (i) For u ∈ Gu there is a natural action of W on Hc∗ (Bu ). (ii) Assume that F is a Frobenius endomorphism; then for u ∈ GFu we have (u) = Trace(wF | Hc∗ (Bu )). QG Tw Note that the right-hand side of the above formula is not a number of fixed points, since W does not act on Bu itself. Note also that the variety Bu depends up to isomorphism only on the G-conjugacy class of u, but the action of F on it, for u ∈ GFu , depends on the GF -class of u. Finally note that we use compact support cohomology Hc (to fit with Chapter 8) while the cited literature uses ordinary cohomology; they coincide since the varieties Bu are projective. Proof (i) was proved in Springer (1976), (ii) of this difficult theorem was conjectured in Springer (1976), proved for p and q large enough in Kazhdan (1977), and finally proved in Lusztig (1986c) and Shoji (1995, II, Theorem 5.5). See Shoji (1987) for the equivalence of the statements in these papers and the statement 13.2.3(ii). 

The Springer Correspondence Until the end of this chapter we assume that F is a Frobenius endomorphism so that 13.2.3(ii) is valid. The group CG (u) acts on Bu . By 8.1.14, the group CG (u) 0 acts trivially on Hc∗ (Bu ), thus the action of CG (u) factors through A(u):= CG (u)/CG (u) 0 . Springer (1976) has shown that the actions of A(u) and of W on Hc∗ (Bu ) commute. Proposition 13.2.4 For u ∈ Gu , all the irreducible components of Bu have the same dimension βu := 12 (dim CG (u) − rank G). Reference

See Spaltenstein (1977).



We will denote by I(Bu ) the set of irreducible components of Bu . It results 2β from 13.2.4 and from 8.1.12(ii) that Hc u (Bu ) has a basis indexed by I(Bu ), on which A(u) acts by permutation.

13.2 Green Functions and the Springer Correspondence

233

Theorem 13.2.5 For φ ∈ Irr(A(u)), the representation χu,φ of W afforded 2β by the φ-isotypic component Hc u (Bu )φ is irreducible or 0. If Irr(A(u)) 0 denotes {φ ∈ Irr(A(u)) | χu,φ  0}, there is a bijection – called the Springer correspondence – between Irr(W) and the set of χu,φ when u runs over representatives of the G-conjugacy classes in Gu , and φ runs over Irr(A(u)) 0 . References

See Springer (1976) and Lusztig (1984b).



Since (by definition) CG (u) acts by inner automorphisms on A(u), the representation χu,φ depends only on the G-conjugacy class C of u. We will denote it by χC,φ from now on. Note that 1 ∈ Irr(A(u) 0 ) since a permutation representation always involves the trivial character – with multiplicity the number of orbits of A(u) on I(Bu ), which is thus equal to χu,1 (1). We will now assume that Z(G) = {1}; this is not a restriction for computing the Green functions since by 11.3.8 they factor through GF /Z(G) F . The advantage is that the groups A(u) are smaller in this case, and it happens more often that Irr(A(u)) 0 = Irr(A(u)). Since Z(G) = {1}, by 2.3.9 (G, F) is a direct product of “descent of scalars”, that is groups of the form G1 × · · · × G1 with n factors permuted cyclically by n F. Since GF  GF1 it is enough to study the characters of (G1 , F n ), so we may and will assume that G is quasi-simple. The Springer correspondence has been computed in good characteristic by Springer (1976, 7.16) for G2 and An , by Shoji (1979) for the other classical groups and Shoji (1982) for F4 , and by Alvis et al. (1982) for E6 , E7 and E8 ; a summary can be found in the tables of Carter (1985, 13.3). It has been computed in bad characteristic – where the description of unipotent classes is different – by Lusztig and Spaltenstein (1985) and Spaltenstein (1985). It gives global information on Hc∗ (Bu ) through the following theorem from Borho and MacPherson (1981). Theorem 13.2.6 (i) χC,φ does not occur in the Q W-module Hci (Bu ) for any u ∈ C, i < 2 βu . (ii) Assume that χC,φ occurs in Hci (Bv ) for some i and v ∈ Gu . Then v ∈ C, the Zariski closure of C. To make the Lusztig–Shoji algorithm work we now just need information on the action of F on A(u) and I(Bu ) for F-stable u; all eigenvalues of F on 2β 2β Hc u (Bu ) are of absolute value q βu and F acts on Hc u (Bu ) as the scalar q βu times a permutation of the basis corresponding to the permutation induced by F on I(Bu ), see 8.1.12(iii).

234

Green Functions

Proposition 13.2.7 Let C ⊂ Gu be an F-stable conjugacy class. There exists always u ∈ CF such that F acts trivially on A(u). Proof For classical adjoint groups, A(u) is always of the form (Z/2Z) i for some i; the result is proved in Shoji (1983, 3.1). For exceptional adjoint groups in good characteristic we always have A(u)  Si for 1 ≤ i ≤ 5. Since these groups have only inner automorphisms, there exists an u in each geometric class such that F acts trivially on A(u). Indeed if F acts on A(u) by ad a then it acts trivially on A( gu) where g−1 Fg ∈ CG (u) is a representative of a−1 . For exceptional adjoint groups in bad characteristic see Liebeck and Seitz (2012, Lemma 20.16).  Proposition 13.2.8 Assume (G, F) split and the characteristic good for G. Let C be an F-stable G-conjugacy class in Gu . Then there is a GF -class C0 ⊂ CF such that F acts trivially on I(Bu ) for u ∈ C0 (and this class is unique), unless G is of type E8 and C is the class called D8 (a3 ) by Mizuno (1980) and called E8 (b6 ) in Carter (1985, §13.1.) We will call such a GF -class split. For u split, the action of F on A(u) is trivial. References This is proved in Springer (1976) for G2 , in Shoji (1982) for F4 , in Shoji (1983) for classical groups and in Beynon and Spaltenstein (1984) for  E6 , E7 and E8 . By definition and 8.1.12, if the GF -class of u is split then F acts by the scalar 2β on Hc u (Bu ). For u ∈ C, where C is the class E8 (b6 ), we have A(u)  S3 and, by Beynon and Spaltenstein (1984), there exists u ∈ CF such that F acts by 2β q βu on Hc u (Bu ) except on the sgn-isotypic component of A(u) when q ≡ −1 (mod 3), in which case F acts by −q βu on that component. Assume now that (G, F) is not split; then if W is the Weyl group of a quasisplit torus T, there exists τ such that F = qτ on X(T). Let C be an F-stable conjugacy class in Gu and let u ∈ CF be such that F acts trivially on A(u) and 2β for φ ∈ Irr(A(u)) 0 consider the φ-isotypic component of Hc u (Bu ). It is an irreducible Q (W × A(u))-module of character χC,φ ⊗ φ; since F commutes with A(u), the action of F on this component is of the form q βu τ1 ⊗I where τ1 defines a certain extension χ˜ C,φ of χC,φ to W  τ. Beynon and Spaltenstein (1984) and Shoji (1983) describe a particular choice of u giving rise to a particular extension; we still say that this u is split for F and we will call the corresponding χ˜ C,φ the almost preferred extension of χC,φ . The reason for this terminology is that Lusztig (1986a, 17.2) defined a “preferred extension” which coincides with our almost preferred extension except in 2E6 for the 3 characters of W (E6 ) such that a χC,φ  βu (mod 2) – see above 14.1.1 for the definition of a χ . We need to choose the almost preferred extension for 13.2.10(iii) to hold. q βu

13.2 Green Functions and the Springer Correspondence

235

Note that since the Green functions take rational values (since by definition QG (u, 1) is a Lefschetz number divided by |TF | so is a rational number, T see 8.1.6), as well as the characters of A(u) (see the proof of 13.2.7) and of W, the almost preferred extension takes rational values. To have uniform notations for the split and non-split cases, when (G, F) is split we will define τ = Id and let χ˜ C,φ = χC,φ . If u is a split element of CF , where C is an F-stable G-conjugacy class of unipotent elements, by 4.2.15(ii) the GF -orbits in CF are indexed by the conjugacy classes of A(u), where we can index u itself by 1. For the class E8 (b6 ) we index the GF -orbits by the F-conjugacy classes of A(u), taking as a reference unipotent the element u described above. Notation 13.2.9 For ι = (C, φ) where C ⊂ Gu is an F-stable G-conjugacy class and φ ∈ Irr(A(u)) we write • Cι for C, • βι for βu , u ∈ C, • Yι for the function on GF which vanishes outside CF and is equal to φ(a) on the GF -class indexed by a ∈ H 1 (F, A(u)), where u ∈ CF is a unipotent element chosen as above (split except for C = E8 (b6 )), • χ˜ ι for χ˜ C,φ with φ ∈ Irr(A(u)) 0 , u ∈ C. We will denote by I the set of all ι = (C, φ) where C runs over the Fstable unipotent classes and φ runs over Irr(A(u)) for u ∈ C, and by I0 the subset where φ ∈ Irr(A(u)) 0 – thus I0 identifies with the image of the Springer correspondence. Note that the Springer correspondence is F-equivariant in that we have F( χC,1 ) = χF(C),1 ; since F induces an inner automorphism of W except for types 3D4 and 2D2n , it follows that except for these cases all classes in Gu are F-stable; see, for example Liebeck and Seitz (2012, Theorem 7.1(i), Theorem 7.3(i) and the tables). Proposition 13.2.10 (i) The set {Yι }ι ∈I is a basis of the GF -class functions on GFu . (ii) For ι ∈ I0 , the restriction of R χ˜ ι to CιF is 0 unless Cι is in the Zariski closure of Cι , where R χ˜ ι is the function defined in 11.6.1 (note that since G is semi-simple, the action of τ on W has same order as τ). (iii) For ι ∈ I0 , the restriction of R χ˜ ι to CιF is equal to ε ι q β ι Yι where ε ι is a sign. In good characteristic, ε ι = 1 except for ι = (E8 (b6 ), sgn) in which case ε ι ≡ q (mod 3). In bad characteristic, ε ι = 1 except possibly for ι = (E8 (b6 ), sgn) where the value of ε ι is still unknown.

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Proof For (i), by the second orthogonality relation for characters of A(u),  where u is a reference unipotent in CF , the function φ ∈Irr(A(u)) φ(a)YC,φ is a multiple of the characteristic function of the GF -class in C parametrised by a ∈ A(u). We prove (ii). For a Q W  F-module M we denote by Trace( F | M) the function w → Trace(wF | M). For u ∈ CιF we have R χ˜ ι (u) = Trace( F | Hc∗ (Bu )), χ˜ ι Wτ by 13.2.3(ii), using that χ˜ ι takes rational values. By 13.2.6(ii) this scalar product is 0 unless Cι is in the Zariski closure of Cι . For (iii), if ua in the GF -class in Cι is parametrised by a ∈ H 1 (F, A(u)) we have R χ˜ ι (ua ) = Trace( F | Hc∗ (Bua )), χ˜ ι Wτ = Trace( F | Hc u (Bua )), χ˜ ι Wτ 2β

where the second equality is by 13.2.6(i). Assume now that the characteristic is good. If Cι is not the class E8 (b6 ), it results from the fact that u is split that F acts as q βu τ1 ⊗ I on the φ-isotypic 2β component Hc u (Bu )φ – see the definition of the almost preferred extension. Thus it results from the definitions that F acts as q βu τ1 ⊗ φ(a) on H 2βu (Bua )φ . It follows that R χ˜ ι (ua ) is equal to q βu φ(a). A variation of the same computation gives the result for the class E8 (b6 ). In bad characteristic a similar argument shows that R χ˜ ι (ua ) differs by a root of unity from q βu φ(a) and then the fact that R χ˜ ι (ua ) is a rational number shows that this root of unity is a sign. This sign has been shown to be 1 by Shoji (2007) in classical groups; it has been determined in most of the remaining cases by Geck (2019); it is still unknown for the class mentioned in type E8 in bad characteristic. 

13.3 The Lusztig–Shoji Algorithm We define a preorder on I by the rule ι ≤ ι if Cι ⊂ Cι (this is a partial order on unipotent classes). We write ι ∼ ι for equivalent elements for this preorder, which happens exactly when Cι = Cι . We define a matrix Ω by Ωι,ι := R χ˜ ι χp , R χ˜ ι χp GF . The following lemma shows that Ω can be computed just from information about the characters of W. Ωι,ι = |GF |p−1 ˜ ι ⊗ χ˜ ι )(q). Feg(sgn ⊗ χ  G , 13.2.2 and the fact ˜ Proof By the definition R χ˜ ι χp = |W | −1 w∈W χ(wτ)Q Tw that the almost preferred extensions take rational values, we get  χ˜ ι (wτ) χ˜ ι (wτ) , Ωι,ι = |W | −1 |TFw | w∈W

Lemma 13.3.1

13.3 The Lusztig–Shoji Algorithm

237

using also that W (Tw ) F  CW (wF). This gives the statement by the second  formula for Feg χ(q) ˜ = R χ˜ (1) in the proof of 13.1.14. Note that the above formula shows that Ω has rational entries. Finally we define entries Pι ,ι by the condition that the decomposition of  R χ˜ ι · χp in the basis Yι is ι ∈I Pι ,ι Yι ; thus computing Green functions is equivalent to knowing the Pι ,ι . Theorem 13.3.2 (Lusztig–Shoji algorithm) (i) Pι ,ι = 0 unless ι ∈ I0 . (ii) We have the matrix equation tPΛP = Ω where P = {Pι,ι }ι,ι ∈I0 and where for ι, ι ∈ I0 we set Λι,ι := Yι , Yι GF . This equation, given Ω, determines uniquely the matrices Λ and P subject to the conditions: • Pι,ι = 0 unless ι < ι or ι = ι ; in this last case Pι,ι = ε ι q β ι where ε ι is as in 13.2.10(iii). • Λι,ι = 0 unless ι ∼ ι ; Λ is invertible. (iii) The matrices P and Λ have rational entries. Condition (i) is by definition and condition (ii) on P reflects 13.2.10(ii) and (iii) and the conditions on Λ hold by definition – it is invertible as the matrix of scalar products between linearly independent functions. Proof Let us write Qι for R χ˜ ι · χp . By Lusztig (1986b, 24.2.9) one can extend ˜ indexed by the whole of I by identifying (up to a scalar) the P to a matrix P, functions {Qι }ι ∈I0 with “characteristic functions of sheaves” IC(C, φ) which are indexed by pairs (C, φ) ∈ I0 and using the fact that the sheaves IC(C, φ) make sense for any ι ∈ I to define functions Qι for any ι ∈ I. We define P˜ ι,ι  by saying that the decomposition of Qι in the basis Yι is ι ∈I P˜ ι ,ι Yι . ˜ ι,ι )ι,ι ∈I = Yι , Yι GF and If we extend Λ and Ω to the whole of I by (Λ ˜ (Ωι,ι )ι,ι ∈I = Qι , Qι GF , then by the definitions we have the matrix equation tP ˜ P˜ = Ω. ˜ ˜Λ Let χ I0 : I → {0, 1} be the characteristic function of I0 . The facts we need from Lusztig (1986b, §24) are:

˜ ι,ι = 0 if χ I (ι)  χ I (ι ). (a) Ω 0 0 (b) The function Qι is supported by Cι and the restriction of Qι to Cι is equal up to a scalar to Yι . (a) is from Lusztig (1986b, 24.3.6); (b), which extends 13.2.10(ii) and (iii), results from the definitions. We prove now by induction on < that P and Λ can be determined from the matrix equation and that (c) P˜ ι,ι = 0 if χ I0 (ι)  χ I0 (ι ).

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˜ ι,ι = 0 if χ I (ι)  χ I (ι ). (d) Λ 0 0 (e) P and Λ are rational. Let us write the matrix equation by blocks indexed by classes, grouping all ˜ ˜ Λ ι with a given Cι to make blocks PC,C , ΛC,C and ΩC,C of the matrices P, ˜ note that by definition Λ ˜ is block-diagonal and that P˜ is upper blockand Ω; triangular if we order the blocks in the same way as the conjugacy classes; we get  t PC,C ΛC,C PC,C = ΩC ,C . (∗) C ≤C ,C ≤C

We will do a double induction, the first over C . Assume that we can determine PC,C0 and ΛC,C for C ≤ C0 < C and that they satisfy (c),(d),(e). We show by a second induction, over C that we can determine PC,C and ΛC ,C and they satisfy (c),(d),(e). Here “determine” is in the following sense: the matrices PC,C and ΛC ,C are themselves by blocks according to the value of χ I0 : we determine completely the value of the block where both indices are in I0 and show it satisfies (e), and we also show (c) and (d); but we do not determine completely (and don’t care about) the block where none of the two indices is in I0 . Assume that we have determined all PC0,C for C0 < C ≤ C ; we show that we can determine PC,C . Equation (∗) applied with C = C can be written  t t PC,C ΛC,C PC,C = ΩC ,C − PC0,C ΛC0,C0 PC0,C . C0 2. J. Algebra, 327, 506–526. Bonnafé, C., and Rouquier, R. 2006. On the irreducibility of Deligne–Lusztig varieties. C. R. Math. Acad. Sci. Paris, 343, 37–39. Borel, A. 1991. Linear Algebraic Groups. Second enlarged ed. Graduate Texts in Mathematics, no. 126. New York, Springer Verlag. Borel, A., and Tits, J. 1965. Groupes réductifs. Publ. Math. Inst. Hautes Études Sci., 27, 55–160. Borel, A., Carter, R., Curtis, C. W., Iwahori, N., Springer, T. A., and Steinberg, R. 1970. Seminar on Algebraic Groups and Related Topics. Lecture Notes in Mathematics, no. 131. New York, Springer Verlag. Borho, W., and MacPherson, R. 1981. Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes. C. R. Math. Acad. Sci. Paris, 292, 707–710. Bourbaki, N. 1968. Groupes et algèbres de Lie, chapters IV, V, VI. Hermann, Paris. Bourbaki, N. 1971. Algèbre, chapters 1–3. Hermann/Addison-Wesley, Paris.

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Index

An (Weyl group), 24 An (root system), 28 2 A , 71 n absolutely semi-simple algebra, 101 a χ , 242 ad g, 5 additive group, 6 adjoint group, 36 Adjoint root datum, 36 affine Weyl group, 170 almost preferred extension, 234 arithmetic Frobenius endomorphism, 60 A(u), 232 automorphism of a root system, 38 Bn (Weyl group), 24 Bn (root system), 29 basis (of a root system), 26 (B, N)-pair, 39 (B, N)-pair (relative), 78 Borel subgroups, 11 Borel subgroups for a (B, N)-pair, 41 braid relations, 19 Bruhat cells, 45 Bruhat decomposition,40 Bruhat decomposition (relative), 44 Bruhat–Chevalley order, 45 Bx , 201 β u , 232 Cn (Weyl group), 24 Cn (root system), 28 C(G), 86 C(H)p , 124 character group, 8 character sheaves, 246 χ p , 124

CK (G), 86 closed root subset, 45 coinvariant algebra, 227 complete flags, 15 coroot, 25 Coxeter generating set, 20 Coxeter group, 20 Coxeter system, 19 crystallographic root datum, 36 crystallographic root system, 25 cuspidal module, 88 cuspidal pair, 88 Dn (Weyl group), 24 Dn (root system), 29 2 D , 71 n 3 D , 71 4 defined over k0 , 59 defined over Fq , 63 Deligne–Lusztig character, 145 Deligne–Lusztig character for disconnected groups, 183 Der(G), 14 descent of scalars, 233 diagonal automorphisms, 184 diagonalisable group, 8 discrete series, 90 dominance order, 239 Drinfeld double, 243 dsG , 157 dual algebraic group, 165 dual isogeny, 165 duality (of representations), 116 E6, E7, E8 (Weyl groups), 24 2 E , 71 6 E (GF , (s)), 170 ε G , 114

255

256

elementary matrix, 15 ε T , 113 extended basis, 169 extended Coxeter diagram, 170 F4 (Weyl group), 24 F-conjugacy classes, 65 F-rank of a torus, 113 F-rank of an algebraic group, 114 fake degree, 228 Feg χ, 228 Feg χ, ˜ 228 finite group of Lie type, 64 flag variety, 15 F on X(T), 63 F+q , 207 Fq -rank of a torus, 113 Frobenius endomorphism, 61 Frobenius root, 64 fundamental invariants, 225 G2 (Weyl group), 24 G2 (root system), 45 Ga , 6 ∗ G∗F -Lusztig series, 214 Gelfand–Graev representation, 208 general linear group, 6 generalised reflection degrees, 228 generated algebraic subgroup, 7 generating reflections, 20 generic degree, 108 geometric conjugacy class, 67 geometric Frobenius endomorphism, 60 geometrically conjugate, 67 geometrically conjugate characters, 161 GLn (k), 6 Gm , 6 good characteristic, 203 Green function, 148 Gu , 11 Harish-Chandra induction, 81 Harish-Chandra restriction, 81 Harish-Chandra series, 88 Hc∗ (X), 131 highest root, 169 hyperoctahedral group, 29 I, 235 I0 , 235 I(Bu ), 232 identity component, 7 1G , 86 inflation, 81 I-reduced-J, 43

Index

I-reduced, 42 Irr(G), 86 Irr(H), 88 Irr(A(u)) 0 , 233 irreducible Coxeter group, 24 irreducible root system, 25 isogeny, 37 Iwahori–Hecke algebra, 98 Iwahori–Hecke algebra (generic), 103 Jordan decomposition, 7 k0 -structure, 59 Lang map, 65 Lefschetz number, 131 length on W, 19 Levi complement, 49 Levi decomposition, 49 Levi subgroup, 49 Levi subgroup of G, 55 linear algebraic group, 7 Lusztig family of characters, 243 Lusztig family of unipotent characters, 243 Lusztig Fourier transform, 244 Lusztig functor, 138 Lusztig quotient, 243 Lusztig series, 170 l(w), 19 Mackey formula, 83 maximal tori for a (B, N)-pair, 41 maximally split torus, 115 μ n , 72 multiplicative group, 6 NFn /F , 143 NGF (L, Λ), 91 NI , 94 norm on a torus, 143 numerical invariant, 101 N(w), 20 On , 17 Ω, 170 one-parameter subgroup, 9 opp , 139 opposed Borel subgroup, 206 opposition automorphism, 37 orthogonal group, 17 parabolic subgroup, 12 parabolic subgroup of a Coxeter group, 42 parabolic subgroups of a (B, N)-pair, 44 parabolic root subset, 47 PGLn , 15 Φ+ , 25

Index

Φ− , 28 Π, 26 ˜ 169 Π, π xH , 159 p-morphism of root data, 37 positive subsystem, 26 principal series, 90 principal series unipotent, 111 pseudo-reflection, 225 PX (g), 226 , 148 QG L QG , 231 T quasi-closed root subsystem, 48 quasi-isolated semi-simple element, 176 quasi-projective variety, 7 quasi-simple group, 32 quasi-split torus, 69 r(G), 116 R(G), 14 radical, 14 rank (of a root system), 26 rank (of a torus), 8 rank (of an algebraic group), 13 rational character, 8 rational over k0 , 59 reduced expression, 19 reduced root datum, 36 reduced root system, 25 reduced-I, 42 reductive group, 14 Ree groups, 74 Ref(W) for a Coxeter group, 20 Ref(W) for a linear group, 225 reflection, 24 reflection degrees, 225 reflections (of a Coxeter group), 20 regG , 86 regular character, 214 regular element, 196 regular linear character, 207 relative position of flags, 42 R χ , R χ˜ , 188 RG , 86, 145 L ∗ RG , 86, 145 L RG , 81, 138 L⊂P ∗ RG , 81, 138 L⊂P root, 25 root datum, 36 root datum of a reductive group, 35 root system, 25 roots of G relative to T, 29

257

RG (s), 167 T∗ (θ), 145 RG T Ru (G), 14 scalar product (on a symmetric algebra), 106 scalar product on a group, 86 Schur element, 106 semi-simple F-rank, 116 semi-simple character, 214 semi-simple element, 7 semi-simple group, 14 semi-simple rank, 87 Semi-simple root datum, 36 Simp(G), 88 Simp(H), 88 Simp(GF | (L, Λ)), 88 simply connected group, 36 Simply connected root datum, 36 SLn , 15 SOn , 18 Sp2n , 16 special character of Weyl group, 243 special linear group, 15 special orthogonal group, 17, 18 special unipotent class, 243 split Frobenius endomorphism, 70 split reductive group, 70 split torus, 69 split unipotent element, 234 Springer correspondence, 233 Springer fibre, 201 standard Levi subgroup, 51 standard parabolic subgroup, 51 standard parabolic subgroup of a Coxeter group, 42 Steinberg character, 126 Steinberg morphisms, 65 Suzuki group, 74 SV, 225 SVW , 227 symmetric algebra of a vector space, 225 symmetric root subset, 45 symplectic bilinear form, 16 symplectic group, 16 Tits’ simple group, 74 torus, 8 trace-special Iwahori–Hecke algebra representation, 107 type of a maximal torus, 70 Uα , 29 uα , 31 uniform functions, 153, 186

258

unipotent characters, 172 unipotent element, 7 unipotent group, 13 unipotent radical, 14 unipotent support, 243 unitary group, 71 UΨ , 50 Uz , 206 W, 31 Weyl group, 24 Weyl group of a (B, N)-pair, 39 Weyl group of a reductive group, 29 WGF (L), 94

Index

WGF (L, Λ), 91 WG (T), 30 W 0 (s), 57 W (s), 57 W (T), 30 ˜ 188 W, XB , 172 X(G), 8 XU , 137 Yι , 235 Y (T), 9 ˆ 60 Z,