The Character Theory of Finite Groups of Lie Type: A Guided Tour (Cambridge Studies in Advanced Mathematics, Band 187) 1108489621, 9781108489621

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C A M B R I D G E S T U D I E S I N A DVA N C E D M AT H E M AT I C S 1 8 7 Editorial Board ´ S , W. F U LTO N , F. K I RWA N , B. BOLLOBA P. S A R NA K , B . S I M O N , B . TOTA RO

THE CHARACTER THEORY OF FINITE GROUPS OF LIE TYPE Through the fundamental work of Deligne and Lusztig in the 1970s, further developed mainly by Lusztig, the character theory of reductive groups over finite fields has grown into a rich and vast area of mathematics. It incorporates tools and methods from algebraic geometry, topology, combinatorics and computer algebra, and has since evolved substantially. With this book, the authors meet the need for a contemporary treatment, complementing in core areas the well-established books of Carter and Digne–Michel. Focusing on applications in finite group theory, the authors gather previously scattered results and allow the reader to get to grips with the large body of literature available on the subject, covering topics such as regular embeddings, the Jordan decomposition of characters, d-Harish-Chandra theory and Lusztig induction for unipotent characters. Requiring only a modest background in algebraic geometry, this useful reference is suitable for beginning graduate students as well as researchers. Meinolf Geck is Professor of Algebra at the University of Stuttgart. He works in the areas of algebraic groups and representation theory of finite groups and has (co-)authored books including An Introduction to Algebraic Geometry and Algebraic Groups (2003), Representations of Hecke Algebras at Roots of Unity (2011) and Representations of Reductive Groups (1998). Gunter Malle is Professor of Mathematics at the University of Kaiserslautern. He works in the area of representation theory of finite groups and he co-authored Linear Algebraic Groups and Finite Groups of Lie Type (2011) and Inverse Galois Theory (1999). Professor Malle received an ERC Advanced Grant on ‘Counting conjectures’ in 2012.

C A M B R I D G E S T U D I E S I N A DVA N C E D M AT H E M AT I C S Editorial Board B. Bollob´as, W. Fulton, F. Kirwan, P. Sarnak, B. Simon, B. Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing, visit www.cambridge.org/mathematics. Already Published 148 S. R. Garcia, J. Mashreghi & W. T. Ross Introduction to Model Spaces and Their Operators 149 C. Godsil & K. Meagher Erd˝os–Ko–Rado Theorems: Algebraic Approaches 150 P. Mattila Fourier Analysis and Hausdorff Dimension 151 M. Viana & K. Oliveira Foundations of Ergodic Theory 152 V. I. Paulsen & M. Raghupathi An Introduction to the Theory of Reproducing Kernel Hilbert Spaces 153 R. Beals & R. Wong Special Functions and Orthogonal Polynomials 154 V. Jurdjevic Optimal Control and Geometry: Integrable Systems 155 G. Pisier Martingales in Banach Spaces 156 C. T. C. Wall Differential Topology 157 J. C. Robinson, J. L. Rodrigo & W. Sadowski The Three-Dimensional Navier–Stokes Equations 158 D. Huybrechts Lectures on K3 Surfaces 159 H. Matsumoto & S. Taniguchi Stochastic Analysis 160 A. Borodin & G. Olshanski Representations of the Infinite Symmetric Group 161 P. Webb Finite Group Representations for the Pure Mathematician 162 C. J. Bishop & Y. Peres Fractals in Probability and Analysis 163 A. Bovier Gaussian Processes on Trees 164 P. Schneider Galois Representations and (ϕ,  )-Modules 165 P. Gille & T. Szamuely Central Simple Algebras and Galois Cohomology (2nd Edition) 166 D. Li & H. Queffelec Introduction to Banach Spaces, I 167 D. Li & H. Queffelec Introduction to Banach Spaces, II 168 J. Carlson, S. M¨uller-Stach & C. Peters Period Mappings and Period Domains (2nd Edition) 169 J. M. Landsberg Geometry and Complexity Theory 170 J. S. Milne Algebraic Groups 171 J. Gough & J. Kupsch Quantum Fields and Processes 172 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Discrete Harmonic Analysis 173 P. Garrett Modern Analysis of Automorphic Forms by Example, I 174 P. Garrett Modern Analysis of Automorphic Forms by Example, II 175 G. Navarro Character Theory and the McKay Conjecture 176 P. Fleig, H. P. A. Gustafsson, A. Kleinschmidt & D. Persson Eisenstein Series and Automorphic Representations 177 E. Peterson Formal Geometry and Bordism Operators 178 A. Ogus Lectures on Logarithmic Algebraic Geometry 179 N. Nikolski Hardy Spaces 180 D.-C. Cisinski Higher Categories and Homotopical Algebra 181 A. Agrachev, D. Barilari & U. Boscain A Comprehensive Introduction to Sub-Riemannian Geometry 182 N. Nikolski Toeplitz Matrices and Operators 183 A. Yekutieli Derived Categories 184 C. Demeter Fourier Restriction, Decoupling and Applications 185 D. Barnes & C. Roitzheim Foundations of Stable Homotopy Theory 186 V. Vasyunin & A. Volberg The Bellman Function Technique in Harmonic Analysis 187 M. Geck & G. Malle The Character Theory of Finite Groups of Lie Type

The Character Theory of Finite Groups of Lie Type A Guided Tour MEINOLF GECK Universit¨at Stuttgart

GUNTER MALLE Technische Universit¨at Kaiserslautern, Germany

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/9781108489621 DOI: 10.1017/9781108779081 © Meinolf Geck and Gunter Malle 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 2020 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library ISBN 978-1-108-48962-1 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface

page vii

1

Reductive Groups and Steinberg Maps 1.1 Affine Varieties and Algebraic Groups 1.2 Root Data 1.3 Chevalley’s Classification Theorems 1.4 Frobenius Maps and Steinberg Maps 1.5 Working with Isogenies and Root Data; Examples 1.6 Generic Finite Reductive Groups 1.7 Regular Embeddings

2

Lusztig’s Classification of Irreducible Characters 2.1 Generalities about Character Tables 2.2 The Virtual Characters of Deligne and Lusztig 2.3 Unipotent Characters and Degree Polynomials 2.4 Towards Lusztig’s Main Theorem 4.23 2.5 Geometric Conjugacy and the Dual Group 2.6 The Jordan Decomposition of Characters 2.7 Average Values and Unipotent Support 2.8 On the Values of Green Functions

92 93 105 120 134 152 167 181 195

3

Harish-Chandra Theories 3.1 Harish-Chandra Theory for BN-Pairs 3.2 Harish-Chandra Theory for Groups of Lie Type 3.3 Lusztig Induction and Restriction 3.4 Duality and the Steinberg Character 3.5 d-Harish-Chandra Theories

211 212 225 236 248 257

4

Unipotent Characters 4.1 Characters of Weyl Groups 4.2 Families of Unipotent Characters and Fourier Matrices

271 272 285

v

1 2 15 27 40 53 68 80

vi

Contents 4.3 4.4 4.5 4.6 4.7 4.8

Unipotent Characters in Type A Unipotent Characters in Classical Types Unipotent Characters in Exceptional Types Decomposition of RLG and d-Harish-Chandra Series On Lusztig’s Jordan Decomposition Disconnected Groups, Groups with Disconnected Centre

Appendix Further Reading and Open Questions References Index

297 301 317 326 343 351 363 371 390

Preface

The subject of this book is the character theory of finite groups of Lie type (or finite reductive groups), following the geometric approach initiated by the fundamental work of Deligne and Lusztig [Lu75], [DeLu76] in the 1970s. Since then, and to its full extent mainly by the monumental work of Lusztig, this has grown into an extremely rich, complex and vast theory, incorporating tools and methods from algebraic geometry, topology, combinatorics and computer algebra. (Lusztig’s papers since 1975 on this subject alone comprise already a few thousand densely written pages.) One of the ultimate aims of this theory is to reduce the computation of character tables of whole series of finite groups of Lie type (e.g., the series of groups E8 (q) where q is any prime power) to purely combinatorial tasks that could, for example, be performed automatically on a computer. For the general linear groups GLn (q) this was already achieved in principle by Green [Gre55] in 1955, but the analogous problem for the closely related special linear and unitary groups SLn (q), SUn (q) is still not completely solved. Within finite group theory, the importance of this subject is highlighted by the classification of finite simple groups: apart from the alternating groups and the 26 sporadic simple groups, all non-abelian finite simple groups are ‘of Lie type’. According to Aschbacher [Asch00], [Asch04] when faced with a problem about finite groups, it nowadays seems best to attempt to reduce the problem or a related problem to a question about simple groups or groups closely related to simple groups. The classification then supplies an explicit list of groups which can be studied in detail using the effective description of the groups. In recent years, this programme has led to substantial advances on various long-standing open problems in the representation theory of finite groups: these were shown to reduce to questions on simple groups which could then be solved by applying the deep results on characters of finite reductive groups; see, for example, the book of Navarro [Na18] and the second author’s survey [Ma17]. Lusztig’s book [Lu84a] is a milestone in the study of representations of finite vii

viii

Preface

groups of Lie type, both in terms of conceptual depth and technical complexity. It brings together various deep and rich theories, culminating in the fundamental ‘Jordan decomposition of characters’. In particular, this provides a classification of the irreducible characters, and formulae for character degrees, in terms of purely combinatorial data. The books by Carter [Ca85] and Digne–Michel [DiMi20] provide more background material and have become influential and highly useful references in this area. Further, more recent texts dealing with more specific aspects are the books by Bonnafé [Bo06] and by Cabanes–Enguehard [CE04]. In our book, the primary focus is on explaining Lusztig’s classification of the irreducible characters, and surrounding topics like complete root data, regular embeddings, degree and character formulae, Lusztig induction and restriction and Jordan decomposition. Thus, we will complement, enhance and go beyond the above texts in several ways: • A substantial part of Chapter 1 is concerned with the discussion of explicit constructions involving root data and Steinberg maps; these are at the basis of efficient algorithms and computer implementations for which the CHEVIE system [GHLMP], [MiChv] is our primary reference. • Chapter 2 provides an introduction (with complete proofs where possible) into the basic formalism of Lusztig’s book [Lu84a] leading to the statement of the ‘Jordan decomposition of characters’, both in the ‘connected centre’ case and in general. We also discuss the computation of Green functions, a problem which is not yet solved in complete generality. • In Chapter 3 we present not only the well-established ordinary Harish-Chandra theory but also d-Harish-Chandra theories defined by means of Lusztig induction, which have proved to be of fundamental importance in block theory of finite reductive groups. • The classification of the all-important unipotent characters of finite reductive groups and various of their properties are described in Chapter 4, including a discussion of Lusztig’s ‘non-abelian Fourier transforms’. We also discuss the decomposition of the Lusztig functor and its commutation with Jordan decomposition for groups with connected centre. Furthermore, we touch upon some topics in the character theory of disconnected reductive groups. • An appendix discusses, in a somewhat informal way, various applications, open problems and connections to related theories, with numerous references to further reading. Throughout, we have tried to design the exposition of the above material with a view towards applications in finite group theory, and to be accessible to a reader with only a modest background in algebraic geometry. In view of the enormous amount of material available in this area, it is clear that we had to make a number

Preface

ix

of choices concerning the topics that we cover in this book. For example, we have decided not to say anything about the classification of the conjugacy classes of finite groups of Lie type (except for some occasional general statements). Furthermore, as a general rule, we only give proofs for statements for which we could not find any convenient reference in the existing literature. (But sometimes we defer from this rule and give a detailed argument when this appears to be a good illustration for the methods developed so far.) On the other hand, we have made a serious attempt to provide precise references, thereby giving something like a guided tour through this vast territory. In short, we hope that this text will be a useful addition to the literature on the character theory of finite groups of Lie type, where the choice of topics and the style of exposition have been strongly influenced, of course, by our own experience with the sometimes difficult task of finding appropriate references, or accommodating the existing literature to specific needs in applications. We are indebted to Marc Cabanes, Bill Casselman, David Craven, Olivier Dudas, Zhicheng Feng, Skip Garibaldi, Jonas Hetz, Jim Humphreys, Radha Kessar, Emil Rotilio, Lucas Ruhstorfer, Jay Taylor for comments on earlier versions. We thank George Lusztig for his interest in this project and for a number of useful conversations about various topics related to it. Kaiserslautern and Stuttgart, October 2019

1 Reductive Groups and Steinberg Maps

This first chapter is of a preparatory nature; its purpose is to collect some basic results about algebraic groups (with proofs where appropriate) which will be needed for the discussion of characters and applications in later chapters. In particular, one of our aims is to arrive at the point where we can give a precise definition of a ‘series of finite groups of Lie type’ {G(q)}, indexed by a parameter q. We also introduce a number of tools which will be helpful in the discussion of examples. For a reader familiar with the basic notions about algebraic groups, root data and Frobenius maps, it might just be sufficient to browse through this chapter on a first reading, in order to see some of our notation. There are, however, a few topics and results that are frequently used in the literature on algebraic groups and finite groups of Lie type, but for which we have found the coverage in standard reference texts (like [Bor91], [Ca85], [DiMi20], [Hum91], [Spr98]) not to be sufficient; these will be treated here in a fairly self-contained manner. Section 1.1 is purely expository: it introduces affine varieties, linear algebraic groups in general, and the first definitions concerning reductive algebraic groups. In Section 1.2, we consider in some detail (abstract) root data, the basic underlying combinatorial structure of the theory of reductive algebraic groups. We present an approach (familiar in the literature concerned with computational aspects, e.g. [CMT04], [BrLu12]) in which root data simply appear as factorisations of the Cartan matrix of a root system. This will be extremely useful for the discussion of examples and the efficient construction of root data from Cartan matrices. Section 1.3 contains the fundamental existence and isomorphism theorems of Chevalley [Ch55], [Ch05] concerning connected reductive algebraic groups. We also state the more general ‘isogeny theorem’ and present some of its basic applications. (There is now a quite short proof available, due to Steinberg [St99].) An important class of homomorphisms of algebraic groups to which this more general theorem applies are the Steinberg maps, to be discussed in detail in Section 1.4. Following [St68], one might just define a Steinberg map of a connected reductive 1

2

Reductive Groups and Steinberg Maps

algebraic group G to be an endomorphism whose fixed point set is finite. But it will be important and convenient to single out a certain subclass of such morphisms to which one can naturally attach a positive real number q (some power of which is a prime power) and such that one can speak of the corresponding finite group G(q). The known results on Frobenius and Steinberg maps are somewhat scattered in the literature so we treat this in some detail here, with complete proofs. In Section 1.5, we illustrate the material developed so far by a number of further basic constructions and examples. In Section 1.6, we show how all this leads to the notion of ‘generic’ reductive groups, in which q will appear as a formal parameter. Finally, Section 1.7 discusses in some detail the first applications to the character theory of finite groups of Lie type: the ‘Multiplicity–Freeness’ Theorem 1.7.15.

1.1 Affine Varieties and Algebraic Groups In this section, we introduce some basic notions concerning affine varieties and algebraic groups. We will do this in a somewhat informal way, assuming that the reader is willing to fill in some details from textbooks like [Bor91], [Ca85], [Ge03a], [Hum91], [MaTe11], [Spr98]. 1.1.1 Affine varieties. Let k be a field and let X be a set. Let A be a subalgebra of the k-algebra A (X, k) of all functions f : X → k. Using A we can try to define a topology on X: a subset X  ⊆ X is called closed if there is a subset S ⊆ A such that X = {x ∈ X | f (x) = 0 for all f ∈ S}. This works well, and gives rise to the Zariski topology on X, if A is neither too small nor too big. The precise requirements are (see [Car55-56]): (1) A is a finitely generated k-algebra and contains the identity of A (X, k); (2) A separates points, that is, given x  x  in X, there exist some f ∈ A such that f (x)  f (x ); (3) any k-algebra homomorphism λ : A → k is given by evaluation at a point (that is, there exists some x ∈ X such that λ( f ) = f (x) for all f ∈ A). A pair (X, A) satisfying the above conditions will be called an affine variety over k; the functions in A are called the regular functions on X. We define dim X to be the supremum of all r  0 such that there exist r algebraically independent elements in A. Since A is finitely generated, dim X < ∞. (See [Ge03a, 1.2.18].) If A is an integral domain, then X is called irreducible. There is now also a natural notion of morphisms. Let (X, A) and (Y, B) be affine varieties over k. A map ϕ : X → Y will be called a morphism if composition with ϕ maps B into A (that is, for all g ∈ B, we have ϕ∗ (g) := g ◦ ϕ ∈ A); in this

1.1 Affine Varieties and Algebraic Groups

3

case, ϕ∗ : B → A is an algebra homomorphism, and every algebra homomorphism B → A arises in this way. The morphism ϕ is an isomorphism if there is a morphism ψ : Y → X such that ψ ◦ ϕ = idX and ϕ◦ψ = idY . (Equivalently: the induced algebra homomorphism ϕ∗ : B → A is an isomorphism.) Starting with these definitions, the basics of (affine) algebraic geometry are developed in [St74], and this is also the approach taken in [Ge03a]. The link with the more traditional approach via closed subsets in affine space (which, when considered as an algebraic set with the Zariski topology, we denote by kn ) is obtained as follows. Let (X, A) be an affine variety over k. Choose a set {a1, . . . , an } of algebra generators of A and consider the polynomial ring k[t1, . . . , tn ] in n independent indeterminates t1, . . . , tn . There is a unique algebra homomorphism π : k[t1, . . . , tn ] → A such that π(ti ) = ai for 1  i  n. Then we have a morphism ϕ : X → kn,

x → (a1 (x), . . . , an (x)),

such that ϕ∗ = π. The image of ϕ is the ‘Zariski closed’ set of kn consisting of all (x1, . . . , xn ) ∈ kn such that f (x1, . . . , xn ) = 0 for all f ∈ ker(π). To develop these matters any further, it is then essential to assume that k is algebraically closed, which we will do from now on. One can go a long way towards those parts of the theory that are relevant for algebraic groups, once the following basic result about morphisms is available (see [St74, §1.13], [Ge03a, §2.2]): Let ϕ : X → Y be a morphism between irreducible affine varieties such that ϕ(X) is dense in Y. Then there is a non-empty open subset V ⊆ Y such that V ⊆ ϕ(X) and, for all y ∈ V, we have dim ϕ−1 (y) = dim X − dim Y. 1.1.2 Algebraic groups. In order to define algebraic groups, we need to know that direct products of affine varieties are again affine varieties. So let (X, A) and (Y, B) be affine varieties over k. Given f ∈ A and g ∈ B, we define the function f ⊗ g : X × Y → k by (x, y) → f (x)g(y). Let A ⊗ B be the subspace of A (X × Y, k) spanned by all f ⊗ g, where f ∈ A and g ∈ B. Then A ⊗ B is a subalgebra of A (X × Y, k) (isomorphic to the tensor product of A, B over k) and the pair (X × Y, A ⊗ B) is easily seen to be an affine variety over k. Now let (G, A) be an affine variety and assume that G is an abstract group where multiplication and inversion are defined by maps μ : G × G → G and ι : G → G. We say that G is an affine algebraic group if μ and ι are morphisms. The first example is the additive group of k which, when considered as an algebraic group, we denote by k+ (with algebra of regular functions given by the polynomial functions k → k). Most importantly, the group GLn (k) (n  1), is an affine algebraic group, with algebra of regular functions given as follows. For 1  i, j  n let fi j : GLn (k) → k be the function that sends a matrix g ∈ GLn (k) to its (i, j)-entry; furthermore, let δ : GLn (k) → k, g → det(g)−1 . Then the algebra of regular functions on GLn (k)

4

Reductive Groups and Steinberg Maps

is the subalgebra of A (GLn (k), k) generated by δ and all fi j (1  i, j  n). In particular, GL1 (k) is an affine algebraic group, which we denote by k× . It is a basic fact that any affine algebraic group G over k is isomorphic to a closed subgroup of GLn (k), for some n  1; see [Ge03a, 2.4.4]. For this reason, an affine algebraic group is also called a linear algebraic group. When we just write ‘algebraic group’, we always mean a linear algebraic group. 1.1.3 Connected algebraic groups. A topological space is connected if it cannot be written as a disjoint union of two non-empty open subsets. A linear algebraic group G can always be written as the disjoint union of finitely many connected components, where the component containing the identity element is a closed connected normal subgroup of G, denoted by G◦ ; see [Ge03a, 1.3.13]. Thus, G is connected if and only if G = G◦ . (Equivalently: G is irreducible as an affine variety; see [Ge03a, 1.1.12, 1.3.1].) What is the significance of this fundamental notion? Every finite group G can be regarded as a linear algebraic group, with algebra of regular functions given by all of A (G, k). Thus, the study of all linear algebraic groups is necessarily more complicated than the study of all finite groups. But, as Vogan [Vo07] writes, “a miracle happens” when we consider connected algebraic groups: things actually become much less complicated. One reason is that a connected algebraic group is almost completely determined by its Lie algebra (see 1.1.5 and also 1.1.12 below), and the latter can be studied using linear algebra methods. Combined with our assumption that k is algebraically closed, this gives us some powerful tools. For example, matrices over algebraically closed fields can be put in triangular form. An analogue of this fact for an arbitrary connected algebraic group is the statement that every element is contained in a Borel subgroup (that is, a maximal closed connected solvable subgroup); see [Ge03a, 3.4.9]. A useful criterion for showing the connectedness of a subgroup of G is as follows. Let {Hi }i ∈I be a family of closed connected subgroups of G. Then the (abstract) subgroup H = Hi | i ∈ I ⊆ G generated by this family is closed and connected; furthermore, H = Hi1 · · · Hin for some n and i1, . . . , in ∈ I. The proof uses the result on morphisms mentioned at the end of 1.1.1; see, e.g., [Ge03a, 2.4.6]. Note that, if U, V are any closed subgroups of G, then the abstract subgroup U, V ⊆ G need not even be closed. For example, if G = SL2 (C), then it is well known that the subgroup SL2 (Z) is generated by two elements, of orders 4 and 6, but this subgroup is certainly not closed in G. However, if V is normalised by U, then U, V = U.V is closed; see [Ch05, §3.3, Corollaire]. We will use without further special mention some standard facts (whose proofs also rely on the above-mentioned result on morphisms). For example, if f : G → G is a homomorphism of linear algebraic groups, then the image f (G) is a closed

1.1 Affine Varieties and Algebraic Groups

5

subgroup of G (connected if G is connected), the kernel of f is a closed subgroup of G and we have dim G = dim ker( f ) + dim f (G). (See, e.g., [Ge03a, 2.2.14].) 1.1.4 Classical groups. These form an important class of examples of linear algebraic groups. They are closed subgroups of GLn (k) defined by certain quadratic polynomials corresponding to a bilinear or quadratic form on the underlying vector space k n . There is an extensive literature on these groups; see, e.g., [Bou07], [Dieu74], [Gro02], [Tay92]. Since our base field k is algebraically closed, the general theory simplifies considerably and we only need to consider three classes of groups, leading to the Dynkin types B, C, D. First, and quite generally, for any invertible matrix Q n ∈ Mn (k), we obtain a linear algebraic group Γ(Q n, k) := { A ∈ Mn (k) | Atr Q n A = Q n }; note that det(A) = ±1 for all A ∈ Γ(Q n, k). Let us now take Q n of the form ⎡ ⎢ ⎢ ⎢ Q n = ⎢⎢ ⎢ ⎢ ⎢ ⎣

0 .. .

··· . ..

0 . ..

0 ±1

±1 0

. .. ···

±1 ⎤⎥ ⎥ 0 ⎥⎥ .. ⎥⎥ ∈ Mn (k) . ⎥ 0 ⎥⎦

(n  2)

where the signs are such that Qtrn = ±Q n . Then Q n is the matrix of a non-degenerate tr symmetric or alternating bilinear form on k n ; furthermore, Q−1 n = Q n and Γ(Q n, k) will be invariant under transposing matrices. If Qtrn = −Q n and n is even, then Γ(Q n, k) will be denoted Spn (k) and called the symplectic group. This group is always connected; see [Ge03a, 1.7.4]. Now assume that Qtrn = Q n and that all signs in Q n are +. Then we also consider the quadratic form on k n defined by the polynomial  2 if n = 2m + 1 is odd, t1 t2m+1 + t2 t2m + . . . + tm tm+2 + tm+1 fn := t1 t2m + t2 t2m−2 + . . . + tm tm+1 if n = 2m is even, (where t1, . . . , tn are indeterminates). This defines a function f n : k n → k, where we regard the elements of k n as column vectors. Using the notation in [MaTe11, §1.2], the general orthogonal group is defined as GOn (k) := { A ∈ Mn (k) | f n (Av) = f n (v) for all v ∈ k n }; furthermore, SOn (k) := GO◦n (k) will be called the special orthogonal group. In each case, we have [GOn (k) : SOn (k)]  2; see [Ge03a, §1.7], [Gro02] for further details. Note also that, if char(k)  2, then GOn (k) = Γ(Q n, k); furthermore, if n is even and char(k) = 2, then GOn (k) will be strictly contained in Γ(Q n, k). (See also Example 1.5.5 for the case where n is odd and char(k) = 2.)

6

Reductive Groups and Steinberg Maps

The particular choices of Q n and fn lead to simple descriptions of a BN-pair in Spn (k) and SOn (k); see, e.g., [Ge03a, §1.7] (and also 1.1.14 below). The Dynkin types and dimensions are given as follows. Group

Type

Dimension

SO2m+1 (k)

Bm

2m2 + m

Sp2m (k)

Cm

2m2 + m

SO2m (k)

Dm

2m2 − m

Later on, if there is no danger of confusion, we shall just write GLn , SLn , Spn , SOn , GOn instead of GLn (k), SLn (k), Spn (k), SOn (k), GOn (k), respectively. 1.1.5 Tangent spaces and the Lie algebra. Let (X, A) be an affine variety over k. Then the tangent space Tx (X) of X at a point x ∈ X is the set of all k-linear maps D : A → k such that D( f g) = f (x)D(g) + g(x)D( f ). (Such linear maps are called derivations.) Clearly, Tx (X) is a subspace of the vector space of all linear maps from A to k. Any D ∈ Tx (X) is uniquely determined by its values on a set of algebra generators of A. Hence, since A is finitely generated, we have dim Tx (X) < ∞. If X ⊆ X is a closed subvariety, we have a natural inclusion Tx (X) ⊆ Tx (X) for any x ∈ X. For example, we can identify Tx (kn ) with k n for all x ∈ kn and so, if X ⊆ kn is a Zariski closed subset, we have Tx (X) ⊆ k n for all x ∈ X (see [Ge03a, 1.4.10]). More generally, any morphism ϕ : X → Y between affine varieties over k naturally induces a linear map dx ϕ : Tx (X) → Tϕ(x) (Y) for any x ∈ X, called the differential of ϕ at x. (See [Ge03a, §1.4].) Now let G be a linear algebraic group and denote L(G) := T1 (G), the tangent space at the identity element of G. Then L(G) = L(G◦ )

and

dim G = dim L(G);

see [Ge03a, 1.5.2]. Furthermore, there is a Lie product [ , ] on L(G) which can be defined as follows. Consider a realisation of G as a closed subgroup of GLn (k) for some n  1. We have a natural isomorphism of L(GLn (k)) onto Mn (k), the vector space of all n × n-matrices over k; see [Ge03a, 1.4.14]. Hence we obtain an embedding L(G) ⊆ Mn (k) where Mn (k) is endowed with the usual Lie product [A, B] = AB − BA for A, B ∈ Mn (k). Then one shows that [L(G), L(G)] ⊆ L(G) and so [ , ] restricts to a Lie product on L(G); see [Ge03a, 1.5.3]. (Of course, there is also an intrinsic description of L(G) in terms of the algebra of regular functions on G which shows, in particular, that the product does not depend on the choice of the realisation of G; see [Ge03a, 1.5.4].) 1.1.6 Quotients. Let G be a linear algebraic group and H be a closed normal subgroup of G. We have the abstract factor group G/H and we would certainly like

1.1 Affine Varieties and Algebraic Groups

7

to know if this can also be viewed as an algebraic group. More generally, let X be an affine variety and H be a linear algebraic group such that we have a morphism H × X → X which defines an action of H on X. The question of whether we can view the set of orbits X/H as an algebraic variety leads to ‘geometric invariant theory’; in general, these are quite delicate matters. Let us begin by noting that there is a natural candidate for the algebra of functions on the orbit set X/H: If A is the algebra of regular functions on X, then AH := { f ∈ A | f (h.x) = f (x) for all h ∈ H and all x ∈ X} can naturally be regarded as an algebra of k-valued functions on X/H. However, the three properties in 1.1.1 will not be satisfied in general. There are two particular situations in which this is the case, and these will be sufficient for most parts of this book: • H is a finite group, and • X = G is an algebraic group and H is a closed normal subgroup (acting by left multiplication). (For the proofs, see [Fo69, 5.25] or [Ge03a, 2.5.12] in the first case, and [Fo69, 2.26] or [Spr98, §5.5] in the second case.) Now let us assume that (X/H, AH ) is an affine variety. Then, first of all, the natural map X → X/H is a morphism of affine varieties. Furthermore, we have the following universal property: If ϕ : X → Y is any morphism of affine varieties that is constant on the orbits of H on X, then there is a unique morphism ϕ¯ : X/H → Y such that ϕ is the composition of ϕ¯ and the natural map X → X/H. (Indeed, if B is the algebra of regular functions on Y, then the induced algebra homomorphism ϕ∗ : B → A has image in AH , hence it factors through an algebra homomorphism ϕ¯∗ : B → AH for a unique morphism ϕ¯ : X/H → Y.) For example, if we are in the second of the above two cases, then the universal property shows that the induced multiplication and inversion maps on G/H are morphisms of affine varieties. Thus, G/H is an affine algebraic group. 1.1.7 Algebraic groups in positive characteristic. The finite groups that we shall study in this book are obtained as GF := {g ∈ G | F(g) = g} where the F : G → G are certain bijective endomorphisms with finitely many fixed points, called ‘Steinberg maps’. (This will be discussed in detail in Section 1.4.) Such maps F will only exist if k has prime characteristic, so we will usually assume that p is a prime number and k = F p is an algebraic closure of the field F p = Z/pZ. Now, algebraic geometry over fields with positive characteristic is, in some respects,

8

Reductive Groups and Steinberg Maps

more tricky than algebraic geometry over C, say (because of the inseparability of certain field extensions; see also 1.1.8 below). However, some things are actually easier. For example, using an embedding of G into some GLn (k) as in 1.1.2, we see that every element g ∈ G has finite order. Thus, we can define g to be semisimple if the order of g is prime to p; we define g to be unipotent if the order of g is a power of p. Then, clearly, any g ∈ G has a unique decomposition g = us = su

where s ∈ G is semisimple and u ∈ G is unipotent,

called the Jordan decomposition of elements. (In characteristic 0, this certainly requires more work; see [Spr98, §2.4].) Another example: An algebraic group G is called a torus if G is isomorphic to a direct product of a finite number of copies of k× . Then G is a torus if and only if G is connected, abelian and consists entirely of elements of order prime to p; see [Ge03a, 3.1.9]. (To formulate this in characteristic 0, one would need the general definition of semisimple elements.) 1.1.8 Some things that go wrong in positive characteristic. Here we collect a few items which show that, when working over k = F p as above, things may not work as one might hope or expect. First note that a bijective homomorphism of algebraic groups ϕ : G1 → G2 need not be an isomorphism. A standard example is the Frobenius map k+ → k+ , x → x p . (Note that, over C, a bijective homomorphism between connected algebraic groups is an isomorphism; see [GoWa98, 11.1.16].) In general, we have (see, e.g., [Ge03a, 2.3.15], [Spr98, 5.3.3]): (a) A bijective homomorphism of linear algebraic groups ϕ : G1 → G2 is an isomorphism if and only if the differential d1 ϕ : T1 (G1 ) → T1 (G2 ) between the tangent spaces is an isomorphism. The next item concerns the Lie algebra of an algebraic group. Let G be a linear algebraic group and U, H be closed subgroups of G. As already noted in 1.1.5, we have natural inclusions of L(U), L(H) and L(U ∩ H) into L(G). It is always true that L(U ∩ H) ⊆ L(U) ∩ L(H). (b) When considering the intersection of closed subgroups U, H of an algebraic group G, it is not always true that L(U ∩ H) = L(U) ∩ L(H). A good example to keep in mind is as follows. Let G = GLn (F p ), H = SLn (F p ) and Z be the centre of G (the scalar matrices in G). Then Z, H are closed subgroups of G. As in 1.1.5, we can identify L(G) = Mn (k); then L(H) consists of all matrices of trace 0 and L(Z) consists of all scalar matrices. (For these facts see, for example, [Ge03a, §1.5].) Assume now that p divides n. Then, clearly, {0}  L(Z) ⊆ L(H), whereas Z ∩ H is finite and so L(Z ∩ H) = L((Z ∩ H)◦ ) = {0}. (This phenomenon cannot happen in characteristic 0; see [Bor91, 6.12] or [Hum91, 12.5].)

1.1 Affine Varieties and Algebraic Groups

9

Closely related to the above item is the next item: semidirect products. Let G be an algebraic group and U, N be closed subgroups such that N is normal, G = U.N and U ∩ N = {1}. Following [Bor91, 1.11], we say that G is the semidirect product (of algebraic groups) of U, N if the natural map U × N → G given by multiplication is an isomorphism of affine varieties. If this holds, we have an inverse isomorphism G → U × N and the first projection will induce an isomorphism of algebraic groups G/N  U. We have the criterion: (c) G is the semidirect product (of algebraic groups) of U, N if and only if L(U) ∩ L(N) = {0}. For example, this holds if U or N is finite. (This easily follows from (a) and the description of the differential of the product map U × N → G; see, e.g., [Ge03a, 1.5.6], [Spr98, 4.4.12].) Take again the above example where G = GLn (F p ), U = SLn (F p ) and N is the centre of G (the scalar matrices in G). Assume now that n = p. Then U, N are closed connected normal subgroups such that G = U.N and U ∩ N = {1}. However, {0}  L(N) ⊆ L(U) and so this is not a semidirect product of algebraic groups! For working with fixed points of groups under automorphisms as in 1.1.7, the following completely general result will be useful on several occasions. Lemma 1.1.9 ([St68, 4.5]) Let A, B be groups and f : A → B be a surjective homomorphism with ker( f ) ⊆ Z(A). Let σ : A → A and τ : B → B be automorphisms such that f ◦ σ = τ ◦ f . Then C := {a−1 σ(a) | a ∈ ker( f )} is a subgroup of ker( f ). Furthermore, let Aσ := {a ∈ A | σ(a) = a}

and

Bτ := {b ∈ B | τ(b) = b}.

Then f (Aσ ) is normal in Bτ and there is a canonical injective homomorphism δ : Bτ / f (Aσ ) → ker( f )/C, with image ({a−1 σ(a) | a ∈ A} ∩ ker( f ))/C. Proof Since ker( f ) ⊆ Z(A) and f ◦ σ = τ ◦ f , it is clear that σ(ker( f )) ⊆ ker( f ) and that C is a subgroup of ker( f ). We define δ  : Bτ → ker( f )/C as follows. Let b ∈ Bτ and choose a ∈ A such that f (a) = b. We have f (σ(a)) = τ( f (a)) = τ(b) = b and so c := a−1 σ(a) ∈ ker( f ). Then set δ (b) = cC ∈ ker( f )/C. One verifies that δ  is well defined and a group homomorphism; furthermore, ker(δ ) = f (Aσ ) and the image of δ  is as stated above. Thus, we obtain an induced map δ with the required properties.  1.1.10 The unipotent radical. Let G be a linear algebraic group over k = F p , where p is a prime. We can now define the unipotent radical Ru (G) ⊆ G, as follows. An abstract subgroup of G is called unipotent if all of its elements are unipotent. Since every element in G has finite order, one easily sees that the product of two normal unipotent subgroups is again a normal unipotent subgroup of G. If G is

10

Reductive Groups and Steinberg Maps

finite, then this immediately shows that there is a unique maximal normal unipotent subgroup in G. (In the theory of finite groups, this is denoted O p (G).) In the general case, we define Ru (G) := subgroup of G generated by all U ∈ Sunip (G), where Sunip (G) denotes the set of all closed connected normal unipotent subgroups of G. It is clear that Ru (G) is an abstract normal subgroup of G. By the criterion in 1.1.3, Ru (G) is a closed connected subgroup of G; furthermore, Ru (G) = U1 . . . Un for some n  1 and U1, . . . , Un ∈ Sunip (G). As already remarked before, this product will consist of unipotent elements. Thus, Ru (G) is the unique maximal closed connected normal unipotent subgroup of G. (The analogous definition also works when k is an arbitrary algebraically closed field, using the slightly more complicated characterisation of unipotent elements in that case.) We say that G is reductive if Ru (G) = {1}. (Thus, connected reductive groups can be regarded as analogues of finite groups G with O p (G) = {1}.) These are the groups that we will be primarily concerned with. In an arbitrary algebraic group G, we always have the closed connected normal subgroups Ru (G) ⊆ G◦ ⊆ G, and G/Ru (G) will be reductive. Note also that, clearly, we have the implication G simple

⇒

G reductive (and connected).

Here, G  {1} is called a simple algebraic group if G is connected, non-abelian and if G has no closed connected normal subgroups other than {1} and G. (For example, SLn (k) is a simple algebraic group, although in general it is not simple as an abstract group; GLn (k) is reductive, but not simple.) Even if one is mainly interested in studying a simple group G, one will also have to look at subgroups with a geometric origin, like Levi subgroups or centralisers of semisimple elements. These subgroups tend to be reductive, not just simple. For example, if G is connected and reductive and s ∈ G is a semisimple element, then the centraliser CG (s) will be a closed reductive (not necessarily connected or simple) subgroup; see [Ca85, 3.5.4]. 1.1.11 Characters and co-characters of tori. The simplest examples of connected reductive algebraic groups are tori, and it will be essential to understand some basic constructions with them. First, a general definition. A homomorphism of algebraic groups λ : G → k× will be called a character of G. The set X = X(G) of all characters of G is an abelian group (which we write additively), called the character group of G. Similarly, a homomorphism of algebraic groups ν : k× → G

1.1 Affine Varieties and Algebraic Groups

11

will be called a co-character of G. If G is abelian, then the set Y = Y (G) of all cocharacters of G also is an abelian group (written additively), called the co-character group of G. Now let G = T be a torus over k; recall that this means that T is isomorphic to a direct product of a finite number of copies of k× . It is an easy exercise to show that every homomorphism of algebraic groups of k× into itself is given by ξ → ξ n for a well-defined n ∈ Z. Thus, X(k× ) = Y (k× )  Z and so X(T)  Y (T)  Zr

where

T  k× × . . . × k×

(r factors).

Hence, X(T) and Y (T) are free abelian groups of the same finite rank. Furthermore, we obtain a natural bilinear pairing  , : X(T) × Y (T) → Z, defined by the condition that λ(ν(ξ)) = ξ λ,ν for all λ ∈ X(T), ν ∈ Y (T) and ξ ∈ k× . This pairing is a perfect pairing, that is, it induces group isomorphisms

∼ λ → ν → λ, ν , X(T) −→ Hom(Y (T), Z),

∼ ν → λ → λ, ν , Y (T) −→ Hom(X(T), Z), (see [MaTe11, 3.6]). Here, Hom just stands for homomorphisms of abstract abelian groups. The pair (X(T), Y (T)), together with the above pairing, is the simplest example of a ‘root datum’, which will be considered in more detail in Section 1.2. The assignment T → X(T) has the following fundamental property: if T is another torus over k, then we have a natural bijection 1−1

{homomorphisms of algebraic groups T → T } ←→ Hom(X(T), X(T)). The correspondence is defined by sending a homomorphism of algebraic groups f : T → T to the map ϕ : X(T) → X(T), λ  → λ  ◦ f . For future reference, we state the following basic properties of this correspondence: (a) f : T → T is a closed embedding (that is, an isomorphism onto a closed subgroup of T) if and only if ϕ : X(T) → X(T) is surjective; in this case, we have a canonical isomorphism ker(ϕ)  X(T/ f (T)). (b) f : T → T is surjective if and only if ϕ : X(T) → X(T) is injective; in this case, the restriction map X(T) → X(ker( f )) is surjective with kernel given by ϕ(X(T)) = {λ  ◦ f | λ  ∈ X(T)}. See [Bor91, Chap. III, §8], [Ch05, §4.3] and [St74, §2.6] for proofs and further details. Furthermore, by [Ca85, §3.1], T can be recovered from X(T) through the isomorphism (c)

∼

T −→ Hom(X(T), k × ),

t → (λ → λ(t)).

12

Reductive Groups and Steinberg Maps

(Here again Hom just stands for abstract homomorphisms of abelian groups.) 1.1.12 Weight spaces. Characters of tori play a major role in the following context. Let G be a linear algebraic group and V be a finite-dimensional vector space over k. Note that V is an affine variety with algebra of regular functions given by the subalgebra generated by the dual space V ∗ = Hom(V, k) ⊆ A (V, k). Assume that we have a representation of G on V, that is, we are given a morphism of affine varieties G × V → V which defines a linear action of G on V. Let T ⊆ G be a maximal torus. (Any torus of maximum dimension is maximal.) For each character λ ∈ X(T) we define the subspace Vλ := {v ∈ V | t.v = λ(t)v for all t ∈ T}. Let Ψ(T, V) be the set of all λ ∈ X(T) such that Vλ  {0}. Since T consists of pairwise commuting semisimple elements, we have  Vλ V= λ∈Ψ(T,V )

(see [Ge03a, 3.1.5]); in particular, this shows that Ψ(T, V) is finite. The characters in Ψ(T, V) are called weights and the corresponding subspaces Vλ called weight spaces (relative to T). Now, we always have the adjoint representation of G on its Lie algebra L(G), defined as follows. For g ∈ G, consider the inner automorphism γg of G defined by γg (x) = gxg −1 . Taking the differential, we obtain a linear map d1 γg : L(G) → L(G), which is a vector space isomorphism. Hence, we obtain a linear action of G on L(G) such that g.v = d1 (γg )(v) for all g ∈ G and v ∈ L(G). (The corresponding map G × L(G) → L(G) is indeed a representation; see, for example, [Hum91, 10.3].) Then the finite set R := Ψ(T, L(G)) \ {0}

⊆ X(T)

is called the set of roots of G relative to T; we have the root space decomposition  L(G) = L(G)0 ⊕ L(G)α . α∈R

This works in complete generality, for any algebraic group G. If G is connected and reductive, then it is possible to obtain much more precise information about the root space decomposition. It turns out that then L(G)0 = L(T),

R = −R

and

dim L(G)α = 1

for all α ∈ R.

So, in this case, the picture is analogous to that in the theory of complex semisimple Lie algebras and, quite surprisingly, it shows that some crucial aspects of the theory do not depend on the underlying field! This fundamental result, first proved in the Séminaire Chevalley [Ch05], will be discussed in more detail in Section 1.3.

1.1 Affine Varieties and Algebraic Groups

13

1.1.13 General structure of connected reductive algebraic groups. Let G be a connected linear algebraic group. Denote by Z = Z(G) the centre of G. Then we have Z◦ = Ru (Z) × S where S is a torus; see [Ge03a, 3.5.3]. Since Ru (Z) is a characteristic subgroup of Z and Z is a characteristic subgroup of G, we see that Ru (Z) is normal in G. Hence, if G is reductive, then Z◦ is a torus. In this case, the above-mentioned results about the root space decomposition lead to the following product decomposition of G (see [MaTe11, §8.4], [Spr98, §8.1]): G = Z◦ .G1 . . . Gn where G1, . . . , Gn are closed normal simple subgroups and Gi , G j pairwise commute with each other for i  j; furthermore, this decomposition of G has the following properties. • The subgroups {G1, . . . , Gn } are uniquely determined in the sense that every closed normal simple subgroup of G is equal to some Gi . • We have G1 . . . Gn = Gder := commutator (or derived) subgroup of G. (Recall from 1.1.10 that simple algebraic groups are assumed to be connected and non-abelian; note also that the commutator subgroup of a connected algebraic group is always a closed connected normal subgroup; see [Ge03a, 2.4.7].) A connected reductive algebraic group G will be called semisimple if Z◦ = {1} (or, equivalently, if the centre of G is finite). Thus, in the above setting, Gder is semisimple. The above product decomposition can be used to prove general statements about connected reductive algebraic groups by a reduction to simple algebraic groups; see, for example, Lemma 1.6.9 and Theorem 1.7.15. 1.1.14 Algebraic BN-pairs (or Tits systems). The concept of BN-pairs has been introduced by Tits [Ti62], and it has turned out to be extremely useful. It applies to connected algebraic groups and to finite groups, and it allows us to give uniform proofs of many results, instead of going through a large number of case-by-case proofs. Recall that two subgroups B, N of an arbitrary (abstract) group G form a BN-pair (or a Tits system) if the following conditions are satisfied. (BN1) G is generated by B and N. (BN2) H := B ∩ N is normal in N and the quotient W := N/H is a finite group generated by a set S of elements of order 2. (BN3) ns Bns  B if s ∈ S and ns is a representative of s in N. (BN4) ns Bn ⊆ Bns nB ∪ BnB for any s ∈ S and n ∈ N. The group W is called the corresponding Weyl group. We have a length function on W, as follows. We set l(1) = 0. If w  1, we define l(w) to be the length of a shortest possible expression of w as a product of generators in S. (Note that we don’t have to take into account inverses, since s2 = 1 for all s ∈ S.) Thus, any w ∈ W can

14

Reductive Groups and Steinberg Maps

be written in the form w = s1 . . . s p where p = l(w) and si ∈ S for all i. Such an expression (which is by no means unique) will be called a reduced expression for w. For any w ∈ W, we set C(w) := Bnw B, where nw ∈ N is a representative of w in N. Since any two representatives of w lie in the same coset of H ⊆ B, we see that C(w) does not depend on the choice of the representative. The double cosets C(w) are called Bruhat cells of G. Then the above axioms imply the fundamental Bruhat decomposition (see [Bou68, Chap. IV, n◦ 2.3]): Bnw B. G= w ∈W

As Lusztig [Lu10] notes, by allowing one to reduce many questions about G to questions about the Weyl group W, the Bruhat decomposition is indispensable for the understanding of both the structure and representations of G. A key role in this context will be played by the Iwahori–Hecke algebra (introduced in [Iw64]); this is a deformation of the group algebra of W whose definition is based on the Bruhat decomposition. (We will come back to this in Section 3.2.) Now let G be a linear algebraic group over k and let B, N be closed subgroups of G which form a BN-pair. Following [Ca85, §2.5], we shall say that this is an algebraic BN-pair if H = B ∩ N is abelian and consists entirely of semisimple elements, and we have an abstract semidirect product decomposition B = U.H where U is a closed normal unipotent subgroup of B such that U ∩ H = {1}. (If B is connected, then this is automatically a semidirect product of algebraic groups as in 1.1.8; see [Spr98, 6.3.5].) We do not assume that G is connected, so the definition can apply in particular to finite algebraic groups. We now have: Proposition 1.1.15 Let B, N be closed subgroups of G that form an algebraic BN-pair in G, where B = U.H as above. Assume that H, U are connected and that CG (H) = H. Then the following hold. (a) G is connected and reductive. (b) B is a Borel subgroup (that is, a maximal closed connected solvable subgroup of G); we have B = NG (U) and [B, B] = U = Ru (B). (c) H is a maximal torus of G and we have N = NG (H). (See [Ca85, §2.5] and [Ge03a, 3.4.6, 3.4.7].) As in [Ge03a, 3.4.5], a BN-pair satisfying the conditions in Proposition 1.1.15 will be called a reductive BN-pair. Much more difficult is the converse of the above result, which comes about as the culmination of a long series of arguments. Namely, if G is a connected reductive algebraic group, then G has a reductive BN-pair in which B is a Borel subgroup and N is the normaliser of a maximal torus contained in B. (We will discuss this in more detail in Section 1.3.) For our purposes here, the realisation of connected reductive algebraic groups in terms of algebraic BN-pairs as above is sufficient for

1.2 Root Data

15

many purposes. For example, if G is a ‘classical group’ as in 1.1.4, then algebraic BN-pairs as above are explicitly described in [Ge03a, §1.7]. In these cases, one can always find an algebraic BN-pair in which B consists of upper triangular matrices and H consists of diagonal matrices. See also the relevant chapters in [GLS94], [GLS96], [GLS98].

1.2 Root Data We now introduce abstract root data and prove some basic properties of them. As we shall see in later sections, these form the combinatorial skeleton of connected reductive algebraic groups, that is, they capture those features which do not depend on the underlying field k. (A reader who wishes to see a much more systematic discussion of root data is referred to [DG70/11, Exposé XXI].) 1.2.1 Let X, Y be free abelian groups of the same finite rank; assume that there is a bilinear pairing  , : X × Y → Z which is perfect, that is, it induces group isomorphisms Y  Hom(X, Z) and X  Hom(Y, Z) (as in 1.1.11). Furthermore, let R ⊆ X and R∨ ⊆ Y be finite subsets. Then the quadruple R = (X, R, Y, R∨ ) is called a root datum if the following conditions are satisfied. (R1) There is a bijection R → R∨ , α → α∨ , such that α, α∨ = 2 for all α ∈ R. (R2) For every α ∈ R, we have 2α  R. (R3) For α ∈ R, we define endomorphisms wα : X → X and wα∨ : Y → Y by wα (λ) = λ − λ, α∨ α

and

wα∨ (ν) = ν − α, ν α∨

for all λ ∈ X and ν ∈ Y . Then we require that wα (R) = R and wα∨ (R∨ ) = R∨ for all α ∈ R. We shall see in 1.2.5 that the concept of root data is, in a very precise sense, an enhancement of the more traditional concept of root systems (related to finite reflection groups; see [Bou68]). First, we need some preparations. The defining formula immediately shows that wα2 = idX and (wα∨ )2 = idY . Hence, we have wα ∈ Aut(X) and wα∨ ∈ Aut(Y ) for all α ∈ R. We set W := wα | α ∈ R ⊆ Aut(X)

and W∨ := wα∨ | α ∈ R ⊆ Aut(Y );

these groups are called the Weyl groups of R and R∨ , respectively. 1 By (R3), we have an action of W on R and an action of W∨ on R∨ . 1

For the time being, we keep a separate notation for these two Weyl groups; in Remark 1.2.12, we will identify them using the isomorphism in Lemma 1.2.3(a).

16

Reductive Groups and Steinberg Maps

1.2.2 Let R = (X, R, Y, R∨ ) and R  = (X , R , Y , R ∨ ) be root data. Let ϕ : X  → X be a group homomorphism. The corresponding transpose map ϕtr : Y → Y  is uniquely defined by the condition that ϕ(λ ), ν = λ , ϕtr (ν) 

for all λ  ∈ X  and ν ∈ Y,

where  , is the bilinear pairing for R and  ,  is the bilinear pairing for R . We say that ϕ is a homomorphism of root data if ϕ maps R  bijectively onto R and ϕtr maps R∨ bijectively onto R ∨ . It then automatically follows that ϕtr (ϕ(β)∨ ) = β∨ for all β ∈ R ; see [DG70/11, XXI, 6.1.2]. If ϕ is a bijective homomorphism of root data, we say that R and R  are isomorphic. Lemma 1.2.3 Let R = (X, R, Y, R∨ ) be a root datum. ∼

(a) There is a unique group isomorphism δ : W → W∨ such that δ(wα ) = wα∨ for all α ∈ R; we have w −1 (λ), ν = λ, δ(w)(ν)

for all w ∈ W, λ ∈ X, ν ∈ Y .

(b) The quadruple (Y, R∨, X, R) is also a root datum, with pairing  , ∗ : Y × X → Z defined by ν, λ ∗ := λ, ν for all ν ∈ Y and λ ∈ X. (c) For any λ ∈ X and w ∈ W, we have λ − w(λ) ∈ ZR. The root datum in (b) is called the dual root datum of R. Proof (a) For any group homomorphism ϕ : X → X, we consider its transpose ϕtr : Y → Y , as defined above. Clearly, we have idtrX = idY and (ϕ ◦ ψ)tr = ψ tr ◦ ϕtr if ψ : X → X is another group homomorphism. Thus, Wtr := {w tr | w ∈ W} is a subgroup of Aut(Y ) and the map δ : W → Wtr , w → (w −1 )tr , is an isomorphism. Now, using the defining formulae in (R3), one immediately checks that wα (λ), ν = λ, wα∨ (ν)

for all α ∈ R, λ ∈ X, ν ∈ Y .

Hence, we have wαtr = wα∨ for all α ∈ R and so Wtr = W∨ . This yields (a). (b) This is a straightforward verification. (c) The defining formula shows that this is true if w = wα for α ∈ R. But then it also follows in general, since W is generated by the wα (α ∈ R).  Lemma 1.2.4 Let R = (X, R, Y, R∨ ) be a root datum. We set X0 := {λ ∈ X | λ, α∨ = 0 for all α ∈ R}. Then X0 ∩ ZR = {0}

and

|X/(X0 + ZR)| < ∞.

Consequently, W is a finite group and the action of W on R is faithful (that is, if w ∈ W is such that w(α) = α for all α ∈ R, then w = 1).

1.2 Root Data

17

Proof Let us extend scalars from Z to Q. We denote XQ = Q ⊗Z X and YQ = Q ⊗Z Y . Then  , extends to a non-degenerate Q-bilinear form on XQ ×YQ which we denote by the same symbol. Since X, Y are free Z-modules, we can naturally regard X as a subset of XQ and Y as a subset of YQ . Similarly, we can regard W as a subgroup of GL(XQ ) and W∨ as a subgroup of GL(YQ ). So, in order to show the statements about X0 and ZR, it is sufficient to show that XQ = X0,Q ⊕ QR

X0,Q := {x ∈ XQ | x, α∨ = 0 for all α ∈ R}.

where

For this purpose, following [DG70/11, XXI, §1.2], we consider the linear map

x → x, α∨ α∨ . f : XQ → YQ, α∈R

Let β ∈ R. Using (R3), Lemma 1.2.3(a) and the fact that (wβ∨ )2 = idY , we obtain



f ◦ wβ (x) = wβ (x), α∨ α∨ = x, wβ∨ (α∨ ) α∨ = (wβ∨ ◦ f )(x) α∈R

α∈R

for all x ∈ XQ . This identity in turn implies that, for any β ∈ R, we have:

β, α∨ wβ∨ (α∨ ) f (β) = − f (wβ (β)) = −wβ∨ ( f (β)) = − =−

∨



α∈R

∨

∨

β, α α − β, α β

α∈R

Noting that β, f (β) =



∨

= − f (β) +



 β, α∨ 2 β∨ .

α∈R α∈R β, α

2 f (β) = β, f (β) β∨

∨ 2,

and

we deduce that β, f (β) > 0

for all β ∈ R.

This shows that f (QR) = QR∨ , and so dim QR  dim QR∨ . By the symmetry expressed in Lemma 1.2.3(b), the reverse inequality also holds and so dim QR = ∼ dim QR∨ . Thus, f restricts to an isomorphism f : QR → QR∨ . Now, we clearly have X0,Q ⊆ ker( f ), whence X0,Q ∩ QR = {0}. Since  , extends to a nondegenerate bilinear form on XQ ×YQ , we have dim XQ = dim X0,Q + dim QR∨ . Since also dim QR∨ = dim QR, we conclude that XQ = X0,Q ⊕ QR, as desired. Now we show that the action of W on R is faithful. Let w ∈ W be such that w(α) = α for α ∈ R. Then w acts as the identity on the subspace QR ⊆ XQ . The defining equation shows that all wα , α ∈ R, act as the identity on X0,Q , so W is trivial on X0,Q . Hence, w = 1 since XQ = X0,Q + QR. Since R is finite, it follows that W must be finite, too.  1.2.5 Let R = (X, R, Y, R∨ ) be a root datum. As in the above proof, we extend scalars from Z to Q and set XQ = Q ⊗Z X. Following [Bou68, Chap. VI, §1, Prop. 3],

18

Reductive Groups and Steinberg Maps

we define a symmetric bilinear form ( , ) : XQ × XQ → Q by setting

x, α∨ y, α∨ for all x, y ∈ XQ . (x, y) := α∈R

Using (R3) and Lemma 1.2.3(a), we see that ( , ) is W-invariant, that is, we have (w(x), w(y)) = (x, y) for all w ∈ W and all x, y ∈ XQ . Clearly, we have (x, x)  0 for all x ∈ XQ ; furthermore, (β, β) > 0 for all β ∈ R (since β, β∨ = 2 > 0). By a standard argument (see [Bou68, Chap. VI, §1, Lemme 2]), this yields that 2

(α, β) = α, β∨ ∈ Z (β, β)

for all α, β ∈ R.

We claim that the restriction of ( , ) to QR × QR is positive-definite. Indeed, assume  that (x, x) = 0 where x ∈ QR. Then 0 = (x, x) = α∈R x, α∨ 2 and so x, α∨ = 0 for all α ∈ R. Hence, Lemma 1.2.4 shows that x = 0. Thus, we see that R is a crystallographic root system in the subspace QR of XQ ; see [Bou68, Chap. VI, §1, Déf. 1]. The Weyl group of R is W; see Lemma 1.2.4. Furthermore, R is reduced, in the sense that R ∩ Qα = {±α}

for all α ∈ R.

(This is an easy consequence of (R2); see e.g. [Bor91, 14.7].) Similarly, R∨ is a reduced crystallographic root system in QR∨ , by the symmetry in Lemma 1.2.3(b). 1.2.6 Keeping the above notation, we now recall some standard results on root systems (see, e.g., the appendices of [St67] or [MaTe11]). (a) There is a subset Π ⊆ R such that Π is linearly independent in QR and every  α ∈ R can be written as α = β ∈Π xβ β, where either xβ ∈ Q0 for all β ∈ Π or xβ ∈ Q0 for all β ∈ Π. We call Π a base for R. The corresponding set of positive roots R+ ⊆ R consists of  those α ∈ R which can be written as α = β ∈Π xβ β, where xβ ∈ Q0 for all β ∈ Π. The roots in R− := −R+ are called negative roots. Furthermore, if (a) holds, then we also have: (b) For every α ∈ R, there exists some w ∈ W such that w(α) ∈ Π. (c) Every α ∈ R is a Z-linear combination of the roots in the base Π. (That is, the coefficients xβ in (a) are always integers.) (d) W is a Coxeter group, with generators {wβ | β ∈ Π} and defining relations (wβ wγ )mβγ = 1 for all β, γ ∈ Π, where mβγ  1 is the order of wβ wγ ∈ W; furthermore, we have 4 cos2 (π/mβγ ) = γ, β∨ β, γ ∨ for all β, γ ∈ Π. Finally, any two bases of R can be transformed into each other by a unique element

1.2 Root Data

19

of W. In particular, r := |Π| is well defined and called the rank of R; furthermore, writing Π = {β1, . . . , βr }, the matrix

C := β j , βi∨ 1i, jr is uniquely determined by R (up to reordering the rows and columns); it is called the Cartan matrix of R. We say that two root data R, R  have the same Cartan type if the corresponding Cartan matrices are the same (up to choosing a bijection between the associated bases Π, Π ). Thus, R and R  have the same Cartan type if and only if R ⊆ XQ and R  ⊆ XQ are isomorphic root systems (see [Bou68, Chap. VI, n◦ 1.5]). We associate with C a Dynkin diagram, defined as follows. It has vertices labelled by the elements in Π = {β1, . . . , βr }. If i  j and |β j , βi∨ |  |βi, β∨j |, then the corresponding vertices are joined by |β j , βi∨ | lines; furthermore, these lines are equipped with an arrow pointing towards the vertex labelled by βi if |β j , βi∨ | > 1. (Note that, in this case, we automatically have |βi, β∨j | = 1 by (e).) We say that C is an indecomposable Cartan matrix if the associated Dynkin diagram is a connected graph; otherwise, we say that C is decomposable. Clearly, any Cartan matrix can be expressed as a block diagonal matrix with diagonal blocks given by indecomposable Cartan matrices. The classification of indecomposable Cartan matrices is well known (see [Bou68, Chap. VI, §4]); the corresponding Dynkin diagrams are listed in Table 1.1. (The Cartan matrices of type An , Bn , Cn , G2 , F4 are printed explicitly in Examples 1.2.19, 1.3.7, 1.5.5 below.) See [Kac85, Chap. 4] for a somewhat different approach to this classification. We have the following general characterisation of Cartan matrices. Proposition 1.2.7 (Cf. [Bou68, Chap. VI, §4]) Let S be a finite set and C = (cst )s,t ∈S be a matrix with integer entries. Then C is the Cartan matrix of a reduced crystallographic root system if and only if the following conditions hold: (C1) We have css = 2 and, for s  t we have cst  0; furthermore, cst  0 if and only if cts  0. (C2) For all s, t ∈ S, we have cst cts = 4 cos2 (π/mst ), where mst ∈ Z1 . (Thus, mss = 1 and mst = mts ∈ {2, 3, 4, 6} for s  t.) The symmetric matrix (− cos(π/mst ))s,t ∈S is positive-definite. Remark 1.2.8 Let C = (cst )s,t ∈S be a Cartan matrix. Let Ω be the free abelian group with basis {ωs | s ∈ S}. Let ZC ⊆ Ω be the subgroup generated by the  columns of C, that is, by all vectors of the form s ∈S cst ωs for t ∈ S. Then Λ(C) := Ω/ZC is called the fundamental group of C. (This agrees with the definitions in [MaTe11,

20

Reductive Groups and Steinberg Maps Table 1.1 Dynkin diagrams of indecomposable Cartan matrices An n1 Dn n3

G2

E7

1

t

2

t

t1 @ 3 @t

3

t

p p p

n

4

p p p

n

t

t

Bn n2

t

Cn

t

2 t

2

3

t

p p p

n

1

3

4

5

t

6

7

8

n2

t2

t

E6

1

2

1

3

t> t t

1

t

1

4

5

t

t

2

t > 3t

t

F4 6

t

7

t

t

4

t

E8

t

t

t t

t2 1

t

3

t

t2

4

t

5

t

6

t

t

t

t2

(This labelling will be used throughout this book; it is the same as in CHEVIE [GHLMP], [MiChv]. Note that B2 = C2 and D3 = A3 , up to relabelling the vertices.)

9.14] or [Spr98, 8.1.11], for example.) If C is indecomposable, then the groups Λ(C) are easily computed and listed in Table 1.2.

Table 1.2 Fundamental groups of indecomposable Cartan matrices Type of C An−1 Bn, Cn Dn G2, F4, E8 E6 E7

Λ(C) Z/nZ Z/2Z Z/2Z ⊕ Z/2Z Z/4Z {0} Z/3Z Z/2Z

(n even) (n odd)

In 1.2.2, we defined what it means for two root data to be isomorphic. We shall also need the following, somewhat more general notion. Definition 1.2.9 Let R = (X, R, Y, R∨ ) and R  = (X , R , Y , R ∨ ) be root data. We fix an integer p such that either p = 1 or p is a prime number. Then a group homomorphism ϕ : X  → X is called a p-isogeny of root data if there exist a bijection R → R , α → α† , and positive integers qα > 0, each an integral power of p, such that ϕ and its transpose ϕtr : Y → Y  satisfy the following conditions. (I1) ϕ and ϕtr are injective.

1.2 Root Data

21

(I2) We have ϕ(α† ) = qα α and ϕtr (α∨ ) = qα (α† )∨ for all α ∈ R. The conditions (I1) and (I2) appear in [Ch05, §18.2]; following Chevalley, we call the numbers {qα } the root exponents of ϕ. Note that α → α† and the numbers {qα } are uniquely determined by ϕ (since R is reduced). Let W ⊆ Aut(X) be the Weyl group of R. Then one easily sees that, for any α ∈ R and w ∈ W, we have qw(α) = qα ; see [Spr98, 9.6.4]. Hence, by 1.2.6(a), the map α → qα is determined by its values on a base of R. We also see that ϕ is an isomorphism of root data if and only if ϕ is a bijective isogeny where qα = 1 for all α ∈ R. Finally note that if p = 1, then qα = 1 for all α ∈ R. A simple example of a p-isogeny of a root datum into itself is given by ϕ : X → X, λ → pλ (scalar multiplication with p); this will be continued in Example 1.3.17. Remark 1.2.10 Keep the notation in the above definition. Let W ⊆ Aut(X) be the Weyl group of R and W ⊆ Aut(X ) be the Weyl group of R . Then one easily sees that a p-isogeny ϕ : X  → X induces a unique group isomorphism σ : W → W

such that

ϕ ◦ σ(w) = w ◦ ϕ

(w ∈ W).

We have σ(wα ) = wα† for all α ∈ R where wα ∈ W is the reflection corresponding to α ∈ R and wα† ∈ W is the reflection corresponding to α† ∈ R . (See [Ch05, §18.3] for further details; see also 1.2.18 below.) In particular, if ϕ : X → X is a p-isogeny of R into itself, then ϕ ◦ W = W ◦ ϕ, and so ϕ normalises W ⊆ Aut(X). Remark 1.2.11 Let R = (X, R, Y, R∨ ) and R  = (X , R , Y , R ∨ ) be root data. Let us fix a base Π of R and a base Π  of R ; see 1.2.6. Let ϕ : X  → X be a group homomorphism that defines a p-isogeny of root data. Then (I2) shows that Π † := {α† | α ∈ Π} also is a base of R , where α → α† denotes the bijection R → R  associated with ϕ. As already mentioned in 1.2.6, there exists a unique w in the Weyl group W of R  such that Π † = w(Π ). Now w ∈ W ⊆ Aut(X ) certainly is an isomorphism of R  into itself. Hence, the composition ϕ  := ϕ ◦ w : X  → X is also a p-isogeny of root data and the bijection R → R  associated with ϕ  will map Π onto Π . This shows that, replacing ϕ by ϕ ◦ w for a suitable w ∈ W if necessary, we can always assume that the bijection R → R  preserves the given bases Π ⊆ R and Π  ⊆ R . Remark 1.2.12 Let R = (X, R, Y, R∨ ) be a root datum and W ⊆ Aut(X) be the corresponding Weyl group. Let Π be a base of R. By 1.2.6, W is a Coxeter group with generating set S = {wα | α ∈ Π}. We denote by l : W → Z0 the corresponding length function. To unify the notation, we shall use S as an indexing set for Π, that is, Π = {αs | s ∈ S} where αs is the root of the reflection s. Using the isomorphism δ in Lemma 1.2.3(a), we can identify W∨ = W = S .

22

Reductive Groups and Steinberg Maps

Under this identification, W will act on both X and Y , and we have w −1 .λ, ν = λ, w.ν

for all w ∈ W, λ ∈ X, ν ∈ Y .

In particular, for any α ∈ R, we have a corresponding reflection wα which acts both on X and on Y : on the one hand, we have wα .λ = λ − λ, α∨ α for all λ ∈ X; on the other hand, we have wα .ν = ν − α, ν α∨ for all ν ∈ Y . We now turn to the question of actually constructing root data and p-isogenies for a Cartan matrix C. The key idea is contained in the following remark; see [CMT04, §2] and [BrLu12, §2.1]. Remark 1.2.13 Let R = (X, R, Y, R∨ ) be a root datum and let Π be a base of R; write Π = {αs | s ∈ S} as above. Let C be the corresponding Cartan matrix; see 1.2.6. Let {λi | i ∈ I} be a Z-basis of X, where I is some finite indexing set. (We have |I |  |S|.) Let {νi | i ∈ I} be the corresponding dual basis of Y (with respect to the pairing  , ). Then we have unique expressions

βs = as,i λi and βs∨ = a˘s,i νi for all s ∈ S, i ∈I

i ∈I

where as,i ∈ Z and a˘s,i ∈ Z for i ∈ I. Consequently, we have a factorisation C = A˘ · Atr

where

A = (as,i )s ∈S,i ∈I

and

A˘ = (a˘s,i )s ∈S,i ∈I .

˘ Note that all of R ⊆ X and R∨ ⊆ Y are uniquely determined by C, A and A. (This follows from 1.2.6(a) and the fact that W = S ; see 1.2.6(d).) Also note that the dual root system (Y, R∨, X, R) (see Lemma 1.2.3) has Cartan matrix C tr , with factorisation C tr = B˘ · Btr where B = A˘ and B˘ = A. Reversing this argument, one sees that every factorisation as above of a Cartan matrix C leads to a root datum. Let us discuss this in some detail. 1.2.14 Let S be a finite set and C = (cst )s,t ∈S be a Cartan matrix, that is, a matrix satisfying the conditions in Proposition 1.2.7. Let I be another finite index set, with |I |  |S|, and assume that we have a factorisation C = A˘ · Atr

where

A = (as,i )s ∈S,i ∈I

and

A˘ = (a˘s,i )s ∈S,i ∈I ;

here, A and A˘ are matrices with integer coefficients of size |S| × |I |. Let X be a free abelian group with Z-basis indexed by I, say {λi | i ∈ I}; also let Y be a free abelian group with Z-basis labelled by I, say {νi | i ∈ I}. We define a bilinear pairing  , : X × Y → Z such that {λi | i ∈ I} and {νi | i ∈ I} are dual bases to each other.

Then C = αt , αs∨ s,t ∈S , where we set

αs := as,i λi and αs∨ := a˘s,i νi for all s ∈ S. i ∈I

i ∈I

1.2 Root Data

23

For s ∈ S, we define endomorphisms ws : X → X and ws∨ : Y → Y by ws (λ) = λ − λ, αs∨ αs

and

ws∨ (ν) = ν − αs, ν αs∨

for all λ ∈ X and ν ∈ Y . Then ws2 = idX and (ws∨ )2 = idY . Hence, we have ws ∈ Aut(X) and ws∨ ∈ Aut(Y ) for all s ∈ S. Finally, let W := ws | s ∈ S ⊆ Aut(X)

and W ∨ := ws∨ | s ∈ S ⊆ Aut(Y );

R := {w(αs ) | w ∈ W, s ∈ S}

and

R∨ := {w ∨ (αs∨ ) | w ∨ ∈ W ∨, s ∈ S}.

Lemma 1.2.15 The quadruple R := (X, R, Y, R∨ ) in 1.2.14 is a root datum with Cartan matrix C, where {αs | s ∈ S} is a base of R and {αs∨ | s ∈ S} is a base of R∨ . Furthermore, we have W = W and W ∨ = W∨ (with the notation of 1.2.1). The bijection R → R∨ , α → α∨ , is determined as follows. If w ∈ W and s ∈ S are such that α = w(αs ), then α∨ = δ(w)(αs∨ ) (with δ as in Lemma 1.2.3(a)). Proof Let V ⊆ Q ⊗Z X be the subspace spanned by {αs | s ∈ S}. Then ws (αt ) = αt − cst αs for all t ∈ S. So ws (V) ⊆ V for all s ∈ S. Let WR ⊆ GL(V) be the group generated by the restrictions ws : V → V; then R = {w(αs ) | w ∈ WR, s ∈ S} ⊆ V. Thus, we are in the setting of [GePf00, §1.1]. The matrix C is symmetrisable, that is, there exist positive numbers {ds | s ∈ S} such that (ds cst )s,t ∈S is a symmetric matrix. (This easily follows from the fact that there are no closed paths in the Dynkin diagram of C; see also [Kac85, §4.6].) Then we can define a WR -invariant symmetric bilinear form on V by (αs, αt ) = ds cst /2 for s, t ∈ S. The WR -invariance implies that cst = 2

(αs, αt ) (αs, αs )

for all s, t ∈ S;

see [GePf00, 1.3.2]. By (C2), this form is positive-definite; furthermore, each map ws : V → V is an orthogonal reflection with root αs . So WR and R are finite; see [Bou68, Chap. V, §8] or [GePf00, 1.3.8]. In fact, R is a root system in V with Weyl group W R , with {αs | s ∈ S} as base and C as Cartan matrix; see [GePf00, 1.1.10]. Also note that R is reduced, that is, (R2) holds; see [GePf00, 1.3.7]. Let R+ be the set of positive roots in R defined by {αs | s ∈ S}. Similarly, let V ∨ ⊆ Q ⊗Z Y be the subspace spanned by {αs∨ | s ∈ S}. Then ws∨ (αt∨ ) = αt∨ − cts αs∨ for all t ∈ S. Let WR∨ ⊆ GL(V ∨ ) be the group generated by the restrictions ws∨ : V ∨ → V ∨ . Again, we are in the setting of [GePf00, §1.1] (with respect to C tr , the transpose of C) and so we can repeat the previous argument. Consequently, WR∨ is also finite; furthermore, R∨ is a reduced root system in V ∨ with Weyl group WR∨ , with {αs∨ | s ∈ S} as base and C tr as Cartan matrix. Let (R∨ )+ be the set of positive roots in R∨ defined by the base {αs∨ | s ∈ S}. Next, we define a linear map f : V → V ∨ by f (αs ) = 12 (αs, αs )αs∨ for all s ∈ S.

24

Reductive Groups and Steinberg Maps

(This is analogous to the definition in the proof of Lemma 1.2.4.) Clearly, f is bijective. One immediately checks that ws∨ ◦ f = f ◦ ws for all s ∈ S. So the map ∼ w → f ◦ w ◦ f −1 defines a group isomorphism δ : WR → WR∨ such that δ(ws ) = ws∨ for all s ∈ S. Consequently, we can define a bijection R → R∨ , α → α∨ , as follows. First, let α ∈ R+ . By definition, α = w(αs ) for some w ∈ W, s ∈ S. Then f (α) = f (w(αs )) = δ(w)( f (αs )) = 12 (αs, αs )δ(w)(αs∨ ). Since αs∨ ∈ R∨ , we have δ(w)(αs∨ ) ∈ R∨ by definition. Thus, f (α) ∈ V ∨ is a positive scalar multiple of some element of R∨ ; since R∨ is reduced, there is a unique positive root with this property, denoted α∨ , and the above computation shows that α∨ = δ(w)(αs∨ ). The definition of α∨ for negative α is analogous; we then have (−α)∨ = −α∨ for all α ∈ R. Consequently, we obtain a map R → R∨ , α → α∨ , which is easily seen to be bijective. Once this is established, the maps wα : X → X and wα∨ : Y → Y are defined for any α ∈ R. The WR -invariance of ( , ) implies that β, α∨ = 2

(α, β) (α, α)

for all α, β ∈ R.

In particular, α, α∨ = 2; thus, (R1) holds. Finally, we show (R3). For each α ∈ R, we claim that wα (R) = R. First one notices that wα (V) ⊆ V. But the above formula for β, α∨ shows that the restriction wα : V → V is the orthogonal reflection with root α. Hence, since W R is the Weyl group of R, we have wα (R) = R, as desired. The argument for wα∨ is analogous.  In view of Remark 1.2.13, the above construction yields all root data up to isomorphism. Thus, given a Cartan matrix C, we can think of the various root data ˘ Atr , where A, A˘ are integer matrices of Cartan type C simply as factorisations C = A· of the same size. (This observation, in this explicit form, appears in [CMT04] and [BrLu12]; it is also implicit in [Lu89, §1].) Example 1.2.16 Let C be a Cartan matrix. We have just seen that any factorisation C = A˘ · Atr as in 1.2.14 gives rise to a root datum R = (X, R, Y, R∨ ). Obviously, there are two natural choices for such a factorisation, namely, • either A is the identity matrix and, hence, A˘ = C; • or A˘ is the identity matrix and, hence, A = C tr . In the first case, we denote the corresponding root datum by R = Rad (C). We have X = ZR in this case; any root datum satisfying X = ZR will be called a root datum of adjoint type. In the second case, we denote the corresponding root datum by R = Rsc (C). We have Y = ZR∨ in this case; any root datum satisfying Y = ZR∨ will be called a root datum of simply connected type. Thus, Rad (C) and Rsc (C) may be regarded as the standard models of root data

1.2 Root Data

25

of adjoint and simply connected types, respectively. (See also Example 1.2.21.) The relevance of these notions will become clearer when we consider semisimple algebraic groups in Section 1.5. Example 1.2.17 There is an obvious notion of a direct product of root data. Indeed, if Ri = (Xi, Ri, Yi, Ri∨ ) for i = 1, . . . , n are root data, then we obtain a new root datum R = (X, R, Y, R∨ ) as follows. We set X := X1 ⊕ · · · ⊕ Xn, Y := Y1 ⊕ · · · ⊕ Yn,

R := R 1 ∪ · · · ∪ R n, R∨ := R ∨ ∪ · · · ∪ R n∨, 1

where, for each i, we let R i ⊆ X denote the image of Ri under the natural embedding Xi → X; similarly, R i∨ ⊆ Y denotes the image of Ri∨ under the natural embedding Yi → Y . Furthermore, the perfect bilinear pairings for the various Ri define a unique perfect bilinear pairing for R in a natural way. Also note that, if Πi is a base

1 ∪···∪Π

n is a base of R. of Ri for i = 1, . . . , n, then Π := Π In terms of the matrix language of 1.2.14, the situation is described as follows. Each Ri is determined by a factorisation Ci = A˘ i · Atri where Ci is the Cartan matrix of Ri with respect to a base Πi of Ri . Then R is determined by the factorisation C = A˘ · Atr , where C, A and A˘ are block-diagonal matrices with diagonal blocks given by the Ci , Ai and A˘ i , respectively. The matrix C is the Cartan matrix of R

n.

1 ∪···∪Π with respect to the base Π = Π We now translate the conditions in Definition 1.2.9 into the matrix language of Remark 1.2.13. This will be an extremely efficient tool for constructing isogenies, as it reduces the conditions to be checked to the verification of simple matrix identities. 1.2.18 Let R = (X, R, Y, R∨ ) and R  = (X , R , Y , R ∨ ) be root data. Assume that X and X  have the same rank and that R and R  have bases indexed by the same set S. Denote these bases by Π = {αs | s ∈ S} and Π  = {βs | s ∈ S}, respectively. Let C and C  be the corresponding Cartan matrices. Let us also fix a Z-basis {λi | i ∈ I} of X and a Z-basis {λ j | j ∈ J} of X . Then R and R  are determined by factorisations as in Remark 1.2.13: C = A˘ · Atr C  = B˘ · Btr

where

A = (as,i )s ∈S,i ∈I

where

B = (bs, j )s ∈S, j ∈J

A˘ = (a˘s,i )s ∈S,i ∈I , and B˘ = (b˘ s, j )s ∈S, j ∈J .

and

(Here, |I | = |J |, since X, X  have the same rank.) Giving a linear map ϕ : X  → X is the same as giving a matrix P = (pi j )i ∈I, j ∈J with integer coefficients:

pi j λi for all j ∈ J. ϕ(λ j ) = i ∈I

Assume now that ϕ :

X

→ X is a linear map which is ‘base preserving’, in the

26

Reductive Groups and Steinberg Maps

sense that there is a permutation S → S, s → s† , such that ϕ(βs† ) = qs αs

where 0  qs ∈ Z for all s ∈ S.

We encode this in a monomial matrix P◦ = (p◦st )s,t ∈S , where p◦ss† = qs for s ∈ S. Let p = 1 or p be a prime number and assume that ϕ is a p-isogeny. Then the conditions in Definition 1.2.9 immediately imply that the following conditions hold. (MI1) P◦ is a monomial matrix whose non-zero entries are all powers of p. (MI2) P is invertible over Q; furthermore, P · Btr = Atr · P◦ , P◦ · B˘ = A˘ · P. Conversely, it is straightforward to check that any pair of integer matrices (P, P◦ ) satisfying (MI1) and (MI2) defines a p-isogeny of root data. The argument is similar to the proof of Lemma 1.2.15; let us just briefly sketch it. Let ϕ : X  → X be the linear map with matrix P. Condition (I1) holds since P is invertible over Q. Since P◦ is monomial, there is a permutation S → S, s → s† , such that qs := p◦ss†  0

for all s ∈ S. ϕtr (αs∨ )

= qs βs∨† for all s ∈ S. Thus, (I2) Then (MI2) means that ϕ(βs† ) = qs αs and holds for simple roots and coroots. To see that (I2) holds for all roots and coroots, note that (MI2) implies that C · P◦ = P◦ · C . Consequently, using 1.2.6(d) for W and for W, there is a unique group isomorphism σ : W → W

such that

wαs → wβs †

(s ∈ S).

Using Lemma 1.2.4, one shows that this implies that ϕ ◦ σ(w) = w ◦ ϕ

for all w ∈ W.

We can now define a bijection R → R , α → α† , with the required properties, as follows. Let α ∈ R and write α = w(αs ) for some w ∈ W and s ∈ S. Then we set α† := σ(w)(βs† ) ∈ R . Now, we have ϕ(α† ) = ϕ(σ(w)(βs† )) = w(ϕ(βs† )) = qs w(αs ) = qs α. Since ϕ is injective, α† is uniquely determined by α (and does not depend on the choice of w and s); furthermore, the first of the two identities in (I2) holds, where qα = qs . The argument for the second identity is similar, using the bijection R → R∨ (see Lemma 1.2.15). Thus, (I2) is seen to hold for all roots and coroots. Example 1.2.19 (Cf. [Ch05, §§21.5, 22.4, 23.7]) Let C = (ci j )1i, jr (r = 2 or 4) be a Cartan matrix of type C2 , G2 or F4 ; see Table 1.1. Explicitly:  C2 :

2 −2

−1 2



 ,

G2 :

2 −1 −3 2

 ,

 2  −1 F4 :   0  0

−1 2 −2 0

0 −1 2 −1

0 0 −1 2

     

1.3 Chevalley’s Classification Theorems

27

(Note that C2 = B2 up to relabelling the two vertices of the Dynkin diagram.) We set p = 2 if C is of type C2 or F4 , and p = 3 if C is of type G2 . Let us consider the corresponding root datum R = Rad (C) = (X, R, Y, R∨ ) as in Example 1.2.16; ˘ For any m  0, we we have C = A˘ · Atr , where A is the identity matrix and C = A. ◦ define two matrices Pm and Pm as follows:   0 2m ◦ , Pm = Pm := C2 : 2m+1 0   0 3m ◦ G2 : , Pm = Pm := 3m+1 0

F4 :

 0  0 ◦ Pm = Pm :=   0 m+1  2

0 0 2m+1 0

0 2m 0 0

2m 0 0 0

  .   

◦ satisfy (MI1), ˘ Then Pm, Pm Now, in the setting of 1.2.18, let C  = C, B = A, B˘ = A. ◦ (MI2) and, hence, the pair (Pm, Pm ) defines a group homomorphism ϕm : X → X which is a p-isogeny of R into itself, such that ϕ2m = p2m+1 idX . See [Ca72, 12.3, 12.4] for a more detailed discussion of these ‘exceptional’ isogenies; they give rise to the finite Suzuki and Ree groups (see Example 1.4.22). Another instance of such an exceptional isogeny will be considered in Example 1.5.5.

Remark 1.2.20 Recall from Definition 1.2.9 that a p-isogeny ϕ : X  → X is an isomorphism of root data if and only if ϕ is bijective and qα = 1 for all α ∈ R. Hence, in the setting of 1.2.18, ϕ is an isomorphism if and only if P◦ is a permutation matrix and P is invertible over Z, where the relations P · Btr = Atr · P◦ , P◦ · B˘ = A˘ · P hold; note that these imply that C · P◦ = P◦ · C . Example 1.2.21 Assume that C is a Cartan matrix of type G2 , F4 or E8 . Then C is invertible over Z and, hence, Rad (C) and Rsc (C) are isomorphic root data. Indeed, Rad (C) corresponds to the factorisation C = A˘ · Atr where A is the identity matrix, while Rsc (C) corresponds to the factorisation C = B˘ · Btr where B˘ is the identity matrix. Then the conditions in Remark 1.2.20 hold, where P = C −1 and P◦ is the identity matrix.

1.3 Chevalley’s Classification Theorems Throughout this section, let k be an algebraically closed field and G be a linear algebraic group over k. We can now explain how one can naturally attach to G a root datum, when G is connected reductive.

28

Reductive Groups and Steinberg Maps

1.3.1 Let G be connected reductive. Let T ⊆ G be a maximal torus, X = X(T) and L(G) be the Lie algebra. Recall from 1.1.12 that there is a finite subset R ⊆ X and a corresponding root space decomposition of L(G):  L(G)α . L(G) = L(G)0 ⊕ α∈R

As already mentioned in 1.1.12, we have L(G)0 = L(T), R = −R and dim L(G)α = 1 for all α ∈ R; in particular, dim G = dim L(G) = dim T + |R|. The roots can be directly characterised in terms of G, as follows. Let α ∈ X. Then α ∈ R if and only if there exists a homomorphism of algebraic groups uα : k+ → G such that uα is an isomorphism onto its image and we have tuα (ξ)t −1 = uα (α(t)ξ)

for all t ∈ T and ξ ∈ k.

Thus, Uα := {uα (ξ) | ξ ∈ k} ⊆ G is a one-dimensional closed connected unipotent subgroup normalised by T. It is uniquely determined by α and called the root subgroup corresponding to α. Conversely, every one-dimensional closed connected unipotent subgroup normalised by T is equal to Uα for some α ∈ R. We have G = T, Uα | α ∈ R . Now consider also the co-character group Y = Y (T); we wish to define a finite subset R∨ ⊆ Y . Recall from 1.1.11 that X, Y are free abelian groups of the same (finite) rank and that there is a natural pairing  , : X × Y → Z. The Weyl group of G with respect to T is defined as W(G, T) := NG (T)/T. Since NG (T) acts on T by conjugation, we have induced actions of W(G, T) on X and on Y via

(w.λ)(t) = λ(w −1 t w)

(w.ν)(ξ) = wν(ξ) w

−1

(λ ∈ X, t ∈ T), (ν ∈ Y, ξ ∈ k × ),

where, for any w ∈ W(G, T), we denote by w a representative in NG (T). Using these actions, we can identify W(G, T) with subgroups of Aut(X) and of Aut(Y ). Now let α ∈ R. Then Gα := CG (ker(α)◦ ) = T, Uα, U−α is a closed connected reductive subgroup of G; its Weyl group W(Gα, T) := NGα (T)/T has order 2. Let wα ∈ W(Gα, T) be the non-trivial element and w α be a representative of wα in NGα (T) ⊆ NG (T). Then there exists a unique α∨ ∈ Y such that wα .λ = λ − λ, α∨ α

for all λ ∈ X.

Following, e.g., [Con14, 1.2.8], this element α∨ can also be determined as follows.

1.3 Chevalley’s Classification Theorems

29

The maps u±α : k+ → U±α can be chosen such that the assignment     1 ξ 1 0 → uα (ξ), → u−α (ξ) (ξ ∈ k) 0 1 ξ 1 defines a homomorphism of algebraic groups ϕα : SL2 (k) → G. Then   ξ 0 ∨ α (ξ) = ϕα ∈T for all ξ ∈ k × . 0 ξ −1 Thus, we obtain a well-defined subset R∨ = {α∨ | α ∈ R} ⊆ Y ; we have W(G, T) = wα | α ∈ R . Complete proofs of the above statements can be found in the textbooks [Bor91], [Hum91], [Spr98] and, of course, the original source [Ch05]. A thorough guide through this argument, with indications of the proofs and many worked-out examples, can be found in [MaTe11, §8]. See also [Al09], [Con14], [Jan03, Chap. II], [DG70/11] for further reading. With this notation, we can now state the following result which shows that we are exactly in the situation described by Proposition 1.1.15. Theorem 1.3.2 The quadruple R = (X(T), R, Y (T), R∨ ) in 1.3.1 is a root datum as defined in 1.2.1, with Weyl group W(G, T) (identified with a subgroup of Aut(X(T)) as above). Furthermore, let R+ ⊆ R be the set of positive roots with respect to a base Π ⊆ R. Then B := T, Uα | α ∈ R+ ⊆ G is a Borel subgroup; the subgroups B and NG (T) form a reductive BN-pair in G, where CG (T) = T = B ∩ NG (T). Proof In its essence, this is due to Chevalley [Ch05], but the notion of BN-pairs was not yet available at that time. A proof of the fact that R is a root datum can be found, for example, in [MaTe11, 9.11], [Spr98, 7.4.3]. The BN-pair axioms are shown in [Bor91, 14.15], [MaTe11, 11.16]. For the equality CG (T) = T, see [MaTe11, 8.13] or [Spr98, 7.6.4].  Theorem 1.3.3 Assume that G is connected reductive. Then G acts transitively (by simultaneous conjugation) on the set of all pairs (T, B) where T ⊆ G is a maximal torus and B ⊆ G is a Borel subgroup such that T ⊆ B. In particular, the root data (as in Theorem 1.3.2) with respect to any two maximal tori of G are isomorphic in the sense of 1.2.2. Proof The conjugacy results are due to Borel [Bor91, 10.6,11.1]; see also [Spr98, 6.2.7, 6.3.5]. (See [PaVi10] for a discussion of earlier work of Morozov on Borel

30

Reductive Groups and Steinberg Maps

subalgebras; we thank George Lusztig for this reference.) A somewhat more elementary proof of the conjugacy of Borel subgroups is given in [St77]. (See also [Ge03a, §3.4].) Once these conjugacy results are shown, the assertion about the isomorphism between root data is clear.  Remark 1.3.4 Let G, T, R, B as in Theorem 1.3.2; let W := W(G, T). Then the set of all Borel subgroups of G containing T is described as follows. Let B1 be any Borel subgroup of G containing T. By Theorems 1.3.2, 1.3.3 and the BN-pair

Now, the base Π ⊆ R used axioms, there is a unique w ∈ W such that B1 = w −1 Bw. to define B is transformed under w to a new base Π1 of R. Consequently, we have B1 = T, Uα | α ∈ R1+ where R1+ ⊆ R is the set of positive roots with respect to Π1 . Further recall from 1.2.6 that any two bases of R can be transformed into each other by a unique element of W. Thus, we obtain bijective correspondences {Borel subgroups containing T}

1−1

←→

W

1−1

←→

{bases of R}.

Remark 1.3.5 Assume that G is connected reductive. In 1.1.13 we have defined G to be semisimple if |Z| < ∞ where Z = Z(G) denotes the centre of G; alternatively, G is semisimple if and only if G = Gder (see 1.1.13). We also have the following characterisation in terms of the root datum R = (X, R, Y, R∨ ) (with respect to a maximal torus T ⊆ G). By [MaTe11, 8.17(h)], [Spr98, 8.1.8], we have Z = {t ∈ T | α(t) = 1 for all α ∈ R}

(a)

and the isomorphism T  Hom(X(T), k × ) in 1.1.11 restricts to Z  Hom(X(T)/ZR, k × ).

(b)

Thus, we obtain the equivalences: |Z| < ∞

⇐⇒

|X/ZR| < ∞

⇐⇒

|Y /ZR∨ | < ∞.

(c)

If we consider the factorisation C = A˘ · Atr determined by R as in Remark 1.2.13, then G is semisimple if and only if A, A˘ are square matrices. Remark 1.3.6 Assume that G is connected reductive. As in 1.1.13, we have G = Z◦ .Gder ; furthermore, Gder = G1 . . . Gn where G1, . . . , Gn are the closed normal simple subgroups of G; they commute pairwise with each other. These subgroups have the following description in terms of the root datum R = (X, R, Y, R∨ ) (with respect to a maximal torus T ⊆ G) and the corresponding root subgroups Uα (α ∈ R). First note that Gder = Uα | α ∈ R , see [MaTe11, 8.21]. Now let C be the Cartan matrix of the root system R, with respect to a base Π of R. Then C can be expressed as a block diagonal matrix where

1.3 Chevalley’s Classification Theorems

31

the diagonal blocks are indecomposable Cartan matrices, C1, . . . , Cn say. (Thus, C1, . . . , Cn correspond to the connected components of the Dynkin diagram of C.) Let Π = Π1  · · ·  Πn be the corresponding partition of Π. Then we also have R = R1  · · ·  Rn where Ri consists of all roots in R which can be expressed as linear combinations of simple roots in Πi . Then we have Gi = Uα | α ∈ Ri ⊆ G

for i = 1, . . . , n.

A maximal torus of Gi is given by Ti := Gi ∩ T where T is a fixed maximal torus of G. (See [Bor91, Chap. IV, §11], [MaTe11, §8.4], [Spr98, §8.1] for further details.) Before continuing with the general theory, we give three concrete examples. We shall see that the point of view in 1.2.14, where root data are described in terms of factorisations of Cartan matrices, provides a particularly efficient and convenient way of encoding the information involved in these examples. Example 1.3.7 Let G = GLn (k). Let B ⊆ G be the subgroup of all upper triangular matrices and N ⊆ G the subgroup of all monomial matrices. It is well known that these groups form a BN-pair; see [Bou68, Chap. IV, n◦ 2.2]. For further details see [Ge03a, 1.6.10, 3.4.5], where it is also shown that this is an algebraic BNpair satisfying the conditions in Proposition 1.1.15; in particular, G is connected reductive. Let us describe the root datum of G with respect to the maximal torus T = B ∩ N consisting of all diagonal matrices in G. It will be convenient to introduce some notation concerning matrices. For 1  i  n − 1, let ni be the matrix which is obtained by interchanging the i th and the (i + 1)th row in the identity matrix, which we denote by In . More generally, if w ∈ Sn is any permutation, let nw be the matrix which is obtained by permuting the rows of In as specified by w. (Thus, if {e1, . . . , en } denotes the standard basis of k n , then nw (ei ) = ew(i) for 1  i  n; we have nww = nw nw for all w, w  ∈ Sn .) Then N = {hnw | h ∈ T, w ∈ Sn } and so we have an exact sequence {1} → T → N → Sn → {1}, where N → Sn sends nw to w. Next, for 1  i, j  n let Ei j be the ‘elementary’ matrix with coefficient 1 at the position (i, j) and 0 otherwise. We define Ui j := {In + ξEi j | ξ ∈ k}

where 1  i, j  n, i  j.

All of these are one-dimensional, closed connected subgroups of G. Finally, if ξ1, . . . , ξn are non-zero elements of k, we denote by h(ξ1, . . . , ξn ) ∈ T the diagonal matrix with ξ1, . . . , ξn along the diagonal. Then the map (k× )n → T,

(ξ1, . . . , ξn ) → h(ξ1, . . . , ξn ),

32

Reductive Groups and Steinberg Maps

is certainly an isomorphism of algebraic groups. Hence X = X(T) is the free abelian group with basis λ1, . . . , λn , where λi (h(ξ1, . . . , ξn )) = ξi for all i. Each subgroup Ui j is normalised by T. Let ui j : k+ → G be the homomorphism given by ui j (ξ) = In +ξEi j for ξ ∈ k+ . Then Ui j is the image of this homomorphism, ui j is an isomorphism onto its image and we have tui j (ξ)t −1 = ui j (ξi ξ j−1 ξ)

where t = h(ξ1, . . . , ξn ) ∈ T and ξ ∈ k+ .

Hence, αi j := λi −λ j ∈ X is a root and Ui j is the corresponding root subgroup. To see that these are all the roots, we can use the formula dim G = dim T+|R| in 1.3.1. Thus, since dim G = n2 and dim T = n, we conclude that R = {αi j | 1  i, j  n, i  j} is the root system of G with respect to T. We also see that Π := {αi,i+1 = λi − λi+1 | 1  i  n − 1} ⊆ R is a base of R and that B is the Borel subgroup associated with this base (as in Remark 1.3.4). Now consider coroots. The dual basis of Y = Y (T) is given by the co-characters ν j : k× → T such that ν j (ξ) is the diagonal matrix with coefficient ξ at position j, and coefficient 1 otherwise. For i  j, we have a unique embedding of algebraic groups ϕi j : SL2 (k) → G such that     1 ξ 1 0 ϕi j = ui j (ξ) and ϕi j = u ji (ξ) for all ξ ∈ k. 0 1 ξ 1 Hence, ϕi j satisfies the condition in 1.3.1 and so we obtain αi∨j ∈ Y such that   ξ 0 ∨ ∈T αi j (ξ) = ϕi j 0 ξ −1 is the diagonal matrix with coefficient ξ at position i and coefficient ξ −1 at position j. Thus, we have R∨ = {αi∨j = νi − ν j | 1  i, j  n, i  j}. We also see that the set ∨ = νi − νi+1 | 1  i  n − 1} ⊆ R∨ Π ∨ := {αi,i+1

is a base of R∨ . The Cartan matrix C given by 2 −1  2  −1    0 −1 C= .  . .. .  .   0 ...  0

= (ci j )1i, jn−1 with respect to this base is 0 −1

... 0

2 .. . 0 ...

−1 .. .

−1 0

0  ... 0  ..  .. . .  . ..  . 0   2 −1  −1 2 

1.3 Chevalley’s Classification Theorems

33

Thus, C is of type An−1 . The factorisation in Remark 1.2.13 is given by

C = A˘ · Atr

where

    A = A˘ =     

1 0 .. . 0 0

−1 1 .. .

0 −1 .. .

... 0 .. .

...

0 ...

1 0

0  ... 0  ..  .. . .  . −1 0  1 −1 

(Here, A = A˘ has n − 1 rows and n columns.) Example 1.3.8 Let n  2 and G = SLn (k), the special linear group. We keep the notation G = GLn (k), Ui j , B, N, T, X = X(T), Y = Y (T) from the previous example. Then an algebraic BN-pair satisfying the conditions in Proposition 1.1.15 is given by the subgroups B := B ∩ G and N := N ∩ G; see [Bou68, Chap. IV, §2, Exercise 10], [Ge03a, 1.6.11, 3.4.5]. Let us describe the root datum of G with respect to the maximal torus T = T ∩ G. Let X  = X(T)

and

Y  = Y (T).

For 1  i, j  n, i  j, the subgroup Ui j of G is already contained in G. So, if αij denotes the restriction of αi j ∈ X to T, then αij ∈ X  and αij is a root of G with corresponding root subgroup Ui j ⊆ G. Since dim G = n2 − 1 and dim T = n − 1, it follows as above that R  = {αij | 1  i, j  n, i  j} is the root system of G with respect to T and that  | 1  i  n − 1} is a base for R  . Π  = {αi,i+1

Furthermore, the image of the embedding ϕi j : SL2 (k) → G is clearly contained in G. Consequently, any coroot αi∨j ∈ Y also is a coroot in Y . Thus, we have R ∨ = R∨ = {αi∨j | 1  i, j  n, i  j} and ∨ | 1  i  n − 1} is a base for R ∨ . Π ∨ = {αi,i+1

In particular, we obtain the same Cartan matrix C of type An−1 as in Example 1.3.7. Now note that we have an isomorphism of algebraic groups

(k× )n−1 → T, (ξ1, . . . , ξn−1 ) → h ξ1, . . . , ξn−1, (ξ1 . . . ξn−1 )−1 (with inverse sending h(ξ1, . . . , ξn ) ∈ T to (ξ1, . . . , ξn−1 ) ∈ (k× )n−1 ). Hence, if we define co-characters ν j : k× → T (for 1  j  n − 1) such that ν j (ξ) is the diagonal  } is a Z-basis of matrix with ξ at position j and ξ −1 at position n, then {ν1, . . . , νn−1  ∨  Y . But then Π also is a Z-basis of Y . If we consider the corresponding dual basis of X , then the factorisation in Remark 1.2.13 is given by C = A˘ · Atr

where A˘ = In−1 and A = C tr .

34

Reductive Groups and Steinberg Maps

Thus, G is semisimple and the root datum of SLn (k) is of simply connected type (see Example 1.2.16). Example 1.3.9 Let G = GLn (k) and Z ⊆ G be the centre of G, consisting of ¯ = PGLn (k) := G/Z, all non-zero scalar matrices. Assume that n  2 and let G the projective general linear group. (This is a linear algebraic group by 1.1.6.) ¯ by g → g. Let us denote the canonical map G → G ¯ In particular, we obtain ¯ ¯ ¯ subgroups B and N of G which form a BN-pair since Z ⊆ B; see [Bou68, Chap. IV, §2, Exercise 2]. One easily checks that this is an algebraic BN-pair satisfying the ¯ with respect conditions in Proposition 1.1.15. Let us describe the root datum of G ¯ ¯ to the maximal torus T of G. Let ¯ X¯ = X(T)

and

¯ Y¯ = Y (T).

For every root α of G, we clearly have Z ⊆ ker(α). So, using the universal property of quotients, there is a well-defined α¯ ∈ X¯ such that α(t) = α( ¯ t¯) for all t ∈ T. ¯ is still ¯ Now, for 1  i, j  n, i  j, the image Ui j of the subgroup Ui j ⊆ G in G + ¯ Hence, α¯ i j is a root with closed, connected, isomorphic to k and normalised by T. ¯ As above, it follows that R¯ = { α¯ i j | 1  ¯ i j ⊆ G. corresponding root subgroup U ¯ with respect to T ¯ and that i, j  n, i  j} is the root system of G ¯ ¯ = { α¯ i,i+1 | 1  i  n − 1} is a base for R. Π ¯ On the other hand, we obtain morphisms of algebraic groups ϕ¯i j : SL2 (k) → G, ¯ Thus, simply by composing ϕi j : SL2 (k) → G with the canonical map G → G. every coroot α∨ of G determines a coroot α¯ ∨ ∈ Y¯ . Consequently, we have R¯ ∨ = { α¯ i∨j | 1  i, j  n, i  j} and ¯ ∨ = { α¯ ∨ | 1  i  n − 1} is a base for R¯ ∨ . Π i,i+1 In particular, we obtain the same Cartan matrix C of type An−1 as in Example 1.3.7. Now consider the homomorphism of algebraic groups T → (k× )n−1,

h(ξ1, . . . , ξn ) → (ξ1 ξn−1, . . . , ξn−1 ξn−1 ).

It has Z in its kernel so there is an induced homomorphism of algebraic groups T¯ → (k× )n−1 . The latter homomorphism is an isomorphism: its inverse is given by sending (ξ1, . . . , ξn−1 ) ∈ (k× )n−1 to the image of the element h(ξ1, . . . , ξn−1, 1) ∈ T ¯ It follows that in T. { α¯ i,n | 1  i  n − 1}

¯ is a Z-basis of X.

¯ If we consider the corresponding dual basis of Y¯ , ¯ also is a Z-basis of X. But then Π then the factorisation in Remark 1.2.13 is given by C = A˘ · Atr

where A˘ = C and A = In−1 .

1.3 Chevalley’s Classification Theorems

35

Thus, PGLn (k) is semisimple and the root datum of PGLn (k) is of adjoint type (see Example 1.2.16). In particular, we see that the root data of PGLn (k) and SLn (k) are not isomorphic. (In the former, ZR = X; in the latter, ZR  X.) So, by Theorem 1.3.3, PGLn (k) and SLn (k) are not isomorphic as algebraic groups (even if Z(SLn (k)) = {1}, in which case these two groups are isomorphic as abstract groups). A key feature of the whole theory is the fact that a connected reductive algebraic group is uniquely determined by its root datum up to isomorphism. This follows from a more general result, the ‘isogeny theorem’. As preparation, we cite the following general results concerning surjective homomorphisms of algebraic groups, which will be useful at several places below. 1.3.10 Let f : G → G be a surjective homomorphism of connected algebraic groups over k. Then we have the following preservation results. (a) f maps a Borel subgroup of G onto a Borel subgroup of G, and all Borel subgroups of G arise in this way; a similar statement holds for maximal tori. (See [Bor91, 11.14].) (b) f maps the unipotent radical of G onto the unipotent radical of G; in particular, if G is reductive, then so is G. (See [Bor91, 14.11].) (c) If G is reductive, then f maps the centre of G onto the centre of G . (This follows from (a) and the fact that the centre of a reductive group is the intersection of all its maximal tori; see [Bor91, 11.11].) (d) Assume that G and G are reductive. Let T be a maximal torus of G; by (a), T := f (T) is a maximal torus of G. Then f induces a surjective homomorphism W(G, T) → W(G, T), and this is an isomorphism if ker( f ) is contained in the centre of G. (See [Bor91, 11.20].) There is a further property which is certainly well known and which we will need in Chapter 2, but it is not easy to find a reference with a proof: (e) If G is reductive and ker( f ) is contained in the centre of G, then we have

f CG◦ (g) = CG◦  ( f (g)) for any g ∈ G.

Here is a proof of (e). Let g  = f (g). It is clear that f CG (g) ⊆ CG (g ) and ◦

f CG (g) ⊆ CG◦  (g ). So it is sufficient to show that the index of f (CG (g)) in CG (g ) is finite. To see this, we apply Lemma 1.1.9 with A = G, B = G, σ(x) = g −1 xg (x ∈ G), τ(x ) = g −1 x  g  (x  ∈ G). Then Aσ = CG (g) and Bτ = CG (g ); since ker( f ) ⊆ Z(A), we have C = {a−1 σ(a) | a ∈ ker( f )} = {1}. So the image of the map δ in Lemma 1.1.9 is given by {a−1 σ(a) | a ∈ A} ∩ ker( f ) ⊆ Gder ∩ Z(G).

36

Reductive Groups and Steinberg Maps

Hence, this image is finite since G is reductive. Consequently, CG (g )/ f (CG (g)) is finite, as required. Thus, (e) is proved. 1.3.11 Let G and G be connected reductive algebraic groups over k. Let f : G → G be an isogeny, that is, a surjective homomorphism of algebraic groups such that ker( f ) is finite. Note that then ker( f ) is automatically contained in the centre of G. Further to the properties in 1.3.10, we can describe quite precisely the effect of f on roots and coroots. For the following discussion, we refer to [Ch05, §18.2], [MaTe11, §11], [Spr98, §9.6], [St99, §1] for further details. Let T be a maximal torus of G; then ker( f ) ⊆ T and T = f (T) is a maximal torus of G. Let (X, R, Y, R∨ ) and (X , R , Y , R ∨ ) be the corresponding root data. The map f induces a homomorphism ϕ : X  → X such that ϕ(λ ) = λ  ◦ f |T for all λ  ∈ X . The transpose map is given by ϕtr : Y → Y , ν → f ◦ ν. Then it follows that ϕ is a p-isogeny of root data as in Definition 1.2.9, where p is the characteristic exponent of k. (Recall that the characteristic exponent of k is 1 in case char(k) = 0 and is equal to char(k) otherwise.) The numbers {qα | α ∈ R} and the bijection R → R , α → α† , in (I2) come about as follows. Let α ∈ R and consider the corresponding root subgroup Uα ⊆ G; see 1.3.1. Then f (Uα ) is a one-dimensional closed connected unipotent subgroup of G normalised by T. Hence, there is a well-defined α† ∈ R  such that f (Uα ) equals the root subgroup Uα † in G. Let uα : k+ → Uα and uα † : k+ → Uα † be the corresponding isomorphisms. Then the map f : Uα → Uα † has the following form. There is some cα ∈ k × such that f (uα (ξ)) = uα † (cα ξ qα )

for all ξ ∈ k+ .

In this situation, the numbers {qα } will also be called the root exponents of f . The above discussion shows that an isogeny of connected reductive groups induces a p-isogeny of root data. Conversely, we have the following fundamental result. Theorem 1.3.12 (Isogeny Theorem) Let G and G be connected reductive algebraic groups over k, let T ⊆ G and T  ⊆ G be maximal tori, and let ϕ : X(T) → X(T) be a p-isogeny of their root data (see Definition 1.2.9), where p is the characteristic exponent of k. Then there exists an isogeny f : G → G which maps T onto T and induces ϕ. If f  : G → G is another isogeny with these properties, then there exists some t ∈ T such that f (g) = f (tgt −1 ) for all g ∈ G. See [St99] for a recent, quite short proof of this fundamental result which, for semisimple groups, is one of the main results of the Séminaire Chevalley; see [Ch05, §18.2]. As a first consequence, we have: Corollary 1.3.13 (Isomorphism Theorem) In the setting of the Isogeny Theorem,

1.3 Chevalley’s Classification Theorems

37

assume that ϕ is an isomorphism of root data. Then the isogeny f : G → G is an isomorphism of algebraic groups. Proof We use the notation in 1.3.11. Since ϕ is an isomorphism, the inverse map ϕ−1 : X → X  also defines an isogeny of root data. By Theorem 1.3.12 there exist isogenies f : G → G and g : G → G corresponding to ϕ and ϕ−1 . Then g ◦ f induces the identity isogeny of the root datum of G and hence equals the inner automorphism ιt for some t ∈ T. Thus g  ◦ f = idG with g  = ι−1 t ◦ g, and then f ◦ g  ◦ f = f and f ◦ g  = idG because f is surjective. Hence f is an isomorphism with g  as its inverse.  The general theory is completed by the following existence result. Theorem 1.3.14 (Existence Theorem) Let R = (X, R, Y, R∨ ) be a root datum. Then there exists a connected reductive algebraic group G over k and a maximal torus T ⊆ G such that R is isomorphic to the root datum of G relative to T. For semisimple groups, this is originally due to Chevalley; see [Ch55] and the comments in [Ch05, §24]. See [Ca72], [St67, §5, Theorem 6] where this is explained in detail, following and extending Chevalley’s original approach; see also [Ge17] where the question of choosing signs in a Chevalley basis for the underlying semisimple Lie algebra is resolved. The general case can be reduced to this one; see [Spr98, §10.1] and [DG70/11, Exposé XXV]. Only recently, Lusztig [Lu09c] found a new approach to the general case based on the theory of ‘canonical bases’ of quantum groups (completing earlier results of Kostant [Ko66]). Example 1.3.15 Let us see what the above results mean in the simplest nontrivial case where R = (X, R, Y, R∨ ) is a root datum of Cartan type A1 . Let G be a corresponding connected reductive algebraic group over k. Now, since C = (2) is the Cartan matrix in this case, R is determined by an equation

2= a˘i ai where d = rank X = rank Y and ai, a˘i ∈ Z for all i; 1id

see Remark 1.2.13. Up to isomorphism (where isomorphisms are determined by an invertible matrix P over Z as in Remark 1.2.20), there are three possible cases: (1) (a1, . . . , ad ) = (2, 0, . . . , 0) and (a˘1, . . . , a˘ d ) = (1, 0, . . . , 0), in which case G  SL2 (k) × (k× )d−1 . (2) (a1, . . . , ad ) = (1, 0, . . . , 0) and (a˘1, . . . , a˘ d ) = (2, 0, . . . , 0), in which case G  PGL2 (k) × (k× )d−1 . (3) d  2, (a1, . . . , ad ) = (1, 1, 0, . . . , 0) and (a˘1, . . . , a˘ d ) = (1, 1, 0, . . . , 0), in which case G  GL2 (k) × (k× )d−2 .

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This is contained in [St99, 2.2]; we leave it as an exercise to the reader. In particular, for d = 1 (that is, G semisimple), we have either G  SL2 (k) or G  PGL2 (k). Besides its fundamental importance for the classification of connected reductive algebraic groups, the Isogeny Theorem is an indispensable tool for showing the existence of homomorphisms with prescribed properties. Here are the first examples. Example 1.3.16 Let G be a connected reductive algebraic group over k and T be a maximal torus of G. Let (X, R, Y, R∨ ) be the corresponding root datum. Then there exists an automorphism of algebraic groups τ : G → G such that τ(t) = t −1

(t ∈ T)

τ(Uα ) = U−α

and

(α ∈ R).

Indeed, ϕ : X → X, λ → −λ, certainly is a p-isogeny, where qα = 1 for all α ∈ R. Hence, since ϕ is bijective, Corollary 1.3.13 shows that there exists an automorphism τ : G → G such that τ(T) = T and such that ϕ is the map induced by τ on X. Now, as discussed in 1.1.11, there is a natural bijection between group homomorphisms of X into itself and algebraic homomorphisms of T into itself. Under this bijection, ϕ clearly corresponds to the map t → t −1 on T. Hence, τ has the required properties. Example 1.3.17 Let p be a prime number and G be a connected reductive algebraic group over k = F p . Let T be a maximal torus of G and (X, R, Y, R∨ ) be the corresponding root datum. Then ϕ : X → X, λ → pλ, certainly is a p-isogeny of root data, where qα = p for all α ∈ R. Hence, by Theorem 1.3.12, there exists an isogeny Fp : G → G such that Fp (T) = T and such that Fp induces ϕ on X. Arguing as in the previous example, it follows that Fp (Uα ) = Uα

(α ∈ R)

and

Fp (t) = t p

(t ∈ T).

p (ai j ),

is an isogeny satisfying the For example, Fp : GLn (k) → GLn (k), (ai j ) → above conditions. We shall see in Section 1.4 that the fixed point set of G under Fp is a finite group. More generally, we shall consider isogenies F : G → G such that F d = Fpm for some d, m  1. The finite groups arising as fixed point sets of connected reductive groups under such isogenies are the finite groups of Lie type; see Definition 1.4.7. Example 1.3.18 Let Ri = (Xi, Ri, Yi, Ri∨ ) (for i = 1, . . . , n) be root data. Let R = (X, R, Y, R∨ ) be the direct product of these root data; see Example 1.2.17. For i = 1, . . . , n, let Gi be a connected reductive algebraic group with root datum isomorphic to Ri (relative to a maximal torus Ti ⊆ Gi ). Then, using Corollary 1.3.13, one easily sees that the direct product G := G1 × · · · × Gn has root datum isomorphic to R (relative to the maximal torus T := T1 × · · · × Tn of G). Example 1.3.19

Let G = GLn (k), with root datum R = (X, R, Y, R∨ ) as in

1.3 Chevalley’s Classification Theorems

39

Example 1.3.7. It is given by a factorisation C = A˘ · Atr where C = (ci j )1i, jn−1 is the Cartan matrix of type An−1 and A = A˘ is a certain matrix of size (n − 1) × n. Then, by the procedure in 1.2.18, we obtain an isogeny ϕ : X → X via the pair of matrices (P, P◦ ) = (−Jn, Jn−1 ) where, for any m  1, we set 0 .. .

··· . ..

1   0  ∈ Mm (k). . ..  .. 0 1 .  1 0 ··· 0  Then ϕ has order 2. So there is a corresponding automorphism of algebraic groups γ : GLn (k) → GLn (k) which maps the maximal torus T into itself and induces ϕ on X. Concretely, the map given by    Jm :=    

0 . ..

γ : GLn (k) → GLn (k),

g → Jn (g tr )−1 Jn,

is an automorphism with this property. Remark 1.3.20 Let f : G → G be an isogeny of connected reductive algebraic groups over k. In the setting of 1.3.11, let {qα | α ∈ R} be the root exponents of f . Following [Spr98, 9.6.3], we say that f is a central isogeny if qα = 1 for all α ∈ R. The terminology is justified as follows. Consider the corresponding homomorphism of Lie algebras d1 f : L(G) → L(G). Then, by [Bor91, 22.4], f is a central isogeny if and only if the kernel of d1 f is contained in the centre of L(G). For example, the isogeny in Example 1.3.16 is central while that in Example 1.3.17 is not. There are extensions of the Isogeny Theorem to the case where we consider homomorphisms whose kernel is still central but not finite: We shall only formulate the following version here. (This will be needed, for example, in Section 1.7.) 1.3.21 Let G, G be connected reductive algebraic groups over k, and f : G → G be a homomorphism of algebraic groups. (a) Following [Bo06, Chap. I, 3.A], we say that f is an isotypy if ker( f ) ⊆ Z(G)  ⊆ f (G). If this is the case, then we have G  = f (G).Z(G  ), f (G ) = G  and Gder der der  . and f restricts to an isogeny Gder → Gder (b) Now let T ⊆ G and T  ⊆ G be maximal tori such that f (T) ⊆ T. Then f induces a group homomorphism ϕ : X(T) → X(T), λ → λ ◦ f |T . In analogy to Remark 1.3.20, we say that f is a central isotypy if ϕ is a homomorphism of root data as in 1.2.2. (Note that, as pointed out in the remarks following [Jan03, II, Prop. 1.14], a central isotypy is automatically an isotypy.) Theorem 1.3.22 (Extended Isogeny Theorem; cf. [Jan03, II, 1.14, 1.15], [St99, §5]) Let G and G be connected reductive algebraic groups over k, let T ⊆ G

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and T ⊆ G be maximal tori, and let ϕ : X(T) → X(T) be a homomorphism of their root data (see 1.2.2). Then there exists a central isotypy f : G → G such that f (T) ⊆ T and f induces ϕ. Furthermore, the following hold. (a) If f  : G → G is another central isotypy inducing ϕ, then there exists some t ∈ T such that f (g) = f (tgt −1 ) for all g ∈ G. (b) If f |T : T → T is an isomorphism, then so is f : G → G. Proof Let Π be a base of the root system R ⊆ X(T). For α ∈ Π, consider the subgroup Gα = T, Uα, U−α ⊆ G defined in 1.3.1. Then Gα ∩ Gβ = T for α  β in Π. As in [Jan03, II, §1.13], one sees that there exists a map  Gα → G  f: α∈Π

which is a homomorphism on each Gα and is such that f maps T into T  and induces ϕ. Now, Uα and U−β certainly commute with each other for all α  β in Π (by Chevalley’s commutator relations; see [MaTe11, 11.8]). Hence, by [St99, Theorem 5.3], f extends to a homomorphism of algebraic groups from G to G. The uniqueness statement in (a) is proved as in the case of Theorem 1.3.12; see [St99, §3]. Finally, (b) holds by [Jan03, II, §1.15]. 

1.4 Frobenius Maps and Steinberg Maps We assume in this section that k = F p is an algebraic closure of the finite field with p elements, where p is a prime number. We consider a particular class of isogenies in this context, the ‘Steinberg maps’. This will be treated in some detail, where one aim is to work out explicitly some useful characterisations of Steinberg maps in terms of isogenies of root data. In particular, in Proposition 1.4.18, we recover the set-up in Example 1.3.17. We also establish a precise characterisation of Frobenius maps among all Steinberg maps; see Proposition 1.4.28. Definition 1.4.1 Let X be an affine variety over k. Let q be a power of p and Fq ⊆ k be the finite subfield with q elements. We say that X has an Fq -rational structure (or that X is defined over Fq ) if there exists some n  1 and an isomorphism of affine varieties ι : X → X where X ⊆ kn is Zariski closed and stable under the standard Frobenius map Fq : kn → kn,

q

q

(ξ1, . . . , ξn ) → (ξ1 , . . . , ξn ).

In this case, there is a unique morphism of affine varieties F : X → X such that ι ◦ F = Fq ◦ ι; it is called the Frobenius map corresponding to the Fq -rational

1.4 Frobenius Maps and Steinberg Maps

41

structure of X. Note that Fq is a bijective morphism whose fixed point set is Fqn . Consequently, F is a bijective morphism such that |XF | < ∞

where

XF := {x ∈ X | F(x) = x}.

Example 1.4.2 Let X ⊆ kn be Zariski closed. Then X is called Fq -closed if X is defined by a set of polynomials in Fq [T1, . . . , Tn ]. If this holds, then X is stable under Fq and so X has an Fq -rational structure, as defined above; the fixed point set XFq consists precisely of those x ∈ X which have all their coordinates in Fq . Conversely, if Fq (X) ⊆ X, then X is Fq -closed. (The proof uses the fact that k ⊇ Fq is a separable field extension; see [Ge03a, 4.1.6], [Bor91, AG.14.4]. In general, the discussion of rational structures is much more complicated.) Remark 1.4.3 Let X be an affine variety over k and assume that X is defined over Fq , with Frobenius map F : X → X. Here are some basic properties of F. First note that F 2, F 3, . . . are also Frobenius maps. Furthermore, for any x ∈ X, we have F m (x) = x for some m  1. Hence,  m m XF where |XF | < ∞ for all m  1. X= m1

(Note that every element of k lies in a finite subfield of k.) Finally, it is also clear that, if X ⊆ X is a closed subset such that F(X) ⊆ X, then X is defined over Fq , with Frobenius map given by the restriction of F to X. Remark 1.4.4 Let X be an affine variety over k and let A be the algebra of regular functions on X. There is an intrinsic characterisation of Frobenius maps in terms of A, as follows. A morphism F : X → X is the Frobenius map corresponding to an Fq -rational structure of X if and only if the following two conditions hold for the associated algebra homomorphism F ∗ : A → A: (a) F ∗ is injective and F ∗ (A) = {aq | a ∈ A}. e (b) For each a ∈ A, there exists some e  1 such that (F ∗ )e (a) = aq . One checks that (a) and (b) are satisfied for the standard Frobenius map Fq : kn → kn . This implies that (a) and (b) hold for any Fq -stable closed subset X ⊆ kn as in Example 1.4.2. The converse requires a bit more work; see [Ge03a, §4.1] or [Sr79, Chap. II] for details. One advantage of this characterisation of Frobenius maps is, for example, that it provides an easy proof of the following statement (see, for example, [Ge03a, Exercise 4.4]): (c) If F is a Frobenius map (with respect to Fq , as above) and γ : X → X is an automorphism of affine varieties of finite order which commutes with F, then γ ◦ F also is a Frobenius map on X (with respect to Fq , same q).

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Reductive Groups and Steinberg Maps

The above characterisation is equivalent to the definition of an ‘abstract affine algebraic (Fq, k)-set’ in [Car55-56]. In the sequel, G will always be a linear algebraic group over k = F p . 1.4.5 Assume that, as an affine variety, G is defined over Fq with corresponding Frobenius map F. Then we say that G (as an algebraic group) is defined over Fq if F is a group homomorphism. In this case, the set of fixed points GF is a finite group. There is a more concrete description, similar to Definition 1.4.1. We have the standard Frobenius map Fq : GLn (k) → GLn (k),

q

(ai j ) → (ai j ).

Then G is defined over Fq if and only if there is a homomorphism ι : G → GLn (k)

(for some n  1)

of algebraic groups such that ι is an isomorphism onto its image and the image is stable under Fq ; in this case, the corresponding Frobenius map F : G → G is defined by the condition that ι ◦ F = Fq ◦ ι. (See [Ge03a, 4.1.11] for further details.) In particular, if G ⊆ GLn (k) is a closed subgroup defined by a collection of polynomials with coefficients in Fq , then Fq restricts to a Frobenius map on G. Example 1.4.6 Let T ⊆ G be an abelian subgroup consisting of semisimple elements (e.g., a torus). We claim that there always exists some Frobenius map F : G → G (with respect to an Fq -rational structure on G) such that T is F-stable and F(t) = t q for all t ∈ T. Indeed, we can realise G as a closed subgroup G ⊆ GLn (k) for some n  1. Since T consists of commuting semisimple elements, we can assume that then T consists of diagonal matrices. Now, the defining ideal of G (as an algebraic subset of GLn (k)) is generated by a finite set of polynomials with coefficients in k. So there is some q = pm (m  1) such that all these coefficients lie in Fq . Then G is stable under the standard Frobenius map Fq on GLn (k). So Fq restricts to a Frobenius map F : G → G. Since any t ∈ T is a diagonal matrix, we have F(t) = t q . Definition 1.4.7 Let F : G → G be an endomorphism of algebraic groups. Then F is called a Steinberg map if some power of F is the Frobenius map with respect to an Fq -rational structure on G, for some power q of p. Note that, in this case, F is a bijective homomorphism of algebraic groups and GF is a finite group. If G is connected and reductive, then GF will be called a finite group of Lie type or a finite reductive group. Note that, if F : G → G is a Steinberg map and H ⊆ G is a closed subgroup such that F(H) ⊆ H, then Remark 1.4.3 implies that F |H : H → H also is a Steinberg map. This will be used frequently without further mention.

1.4 Frobenius Maps and Steinberg Maps

43

The following result is the key tool to pass from properties of G to properties of the finite group GF . Theorem 1.4.8 ([La56], [St68, 10.1]) Assume that G is connected. Let F : G → G be a Steinberg map (or, more generally, any endomorphism such that |GF | < ∞). Then the map L : G → G, g → g −1 F(g), is surjective. Proof If F is a Steinberg map (and this is the case that we are mainly interested in), then [Mu03] gives a quick proof, as follows. The group G acts on itself (on the right) where g ∈ G sends x ∈ G to g −1 xF(g). Any action of an algebraic group on an affine variety has a closed orbit; see [Ge03a, 2.5.2]. Let Ω be such a closed orbit and let x ∈ Ω. Since G is connected, it will be sufficient to show that dim G = dim Ω (because then G = Ω and so 1 ∈ Ω). We have dim Ω = dim G − dim StabG (x) (see [Ge03a, 2.5.3]), so it will be sufficient to show that StabG (x) is finite. Now, an element g ∈ G belongs to this stabiliser if and only if g −1 xF(g) = x, which is equivalent to f (g) = g, where f (g) := xF(g)x −1 . Let m  1 be such that F m is a Frobenius map and F m (x) = x (see Remark 1.4.3). Let r  1 be the order of the element xF(x)F 2 (x) . . . F m−1 (x) ∈ G. Then f mr (g) = F mr (g) for all g ∈ G. So f mr (g) = g has only finitely many solutions in G, hence f (g) = g has only finitely many solutions in G.  For various parts of the subsequent discussion it would be sufficient to work with endomorphisms of G whose fixed point set is finite. However, we will just formulate everything in terms of Steinberg maps, as defined above. We note that the discussion in [St68, 11.6] in combination with Proposition 1.4.18 below implies that an endomorphism of a simple algebraic group with a finite fixed point set is automatically a Steinberg map; see also Example 1.4.20 below. Here is the prototype of an application of the above theorem. Proposition 1.4.9 Assume that G is connected and let F : G → G be a Steinberg map. Let X   be a set on which G acts transitively; let F  : X → X be a map such that F (g.x) = F(g).F (x) for g ∈ G and x ∈ X. (a) There exists some x0 ∈ X such that F (x0 ) = x0 . (b) If x0 is as in (a) and StabG (x0 ) ⊆ G is closed and connected, then the set {x ∈ X | F (x) = x} is a single GF -orbit. Proof (a) Take any x ∈ X. Since G acts transitively, we have F (x) = g −1 .x for some g ∈ G. By Theorem 1.4.8, we can write g = h−1 F(h). Then one immediately checks that x0 := h.x is fixed by F . (b) Let H := StabG (x0 ). Since F (x0 ) = x0 , we have F(H) ⊆ H and so F restricts to a Steinberg map on H. Let x ∈ X be such that F (x) = x and write x = g.x0 for some g ∈ G. Then g.x0 = x = F (x) = F (g.x0 ) = F(g).F (x0 ) = F(g).x0 and so

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Reductive Groups and Steinberg Maps

g −1 F(g) ∈ H. By Theorem 1.4.8 (applied to H), we can write g −1 F(g) = h−1 F(h) for some h ∈ H. Then gh−1 ∈ GF and x = gh−1 .x0 as desired.  Example 1.4.10 Assume that G is connected and let F : G → G be a Steinberg map. Let C be a conjugacy class of G such that F(C) = C. Then G acts transitively on C by conjugation; let F  be the restriction of F to C. Applying Proposition 1.4.9 yields that there exists an element x ∈ C such that F(x) = x. Furthermore, if CG (x) is connected, then C F is a single GF -conjugacy class. Similarly, if G is reductive, there exists a pair (T, B) with T an F-stable maximal torus of G and B an F-stable Borel subgroup with T ⊆ B. (Just note that, by Theorem 1.3.3, all these pairs are conjugate in G and, by 1.3.10(a), F preserves the set of all these pairs.) An F-stable maximal torus of G which is contained in an F-stable Borel subgroup of G will be called a maximally split torus. Example 1.4.11 Let F : G → G be a Steinberg map. Let U, V ⊆ G be F-stable closed subgroups. (a) Suppose that G = U.V and U ∩ V is connected. Then GF = UF .VF . Indeed, let g ∈ GF and write g = uv where u ∈ U and v ∈ V. Then uv = F(uv) = F(u)F(v) and so x := u−1 F(u) = vF(v)−1 ∈ U ∩ V. By Theorem 1.4.8 (applied to U ∩ V) we can write x = y −1 F(y) for some y ∈ U ∩ V. It follows that F(uy −1 ) = uy −1 and F(yv) = yv. Hence, we have g = (uy −1 )(yv) ∈ UF .VF . (b) Assume that U is connected and U ⊆ V. Let v ∈ V be such that the coset Uv is F-stable. Then there exists some u ∈ U such that F(uv) = uv. (Indeed, U acts transitively on X := Uv by left multiplication; so we can just apply Proposition 1.4.9(a).) Furthermore, if U is normal in V, then F induces an abstract automorphism of V/U (which we denote by the same symbol) and we conclude that VF /UF is isomorphic to the group of fixed points of F on V/U. Proposition 1.4.12 (Cf. [St68, 10.10]) Let G be connected reductive and F : G → G be a Steinberg map. Then all maximally split tori of G are GF -conjugate. More precisely, all pairs (T, B) consisting of an F-stable Borel subgroup B and an Fstable maximal torus T ⊆ B are conjugate in GF . Proof Let (T, B) and (T1, B1 ) be two pairs as above. By Theorem 1.3.3, there exists some x ∈ G such that xBx −1 = B1 and xTx −1 = T1 . Since B, B1 are F-stable, this implies that x −1 F(x) ∈ NG (B) = B, where the last equality holds by [Bor91, 11.16] or [Spr98, 6.4.9]. Similarly, since T, T1 are F-stable, we have x −1 F(x) ∈ NG (T). Hence, x −1 F(x) ∈ B ∩ NG (T) = T (see Theorem 1.3.2). Applying Theorem 1.4.8 to the restriction of F to T, we obtain an element t ∈ T such that x −1 F(x) = t −1 F(t). Then g := xt −1 ∈ GF and g simultaneously conjugates B to B1 and T to T1 . 

1.4 Frobenius Maps and Steinberg Maps

45

The following result deals with a subtlety concerning Steinberg maps: A surjective homomorphism of algebraic groups will not necessarily induce a surjective map on the level of the fixed point sets under Steinberg maps. But one can say precisely what happens in this situation: Proposition 1.4.13 (Cf. [St68, 4.5]) Let f : G → G be a surjective homomorphism of connected algebraic groups such that K := ker( f ) is contained in the centre of G. Let F : G → G and F  : G → G be Steinberg maps such that F  ◦ f = f ◦ F.  We denote G = GF and G  = GF . (a) Let L : G → G, g → g −1 F(g). Then L (K) is a normal subgroup of K. (b) f (G) ⊆ G  is a normal subgroup and G / f (G)  K/L (K). In particular, if K is connected, then L (K) = K and f (G) = G . (c) If K is finite (that is, f is an isogeny), then |G| = |G  |. Proof We apply Lemma 1.1.9 with A = G, B = G, σ = F, τ = F . Then C = {g−1 F(g) | g ∈ K} = L (K); thus, (a) holds. Since G is connected, we have L (G) = G by Theorem 1.4.8; this yields (b). Furthermore, we have ker(L |K ) = {z ∈ K | z−1 F(z) = 1} = KF = ker( f |G ). Now assume that K is finite. Then |K/L (K)| = | ker( f |G )| and, hence, |G| = | f (G)||K/L (K)|. But, by (b), K/L (K) and G / f (G) have the same order and so |G| = |G  |, that is, (c) holds.  Lemma 1.4.14 (Cf. [St68, 10.9]) Let G be connected and F : G → G be a Steinberg map. Let y ∈ G and define F  : G → G by F (g) = yF(g)y −1 for all  g ∈ G. Then F  is a Steinberg map and we have GF  GF . Furthermore, if F is a Frobenius map corresponding to an Fq -rational structure, then so is F  (with the same q). Proof Since G is connected, Theorem 1.4.8 shows that we can write y = x −1 F(x) for some x ∈ G. Then F (g) = x −1 F(xgx −1 )x for all g ∈ G. Thus, we have F  = ι−1 x ◦ F ◦ ι x where ι x denotes the inner automorphism of G defined by x. This formula shows that F (g) = g if and only if xgx −1 ∈ GF . Hence, conjugation with  x defines a group isomorphism GF  GF . Now we show that F  is a Steinberg map. For m  1, we have (F )m (g) = x −1 F m (xgx −1 )x for all g ∈ G. By Remark 1.4.3 (and the definition of Steinberg maps), there exists some m  1 such that F m (x) = x. For this m, we have (F )m (g) = F m (g) for all g ∈ G. Thus, F  is a Steinberg map. Finally, assume that F is a Frobenius map. We use the characterisation in Remark 1.4.4 to show that F  is a Frobenius map. Since F  is the conjugate of F by an automorphism of G, we have that F ∗ is the conjugate of F ∗ by an algebra

46

Reductive Groups and Steinberg Maps

automorphism of A. So F ∗ is injective and F ∗ (A) = {aq | a ∈ A}. On the other e hand, (F )m = F m . So, if a ∈ A and e  1 are such that (F ∗ )e (a) = aq , then em  (F ∗ )em (a) = (F ∗ )em (a) = aq , as required. Lemma 1.4.15 Assume that G is connected reductive. Let F : G → G be a Steinberg map and T be an F-stable maximal torus of G. Let F  : G → G be another isogeny of G such that F (T) = T. If F and F  induce the same map on X(T), then there exists some y ∈ T such that F (g) = yF(g)y −1 for all g ∈ G. In particular, the conclusions of Lemma 1.4.14 apply to F . Proof Since F, F  induce the same map on X(T), Theorem 1.3.12 implies that there exists some y ∈ T such that F (g) = yF(g)y −1 for all g ∈ G.  Example 1.4.16 Assume that G is connected reductive and let T ⊆ G be a maximal torus, with associated root datum R = (X, R, Y, R∨ ). Let Fp : G → G be an isogeny as in Example 1.3.17, such that Fp (Uα ) = Uα

(α ∈ R)

and

Fp (t) = t p

(t ∈ T).

(Note that Fp is only unique up to composition with inner automorphisms defined by elements of T.) Now, there is a stronger version of Theorem 1.3.14 (involving fields of definition), which shows that an isogeny Fp as above is the Frobenius map with respect to an F p -rational structure on G; see [Spr98, 16.3.3], [DG70/11, Exposé XXV]. (A more direct argument is given by [Klu16, 2.93]; alternatively, one could use Lusztig’s approach [Lu09c], as pointed out in [DG70/11, Exposé XXV, footnote 1].) If G is semisimple, then this is also contained in [St67, Theorem 6 (p. 58)], [Bor70, Part A, §3.3 and §4.3]. Note that, once some Fp as above is known to be a Frobenius map, then Lemma 1.4.15 shows that any Fp satisfying the above conditions is a Frobenius map. We also point out that, in any case, it is easily seen that Fp is a Steinberg map. Indeed, by Example 1.4.6, there exists a Frobenius map F : G → G such that F(t) = t q for all t ∈ T, where q = pm for some m  1. Then F induces multiplication with q on X. Hence, F induces the same map on X as Fpm . So Lemma 1.4.15 shows that Fp is a Steinberg map. Lemma 1.4.17 Let T be a torus over k and F : T → T be the Frobenius map corresponding to an Fq -rational structure on T, where q is a power of p. Then the map induced by F on X = X(T) is given by qψ0 where ψ0 : X → X is an invertible endomorphism of finite order. Proof (Cf. [DiMi20, Prop. 4.2.3], [Sa71, §I.2.4].) Let A be the algebra of regular functions on T. Let λ ∈ X. Composing λ with the inclusion k × → k, we can regard λ as a regular function on T, that is, λ ∈ A. By Remark 1.4.4, we have

1.4 Frobenius Maps and Steinberg Maps

47

F ∗ (A) = {aq | a ∈ A}. Hence, λ q = F ∗ (λ• ) for some λ• ∈ A. Then λ• (F(t)) = λ(t)q = λ(t q )

for all t ∈ T.

(∗)

Since F : T → T is a bijective group homomorphism, λ• is uniquely determined by (∗); furthermore, λ• (T) ⊆ k× and λ• : T → k× is a group homomorphism. Hence, λ• ∈ X. We also see that the map ψ : X → X, λ → λ• , is linear. Finally, (∗) implies

m that ψ m (λ) (F m (t)) = λ(t q ) for all m  1. Now, by Example 1.4.6, we can find m some m  1 such that F m (t) = t q for all t ∈ T. For any such m, we then have ψ m (λ) = λ for all λ ∈ X. Hence, ψ is an endomorphism of X of order dividing m.  Setting ψ0 := ψ −1 , the map on X induced by F is given by qψ0 . We now obtain the following characterisation of Steinberg maps. Proposition 1.4.18 Let G be connected reductive, F : G → G be an isogeny and T ⊆ G be an F-stable maximal torus. Then the following are equivalent. (i) F is a Steinberg map. (ii) There exist integers d, m  1 such that the map induced by F d on X = X(T) is given by scalar multiplication with pm . (iii) There is an isogeny Fp as in Example 1.4.16 such that some positive power of F equals some positive power of Fp . Proof ‘(i) ⇒ (ii)’ Let d1  1 be such that F d1 is a Frobenius map with respect to some Fq0 -rational structure on G where q0 is a power of p. Let ϕ : X → X be the map induced by F. By Lemma 1.4.17, we have ϕ d1 = q0 ψ0 where ψ0 : X → X has finite order, e  1 say. Then ϕ d1 e = q0e idX . ‘(ii) ⇒ (iii)’ Assume that the map induced by F d on X is given by scalar multiplication with pm . Let Fp be as in Example 1.4.16. Then F d and Fpm induce the same map on X and so there is some y ∈ T such that F d (g) = yFpm (g)y −1 for all g ∈ G; see Lemma 1.4.15. By Theorem 1.4.8, we can write y = x −1 Fpm (x) for some m x ∈ T. As in the proof of Lemma 1.4.14, we have F d = ι−1 x ◦ Fp ◦ ι x . But then we d −1 m  also have F = (ιx ◦ Fp ◦ ιx ) and it remains to note that Fp := ι−1 x ◦ Fp ◦ ι x is a map satisfying the conditions in Example 1.4.16. ‘(iii) ⇒ (i)’ This is clear by definition, since Fp is known to be a Steinberg map (see Example 1.4.16).  Proposition 1.4.19 Assume that G is connected. Let F : G → G be a Steinberg map. Let q be the positive real number defined by q d = q0 , where d  1 is an integer such that F d is a Frobenius map with respect to some Fq0 -rational structure on G (where q0 is a power of p). Then q does not depend on d, q0 . Furthermore, the following hold for every F-stable maximal torus T of G.

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(a) We have det(ϕ) = ±qrank X where ϕ : X(T) → X(T) is the linear map induced by F. (b) The map induced by F on XR := R ⊗Z X(T) is of the form qϕ0 where ϕ0 ∈ GL(XR ) has finite order. Proof The independence of q from d, q0 is clear, once (a) is established. So let T be any F-stable maximal torus of G (which exists by Example 1.4.10). Let X = X(T) and ϕ : X → X be the linear map induced by F. (a) By Remark 1.4.3, the restriction of F d to T is a Frobenius map with respect to an Fq0 -rational structure on T. So, by Lemma 1.4.17, we have ϕ d = q0 ψ0 where ψ0 : X → X has finite order. Then det(ϕ)d = q0rank X det(ψ0 ). Since det(ϕ) is an integer and det(ψ0 ) is a root of unity, we must have det(ϕ)d = ±q0rank X and, hence, det(ϕ) = ±qrank X . (b) Denote by ϕR the canonical extension of ϕ to XR . Then ϕ0 := q−1 ϕR is a linear map such that ϕ0d = ψ0 . Hence, ϕR = qϕ0 where ϕ0 has finite order.  Having defined q, one may also write G(q) instead of GF if there is no danger of confusion. An alternative characterisation of q will be given in Remark 1.6.8(a). The defining formula in Proposition 1.4.19 shows that q is a dth root of a prime power. The examples below will show that all such roots do actually occur. Example 1.4.20 This example is just meant to give a simple illustration of the difference between Steinberg maps and arbitrary isogenies with a finite fixed point set. Let q, q  be two distinct powers of p. Let G = SL2 (k) × SL2 (k) and define F : G → G by F(x, y) = (Fq (x), Fq (y)) where Fq and Fq denote the standard Frobenius maps with respect to q and q , respectively. Then F is a bijective homomorphism of algebraic groups and GF = SL2 (q) × SL2 (q ) certainly is finite. Let T  k× be the standard maximal torus of SL2 (k). Then T × T is an F-stable maximal torus of G and we can identify X(T × T) with Z2 . Under this identification, the map induced by F is given by (n, m) → (qn, q  m) for all (n, m) ∈ Z2 . Thus, Proposition 1.4.18(ii) shows that F is not a Steinberg map. Example 1.4.21 Assume that G is connected reductive. Let T ⊆ G be a maximal torus and R = (X, R, Y, R∨ ) be the root datum relative to T. Assume that we have an automorphism ϕ0 : X → X of finite order such that ϕ0 (R) = R and ϕ0tr (R∨ ) = R∨ . (In particular, ϕ0 is an isogeny of root data with all root exponents equal to 1.) Let q = pm for some m  1. Then qϕ0 is a p-isogeny and so, by Theorem 1.3.12, there is a corresponding isogeny F : G → G such that F(T) = T. Now F is a Steinberg map by Proposition 1.4.18; the number q = pm satisfies the conditions in Proposition 1.4.19. If G is semisimple, then G = GF is an untwisted (ϕ0 = idX ) or twisted Chevalley group; see Steinberg’s lecture notes [St67, §11] for further details. We discuss the various possibilities in more detail in Section 1.6.

1.4 Frobenius Maps and Steinberg Maps

49

Let us just give one concrete example. Let G = GLn (k). If ϕ0 = idX , then we obtain a ‘standard’ Frobenius map F : GLn (k) → GLn (k),

q

(ai j ) → (ai j ),

such that GLn (k)F = GLn (q), the finite general linear group over Fq . On the other hand, the automorphism ϕ0 : X → X of order 2 in Example 1.3.19 also satisfies the above conditions. The corresponding isogeny F  = F ◦ γ : GLn (k) → GLn (k) is a  Steinberg map such that GUn (q) := GLn (k)F is the finite general unitary group.  Similarly, we have SLn (k)F = SLn (q) and SLn (k)F = SUn (q). Example 1.4.22 Assume that G is connected reductive and that the root datum R = (X, R, Y, R∨ ) relative to a maximal torus T ⊆ G is as in Example 1.2.19, where p = 2 or 3. For any m  0, we have a p-isogeny ϕm on X such that ϕ2m = p2m+1 idX . Let F : G → G be the corresponding isogeny such that F(T) = T. Then Proposition 1.4.18 shows that F is a Steinberg map; the number q in √ Proposition 1.4.19 is given by q = p2m+1 . In these cases 2 , G = GF is the Suzuki group 2B2 (q2 ) = 2C 2 (q2 ), the Ree group 2G2 (q2 ) or the Ree group 2F 4 (q2 ), respectively. See [Ca72, Chap. 13] or Steinberg’s lecture notes [St67, §11] for further details. Example 1.4.23 Let F : G → G be the Frobenius map corresponding to some Fq0 -rational structure on G where q0 is a power of p. Consider the direct product G = G × · · · × G (with r factors) and define a map F  : G → G,

(g1, g2, . . . , gr ) → (F(gr ), g1, . . . , gr−1 ).

Then F  is a homomorphism of algebraic groups and we have (F )r (g1, . . . , gr ) = (F(g1 ), F(g2 ), . . . , F(gr )) for all gi ∈ G. Clearly, the latter map is a Frobenius map on G. Thus, F  is a √ Steinberg map. The number q in Proposition 1.4.19 is given by q = r q0 . Note also that we have a group isomorphism 

∼

GF −→ GF ,

(g1, g2, . . . , gr ) → g1 .

(This example is mentioned in [DeLu76, §11].) We have the following extension of the Isogeny Theorem 1.3.12, taking into account the presence of Steinberg maps. 2

In finite group theory, it is common to write 2B 2 (q 2 ) etc., although this is not entirely consistent with the general setting of algebraic groups where the notation should be 2B2 (q). See also [DN19] for an interpretation as algebraic groups over F√ p .

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Reductive Groups and Steinberg Maps

Lemma 1.4.24 In the set-up of Theorem 1.3.12 assume, in addition, that there are Steinberg maps F : G → G and F  : G → G such that F(T) = T, F (T) = T and Φ ◦ ϕ = ϕ ◦ Φ where Φ : X(T) → X(T) and Φ : X(T) → X(T) are the maps induced by F and F . Then there exists an isogeny f : G → G that maps T onto T and induces ϕ, such that f ◦ F = F  ◦ f . Proof By Theorem 1.3.12, there exists an isogeny f  : G → G that maps T onto T and induces ϕ. Then f  ◦ F and F  ◦ f  both induce Φ ◦ ϕ = ϕ ◦ Φ. Hence, by Theorem 1.3.12, there exists some t ∈ T such that (F  ◦ f )(g) = ( f  ◦ F)(t −1 gt) for all g ∈ G. Then f (F(t)) ∈ T and so, by Theorem 1.4.8, we can write f (F(t)) = x −1 F (x) for some x ∈ T. We define f : G → G by f (g) = x f (g)x −1 for all g ∈ G. Then f is an isogeny that maps T onto T and also induces ϕ. Furthermore, (F  ◦ f )(g) = F (x)( f  ◦ F)(t −1 gt)F (x)−1 = x( f  ◦ F)(g)x −1 = ( f ◦ F)(g) for all g ∈ G, as required.



Example 1.4.25 Assume that G is connected reductive. Let F : G → G be a Steinberg map and T be an F-stable maximal torus of G. (a) Lemma 1.4.24 immediately shows that an automorphism τ : G → G as in Example 1.3.16 can be chosen such that we also have τ ◦ F = F ◦ τ. (b) Consider an isogeny Fp : G → G as in Example 1.4.16 and let ϕ be the map induced on X = X(T) by F. Since Fp is a Steinberg map, Lemma 1.4.24 shows that there is an isogeny F  : G → G which maps T onto T and induces ϕ, and such that F  ◦ Fp = Fp ◦ F . Since F, F  induce the same map on X, we have that F  is a  Steinberg map such that GF  GF ; see Lemma 1.4.15. (Thus, replacing F by F  if necessary, we can always assume that F ◦ Fp = Fp ◦ F, that is, we are in the setting of [Lu84a, §2.1].) Lemma 1.4.26 Assume that G is connected reductive. Let K be a closed normal ¯ := G/K is connected and reductive. If, subgroup of G. Then K is reductive and G furthermore, F : G → G is a Steinberg map such that F(K) = K, then the map ¯ →G ¯ induced by F is a Steinberg map. F¯ : G Proof Since the unipotent radical in a linear algebraic group is invariant under any automorphism of algebraic groups, it is clear that every closed normal subgroup of ¯ = G/K. First recall from 1.1.6 that G ¯ is a linear G is reductive. Now consider G algebraic group; it is also connected since it is the quotient of a connected group. ¯ is reductive by 1.3.10(b). Finally, G Now let F : G → G be a Steinberg map and assume that F(K) ⊆ K. Then we ¯ → G, ¯ which is bijective. obtain an induced (abstract) group homomorphism F¯ : G ¯ By the universal property of quotients, F is a homomorphism of algebraic groups. ¯ By 1.3.10(a), ¯ be the image of T in G. Let T ⊆ G be an F-stable maximal torus. Let T

1.4 Frobenius Maps and Steinberg Maps

51

¯ we also have F( ¯ is a maximal torus of G; ¯ = T. ¯ Let X = X(T), X¯ = X(T) ¯ and ¯ T) T ¯ note that ϕ is ϕ : X¯ → X be the map induced by the canonical map f : G → G; ¯ Since injective. Let ψ : X → X and ψ¯ : X¯ → X¯ be the maps induced by F and F. m m ¯ ¯ ¯ F ◦ f = f ◦ F, we have ϕ ◦ ψ = ψ ◦ ϕ and so ϕ ◦ ψ = ψ ◦ ϕ for all m  1. Now there is some m  1 such that ψ m is given by scalar multiplication with a power of p. Since ϕ is injective, this implies that ψ¯ m is also given by scalar multiplication with a power of p. Hence, F¯ is a Steinberg map by Proposition 1.4.18.  Finally, we address the question of characterising Frobenius maps among all Steinberg maps on G. The results are certainly well known to the experts and are contained in more advanced texts on reductive groups (like [BoTi65], [Sa71]), where they appear as special cases of general considerations of rationality questions. Since in our case the rational structures are given by Frobenius maps, one can give more direct proofs. The key property is contained in the following result. Lemma 1.4.27 Let G be connected reductive and F : G → G be a Frobenius map with respect to some Fq -rational structure on G. Let T be an F-stable maximal torus. Then the root exponents of F (relative to T) are all equal to q. Proof (Cf. [BoTi65, 6.2], [Sa71, §II.2.1].) Let R be the set of roots with respect to T. Let α ∈ R and uα : k+ → G be the corresponding homomorphism with image Uα ⊆ G. We have F(Uα ) = Uα† , where α† ∈ R; see 1.3.11. In order to identify the root exponents, we need to exhibit a homomorphism uα† : k+ → G whose image is Uα† and such that uα† is an isomorphism onto its image. This is done as follows. Let A be the algebra of regular functions on G. The algebra of regular functions on k+ is given by the polynomial ring k[c] where c is the identity function on k+ . Since uα is an isomorphism onto its image, the induced algebra homomorphism uα∗ : A → k[c] is surjective (see [Ge03a, 2.2.1]). Now consider the standard Frobenius map F1 : k+ → k+ , ξ → ξ q . By Remark 1.4.4, we have F ∗ (A) = {aq | a ∈ A} and F1∗ (k[c]) = k[cq ]. Hence, the composition uα∗ ◦ F ∗ sends A onto k[cq ]. Since F1∗ : k[c] → k[cq ] is an isomorphism, we obtain an algebra homomorphism γ : A → k[c] by setting γ := (F1∗ )−1 ◦ uα∗ ◦ F ∗ ; note that γ is surjective. Let uα† : k+ → G be the morphism of affine varieties such that uα∗ † = γ. Then F ◦ uα = uα† ◦ F1 and so uα† (ξ q ) = (uα† ◦ F1 )(ξ) = (F ◦ uα )(ξ) = F(uα (ξ))

for all ξ ∈ k.

This shows, first of all, that uα† is a group homomorphism with image F(Uα ) = Uα† . Furthermore, since γ is surjective, uα† is an isomorphism onto its image (see again [Ge03a, 2.2.1]). For all t ∈ T and ξ ∈ k, we have F(t)uα† (ξ q )F(t)−1 = F(tuα (ξ)t −1 ) = F(uα (α(t)ξ)) = uα† (α(t)q ξ q ),

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Reductive Groups and Steinberg Maps

which shows that α† (F(t)) = α(t)q for all t ∈ T, as desired.



Proposition 1.4.28 Assume that G is connected reductive. Let F : G → G be an isogeny and T ⊆ G be an F-stable maximal torus. Let ϕ be the map induced by F on X = X(T). Then the following conditions are equivalent. (i) F is a Frobenius map (corresponding to a rational structure on G over a finite subfield of k). (ii) We have ϕ = pm ϕ0 where m ∈ Z1 and ϕ0 : X → X is an automorphism of finite order such that ϕ0 (R) = R and ϕ0tr (R∨ ) = R∨ . (In particular, ϕ0 is an isogeny of root data with all root exponents equal to 1.) (iii) There exists an automorphism of algebraic groups γ : G → G of finite order   such that γ(T) = T and some m   1 such that F = γ ◦ Fpm = Fpm ◦ γ, where Fp is an isogeny as in Example 1.4.16. If these conditions hold, then m (as in (ii)) equals m  (as in (iii)) and F is the Frobenius map with respect to an Fq -rational structure where q = pm . Furthermore, all root exponents of F are equal to q and q is the number defined in Proposition 1.4.19. Proof ‘(i) ⇒ (ii)’ Let F be a Frobenius map corresponding to an Fq -rational structure on G where q = pm for some m  1. By assumption, T is F-stable so T is also defined over Fq ; see Remark 1.4.3. Hence, we can apply Lemma 1.4.17 and so ϕ = qϕ0 where ϕ0 : X → X has finite order. Furthermore, using Lemma 1.4.27, one sees that ϕ0 (α† ) = α and ϕ0tr (α∨ ) = (α† )∨ for all α ∈ R. Thus, (I1) and (I2) hold for ϕ0 , where the root exponents of ϕ0 are all equal to 1. ‘(ii) ⇒ (iii)’ Let Fp : G → G be as in Example 1.4.16. Then Fpm is a Steinberg map and the map induced by Fpm on X is scalar multiplication with pm . So, by Lemma 1.4.24, there exists an isogeny f : G → G which maps T onto itself and induces ϕ0 , and such that f ◦ Fpm = Fpm ◦ f . Now ϕ0 has finite order, say d. Then f d induces the identity on X. Hence, by Theorem 1.3.12, there exists some t ∈ T such that f d (g) = tgt −1 for all g ∈ G. Since t also has finite order, we conclude that some positive power of f d is the identity. Hence, f itself has finite order. Now, F and F  := f ◦ Fpm are isogenies which induce the same map on X. Hence, by Theorem 1.3.12, there exists some y ∈ T such that F (g) = yF(g)y −1 for all g ∈ G. As in the proof of Lemma 1.4.14, there exists some x ∈ T such that F  = ι−1 x ◦ F ◦ ιx , where ιx denotes the inner automorphism of G defined by x. Then −1 −1 m F = ιx ◦ F  ◦ ι−1 x = (ι x ◦ f ◦ ι x ) ◦ (ι x ◦ Fp ◦ ι x )

(and the two factors still commute). Now, since x ∈ T, the isogeny Fp := ιx ◦ Fp ◦ ι−1 x also satisfies the conditions in Example 1.4.16. Furthermore, γ := ιx ◦ f ◦ ι−1 x is an automorphism of finite order such that γ(T) = T. Thus, (iii) holds.

1.5 Working with Isogenies and Root Data; Examples

53

‘(iii) ⇒ (i)’ As discussed in Example 1.4.16, Fp is the Frobenius map corresponding to an F p -rational structure on G. Then Fpm is the Frobenius map corresponding to an Fq -rational structure on G where q = pm . Hence so is F = γ ◦ Fpm by Remark 1.4.4(c). Finally, assume that (i), (ii), (iii) hold. Then the above arguments show that m = m . Furthermore, (ii) shows that det(ϕ) = ±(pm )rankX and so q = pm satisfies the conditions in Proposition 1.4.19.  The following example indicates that Steinberg maps can be much more complicated than Frobenius maps. Example 1.4.29 (a) In the setting of Example 1.4.23, one easily sees that neither the conclusion of Lemma 1.4.17 nor that of Lemma 1.4.27 hold for F . Hence, although F is a Frobenius map, the map F  is not. (b) Let G = SL2 (k) × PGL2 (k) and p = 2. Then G is semisimple of type A1 × A1 , with Cartan matrix C = 2I2 where I2 denotes the identity matrix. The root datum of G is determined by the factorisation     2 0 1 0 tr ˘ ˘ and A = . C = A · A where A = 0 1 0 2 For a fixed m  1, we define   0 2m P= 2m 0

and

P◦ =



0 2m+1

2m−1 0

 .

Then the pair (P, P◦ ) satisfies the conditions in 1.2.18 and so there is a corresponding isogeny F : G → G, with root exponents 2m+1 , 2m−1 . Since P2 = 4m I2 , we know that F is a Steinberg map (see Proposition 1.4.18). Furthermore, we have P = 2m P0 where P0 ∈ M2 (Z) has order 2; thus, the conclusion of Lemma 1.4.17 holds for F where q = 2m . Note also that the two projections (on the first and on the second factor) define isomorphisms of finite groups GF  SL2 (q)  PGL2 (q). On the other hand, since not all root exponents are equal, Lemma 1.4.27 shows that F is not a Frobenius map! One easily checks directly that there is no matrix P0◦ such that the pair (P0, P0◦ ) satisfies the conditions in 1.2.18. Thus, P0 does not come from an isogeny of root data.

1.5 Working with Isogenies and Root Data; Examples We now discuss some applications and examples of the theory developed so far. We start with some basic material about semisimple groups. Up until Proposition 1.5.10, k may be any algebraically closed field.

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Reductive Groups and Steinberg Maps

1.5.1 Let us fix a Cartan matrix C = (cst )s,t ∈S . Let Λ(C) be the finite abelian group defined in Remark 1.2.8. Then the semisimple algebraic groups with a root datum of Cartan type C are classified in terms of subgroups of Λ(C). This works as follows. We have Λ(C) := Ω/ZC where Ω is the free abelian group with basis  {ωs | s ∈ S} and ZC is the subgroup generated by { s ∈S cst ωs | t ∈ S}. Thus, giving a subgroup of Λ(C) is the same as giving a lattice L such that ZC ⊆ L ⊆ Ω. Such a lattice L is free abelian of the same rank as Ω. We choose a set of free generators {xs | s ∈ S} of L. Since ZC ⊆ L, we have unique expressions

cst ωs = atu xu (t ∈ S), where A = (atu )t,u ∈S (∗) s ∈S

u ∈S

 and A is a square matrix with integer coefficients. We also write xu = s ∈S a˘su ωs where A˘ = (a˘su )s,u ∈S is a square matrix with integer coefficients. Substituting this into (∗) and comparing coefficients, we obtain C = A˘ · Atr . As in 1.2.14, such a factorisation of C determines a root datum R L = (X, R, Y, R∨ ) of Cartan type C,  where R has a base given by αt := s ∈S ats xs for t ∈ S. We have |X/ZR| < ∞ since A, A˘ are square matrices. If we choose a different set of generators of L, say {yt | t ∈ S}, then we obtain another factorisation C = B˘ · Btr where B, B˘ are square  integer matrices. Writing yt = u ∈S put xu where P = (put )u,t ∈S is invertible over Z, we have P · Btr = Atr and B˘ = A˘ · P. Hence, the root data defined by C = A˘ · Atr and by C = B˘ · Btr are isomorphic; see Remark 1.2.20. Thus, every lattice L such that ZC ⊆ L ⊆ Ω determines a root datum R L as above, which is unique up to isomorphism. By Theorem 1.3.14, there exists a corresponding connected reductive algebraic group G L over k (unique up to isomorphism by Corollary 1.3.13). The group G L is semisimple since |X/ZR| < ∞; see Remark 1.3.5. Proposition 1.5.2 Let G be a semisimple algebraic group over k with root datum R = (X, R, Y, R∨ ) (relative to some maximal torus of G). Let C = (cst )s,t ∈S be the Cartan matrix of R. Then there exists a lattice L as in 1.5.1 such that G  G L . We have X/ZR  L/ZC and, hence, Z(G)  Hom(L/ZC, k × ). Proof Let Π be a base of R; we have |Π| = rank X since G is semisimple and, hence, X/ZR is finite. Also choose a Z-basis of X. By Remark 1.2.13, we have a corresponding factorisation C = A˘ · Atr where A, A˘ are square integral matrices. In particular, we can use S as an indexing set for both the rows and the columns of these matrices. Then let L be the sublattice of Ω spanned by the elements

xt := a˘st ωs (t ∈ S), where A˘ = (a˘st )s,t ∈S . s ∈S

We have ZC ⊆ L ⊆ Ω, since C = A˘ · Atr . Applying the construction in 1.5.1 to L, we obtain a group G L . Then Corollary 1.3.13 shows that G  G L . Finally, 1.5.1(∗) implies that X/ZR  L/ZC and this yields Z(G); see Remark 1.3.5. 

1.5 Working with Isogenies and Root Data; Examples

55

Example 1.5.3 Let C = (cst )s,t ∈S be a Cartan matrix and consider the group Λ(C) = Ω/ZC, as above. (a) If L = ZC, then we choose the generators {xs | s ∈ S} of L to be the given generators of ZC. So A in 1.5.1(∗) is the identity matrix and A˘ = C. Thus, R L is the root datum Rad (C) in Example 1.2.16. The corresponding group G L will be denoted by Gad ; we have Z(Gad ) = {1}. (b) If L = Ω, then we can take xs = ωs for all s ∈ S. So A = C tr and A˘ is the identity matrix. Hence, in this case, R L is the root datum Rsc (C) in Example 1.2.16. The corresponding group G L will be denoted by Gsc ; we have Z(Gsc )  Hom(Λ(C), k × ). The groups Gsc and Gad have some important universal properties that will be discussed in further detail below. We shall call Gsc the semisimple group of simply connected type C and Gad the semisimple group of adjoint type C. Example 1.5.4 Assume that C is an indecomposable Cartan matrix. The isomorphism types of Λ(C) are listed in Remark 1.2.8. (a) If C is of type An−1 , then Λ(C)  Z/nZ. Hence, for each divisor d of n, we have a unique lattice Ld ⊆ Ω such that |Ld /ZC| = d; let G(d) be the corresponding group. We have G(1) = Gad  PGLn (k) and G(n) = Gsc  SLn (k); see Examples 1.3.9 and 1.3.8. The remaining groups G(d) are explicitly constructed in [Ch05, §20.3], as images of SLn (k) under certain representations. (b) If C is of type Bn , Cn , E6 or E7 , then Λ(C) is cyclic of prime order. Hence, either L = ZC or L = Ω. In this case, the only possible groups are Gad and Gsc . In type Bn , we have Gad  SO2n+1 (k) and Gsc  Spin2n+1 (k). In type Cn , we have Gad  PCSp2n (k) and Gsc  Sp2n (k). (See the references in 1.1.4 for the precise definitions of these groups.) (c) If C is of type Dn , then Λ(C) has order 4 and there are 3 (for n odd) or 5 (for n even) possible lattices L. We have Gad  PCO◦2n (k) and Gsc  Spin2n (k). Using the labelling in Table 1.1, the group SO2n (k) corresponds to the unique L of index 2 in Ω that is invariant under the involution of Ω obtained by exchanging ω1 and ω2 . In terms of our matrix language in Section 1.2, the root datum of SO2n (k) is given by the factorisation C = A˘ · Atr where 1   −1    0 A = A˘ =  .  .  .   0  0

1 1

0 0

... 0

−1 .. .

1 .. . 0 ...

0 .. . −1 0

...

0  ... 0  .  .. . ..  . ..  . 0   1 0  −1 1 

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Reductive Groups and Steinberg Maps

(See [Spr98, Exercise 7.4.7].) If n is even, then there are two further lattices of index 2, which both give rise to the half-spin group HSpin2n (k). (d) Finally, if C is of type G2 , F4 or E8 , then Λ(C) = {0}. So, in this case, all semisimple algebraic groups over a fixed field k with a root datum of Cartan type C are isomorphic to each other; in particular, Gsc  Gad . We refer to [Gro02], [Ge03a, §1.7], [Spr98, §7.4], for further details about the various groups of classical type Bn , Cn and Dn . Example 1.5.5 given by

The Cartan matrices of type Bn and Cn are the n × n-matrices

2 −1 0 . . . 0 2 −2 0 . . . 0    −1 2 −1 0 . . . 0   −2 2 −1 0 . . . 0    ..  ..  . .  . 0 −1 2 −1 . . .  0 −1 2 −1 . . ,  , Cn :  . .. . . . . . . . .  . .. .. .. ..   . . . 0 . . . 0  . . . .    0 . . . 0 −1 2 −1   0 . . . 0 −1 2 −1  . . . 0 −1 2  0 . . . 0 −1 2   0  respectively. Let C denote the second matrix, and C the first. Let P◦ be the diagonal matrix of size n with diagonal entries 1, 2, 2, . . . , 2. Then CP◦ = P◦ C . Thus, if we also set P = P◦ , then the two conditions in 1.2.18 are satisfied and so the pair (P, P◦ ) defines a 2-isogeny from Rsc (C ) to Rsc (C) (see Example 1.2.16). Let k be  be the semisimple an algebraically closed field of characteristic 2. Let Gsc and Gsc  algebraic groups over k corresponding to Rsc (C) and Rsc (C ), respectively. We have   Spin Gsc  Sp2n (k) and Gsc 2n+1 (k). Then Theorem 1.3.12 yields the existence  of an isogeny f : Gsc → Gsc . This is one of Chevalley’s exceptional isogenies considered at the end of [Ch05, §23.7].      Bn :      

Example 1.5.6 Let G be connected reductive over k. Let R = (X, R, Y, R∨ ) be the root datum relative to a maximal torus T of G. Dual to the isomorphism in 1.1.11(c) we have a canonical isomorphism of abelian groups (see [Ca85, §3.1]): ∼

k × ⊗Z Y (T) −→ T,

ξ ⊗ ν → ν(ξ).

Hence, if {ν1, . . . , νn } is a Z-basis of Y (T), then every element t ∈ T can be written uniquely in the form t = ν1 (ξ1 ) · · · νn (ξn ) where ξ1, . . . , ξn ∈ k × . Now assume that G is semisimple of simply connected type. Then Y (T) = ZR∨ . Let Π = {α1, . . . , αn } be a base for R and {α1∨, . . . , αn∨ } be the corresponding base for R∨ . Hence, we have T = {h(ξ1, . . . , ξn ) := α1∨ (ξ1 ) · · · αn∨ (ξn ) | ξ1, . . . , ξn ∈ k × }.

1.5 Working with Isogenies and Root Data; Examples

57

In this setting, one can now explicitly determine the centre of G as a subset of T. Indeed, using Remark 1.3.5 and the above description of T, we obtain α j ,α1∨

Z(G) = {h(ξ1, . . . , ξn ) ∈ T | ξ1

α j ,αn∨

· · · ξn

= 1 for 1  j  n}.

Now the numbers ci j = α j , αi∨ (1  i, j  n) are just the entries of the Cartan matrix of R, so this yields an explicit system of n equations which we need to solve for ξ1, . . . , ξn . Let us describe this explicitly in all cases, where we refer to the labelling of the simple roots in Table 1.1; in each case, we also describe a subtorus S ⊆ T such that Z(G) ⊆ S. An : G  SLn+1 (k) and Z(G) = {h(ξ, ξ 2, ξ 3, . . . , ξ n ) | ξ n+1 = 1}; this is contained in the subtorus S := {h(ξ, ξ 2, ξ 3, . . . , ξ n ) | ξ ∈ k × } ⊆ T. Bn : G  Spin2n+1 (k). If n  2 is even, then Z(G) = {h(1, ξ, 1, ξ, 1, ξ, 1, . . .) | ξ 2 = 1}, and we may take S := {h(1, ξ, 1, ξ, 1, ξ, 1, . . .) | ξ ∈ k × }. For n  3 odd, we have Z(G) = {h(ξ, 1, ξ, 1, ξ, 1, ξ, . . .) | ξ 2 = 1}, and this is contained in the subtorus S := {h(ξ, 1, ξ, 1, ξ, 1, ξ, . . .) | ξ ∈ k × }. Cn : G  Sp2n (k) and Z(G) = {h(ξ, 1, 1, 1, . . .) | ξ 2 = 1}; this is contained in the subtorus S := {h(ξ, 1, 1, 1, . . .) | ξ ∈ k × }. Dn : G  Spin2n (k). If n  4 is even, then Z(G) = {h(ξ1, ξ2, 1, ξ1 ξ2, 1, ξ1 ξ2, 1, ξ1 ξ2, . . .) | ξ12 = ξ22 = 1}; this is contained in S := {h(ξ1, ξ2, 1, (ξ1 ξ2 )−1, 1, ξ1 ξ2, . . .) | ξ1, ξ2 ∈ k × } (where we follow [Mas10, Example 4.2]). If n  3 is odd, then Z(G) = {h(ξ, ξ −1, ξ 2, 1, ξ 2, 1, ξ 2, 1, ξ 2, . . .) | ξ 4 = 1}; G2 : F4 : E6 : E7 : E8 :

this is contained in S := {h(ξ, ξ −1, ξ 2, 1, ξ 2, 1, ξ 2, . . .) | ξ ∈ k × }. Since det(ci j ) = 1, we have Z(G) = {1}; we may take S := {1}. Since det(ci j ) = 1, we have Z(G) = {1}; we may take S := {1}. We have Z(G) = {h(ξ, 1, ξ −1, 1, ξ, ξ −1 ) | ξ 3 = 1}; this is contained in the subtorus S := {h(ξ, 1, ξ −1, 1, ξ, ξ −1 ) | ξ ∈ k × }. We have Z(G) = {h(1, ξ, 1, 1, ξ, 1, ξ) | ξ 2 = 1}; this is contained in the subtorus S := {h(1, ξ, 1, 1, ξ, 1, ξ) | ξ ∈ k × }. Since det(ci j ) = 1, we have Z(G) = {1}; we may take S := {1}.

We have dim S  1 except for type Dn with n  4 even, in which case dim S = 2. In order to obtain these descriptions, we did not have to rely on explicit realisations of groups of classical type as matrix groups: the abstract setting in terms of root data has actually been more efficient in this context.

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Reductive Groups and Steinberg Maps

For the construction of isogenies between groups of the same Cartan type, the following remarks will be useful. 1.5.7 Let G1 , G2 be connected reductive algebraic groups over k. For i = 1, 2, let Ri = (Xi, Ri, Yi, Ri∨ ) be the corresponding root datum, relative to a maximal torus Ti ⊆ Gi . Furthermore, let us choose Borel subgroups Bi ⊆ Gi such that Ti ⊆ Bi . By Remark 1.3.4, this is equivalent to choosing bases Πi ⊆ Ri . We assume that X1, X2 have the same rank and that R1, R2 have the same Cartan matrix C = (cst )s,t ∈S . If we also choose Z-bases of X1 and X2 , then R1 and R2 are determined by factorisations as in Remark 1.2.13: R1 :

C = A˘ 1 · Atr1

and

R2 :

C = A˘ 2 · Atr2,

where A1 , A2 , A˘ 1 , A˘ 2 are integer matrices, all of the same size. Now, this setting gives rise to a correspondence:     Isogenies f : G1 → G2 with Pairs of integer matrices (P, P◦ ) ↔ f (T1 ) = T2 , f (B1 ) = B2 satisfying (MI1), (MI2) in 1.2.18 (Note that, here, the relations in (MI2) read P · Atr2 = Atr1 · P◦ and P◦ · A˘ 2 = A˘ 1 · P.) Indeed, by 1.3.11, each isogeny of groups on the left determines a p-isogeny of root data; since f (B1 ) = B2 , this p-isogeny is ‘base preserving’ as in 1.2.18 and, hence, it determines a unique pair of matrices on the right. Conversely, a pair of matrices on the right determines a p-isogeny of root data by 1.2.18; by the Isogeny Theorem 1.3.12, there is a corresponding isogeny of groups on the left, which is unique up to inner automorphisms given by elements of T1 . Proposition 1.5.8 Let G be semisimple and assume that the root datum of G (relative to some maximal torus T and some Borel subgroup B containing T) is of Cartan type C. Let Gsc and Gad be of Cartan type C, as in Example 1.5.3, relative ˜ ⊆B ˜ ⊆ Gsc and T ⊆ B ⊆ Gad . Then there exist central isogenies to T f˜ : Gsc −→ G

and

f  : G −→ Gad,

˜ = T, f˜(B) ˜ = B, f (T) = T, f (B) = B. such that f˜(T) Proof

First consider Gsc . We place ourselves in the setting of 1.5.7, where ˜ B) ˜ (G1, T1, B1 ) = (Gsc, T,

and

(G2, T2, B2 ) = (G, T, B).

The root datum of G1 is given by a factorisation of C as above where A1 = C tr and A˘ 1 = I (identity matrix). The only extra information about the root datum of G2 is that, in the factorisation C = A˘ 2 · Atr2 , both A2 , A˘ 2 are square matrices (since G is semisimple). In order to find f˜ : G1 → G2 , we need to specify a pair of (square) ˜ P˜◦ ) where P˜ · Atr = C · P˜◦ , P˜◦ · A˘ 2 = P˜ and P˜◦ is a monomial integral matrices (P, 2

1.5 Working with Isogenies and Root Data; Examples

59

matrix whose non-zero entries are powers of p. (Then P˜ is automatically invertible ˜ P˜◦ ) := ( A˘ 2, I). Thus, over Q.) There is a natural choice for such a pair, namely, (P, the correspondence in 1.5.7 yields the existence of f˜. The root exponents of f˜ (which are the non-zero entries of P˜◦ = I) are all equal to 1, hence f˜ is a central isogeny. Now consider Gad . We argue as above, where now (G1, T1, B1 ) = (G, T, B) and (G2, T2, B2 ) = (Gad, T, B). The root datum of G2 is given by C = A˘ 2 · Atr2 where A2 = I and A˘ 2 = C. We need to specify a pair of (square) integral matrices (P , P ◦ ) where P  = Atr1 · P ◦ , P ◦ · C = A˘ 1 · P  and P ◦ is a monomial matrix whose non-zero entries are powers of p. Again, there is a natural choice for such a pair, namely, (P , P ◦ ) := (Atr1, I). As above, this yields the existence of f .  Proposition 1.5.9 (Steinberg [St68, 9.16]) Let G be semisimple and consider central isogenies f˜ : Gsc → G and f  : G → Gad as in Proposition 1.5.8. Assume, furthermore, that F : G → G is an isogeny. Then the following hold. (a) The isogeny F lifts to Gsc ; more precisely, there is a unique isogeny F˜ : Gsc → ˜ Gsc such that F ◦ f˜ = f˜ ◦ F. (b) The isogeny F descends to Gad ; more precisely, there is a unique isogeny F  : Gad → Gad such that F  ◦ f  = f  ◦ F. In both cases, the root exponents of F˜ and of F  are equal to those of F. If, moreover, ˜ F is a Steinberg map, then so are F  and F. Proof First note that F , if it exists, is clearly unique. The uniqueness of F˜ (if it exists) is shown as follows. Let F1 : Gsc → Gsc be another isogeny such that −1 is a homomorphism ˜ F ◦ f˜ = f˜ ◦ F1 . Then the map sending g ∈ Gsc to F(g)F 1 (g) of algebraic groups from Gsc to the centre of Gsc . Since Gsc is connected and the −1 = 1 for all ˜ centre of Gsc is finite, that map must be constant and so F(g)F 1 (g) g ∈ Gsc . We now turn to the problem of showing the existence of F˜ and F . ˜ ⊆ Gsc , T ⊆ B ⊆ Gad be as in Proposition 1.5.8. Let T ⊆ B ⊆ G, T˜ ⊆ B

We consider the corresponding root data of G, Gsc , Gad , and write X = X(T), ˜ X  = X(T). Then f˜ induces a p-isogeny ϕ˜ : X → X˜ and f  induces X˜ = X(T), a p-isogeny ϕ  : X  → X. Thus, we are in the setting of 1.5.7. Now consider the isogeny F : G → G. As already pointed out in the proof of [St68, 9.16], one easily sees that (a) and (b) hold for F if and only if (a) and (b) hold for ιg ◦ F, where ιg is an inner automorphism of G (for any g ∈ G). Hence, using Theorem 1.3.3 and replacing F by ιg ◦ F for a suitable g, we may assume without loss of generality that F(T) = T and F(B) = B. Then F induces a p-isogeny Φ : X → X and, again, we are ˜ : X˜ → X˜ in the setting of 1.5.7. Now, if we can show that there exist p-isogenies Φ

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Reductive Groups and Steinberg Maps

and Φ : X  → X  such that ˜ ◦ ϕ˜ ϕ˜ ◦ Φ = Φ

and

ϕ  ◦ Φ = Φ ◦ ϕ ,

then the existence of F˜ and F  follows from a general result about isogenies, which can already be found in [Ch05, §18.4] and which is a step in the proof of the Isogeny ˜ and Φ can be Theorem 1.3.12 (see also [St99, 3.2, 3.3]). In order to see how Φ constructed, we use the correspondence in 1.5.7 to describe everything in terms of pairs of square integral matrices. (The following part of the proof is somewhat different from the original proof ˜ P˜◦ ) and (P , P ◦ ) be the pairs of matrices corresponding to ϕ˜ in [St68].) Let (P,  and ϕ , respectively. Furthermore, let (Q, Q◦ ) be the pair of matrices corresponding to Φ. Now recall that the root data of Gsc and Gad are given by the factorisations C = I · (C tr )tr and C = C · I tr , respectively. Let A, A˘ be square integral matrices such that the root datum of G is given by the factorisation C = A˘ · Atr . Then the conditions in 1.2.18 imply that ˘ P˜ = P˜◦ · A,

P  = Atr · P ◦,

Q · Atr = Atr · Q◦,

Q◦ · A˘ = A˘ · Q,

where P˜◦ and P ◦ are monomial matrices all of whose non-zero entries are equal to 1 (since f˜ and f  are central). ˜ Q˜ ◦ ) be the corresponding pair of matrices. ˜ exists and let (Q, Assume first that Φ Since the root datum of Gsc is given by the factorisation C = I ·(C tr )tr , the conditions ˜ ˜ ◦ ϕ, ˜ we must have P˜ · Q = Q˜ · P. in 1.2.18 imply that Q˜ = Q˜ ◦ . Since ϕ˜ ◦ Φ = Φ Using the above relations, we deduce that Q˜ = Q˜ ◦ = P˜◦ · Q◦ · (P˜◦ )−1 . ˜ Q˜ ◦ ) is determined by Q◦ and P˜◦ ; in particular, the root ˜ exists, then (Q, Thus, if Φ ˜ are equal to those of Φ. Conversely, it is straightforward to check exponents of Φ ˜ : X˜ → X˜ defined by the matrix Q˜ given by the above formula has that the map Φ the required properties. Thus, (a) is proved. Similarly, assume first that Φ  exists and let (Q , Q ◦ ) be the corresponding pair of matrices. Then one deduces that Q  = Q ◦ = (P ◦ )−1 · Q◦ · P ◦ . Conversely, one checks that the map Φ : X  → X  defined by the matrix Q  given by the above formula has the required properties. Thus, (b) is proved. Finally, if F is a Steinberg map, then (using the characterisation in Proposi tion 1.4.18) one easily sees that F  and F˜ are also Steinberg maps. Proposition 1.5.10 (Cf. [St67, p. 46])

Let C be a Cartan matrix and Gsc , Gad be

1.5 Working with Isogenies and Root Data; Examples

61

as in Example 1.5.3. Assume that C is a block diagonal matrix, with diagonal blocks C1, . . . , Cn . Then we have ˜1 ×···×G ˜n Gsc = G

and

Gad = G1 × · · · × Gn ,

˜ i ⊆ Gsc and G ⊆ Gad are the normal subgroups corresponding to the where G i ˜ i is simple various diagonal blocks Ci , as in Remark 1.3.6. For each i, the group G of simply connected type Ci and Gi is simple of adjoint type Ci . Proof The definition of Rsc (C) shows that this root datum is the direct sum of Rsc (C1 ), . . . , Rsc (Cn ). Hence, the assertion concerning Gsc immediately follows  from Remark 1.3.6. The argument for Gad is analogous. 1.5.11 We shall assume from now on that G is connected reductive over k = F p (where p is a prime number) and F : G → G is a Steinberg map. Let Z be the centre of G and Gder be the derived subgroup of G. Clearly, we have F(Z) = Z and F(Gder ) = Gder . Since G = Z◦ .Gder and Z◦ ∩ Gder is finite, we obtain isogenies Z◦ × Gder

−→

G

(z, g)

→

zg

and

G

−→

G/Gder × G/Z◦

g

→

(gGder, gZ◦ ).

(Note that these are maps between groups of the same dimension; the first map is clearly surjective and, hence, has a finite kernel; the second map has a finite kernel and, hence, is surjective.) Recall from 1.1.13 that Gder is semisimple. The group G/Gder is a torus. (For, it is connected, abelian and consists of elements of order prime to p; see 1.1.7.) Furthermore, Gss := G/Z◦ is reductive (see Lemma 1.4.26) with a finite centre and, hence, is semisimple. Using also the isogenies in Proposition 1.5.8, we obtain isogenies Z◦ × (Gder )sc −→ G −→ G/Gder × (Gss )ad . Now, by Lemma 1.4.26, we have induced Steinberg maps on Gss and on G/Gder . By Proposition 1.5.9, there are also induced Steinberg maps on (Gder )sc and on (Gss )ad . Since all these maps are induced and uniquely determined by F, we will now simplify our notation and denote all these induced maps by F as well. Using F | = |(G )F |, |GF | = |(G )F | and Proposition 1.4.13(c), we conclude that |Gder der sc ss ad ss F F | = |(G/Gder )F ||Gss |. |GF | = |(Z◦ )F ||Gder F (since Also note that the natural map G → Gss induces a surjective map GF → Gss the kernel of G → Gss is connected).

Remark 1.5.12 By a slight abuse of notation, we shall denote (Gss )ad simply by Gad . Thus, as above, we obtain a central isogeny Gss → Gad which commutes

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Reductive Groups and Steinberg Maps

with the action of F on both sides. Composing this isogeny with the natural map G → Gss = G/Z◦ , we obtain a surjective, central isotypy πad : G → Gad

with

ker(πad ) = Z,

which commutes with the action of F on both sides and which we call an adjoint F but, in general, quotient of G. We certainly have an inclusion πad (GF ) ⊆ Gad F F equality will not hold. By Proposition 1.4.13, πad (G ) is a normal subgroup of Gad and we have an isomorphism F /πad (GF )  Z/L (Z), Gad

where L : G → G, g → g−1 F(g). One easily sees that Z/L (Z) = (Z/Z◦ )F , where the subscript F denotes ‘F-coinvariants’, that is, the largest quotient on which F F to acts trivially. The above isomorphism is explicitly obtained by sending g ∈ Gad −1

∈ Z where g ∈ G is any element satisfying πad (g)

= g. g F(g) F defines an automorphism α : GF → GF , g → Also note that each g ∈ Gad g 1

= g; the map αg obviously does

1 g −1 (where, as above, g ∈ G is such that πad (g) gg

In this way, we obtain a group homomorphism not depend on the choice of g). F → Aut(GF ), Gad

g → αg .

The automorphisms αg are called diagonal automorphisms of GF . Remark 1.5.13 Again, by a slight abuse of notation, we shall denote (Gder )sc simply by Gsc . Thus, as above, we obtain a central isotypy πsc : Gsc → G

with

πsc (Gsc ) = Gder,

which commutes with the action of F on both sides and which we call a simply connected covering of the derived subgroup of G. A special feature of groups of simply connected type is that F = u ∈ GF | u unipotent ; see [St68, 12.4]. (a) Gsc sc F ) ⊆ GF but, again, equality will not hold in general. In fact, We have πsc (Gsc der applying Proposition 1.4.13 to πsc : Gsc → Gder , one easily sees that F ) = u ∈ GF | u unipotent ⊆ GF ; see [St68, 12.6]. (b) πsc (Gsc der −1 (T ) ⊆ G , then the Finally, if T1 ⊆ G is any F-stable maximal torus and T˜ 1 = πsc 1 sc inclusion T1 ⊆ G induces an isomorphism F ) (see [DeLu76, 1.23]). ˜ F )  GF /πsc (Gsc (c) T1F /πsc (T 1 F ) is a characteristic subgroup of GF ; furthermore, we have the Note that πsc (Gsc F F ) ⊆ GF but these may be strict, as can be seen inclusions [G , GF ] ⊆ πsc (Gsc der already in the example where G = Gder = PGL2 (k).

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63

Following [St67, p. 45], [St68, 9.1], we call ker(πsc ) the fundamental group of G. For example, if G is simple of simply connected type, then the fundamental group of G is trivial (but the converse is not necessarily true). If G is simple of adjoint type, with corresponding Cartan matrix C, then ker(πsc )  Hom(Λ(C), k × ) where Λ(C) is the fundamental group of C; see Remark 1.2.8 and Example 1.5.3. 1.5.14 We keep the above notation. As already stated in 1.1.13, we have Gder = G1 . . . Gn where G1, . . . , Gn are the closed normal simple subgroups of G. (They elementwise commute with each other.) Now, one complication of the theory arises from the fact that, in general, this product decomposition is not stable under the Steinberg map F : Gder → Gder . What happens is the following. Consider the set-up in Remark 1.3.6, with partitions Π = Π1  . . .  Πn and R = R1  . . .  Rn ; then Gi = Uα | α ∈ Ri for i = 1, . . . , n. Now, F will permute the simple subgroups Gi and, correspondingly, the permutation α → α† of R (induced by F) will permute the subsets Ri . Hence, there is an induced permutation ρ of {1, . . . , n} such that, for all i = 1, . . . , n, we have Rρ(i) = {α† | α ∈ Ri } and F(Gi ) = F(Uα ) | α ∈ Ri = Uα† | α ∈ Ri = Gρ(i) .

(a)

Following [St68, p. 78], we say that Gder is F-simple if ρ is a cyclic permutation (it has a single orbit). Thus, by grouping together the factors in the various ρorbits on {1, . . . , n}, we can write Gder as a product of various F-stable and Fsimple semisimple groups. An analogous statement holds for Gss . Indeed, for each i, ¯ i be the image of Gi under the canonical map G → Gss . Then we have let G ¯1···G ¯ n and the induced Steinberg map F : Gss → Gss permutes the Gss = G ¯ factors Gi according to the permutation ρ. Now consider the isogenies in 1.5.11 and the groups Gsc , Gad . By Proposition 1.5.10, we have direct product decompositions ˜1 ×···×G ˜n Gsc = G

and

Gad = G1 × · · · × Gn

(b)

˜ i is mapped to Gi and, under such that, under the isogeny Gsc → Gder , the factor G ¯ i is mapped to G . By the compatibility of all the isogeny Gss → Gad , the factor G i of the above isogenies with the various Steinberg maps involved, it follows that ˜ ρ(i) ˜ i) = G F(G

and

 F(Gi ) = Gρ(i)

for i = 1, . . . , n.

(c)

The following result deals with F-simple semisimple groups. (If F is a Frobenius map with respect to some Fq -rational structure, then the construction below is related to the operation ‘restriction of scalars’; see [Spr98, §11.4].) Lemma 1.5.15 Assume that G is semisimple and F-simple, as defined in 1.5.14. Then G = Gder = G1 · · · Gn as above. Assume that this product is an abstract direct

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product. Then F n (G1 ) = G1 and ι : G1 → G,

g → gF(g) · · · F n−1 (g),

is an injective homomorphism of algebraic groups which restricts to an isomorphism n G1F  GF . Furthermore, if q is the positive real number attached to (G, F) (see Proposition 1.4.19), then q n is the positive real number attached to (G1, F n ). Proof By assumption, F cyclically permutes the factors Gi . So we can choose the labelling such that F i (G1 ) = Gi+1 for i = 1, . . . , n − 1 and F n (G1 ) = G1 . The map ι clearly is a morphism of affine varieties, which is injective since the product is direct. This map is a group homomorphism because the groups Gi elementwise commute n with each other. For the same reason, we have ι(G1F ) ⊆ GF . Since we have a direct product, every g ∈ G can be written uniquely as g = g1 · · · gn with gi ∈ Gi . Then F(g) = g if and only if F i (g1 ) = gi+1 for i = 1, . . . , n − 1 and F n (g1 ) = g1 . Thus, F(g) = g if and only if g = ι(g1 ) and F n (g1 ) = g1 . The statement concerning q, q n immediately follows from the definition of these numbers in Proposition 1.4.19.  Corollary 1.5.16 Assume that G is connected reductive; let G = Z◦ .Gder and Gder = G1 · · · Gn , as above. Let F : G → G be a Steinberg map and I ⊆ {1, . . . , n} be a set of representatives of the ρ-orbits on {1, . . . , n}, with ρ as in 1.5.14(a). For each i ∈ I, let ni be the length of the corresponding ρ-orbit. Then we have isomorphisms (of abstract finite groups)   ni F F ˜ F ni  and Gad  GiF , G Gsc i i ∈I

i ∈I

˜1 ×···×G ˜ n and Gad = G × · · · × Gn are as in 1.5.14(b). where Gsc = G 1 Proof This is immediate from 1.5.14 and Lemma 1.5.15. Also recall our identification in 1.5.11 of the various Steinberg maps involved.  See Example 1.4.29 for a good illustration of the above result; the group G = SL2 (k) × PGL2 (k) (with char(k) = 2) considered there is F-simple! In particular, the simple factors of an F-simple semisimple group need not all be isomorphic to each other as algebraic groups. (They are isomorphic as abstract groups.) To complete this section, we discuss another fundamental construction involving the formalism of root data: ‘dual groups’. Definition 1.5.17 ([DeLu76, 5.21]; see also [Ca85, §4.3], [Lu84a, 8.4]) Consider two pairs (G, F) and (G∗, F ∗ ) where G, G∗ are connected reductive and F : G → G, F ∗ : G∗ → G∗ are Steinberg maps. We say that (G, F) and (G∗, F ∗ ) are in duality if there is a maximally split torus T0 ⊆ G and a maximally split torus T∗0 ⊆ G∗ such that the following conditions hold, where R = (X, R, Y, R∨ ) is the root datum

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65

of G (with respect to T0 ) and R ∗ = (X ∗, R∗, Y ∗, R∗ ∨ ) is the root datum of G∗ (with respect to T∗0 ): (a) There is an isomorphism δ : X → Y ∗ such that δ(R) = R∗ ∨ and λ, α∨ = α∗, δ(λ)

for all λ ∈ X(T0 ) and α ∈ R,

where α∗ ∈ R∗ is defined by δ(α) = α∗ ∨ . (b) We have δ(λ ◦ F |T0 ) = F ∗ |T∗0 ◦ δ(λ) for all λ ∈ X. Thus, if λ : T0 → k× (an element of X = X(T0 )) and ν : k× → T∗0 (an element of Y ∗ = Y (T∗0 )) correspond to each other under δ, then λ ◦ F : T0 → k× and F ∗ ◦ ν : k× → T∗0 also correspond to each other under δ. The relation of being in duality is symmetric: the above conditions on δ : X → Y ∗ are equivalent to analogous conditions concerning a map ε : Y → X ∗ obtained by transposing δ; see [Ca85, 4.2.2, 4.3.1]. In particular, δ defines an isomorphism of root data between R and the dual of R ∗ (where the dual root datum is given as in Lemma 1.2.3(b)). Thus, connected reductive groups in duality have dual root data. Note also that the dual of (G∗, F ∗ ) can be naturally identified with (G, F). 1.5.18 It may be worthwhile to reformulate the above definition in terms of the matrix language of Section 1.2; the following discussion will also show that, for any given pair (G, F), there exists a corresponding dual pair. So let G be connected reductive and F : G → G be a Steinberg map. Let T0 ⊆ G be a maximally split torus and B0 ⊆ G be an F-stable Borel subgroup such that T0 ⊆ B0 . Let R = (X, R, Y, R∨ ) be the root datum of G with respect to T0 ; let Π be the base for R determined by B0 (see Remark 1.3.4). As in Remark 1.2.13, we choose a Z-basis of X. Then R determines a factorisation C = A˘ · Atr

where

C = Cartan matrix of R.

Furthermore, as discussed in 1.5.7, the isogeny F : G → G determines a pair of integer matrices (P, P◦ ) satisfying the conditions (MI1), (MI2) in 1.2.18. Now we notice that C tr is also a Cartan matrix and the corresponding factorisation C tr = B˘ · Btr

where

B˘ := A and

B := A˘

gives rise to the root datum R ∗ := (Y, R∨, X, R) which is dual to R; see Remark 1.2.13. Here, Π ∨ is a base for R∨ . Furthermore, setting Q := Ptr and Q◦ := (P◦ )tr , the pair (Q, Q◦ ) satisfies the conditions (MI1), (MI2) with respect to C tr = B˘ · Btr . Hence, performing the above argument backwards, there exists a connected reductive group G∗ such that R ∗ is isomorphic to the root datum of G∗ with respect to a maximal torus T∗0 ⊆ G∗ ; the base Π ∨ for R∨ determines a Borel subgroup B∗0 ⊆ G∗ containing T∗0 . The pair of matrices (Q, Q◦ ) determines

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an isogeny F ∗ : G∗ → G∗ such that T∗0 and B∗0 are F-stable. Since F is a Steinberg map, the characterisation in Proposition 1.4.18 immediately shows that F ∗ also is a Steinberg map. Thus, we conclude that the pairs (G, F) and (G∗, F ∗ ) are in duality; the number q in Proposition 1.4.19 is the same for F and for F ∗ . Remark 1.5.19 In the setting of 1.5.18, let W = NG (T0 )/T0 be the Weyl group of G and W∗ = NG∗ (T∗0 )/T∗0 be the Weyl group of G∗ . The choice of a base Π for R determines a set S of Coxeter generators for W; thus, using the notational convention in Remark 1.2.12, we have W = S and Π = {αs | s ∈ S}. We have seen above that Π determines a base Π ∗ for R∗ ; in fact, we have Π ∗ = {αs∗ | s ∈ S}, where we use the bijection R ↔ R∗ , α ↔ α∗ , in Definition 1.5.17(a). (Recall that δ(α) = α∗ ∨ for ∼ all α ∈ R.) Now, by [Ca85, 4.2.3], there is a unique group isomorphism W −→ W∗ , w → w ∗ , such that: (a) We have δ(w.λ) = w ∗ .δ(λ) for all λ ∈ X(T0 ). (b) If α ∈ R, then wα∗ ∈ W∗ is the reflection in the root α∗ ∨ ∈ R∗ ∨ . In particular, S ∗ = {s∗ | s ∈ S} is the set of Coxeter generators for W∗ determined by Π ∗ . This also implies that l(w) = l ∗ (w ∗ ) for all w ∈ W, where l : W → Z0 is the length function with respect to S and l ∗ : W∗ → Z0 is the length function with respect to S ∗ . Example 1.5.20 (a) Assume that G is semisimple of adjoint type. Thus, in the setting of 1.5.18, we have C = A˘ · Atr where A is the identity matrix and A˘ = C. Hence, C tr = B˘ · Btr where B˘ is the identity matrix and B = C. So G∗ is seen to be semisimple of simply connected type. If C is symmetric, then Proposition 1.5.8 yields a central isogeny G∗ → G. Similarly, if G is semisimple of simply connected type, then G∗ is seen to be semisimple of adjoint type. (b) The examples in (a) seem to indicate that dual groups are related in quite a strong way. However, as pointed out in the introduction of [Lu09d], dual groups in general are related only through a very weak connection (via their root data); in particular there is no direct, elementary construction which produces G∗ from G. Perhaps the most striking example is the case where G = SO2n+1 (k). Then G is simple of adjoint type, with Cartan matrix C of type Bn . As in (a), G∗ will be simple of simply connected type. However, since C tr has type Cn , we see that G∗  Sp2n (k). If char(k)  2, then there is no isogeny from G∗ to G! Example 1.5.21 Let G be connected reductive and F : G → G be a Steinberg map. It may well happen that (G, F) is dual to itself. To see this, let T0 ⊆ G be an F-stable maximal torus and B0 ⊆ G be an F-stable Borel subgroup such that T0 ⊆ B0 . Let C = (cst )s,t ∈S be the corresponding Cartan matrix. As in 1.5.18, the root datum R = (X, R, Y, R∨ ) of G with respect to T0 gives rise to a factorisation

1.5 Working with Isogenies and Root Data; Examples

67

C = A˘ · Atr . The map F gives rise to a pair of integer matrices (P, P◦ ) satisfying the conditions (MI1), (MI2) in 1.2.18. Now, an isomorphism between the dual of R and R itself is given by a group isomorphism δ : X → Y as in Definition 1.5.17. In the setting of 1.2.18, such an isomorphism δ is specified in terms of a permutation ◦) matrix D◦ = (dst s,t ∈S and an integer matrix D (square and invertible over Z, of the appropriate size) such that D◦ · A˘ = A · D,

D◦ · C = C tr · D◦,

D · P = Ptr · D.

(∗)

(Note that the first two conditions imply that D · Atr = A˘ tr · D◦ ; the third condition ensures that the compatibility in Definition 1.5.17(b) holds.) Thus, if (∗) holds, then (G, F) is in duality with itself; in particular, we obtain a group isomorphism ∼

W −→ W,

w → w ∗,

such that S ∗ = {s∗ | s ∈ S} = S. If s ∈ S, then s∗ ∈ S is uniquely determined by the condition that ds◦∗ s = 1. Let us consider a few concrete examples. (a) Let G = GLn (k) and F be one of the two Steinberg maps in Example 1.4.21. Then G has Cartan type An−1 and the Cartan matrix (which is symmetric in this ˘ see Example 1.3.7. Furthermore, P is q case) factorises as C = A˘ · Atr where A = A; times the identity matrix or q times a permutation matrix of order 2; in both cases, P is symmetric. Hence, (∗) holds if we take for D the identity matrix of size n × n and for D◦ the identity matrix of size (n − 1) × (n − 1). So (G, F) is dual to itself; the map w → w ∗ is the identity. (b) Let G be simple of adjoint type G2 or F4 ; let F : G → G be such that F(t) = t q for all t ∈ T0 . Then both P and P◦ are q times the identity matrix (of the appropriate size). Furthermore, C = A˘ · Atr where A is the identity matrix. Then (∗) holds if we take D = D◦ · C and 0 0 0 1    0 0 1 0  0 1 ◦ ◦  D = 1 0 (type G2 ), D =  0 1 0 0  (type F4 ).  1 0 0 0  Thus, (G, F) is dual to itself but something non-trivial is going on: in this case, w → w ∗ is the non-trivial graph automorphism of W. We shall have to say more about groups in duality in later sections. It will be useful and important to know how properties of G translate or connect to properties of G∗ . The lemma below contains just one example. (See Lusztig [Lu09d] where a number of such ‘bridges’ of a much deeper nature are discussed.) Lemma 1.5.22 Let (G, F) and (G∗, F ∗ ) be two pairs in duality, as in Definition 1.5.17. Then the following conditions are equivalent.

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(i) The centre of G is connected. (ii) The abelian group X/ZR has no p-torsion. (iii) The fundamental group of G∗ (see Remark 1.5.13) is trivial. Proof For the equivalence of (i) and (ii), see [Ca85, 4.5.1]. The equivalence of (ii) and (iii) is shown in [Ca85, 4.5.8]. 

1.6 Generic Finite Reductive Groups Recall from Definition 1.4.7 that a finite group of Lie type is a finite group of the form G = GF , where G is a connected reductive algebraic group over k = F p and F : G → G is a Steinberg map. Then it is common to speak of the (twisted or untwisted) ‘type’ of GF : for example, we say that the finite general linear groups are of untwisted type An−1 , the finite unitary groups are of twisted type An−1 (denoted 2A n−1 ; see Example 1.4.21), or that the Suzuki groups are of ‘very twisted’ type B2 (denoted 2B2 ; see Example 1.4.22). Here, the superscript (as in 2An−1 ) indicates the order of the automorphism of the Weyl group of G which is induced by F; in particular, GF is of ‘untwisted’ type if F induces the identity map on the Weyl group. Using the machinery developed in the previous sections, we can now give a somewhat more precise definition, as follows. 1.6.1 Assume that G is connected reductive and let F : G → G be a Steinberg map. Then we can canonically attach to G and F a pair C (G, F) := (C, P◦ ), where C = (cst )s,t ∈S is a Cartan matrix and P◦ = (pst )s,t ∈S is a monomial matrix whose non-zero entries are positive powers of p and such that CP◦ = P◦ C. Let us recall how this is done. First, we choose a maximally split torus T0 ⊆ G. Recall from Example 1.4.10 that this means that T0 is an F-stable maximal torus of G which is contained in an F-stable Borel subgroup B0 ⊆ G. (By Proposition 1.4.12, the pair (T0, B0 ) is unique up to conjugation by elements of GF .) Let R = (X, R, Y, R∨ ) be the root datum of G relative to T0 and ϕ : X → X be the p-isogeny induced by F. By Remark 1.3.4, there is a unique base Π of R such that B0 = T0, Uα | α ∈ R+ where R+ are the positive roots with respect to Π. Let us write Π = {αs | s ∈ S} and let C = (cst )s,t ∈S be the corresponding Cartan matrix. Since ϕ is a p-isogeny, there is a permutation α → α† of R such that ϕ(α† ) = qα α for all α ∈ R. The fact that B0 is F-stable implies that this permutation leaves R+ invariant. Hence, this permutation will also leave the base Π invariant and so there is an induced permutation S → S, s → s† , such that αs† = αs† for all s ∈ S. Thus, ϕ is ‘base preserving’ as in 1.2.18 and we have a corresponding monomial matrix

1.6 Generic Finite Reductive Groups

69

P◦ = (p◦st )s,t ∈S whose non-zero entries are given by p◦ss† = qs := qαs for all s ∈ S. The condition (MI2) implies that CP◦ = P◦ C, which means that qt cst = qs cs† t †

for all s, t ∈ S.

(a)

Since all pairs (T0, B0 ) as above are conjugate by elements of GF , the pair (C, P◦ ) is uniquely determined by G, F up to relabelling the elements of S. Now consider the Weyl group W of R. Recall from Remark 1.2.12 that we identify S with a subset of W via s ↔ wαs ; thus, we have W = S . Let l : W → Z0 be the corresponding length function. By Remark 1.2.10, the p-isogeny ϕ induces a group automorphism σ : W → W such that σ(s) = s†

(s ∈ S)

and

ϕ ◦ σ(w) = w ◦ ϕ

(w ∈ W).

(b)

By Theorem 1.3.2, we can naturally identify W = NG (T0 )/T0 . (Under this identification, the reflection wα ∈ W corresponds to the element w α ∈ NG (T0 ) in 1.3.1.) Since T0 and, hence, NG (T0 ) are F-stable, F naturally induces an automorphism σF : W → W, gT0 → F(g)T0 (g ∈ NG (T0 )). It is straightforward to check that all of the above constructions and identifications are compatible, that is, we have σF (w) = σ(w) for all w ∈ W. Finally, the numbers {qs } satisfy the following conditions. If S1, . . . , Sr are the orbits of the permutation s → s† on S, then  qs (i = 1, . . . , r), (c) q |Si | = s ∈Si

where q ∈ R>0 is defined in Proposition 1.4.19. (This easily follows from the equation ϕ(αs† ) = qs αs for all s ∈ S; see [St68, 11.17].) Hence, we also have  q |S | = s ∈S qs , which provides an alternative characterisation of q. 1.6.2 As in [Lu84a, 3.1], we say that σ : W → W is an ordinary automorphism if the following condition is satisfied: whenever s  t in S are in the same †-orbit on S, then the order of the product st is 2 or 3. With this notion, we have the following distinction of cases. The group GF is • either ‘untwisted’, that is, σ is the identity (and qs = q for all s ∈ S); • or ‘twisted’, that is, σ is not the identity but ordinary (as defined above); • or ‘very twisted’, otherwise. The typical examples to keep in mind are: the finite general linear groups (untwisted), the finite general unitary groups (twisted) and the finite Suzuki groups (very twisted). Note that these are notions which depend on G and F used to define GF , not just on the finite group GF : In Example 1.4.29, there is a realisation of SL2 (q) (where q is a power of 2) as a twisted (but not very twisted) group.

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Remark 1.6.3 Assume that F is a Frobenius map, with respect to some Fq rational structure on G. Then, as pointed out by Lusztig [Lu84a, 3.4.1], the induced automorphism σ : W → W is ordinary in the sense defined above. This is seen as follows. Let s  t in S be in the same †-orbit. Assume that cst  0. Replacing F by a power of F if necessary, we can assume without loss of generality that t = s† . Now, applying † repeatedly to {s, t}, we obtain a whole †-orbit of pairs {s , t  } where s   t  are in S. Then 1.6.1(a) shows that cs t   0 for all these pairs. So, if this †-orbit of pairs had more than one element, then we would obtain a closed path in the Dynkin diagram of C, which is impossible (see Table 1.1, p. 20). Hence, we must have t = s† and s = t † . But then 1.6.1(a) implies that qt cst = qs cs† t † = qs cts . However, by Lemma 1.4.27, we have qs = qt = q. Hence, cst = cts and so cst = cts = −1, which means that st has order 3, as required. 1.6.4 Let T ⊆ G be an F-stable maximal torus. We know that T is conjugate in G to our reference torus T0 but it is not immediately clear how the structure of TF can be described. For this purpose, one uses the following construction. Let T = gT0 g −1 where g ∈ G. Then F(g)T0 F(g)−1 = F(T) = T = gT0 g −1 and so g −1 F(g) ∈ NG (T0 ). Hence, g −1 F(g) = w for some w ∈ W. In this situation, we say that T is a torus of type w. Now define F  : G → G,

x → wF(x) w −1 .

By Lemma 1.4.14, F  is a Steinberg map and conjugation with g defines an iso morphism GF  GF . Note that F (T0 ) = T0 . For t ∈ T0 , we have: 

gtg −1 ∈ TF ⇔ F(gtg −1 ) = gtg −1 ⇔ g wF(t) w −1 g −1 = gtg −1 ⇔ t ∈ T0F . Thus, we have TF = gT0 [w]g −1 where we define 

T0 [w] := T0F = {t ∈ T0 | F(t) = w −1 t w}. Note that T0 [w] is a finite subgroup of T0 that only depends on w, but not on

(Another common notation for this subgroup is T0wF .) In the representative w.  F particular, |T | = |T0 [w]| = |T0F |. Recall from 1.6.1 that we denote by ϕ : X → X  the map induced by F. Let ϕ : X → X, λ → λ ◦ F  |T0 , be the map induced by F . Then, for λ ∈ X and t ∈ T0 , we have



ϕ (λ)(t) = λ wF(t)

w −1 =(w −1 .λ) F(t)



=ϕ w −1 .λ (t) = (ϕ◦w −1 )(λ) (t) and so ϕ  = ϕ◦w −1 (as maps X → X); see also [Ca85, Prop. 3.3.4(i)]. Before we continue with the general discussion, we give an example which illustrates that the above constructions indeed yield an explicit description of the groups TF . (For more substantial examples, see [Der84], [Miz77].)

1.6 Generic Finite Reductive Groups

71

Example 1.6.5 Let G = GLn (k), with root datum as in Example 1.3.7, relative to the maximal torus T0 = {h(ξ1, . . . , ξn ) | ξi ∈ k × }, where we let h(ξ1, . . . , ξn ) denote the diagonal matrix with diagonal entries ξ1, . . . , ξn . Let F : G → G be the ‘standard’ Frobenius map, such that GF = GLn (q) (as in Example 1.4.21). We can identify W with the group of permutation matrices in G. Let wc ∈ W be the permutation matrix corresponding to the n-cycle (1, 2, . . . , n) ∈ Sn . Then h(ξ1, . . . , ξn ) ∈ T0 [wc ] if and only if h(ξ1 , . . . , ξn ) = F(h(ξ1, . . . , ξn )) = wc−1 h(ξ1, . . . , ξn )wc = h(ξ2, . . . , ξn, ξ1 ). q

q

This yields the following explicit description: 2

T0 [wc ] = {h(ξ, ξ q, ξ q , . . . , ξ q

n−1

) | ξ ∈ k ×, ξ q = ξ}  F×q n , n

which shows that T0 [wc ] is cyclic of order q n − 1. Thus, if T ⊆ G is an F-stable maximal torus of type wc , then TF  T0 [wc ] is a subgroup of GF which is usually called a Singer cycle in finite group theory (see, e.g., [Hup67, Satz II.7.3]). Lemma 1.6.6 In the setting of 1.6.4, we extend scalars to R and write ϕR = qϕ0 where ϕ0 ∈ GL(XR ) is a linear map of finite order (see Proposition 1.4.19). Then |TF | = |T0 [w]| = ± det(ϕ◦w −1 − idX ) = det(q idXR − ϕ0 ◦w −1 ), and this equals det(q idXR − w −1 ◦ϕ0 ). In particular, |T0F | = det(q idXR − ϕ0 ). Proof Consider the algebraic homomorphism f  : T0 → T0 , t → t −1 F (t), with  ker( f ) = T0F . By the Lang–Steinberg theorem, f  is surjective. So, by 1.1.11(b),  the restriction map X → Hom(T0F , k × ) is surjective with kernel {λ ◦ f  | λ ∈ X } = (ϕ  − idX )(X) (where ϕ  : X → X is the map induced by F , as in 1.6.4). Hence, we have 



|X/(ϕ  − idX )(X)| = |Hom(T0F , k × )| = |T0F | 

where the second equality holds since T0F has order prime to p. By considering the elementary divisors of ϕ  − idX : X → X, the above cardinalities are also equal to ± det(ϕ  − idX ). So we conclude that 



|TF | = |T0 [w]| = |T0F | = |Irr(T0F )| = ± det(ϕ  − idX ). Extending scalars to R, we have ϕR = ϕR ◦ w −1 = qϕ0 ◦w −1 , which yields: det(ϕ −idX ) = det(q ϕ0 ◦w −1 −idXR ) = det(ϕ0 ◦w −1 ) det(q idXR −w◦ϕ0−1 ). Now the formula in 1.6.1(b) implies an analogous formula with ϕ replaced by ϕ0 ; thus, ϕ0 normalises W. Hence, since ϕ0 has finite order, it follows that the map w◦ϕ0−1 : XR → XR has finite order. But then the characteristic polynomials of

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w◦ϕ0−1 and its inverse ϕ0 ◦w −1 are equal; furthermore, det(ϕ0 ◦w −1 ) = ±1. Hence, |TF | = ± det(q idXR − ϕ0 ◦w −1 ). Finally, arguing as in [Ca85, Prop. 3.3.5], one sees that the latter determinant is a strictly positive real number. Hence, the sign must be +1. Clearly, ϕ0 ◦w −1 and  w −1 ◦ϕ0 have the same characteristic polynomial. We now have all the ingredients to state the order formula for GF . Since B0 is F-stable, its unipotent radical U0 = Ru (B0 ) is also F-stable. Since B0 = U0 .T0 where U0 ∩ T0 = {1}, we obtain |GF | = |U0F | · |T0F | · |GF /B0F |. The factor |T0F | is given by Lemma 1.6.6. A further evaluation of the remaining two factors in the above expression for |GF | leads to the following formula, due to Chevalley [Ch55] (in the case where G is semisimple and † is the identity) and Steinberg [St68, §11] (in general). Theorem 1.6.7 (Order formula) With the above notation, we have

ql(w), |GF | = q |R |/2 det(q idXR − ϕ0 ) w ∈W σ

where Wσ = {w ∈ W | σ(w) = w} (which is a finite Coxeter group). When G is simple, formulae for the various possibilities are given in Table 1.3. The fact that the list in Table 1.3 exhausts all the possible pairs (G, F) where G is simple and F : G → G is a Steinberg map is shown in [St68, §11.6]; note that, in F | = |GF | (see 1.5.11). this case, we have |GF | = |Gad sc Remark 1.6.8 (a) The formula shows that q |R |/2 is the p-part of the order of GF ; this provides a further characterisation of the number q. We also note that the above expression for |GF | can be interpreted as a polynomial in one variable evaluated at q, where the polynomial only depends on the root datum of G and the maps ϕ0 , σ derived from F. This will be formalised in Definition 1.6.10 below. (b) Steinberg [St68, 14.14] shows that the number of F-stable maximal tori of G is equal to q |R | . This equality is in fact equivalent to the following Molien series identity which yields another expression for the order of GF :  1  −1 1  −1 |R |  1 1 =q |GF| = q |R | |W| w ∈W |T0 [w]| |W| w ∈W det(q idXR − ϕ0 ◦w −1 ) see [Ca85, §3.4], [MaTe11, Exc. 30.15] for further details. One advantage of this expression is that it does not involve the length function on W or the induced automorphism σ of W. The inverse of the sum on the right-hand side can be further expressed as a product of various cyclotomic polynomials; in this way, one obtains the familiar formulae for the order of GF when G is simple; see Table 1.3. Also note

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73

Table 1.3 Order formulae for |GF | when G is simple Type

|GF |

An−1

q n(n−1)/2 (q2 −1)(q3 −1) . . . (q n −1)

Bn Cn

q n (q2 −1)(q4 −1) . . . (q2n −1) same as Bn

Dn G2 F4 E6 E7 E8

q n −n (q2 −1)(q4 −1) . . . (q2n−2 −1)(q n −1) q6 (q2 −1)(q6 −1) q24 (q2 −1)(q6 −1)(q8 −1)(q12 −1) q36 (q2 −1)(q5 −1)(q6 −1)(q8 −1)(q9 −1)(q12 −1) q63 (q2 −1)(q6 −1)(q8 −1)(q10 −1)(q12 −1)(q14 −1)(q18 −1) q120 (q2 −1)(q8 −1)(q12 −1)(q14 −1)(q18 −1)(q20 −1)(q24 −1)(q30 −1)

2A n−1 2D n 3D 4 2E 6 2B 2 2G 2 2F 4

2

2

q n(n−1)/2 (q2 −1)(q3 +1) . . . (q n −(−1)n ) q n −n (q2 −1)(q4 −1) . . . (q2n−2 −1)(q n +1) q12 (q2 −1)(q6 −1)(q8 +q4 +1) q36 (q2 −1)(q5 +1)(q6 −1)(q8 −1)(q9 +1)(q12 −1) 2

q4 (q2 −1)(q4 +1) q6 (q2 −1)(q6 +1) q24 (q2 −1)(q6 +1)(q8 −1)(q12 +1)

√ 2m+1 (q = 2 ) √ 2m+1 (q = 3 ) √ 2m+1 (q = 2 )

(The first 9 are ‘untwisted’, the next 4 ‘twisted’, the last 3 ‘very twisted’.)

that, instead of considering the maps ϕ0 ◦w −1 : XR → XR in the above formula, one can take their transposes (ϕ0 ◦w −1 )tr : YR → YR where YR = R ⊗Z Y : the characteristic polynomials will certainly remain the same. (c) If F is a Frobenius map, then Theorem 1.6.7 can be proved by a general argument; see [Ge03a, 4.2.5]. The general case (where the root exponents qs may not all be equal) is treated in [St68, §11] (see also [MaTe11, §24.1]), assuming that G is semisimple and ‘F-simple’ (see 1.5.14). But, as already noted in [St68, p. 78], the case of an arbitrary connected reductive G can be easily recovered from this case. As an illustration of the methods developed in the previous section, let us explicitly work out the reduction argument. Lemma 1.6.9 Suppose that the formula in Theorem 1.6.7 is known to hold when G is simple of adjoint type. Then the formula holds in general. Proof Since the formula for |T0F | in Lemma 1.6.6 is already known to hold in general, it will be sufficient to consider the cardinality of GF /T0F . It will be convenient to slightly rephrase this as follows. Let us consider the set of cosets G/T0 = {gT0 | g ∈ G} (just as an abstract set, we don’t need the notion of a quotient variety here). Since T0 is F-stable, we have an induced action of F on G/T0 .

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Consequently, we have a natural injective map GF /T0F → (G/T0 )F , gT0F → gT0 . Now the connected group T0 acts transitively on gT0 by right multiplication. Hence, if gT0 is F-stable, then Proposition 1.4.9 shows that gT0 contains a representative fixed by F. It follows that the above map is surjective and so |GF /T0F | = |(G/T0 )F |. Thus, it will now be sufficient to consider the identity:

|(G/T0 )F | = O(W, σ, q) where O(W, σ, q) := q |R |/2 ql(w) . (∗) w ∈W σ

Since the formula in Theorem 1.6.7 is assumed to hold when G is simple of adjoint type, the same is true of the formula (∗). We must deduce from this that (∗) holds in general. We do this in two steps. (1) First assume that G is semisimple of adjoint type. As in 1.5.14(b), we have a direct product decomposition G = G1 × · · · × Gn where each Gi is simple of adjoint type. Furthermore, there is a permutation ρ of {1, . . . , n} such that F(Gi ) = Gρ(i) for i = 1, . . . , n. Now note that T0 = T1 × · · · × Tn where each Ti (for i = 1, . . . , n) is a maximal torus of Gi such that F(Ti ) = Tρ(i) . Hence, we can also identify G/T0 with G1 /T1 × · · · × Gn /Tn . Furthermore, if I ⊆ {1, . . . , n} and ni (i ∈ I) are as in Corollary 1.5.16, then F ni (Gi /Ti ) = Gi /Ti for all i ∈ I and  ni |(G/T0 )F | = |(Gi /Ti )F |. (1a) i ∈I

It remains to show that there is a similar factorisation of the right-hand side of (∗). Recall from 1.5.14 that we have a partition R = R1  · · ·  Rn . Consequently, we also have a direct product decomposition W = W1 × · · · × Wn

where

Wi := wα | α ∈ Ri ;

furthermore, σ(Wi ) = Wρ(i) for i = 1, . . . , n. Here, Wi is the Weyl group of the factor Gi (relative to Ti ⊆ Gi ). Now, if w ∈ W and w = w1 · · · wn with wi ∈ Wi for all i, then l(w) = l(w1 ) + · · · + l(wn ). Using this formula, it is straightforward to verify that the expression for O(W, σ, q) is compatible with the above product decomposition, that is, we have σ ni (Wi ) = Wi for all i ∈ I and  O(W, σ, q) = O(Wi, σ ni , q ni ). (1b) i ∈I n

By assumption and Lemma 1.5.15, we have |(Gi /Ti )F i | = O(Wi, σ ni , q ni ) for i ∈ I. Comparing (1a) and (1b), we see that (∗) holds for G as well. (2) Now let G be arbitrary (connected and reductive). As in Remark 1.5.12, we consider an adjoint quotient πad : G → Gad with kernel Z = Z(G). By Remark 1.3.5(a), we have Z ⊆ T0 ; furthermore, T := πad (T0 ) is an F-stable maximal torus of Gad (see 1.3.10(a)). So we get a bijective map G/T0 → Gad /T,

gT0 → πad (g)T,

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75

which is compatible with the action of F on G/T0 and on Gad /T. In particular, |(G/T0 )F | = |(Gad /T)F | and so the left-hand side of (∗) does not change when we pass from G to Gad . On the other hand, by 1.3.10(d), πad induces an F-equivariant isomorphism from the Weyl group of G (relative to T0 ) onto the Weyl group of Gad (relative to T). Hence, the right-hand side of (∗) does not change either when we pass from G to Gad . Thus, if (∗) holds for Gad , then (∗) also holds for G.  Following [BrMa92], we now formally introduce ‘series of finite groups of Lie type’. This relies on the following definition, which is a slight modification of [BrMa92, §1]. (See Example 1.6.11 below for further comments.) Definition 1.6.10 Let R = (X, R, Y, R∨ ) be a root datum, with Weyl group W ⊆ Aut(X). We set XR := R ⊗Z X. We can canonically regard X as a subset of XR ; we also regard W as a subgroup of GL(XR ). Let ϕ0 ∈ GL(XR ) be an invertible linear map of finite order which normalises W, and assume that     qϕ0 (X) ⊆ X and the corresponding map P := q ∈ R>0  qϕ0 : X → X is a p-isogeny for some prime p is non-empty. We form the coset ϕ0 W = {ϕ0 ◦w | w ∈ W} ⊆ GL(XR ). Then

G = (X, R, Y, R∨ ), ϕ0 W is called a complete root datum or a generic finite reductive group; we also write P = PG in this case. We define a corresponding rational function |G| ∈ R(q) (where q is an indeterminate) by  1  −1 1 . |G| = q |R | |W| w ∈W det(q idXR − ϕ0 ◦w −1 ) We call |G| the order polynomial of G; this will be justified in Remark 1.6.15 below. Note that P is an infinite set: If q ∈ P and qϕ0 is a p-isogeny (where p is a prime), then pm q ∈ P for all integers m  1. Example 1.6.11 Let G be connected reductive and F : G → G be a Steinberg map. Then we obtain a corresponding complete root datum by taking the root datum of G (relative to a maximally split torus T0 ⊆ G as in 1.6.1) together with the linear map ϕ0 defined in Proposition 1.4.19(b) (such that ϕR = q ϕ0 ). In particular, this includes all the cases discussed in Examples 1.4.21, 1.4.22, 1.4.23. This shows that the above Definition 1.6.10 is somewhat more general than that in [BrMa92], in which cases like those in Example 1.4.23 are not included.

Let us now fix a complete root datum G = (X, R, Y, R∨ ), ϕ0 W . Remark 1.6.12 Let q ∈ P and set ϕ := qϕ0 . Then ϕ is a p-isogeny (for some prime p) and we have ϕ d = q d idX , where d  1 is the order of ϕ0 . In particular,

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Reductive Groups and Steinberg Maps

this implies that q d = pm for some m  1. Let G be a connected reductive algebraic group over k = F p whose root datum (relative to a maximal torus T ⊆ G) is isomorphic to (X, R, Y, R∨ ). Then ϕ gives rise to an isogeny F : G → G which is a Steinberg map by Proposition 1.4.18. We write G(q) := GF . Thus, we obtain a family of finite groups {G(q) | q ∈ P } which we call the series of finite groups of Lie type defined by G. There are some choices involved in the definition of G(q) but we shall see in the remarks below that different choices lead to isomorphic finite groups. Remark 1.6.13 Since the map ϕ0 ∈ GL(XR ) normalises W, we obtain a group automorphism σ : W → W such that σ(w) = ϕ0−1 ◦ w ◦ ϕ0

for all w ∈ W.

Note that this is compatible with Remark 1.2.10: For any q ∈ P, the automorphism of W induced by the p-isogeny qϕ0 (where p is a prime) is given by σ. (The notation is also compatible with 1.6.1(b).) Let us now see what happens when we replace ϕ0 by another map in ϕ0 W. First note that ϕ0 ◦w has finite order for any w ∈ W. Furthermore, if q ∈ P and qϕ0 is a p-isogeny (where p is a prime), then (qϕ0 )◦w also is a p-isogeny. Thus, if ϕ0 satisfies the defining conditions for a complete root datum, then so does ϕ0 ◦w for any w ∈ W. We also note the following identity:

for all m  1. (ϕ0 ◦w)m = ϕ0m ◦ σ m−1 (w) · · · σ 2 (w)σ(w)w Now let q ∈ P and let G, T, F be as in Remark 1.6.12. Then ϕ := qϕ0 is the linear map induced on X  X(T) by F. Let w ∈ W and w be a representative of w in NG (T). We define F  : G → G by F (g) := w −1 F(g)w for g ∈ G. By Lemma 1.4.14,  F  also is a Steinberg map and we have GF  GF . Now T is F -stable and one easily sees that ϕ◦w : X → X is the linear map induced by F . (See once more 1.6.4.) This shows that, if we replace ϕ0 by ϕ0 ◦w for some w ∈ W, then F changes to F  but we obtain isomorphic finite groups. Remark 1.6.14 Let q ∈ P and set ϕ := qϕ0 . Then ϕ is a p-isogeny (where p is a prime) and so there is a corresponding permutation α → α† of R; we have ϕ(α† ) = qα α for all α ∈ R, where {qα } are the root exponents of ϕ. Let σ : W → W be the group automorphism in Remark 1.6.13. By Remark 1.2.10, we have σ(wα ) = wα† for all α ∈ R; also recall that the root exponents are positive. Hence, we conclude that the permutation α → α† only depends on ϕ0 , but not on q. Remark 1.6.15 Let us fix a base Π of R. Since any two bases can be transformed into each other by a unique element of W, there is a unique w ∈ W such that, if we replace ϕ0 by ϕ0 := ϕ0 ◦w, then Π † = Π where α → α† is the permutation

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77

induced by ϕ0 ; see Remarks 1.2.11 and 1.6.14. Assume now that this is the case. Let q ∈ P and G, T, F be as in Remark 1.6.12. Then T lies in the F-stable Borel subgroup B = T, Uα | α ∈ R+ (where R+ are the positive roots with respect to Π) and we are in the setting of 1.6.1, where T0 := T. So, by Remark 1.6.8, we have |GF | = |G|(q) and this is also equal to the expression in Theorem 1.6.7. Since this holds for all q ∈ P, we obtain an identity of rational functions in q:

ql(w) . (a) |G| = q |R |/2 det(q idXR − ϕ0 ) w ∈W σ

Thus, the rational function |G| actually is a polynomial in q such that |GF | = |G|(q). This provides the justification for calling |G| the order polynomial of G (see also [BrMa92, 1.12]). Note that det(q idXR − ϕ0 ) has degree dim T and the polynomial  l(w) has degree |R|/2. Hence, we conclude that w ∈W σ q |G| ∈ R[q] has degree dim G = |R| + dim T.

(b)

Now let K ⊆ R be a subfield such that the polynomial det(q idXR −ϕ0 ) has coefficients in K. Since ϕ0 has finite order, all roots of this polynomial are roots of unity. By  [St68, 2.1], an analogous result is also true for the term w ql(w) in (a). So there is a factorisation |G| = q |R/2 | × (product of cyclotomic polynomials in K[q]).

(c)

If G is simple, then such factorisations can be seen explicitly in Table 1.3 (p. 73). If F : G → G is a Frobenius map, then we can take K = Q. Remark 1.6.16 Let Π = {αs | s ∈ S} be a base of R and C = (cst )s,t ∈S be the corresponding Cartan matrix; also choose a Z-basis of X. Then R is determined by a factorisation C = A˘ · Atr as in Remark 1.2.13. Assume that ϕ0 is chosen so that the permutation of R induced by ϕ0 leaves Π invariant (which is possible by Remark 1.6.15). Let Q be the matrix of ϕ0 : XR → XR (with respect to the chosen basis of X). If q ∈ P, then qϕ0 is a p-isogeny (for some prime p) and the conditions ˘ where P◦ is a (MI1), (MI2) in 1.2.18 show that qQ Atr = Atr P◦ and P◦ A˘ = q AQ, monomial matrix whose non-zero entries are all powers of p. It follows that Q Atr = Atr Q◦

and

˘ Q◦ A˘ = AQ,

where Q◦ := q−1 P◦ is a monomial matrix; each non-zero entry of Q◦ is a positive real number such that some positive power of it is an integral power of p. Now note that, although the pair (P, P◦ ) is used in the construction, Q◦ is uniquely determined by Q and, hence, independent of (P, P◦ ). There are two cases: (I) All non-zero entries of Q◦ are equal to 1. Then ϕ0 is a 1-isogeny, as in Example 1.4.21. Consequently, the set P consists of all prime powers. This is what is called the ‘cas général ’ in [BrMa92, §1].

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(II) Otherwise, there is a unique prime number p such that each non-zero entry of Q◦ has the property that some positive integral power of it is an integral power of p. In this case, P will only consist of positive real numbers q such that qϕ0 is a p-isogeny for this prime p. In case (I), we will also say that G is ‘ordinary’. Note that then the induced map σ : W → W is ordinary in the sense of 1.6.2. Here is an example where we are in case (II). Consider the root datum of Cartan type A1 × A1 in Example 1.4.29(b), where ϕ0 is determined by a certain matrix of order 2, denoted P0 . Then     0 1 0 1/2 ◦ and Q = . Q = P0 = 1 0 2 0 So we are in case (II) where p = 2 and P = {2m | m  1}. Similarly, the complete root data of the Suzuki and Ree groups in Example 1.4.22 are of type (II), where √ 2m+1 √ 2m+1 | m  0} (for 2B2 , 2F 4 ) or P = { 3 | m  0} (for 2G2 ). P={ 2 Definition 1.6.17 (See [BrMa92, p. 250], [BMM93, 1.5]) The Ennola dual of a

complete root datum G = (X, R, Y, R∨ ), ϕ0 W is defined by

G− := (X, R, Y, R∨ ), −ϕ0 W . (Note that G− is a complete root datum since, for any p-isogeny of root data ϕ : X → X, the map −ϕ also is a p-isogeny of root data; in particular, PG− = PG .) In this situation, we write G(−q) := G− (q) for any q ∈ PG . We have |G− |(q) = (−1)rankX |G|(−q). For the origin of the name ‘Ennola dual’, see Example 1.6.18 below. Example 1.6.18 (a) Assume that −idX ∈ W. Then, clearly, we have G− = G and |G− | = |G| ∈ R[q]. (b) Let G = GLn (k) and T0 ⊆ G be the maximal torus consisting of the diagonal matrices in G. We have described the corresponding root datum in Example 1.3.7. If ϕ0 := idXR and G is the corresponding complete root datum, then G(q)  GLn (q) for all prime powers q. We claim that G(−q)  GUn (q)

for all q ∈ PG .

This is seen as follows. Let τ : G → G be the automorphism which sends an invertible matrix to its transpose inverse. Then τ(T0 ) = T0 and the induced map on X is −idX . (Thus, τ is a concrete realisation of the isogeny in Example 1.3.16.) Let Fq : G → G be the standard Frobenius map (raising every matrix entry to its qth power). Then τ commutes with Fq and so F  = τ ◦ Fq is a Frobenius map on G; see Remark 1.4.4(c). We have F (T0 ) = T0 and the induced map on X is given by

1.6 Generic Finite Reductive Groups

79

−q idX . Thus, (G, F ) gives rise to the complete root datum G− ; finally, note that  GF  GUn (q). (The difference between this realisation of GUn (q) and the one in Example 1.4.21 is that, here, T0 is not a maximally split torus for F .) Now the identity |G− |(q) = (−1)rankX |G|(−q) gives an a priori explanation for the fact that the order formula for GUn (q) in Table 1.3 (p. 73) is obtained from that of GLn (q) by simply changing q to −q (and fixing the total sign). Ennola [Enn63] observed that a similar statement should even be true for the irreducible characters of these groups; we will discuss this in further detail at the end of Section 2.8. Example 1.6.19 We define the dual complete root datum of the root datum G =

(X, R, Y, R∨ ), ϕ0 W by

G∗ := (Y, R∨, X, R), ϕ0tr W , where ϕ0tr : YR → YR is the transpose map defined, as in 1.2.2, through the canonical extension of the pairing  , : X × Y → Z to a pairing XR × YR → R. Here, we also use the identification of W∨ ⊆ Aut(Y ) with W, as in Remark 1.2.12. We have PG∗ = PG . Now, for each q ∈ PG , we obtain a finite group G(q) (arising from a pair (G, F) as in Remark 1.6.12) and a finite group G∗ (q) (arising from an analogous pair (G∗, F ∗ )). We then see that (G, F) and (G∗, F ∗ ) are in duality as in Definition 1.5.17. By Theorem 1.6.7 and the argument in [Ca85, 4.4.4], we have |G∗ |(q) = |G|(q)

|G(q)| = |G∗ (q)| for all q ∈ PG .

Definition 1.6.20 (See [BrMa92, 1.1]) Let G = (X, R, Y, R∨ ), ϕ0 W be a complete root datum. For any w ∈ W, the complete root datum

Tw := (X, , Y, ), ϕ0 ◦w −1 and

is called a maximal toric sub-datum of G. (We take ϕ0 ◦w −1 here in order to have consistency with 1.6.4; in [BrMa92], the Weyl group W acts on the right on X so that w −1 is replaced by w there.) In general, G is said to be a toric datum if R = ; in this case, a corresponding connected reductive algebraic group is a torus.

1.6.21 Let G = (X, R, Y, R∨ ), ϕ0 W be a complete root datum. We assume that ϕ0 is chosen such that the permutation of R induced by ϕ0 leaves a base of R invariant (which is possible by Remark 1.6.15). Thus, if q ∈ P and G, T, F are as in Remark 1.6.12, then we are in the setting in 1.6.1, where T0 := T. Now let w ∈ W and consider the maximal toric sub-datum Tw as in Definition 1.6.20. The corresponding order polynomial is just given by |Tw | = det(q idXR − ϕ0 ◦w −1 ) ∈ R[q]. Let q ∈ P and G, T0, F be as above. Let w be a representative of w in NG (T0 ). By Theorem 1.4.8 (Lang–Steinberg), we can write w = g −1 F(g) for some g ∈ G. Then

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T := gT0 g −1 is an F-stable maximal torus of G; in fact, T is a torus of type w, as in 1.6.4. By Lemma 1.6.6, we have |TF | = |T0 [w]| = |Tw |(q) = det(q idXR − ϕ0 ◦w −1 ). Thus, TF  T0 [w] is a member of the series of finite groups of Lie type defined by the complete root datum Tw . Since |TF | divides |GF | and since this holds for all q ∈ P, we conclude that |Tw |

divides

|G|

in R[q].

In fact, if N = |R|/2 denotes the number of positive roots of G, then q N |Tw | divides |G| in R[q] since q N divides |G| and q does not divide |Tw |. Remark 1.6.22 Assume that G is ‘ordinary’, as in Remark 1.6.16. Then ϕ0 : XR → XR is an automorphism of finite order which leaves X ⊆ XR invariant. It follows that the order polynomials |G| and |Tw | (w ∈ W) are actually polynomials in Z[q] in this case (not just in R[q]). In later chapters, we will see that various other classes of subgroups of GF fit into the framework of complete root data. The general formalism is further developed in [BMM93], [BMM99], [BMM14]. Note that, even with our slightly more general definition, any complete root datum as above defines a reflection datum as in [BMM14, Def. 2.6], over a suitable subfield K ⊆ R.

1.7 Regular Embeddings Lusztig’s work [Lu84a], [Lu88] (to be discussed in more detail in later chapters) shows that the character theory of finite groups of Lie type is considerably easier when the centre of the underlying algebraic group is connected. Thus, when trying to prove a result about a general finite group of Lie type, it often happens that one first tries to establish an analogous result in the case where the centre is connected. The concept of ‘regular embedding’ provides an efficient technical tool for passing back and forth between groups with a connected centre and a non-connected centre. ˜ be connected reductive algebraic groups over k = F p Definition 1.7.1 Let G, G ˜ ˜ ˜ be a homomorphism ˜ and F : G → G, F : G → G be Steinberg maps. Let i : G → G of algebraic groups such that i ◦ F = F˜ ◦ i. Following [Lu88, §7], we say that i is ˜ has a connected centre, i is an isomorphism of G with a a regular embedding if G ˜ ˜ have the same derived subgroup. closed subgroup of G and i(G), G ˜ with G/i(G) ˜ ˜ abelian. Then Note that Gder ⊆ i(G) and so i(G) is normal in G ˜ F F the finite group i(G ) = i(G) contains the derived subgroup of the finite group ˜ F˜ with G ˜ F˜ /i(GF ) abelian. Thus, as far as the ˜ F˜ and so i(GF ) is normal in G G

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81

˜ F˜ is concerned, we are in a situation where representation theory of GF and of G Clifford theory (with abelian factor group) applies. (See also [Ta19] for a refinement of the notion of regular embeddings.) Example 1.7.2 (a) Let G be a connected reductive algebraic group with a connected centre and F : G → G be a Steinberg map. Then Gder is semisimple and, clearly, Gder ⊆ G is a regular embedding. A standard example is given by G = GLn (k) where Gder = SLn (k); note that this works for both the Frobenius maps in Example 1.4.21, where either GF = GLn (q) or GF = GUn (q). (b) Let G = SLn (k) and F : G → G be a Frobenius map. We can also construct ˜ as follows. Let a regular embedding i : G → G, ˜ := {(A, ξ) ∈ Mn (k) × k× | ξ det(A) = 1}; G ˜ = 1. For A ∈ G we ˜ = {(ξ In, ξ −n ) | ξ ∈ k × } is connected and dim Z(G) then Z(G) ˜ ˜ →G ˜ set i(A) := (A, 1) ∈ G; then i is a closed embedding. A Frobenius map F˜ : G q F −q ˜ ˜ is defined by F(A, ξ) = (F(A), ξ ) if G = SLn (q), and by F(A, ξ) = (F(A), ξ ) if GF = SUn (q). (c) Let n  2 and G ⊆ GLn (k) be one of the classical groups in 1.1.4. Then one can apply a similar construction as in (b). In each case, Z(G) consists of the scalar matrices in G. If G = SO2 (k), then G = Z(G)  k× ; otherwise, we have Z(G) = {±In } where In is the identity matrix. So let us now assume that char(k)  2 and Z(G) = {±In }. Then G = Γ(Q n, k) where Qtrn = ±Q n . We set ˜ = CΓ(Q n, k) := {(A, ξ) ∈ Mn (k) × k× | Atr Q n A = ξQ n }; G ˜ this is called the conformal group corresponding to the classical group G. Then G 2 × ˜ is a linear algebraic group such that Z(G) = {(ξ In, ξ ) | ξ ∈ k } is connected and ˜ = 1. Consider the closed subgroup G1 = {(A, 1) | A ∈ G} ⊆ G. ˜ Then we dim Z(G) have an injective homomorphism of algebraic groups i : G → G1 , A → (A, 1), with inverse given by (A, 1) → A. Hence, i is a closed embedding. Let F : GLn (k) → GLn (k) be the standard Frobenius map (raising each entry of a matrix to its qth power). Then F restricts to a Frobenius map on G. The map ˜ → G, ˜ (A, ξ) → (F(A), ξ q ), is easily seen to be a Frobenius map such that F˜ : G i ◦ F = F˜ ◦ i. Thus, i is a regular embedding. If n = 2m is even and G = SOn (k), then we also have a ‘twisted’ Frobenius map F1 : G → G, A → tn−1 F(A)tn , where ⎡ Im−1 ⎢ ⎢ ⎢ tn := ⎢⎢ 0 ⎢ ⎢ ⎢ 0 ⎣

0

0

0

1

1

0 0

0 Im−1

⎤ ⎥ ⎥ ⎥ ⎥ ∈ GO2m (k). ⎥ ⎥ ⎥ ⎥ ⎦

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This gives rise to the finite ‘non-split’ orthogonal group GF1 = SO−n (q); see, e.g., [Ge03a, 4.1.10(d)]. We have a Frobenius map ˜ → G, ˜ F˜1 : G

(A, ξ) → (F1 (A), ξ q ),

such that i ◦ F1 = F˜1 ◦ i. Thus, i also is a regular embedding with respect to F1 . (These examples already appeared in [Lu77a, §8.1].) Lemma 1.7.3 (Cf. [DeLu76, 1.21]) Let G be connected reductive and F : G → G be a Steinberg map. Let Z be the centre of G and S ⊆ G be an F-stable torus such ˜ that Z ⊆ S. (For example, one could take any F-stable maximal torus of G.) Let G −1 be the quotient of G × S by the closed normal subgroup {(z, z ) | z ∈ Z}. Let S˜ ˜ Then the map F˜ : G ˜ →G ˜ induced by F is be the image of {1} × S ⊆ G × S in G. ˜ induced by G → G × S, g → (g, 1), is a a Steinberg map and the map i : G → G ˜ regular embedding, where S˜ is the centre of G. ˜ is reductive and F˜ : G ˜ → G ˜ is a Steinberg map. Proof By Lemma 1.4.26, G ˜ ˜ Furthermore, one easily sees that i is injective, that i ◦ F = F ◦ i and that S˜ = Z(G). −1 ˜ ˜ (Thus, the centre of G indeed is connected.) Let Z := {(z, z ) | z ∈ Z} and ˜ We have G ˜ = H.S. ˜ Since S˜ = Z(G), ˜ it follows that H := i(G) = (G × Z)/Z˜ ⊆ G. ˜ der = Hder = i(Gder ). G We claim that i1 : G → H, g → i(g), is an isomorphism of algebraic groups. To see this, consider the homomorphism π : G × Z → G, (g, z) → gz. Since ˜ ⊆ ker(π), we have an induced homomorphism of algebraic groups π¯ : H → G, Z which is obviously inverse to i1 : G → H. Thus, the claim is proved, and it follows that i is a regular embedding.  Example 1.7.4 Let G be simple of simply connected type and F : G → G be a Steinberg map. Assume that Z = Z(G) is non-trivial. Let T0 ⊆ G be a maximally split torus and S ⊆ T0 be a subtorus as in Example 1.5.6. Then Z ⊆ S and we would like to perform the construction in Lemma 1.7.3 using S. For this purpose, we need to check that S is F-stable. Let us check when this happens. Let R = (X, R, Y, R∨ ) be the root datum of G with respect to T0 . Let Π = {α1, . . . , αn } be a base for R, with a labelling as in Table 1.1. Then Y = ZR∨ and {α1∨, . . . , αn∨ } is a Z-basis of Y . Now F induces a linear map ϕ : X → X which is a p-isogeny of R. Hence, there is a permutation s → s  of {1, . . . , n} and there are integers qs > 0 (each an integral power of p) such that ϕtr (αs∨ ) = qs αs∨

for 1  s  n.

(Note: The permutation αs → αs is inverse to the permutation αs → αs† in 1.6.1.) Now consider the isomorphism k × ⊗Z Y → T0 , ξ ⊗ ν → ν(ξ). Then, for each ξ ∈ k × and ν ∈ Y , the element ξ ⊗ ϕtr (ν) corresponds to F(ν(ξ)). (See [Ca85, §3.2].) Thus,

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83

in terms of the notation T0 = {h(ξ1, . . . , ξn ) | ξ1, . . . , ξn ∈ k × }

(see Example 1.5.6),

the action of F on T0 is given by

q  q q F h(ξ1, ξ2, . . . , ξn ) = h(ξ11 , ξ22 , . . . , ξnn )

for all ξ1, . . . , ξn ∈ k × .

We do not need to consider the case where F is a Steinberg map but not a Frobenius map. For, this case only occurs in types B2 , G2 , F4 and, in all three cases, we have Z = {1} (since char(k) = 2 in type B2 ). So let now F be a Frobenius map. Then all qs are equal, to q say, and s → s  determines a symmetry of the Dynkin diagram of R. If s → s  is the identity, then

q q q for all ξs ∈ k × . F h(ξ1, ξ2, . . . , ξn ) = h(ξ1 , ξ2 , . . . , ξn ) The cases where there exists a non-trivial permutation s → s  are as follows. An (n  2): s  = n + 1 − s for 1  s  n and so

q q q F h(ξ1, ξ2, . . . , ξn ) = h(ξn , ξn−1, . . . , ξ1 )

for all ξs ∈ k × .

In this case, the subtorus S ⊆ T0 is not necessarily F-stable. Dn (n  3): 1 = 2, 2 = 1, s  = s for 3  s  n and so

q q q q for all ξs ∈ k × . F h(ξ1, ξ2, . . . , ξn ) = h(ξ2 , ξ1 , ξ3 , . . . , ξn ) In this case, the subtorus S ⊆ T0 is F-stable. D4 : 1 = 2, 2 = 4, 3 = 3, 4 = 1 and so

q q q q F h(ξ1, ξ2, ξ3, ξ4 ) = h(ξ2 , ξ4 , ξ3 , ξ1 )

for all ξs ∈ k × .

In this case, the subtorus S ⊆ T0 is F-stable. E6 : 1 = 6, 2 = 2, 3 = 5, 4 = 4, 5 = 3, 6 = 1 and so

q q q q q q F h(ξ1, ξ2, ξ3, ξ4, ξ5, ξ6 ) = h(ξ6 , ξ2 , ξ5 , ξ4 , ξ3 , ξ1 )

for all ξs ∈ k × .

In this case, the subtorus S ⊆ T0 is F-stable. Proposition 1.7.5 (Cf. [Lu84a, §14.1], [Lu88, §10], [Lu08a, 5.3]) Let G be simple of simply connected type and F : G → G be a Steinberg map. Then there exists a ˜ with the following properties. regular embedding i : G → G (a) If G is of type Dn with n even, char(k)  2 and GF is ‘untwisted’ (i.e., the above ˜ = 2 and there is a surjective permutation s → s  is the identity), then dim Z(G) ˜ F F ˜ /i(G ) → Z/2Z × Z/2Z. map G ˜ F˜ /i(GF ) is cyclic. (b) In all other cases, G

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Reductive Groups and Steinberg Maps

˜ = G and i the Proof We begin by noting that, if Z(G) = {1}, then we can take G ˜ F F ˜ = i(G ) in this case. So, for the remainder of the proof, we can identity; we have G ˜ := GLn (k) assume that Z(G)  {1}. If G = SLn (k), then the inclusion i : G → G is a regular embedding with the required properties. We can now assume that G is not of type An . Let S ⊆ G be the torus in Example 1.5.6. We have Z(G) ⊆ S and S is F-stable by the discussion in Example 1.7.4. So, applying Lemma 1.7.3 with S, ˜ Since G = Gder , we have G ˜ der = i(G). we obtain a regular embedding i : G → G. ˜ ˜ F F F ˜ Then i(G ) = i(G) = Gder and so Example 1.4.11(b) implies that ˜ F˜ /i(GF )  KF˜ G

where

˜ G ˜ der . K := G/

˜ = i(G).S˜ and (We have an induced action of F˜ on K by Lemma 1.4.26.) Since G ˜S = Z(G), ˜ the inclusion S˜ → G ˜ induces an isogeny S˜ → K. Composition with the map S → S˜ from Lemma 1.7.3 yields an isogeny f : S → K such that f ◦ F = F˜ ◦ f and ker( f ) = Z(G). In particular, K is a torus and 1  dim K = dim S  2. ˜ If dim K = 1, then K  k× and so KF is isomorphic to a finite subgroup of k× ; ˜ hence, KF is cyclic in this case. Finally, assume that dim K = 2. This case only occurs in type Dn with n even, where G = Spin2n (k) and Z(G) = {t ∈ S | t 2 = 1}. Since Z(G)  {1}, we have char(k)  2 and Z(G)  Z/2Z × Z/2Z. If F is ‘untwisted’, then Remark 1.7.6(b) ˜ F˜ /i(GF ) has a factor group isomorphic to Z/2Z × Z/2Z. below will show that G This completes the proof of (a). It remains to consider the case where F is ‘twisted’, with all root exponents equal to q. By the description in Example 1.7.4, there is an isomorphism of algebraic groups S  k× × k× such that the action of F on S corresponds to the map q q (s1, s2 ) → (s2 , s1 ) on k× × k× . Consequently, SF  F×q2 is cyclic. We want to show that a similar argument works for K. To see this, let {ε1, ε2 } be a Z-basis of X(S), such that F induces the linear map ϕ : X(S) → X(S) with ϕ(ε1 ) = qε2 and ϕ(ε2 ) = qε1 . Now consider the isogeny f : S → K mentioned above. Since it has kernel Z(G), and since char(k)  2, the correspondences in 1.1.11 show that X(S)/ f ∗ (X(K))  X(Z(G))  Z/2Z × Z/2Z. So we have f ∗ (X(K)) = 2X(S). For i = 1, 2, let δi ∈ X(K) be such that f ∗ (δi ) = 2εi . Then {δ1, δ2 } is a Z-basis of X(K). Let β : X(K) → X(K) be the linear map induced ˜ Since f ◦ F = F˜ ◦ f , we also have ϕ ◦ f ∗ = f ∗ ◦ β. Hence, β(δ1 ) = qδ2 and by F. β(δ2 ) = qδ1 . Then K → k× × k× , t → (δ1 (t), δ2 (t)), is an isomorphism of algebraic groups such that the action of F˜ on K corresponds to the isogeny k× × k× → k× × k×, Consequently, KF  F×q2 is also cyclic. ˜

q

q

(t1, t2 ) → (t2 , t1 ). 

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85

The following remarks contain a number of useful, purely group-theoretical properties of a regular embedding. (Most of these are taken from [Leh78, §1].) ˜ be a regular embedding. To simplify the notation, Remark 1.7.6 Let i : G → G ˜ der . Let Z denote the ˜ Thus, G ⊆ G ˜ and Gder = G we identify G with its image in G. ˜ ˜ ˜ centre of G and Z denote the centre of G. Now, since Gder = Gder ⊆ G, we certainly ˜ = G.Z. ˜ F˜ . Furthermore, the ˜ This implies that Z = Z ˜ ∩ G and ZF = GF ∩ Z have G ˜ induces bijective homomorphisms G/Z  G/ ˜ Z˜ and inclusion G ⊆ G ˜ Z) ˜ F˜ /Z˜ F˜ ˜ F˜  G (G/Z)F  (G/

(a)

(where the last isomorphism holds by Example 1.4.11(b), using that Z˜ is connected). ˜ given by multipli˜ and consider the natural map f : G → G Now let G := G × Z −1 cation; note that f is surjective and ker( f ) = {(z, z ) | z ∈ Z} ⊆ Z(G). Applying ˜ τ = F), ˜ we obtain a canonical isomorphism Lemma 1.1.9 to f (and σ = F × F, ˜ F˜ /(GF .Z ˜ F˜ )  (Z/Z◦ )F G

(b)

where (Z/Z◦ )F is defined in Remark 1.5.12; explicitly, the above isomorphism is ˜ F˜ to g −1 F(g) ∈ G ∩ Z ˜ = Z where g ∈ G is such that induced by sending g˜ ∈ G ˜ F˜ = GF .Z˜ F˜ . ˜ In particular, if Z is connected, then G g ∈ g˜ Z. Lemma 1.7.7 In the above setting, let T be an F-stable maximal torus of G. Then ˜ and every F-stable ˜ ˜ T˜ := T.Z˜ is an F-stable maximal torus of G, maximal torus of ˜ G is of this form. In this situation, we have ˜ T = G ∩ T,

˜ F˜ = GF .T ˜ F˜ , G

˜

˜ F. TF = GF ∩ T

˜ induces an isomorphism NG (T)/T  N ˜ (T)/ ˜ T˜ Furthermore, the inclusion G ⊆ G G ˜ which is compatible with the actions of F, F. ˜ given ˜ →G Proof As in Remark 1.7.6, we have a natural surjective map f : G × Z ˜ it follows that T ˜ = T.Z˜ by multiplication. Since T × Z˜ is a maximal torus of G × Z, ˜ (see 1.3.10(a)); clearly, T ˜ is F-stable. ˜ But, ˜ is a maximal torus of G Now T ⊆ G ∩ T. ˜ since G is reductive, we have CG (T) = T and so T = G ∩ T. This also implies that ˜ = G.T. ˜ F˜ . Since Z ˜ ⊆ T, ˜ we have G ˜ Furthermore, since T = G ∩ T˜ is TF = GF ∩ T ˜ F˜ = GF .T ˜ F˜ ; see Example 1.4.11(a). connected, it follows that G ˜ and so we obtain an Now consider the Weyl groups. We have NG (T) ⊆ NG˜ (T) ˜ ˜ injective homomorphism W(G, T) → W(G, T). On the other hand, by 1.3.10(d), ˜ T) ˜ is isomorphic to W(G × Z, ˜ T × Z) ˜  W(G, T). Hence, the injection W(G, ˜ T) ˜ is also surjective. W(G, T) → W(G, ˜ be an arbitrary F-stable ˜ Finally, let T˜  ⊆ G maximal torus. Then T˜  = g˜ T˜ g˜ −1 ˜ for some g˜ ∈ G. Since f is surjective, we can write g˜ = g z˜ where g ∈ G and ˜ Hence, setting T := gTg −1 , we have T˜  = T .Z. ˜ But then we have again z˜ ∈ Z.    ˜ and so T is F-stable.  T =G∩T

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Lemma 1.7.8 In the setting of Remark 1.7.6, let H ⊆ G be an F-stable closed ˜ := H.Z˜ is an F-stable ˜ connected subgroup such that Z◦ ⊆ H. Then H closed ˜ and connected subgroup of G ˜ = dim H + dim Z˜ − dim Z◦ dim H

and

˜ F | = |HF || Z˜ F |/|Z◦ F |. |H ˜

˜

◦ is finite; consequently, ˜ ⊆ G∩ Z ˜ = Z and so (H∩ Z)/Z ˜ Proof We have Z◦ ⊆ H∩ Z ◦ ◦ ◦ ˜ ˜ → H/Z given by multiplication is an isogeny. the natural map H/Z × Z/Z This yields the dimension formula. Furthermore, Proposition 1.4.13(c) shows that ˜ ◦ )F˜ | = |(H/Z ˜ ◦ )F˜ |. So the formula for the order of H ˜ F˜ follows from |(H/Z◦ )F ||(Z/Z Example 1.4.11(b). 

In order to obtain further properties of regular embeddings, it will be useful to characterise these maps entirely in terms of root data. In particular, this will allow us to show how regular embeddings relate to dual groups. Lemma 1.7.9 Let G, G be connected reductive groups over k and f : G → G be an isotypy (see 1.3.21). Let T ⊆ G and T ⊆ G be maximal tori such that f (T) ⊆ T. Let ϕ : X(T) → X(T), χ  → χ  ◦ f |T , be the induced homomorphism. Then the following two conditions are equivalent. (i) f is an isomorphism of G onto a closed subgroup of G. (ii) f is a central isotypy and ϕ is surjective. Proof First note that the assumptions imply that G  = f (G).Z(G) and f (Gder ) =  . Let G := f (G) ⊆ G  ; this is a closed subgroup which is connected and Gder 1  = f (G ) = (G ) . Let T := reductive (see Lemma 1.4.26); furthermore, Gder der 1 der 1  f (T) ⊆ T ; then T1 is a maximal torus of G1 (see 1.3.10(a)) and we have T1 = G1 ∩ T. (We have G1 ∩ T ⊆ CG1 (T1 ) = T1 where the last equality holds since G1 is connected reductive; the reverse inclusion is clear.) Thus, we have f = i ◦ f1 where f1 : G → G1 is the restricted map and i : G1 → G is the inclusion; it is clear that i is a central isotypy. (Note that (G1 )der contains all the root subgroups of G1 ; see Remark 1.3.6.) Correspondingly, we have a factorisation ϕ = ϕ1 ◦ ε where ϕ1 : X(T1 ) → X(T) is induced by f1 and ε : X(T ) → X(T1 ) is given by restriction. Note that ε is surjective; see 1.1.11. Now suppose that (i) holds, that is, f1 : G → G1 is an isomorphism of algebraic groups. Then the composition f = i ◦ f1 will be a central isotypy. Furthermore, ϕ1 : X(T1 ) → X(T) is an isomorphism of abelian groups. Since ε is surjective, it follows that ϕ = ϕ1 ◦ ε must be surjective. Thus, (ii) holds. Conversely, assume that (ii) holds. Then ϕ1 is also surjective. So the correspondences in 1.1.11 show that f1 : T → T1 is a closed embedding. But we also have T1 = f1 (T), and so f1 : T → T1 is an isomorphism. Then Theorem 1.3.22(b) shows  that f1 : G → G1 also is an isomorphism.

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Corollary 1.7.10 Let G and G be connected reductive. Let F : G → G and F  : G → G be Steinberg maps. Let i : G → G be a homomorphism such that i ◦ F = F  ◦ i and i(T) ⊆ T, where T is an F-stable maximal torus of G and T is an F -stable maximal torus of G. Then i is a regular embedding if and only if the following three conditions hold. (1) i is a central isotypy, i.e., the induced map ϕ : X(T) → X(T) is a homomorphism of root data; (2) the map ϕ : X(T) → X(T) is surjective; and (3) X(T)/ZR  has no p-torsion, where R  are the roots relative to T.  , we have Proof Suppose that i is a regular embedding. Since i(Gder ) = Gder G = i(G).Z(G). So the general assumptions of Lemma 1.7.9 plus condition (i) are satisfied. Hence, the first two conditions hold; the third one holds because Z(G) is connected (see Lemma 1.5.22). Conversely, if the above three conditions are satisfied, then Z(G) is connected and Lemma 1.7.9 shows that i is an isomorphism of G onto a closed subgroup of G. Since i is central, we have G = i(G).Z(G)  = i(G ). Hence, i is a regular embedding.  which implies that Gder der

1.7.11 Assume that (G, F), (G∗, F ∗ ) are in duality (see Definition 1.5.17), with respect to maximally split tori T0 ⊆ G and T∗0 ⊆ G∗ . Furthermore, assume that (G, F ) and (G∗, F ∗ ) are in duality, with respect to maximally split tori T0 ⊆ G and T0∗ ⊆ G∗ . Thus, we are given isomorphisms ∼

δ : X(T0 ) → Y (T∗0 )

and

∼

δ  : X(T0 ) → Y (T0∗ )

satisfying the above conditions. Now let f : G → G be a central isotypy such that f ◦ F = F  ◦ f and f (T0 ) ⊆ T0 . Thus, the induced map ϕ : X(T0 ) → X(T0 ) is a homomorphism of root data as in 1.2.2. But then the transpose map ϕtr : Y (T0 ) → Y (T0 ) defines a morphism of the dual root data. Using the transposed isomorphisms δtr : X(T∗0 ) → Y (T0 ) and δ tr : X(T0∗ ) → Y (T0 ), we obtain a map ϕˆ = (δ tr )−1 ◦ ϕtr ◦ δtr : X(T∗0 ) → X(T0∗ ) which is a homomorphism between the root data of G∗ and G∗ . Now, by Theorem 1.3.22 (extended isogeny theorem), there exists a central isotypy f ∗ : G∗ → ˆ Arguing as in Lemma 1.4.24, one shows G∗ which maps T0∗ into T∗0 and induces ϕ. ∗ ∗ ∗ that f can be chosen such that f ◦ F = F ∗ ◦ f ∗ . We have ϕˆtr ◦ δ  = δ ◦ ϕ. So, for any λ  ∈ X(T0 ), we obtain the compatibility relation:





f ∗ ◦ δ (λ ) = ϕˆtr δ (λ ) = ϕˆtr ◦ δ  (λ ) = δ ◦ ϕ (λ ) = δ(λ  ◦ f ) ∈ Y (T∗0 ). In this situation, we say that the two central isotypies f : G → G and f ∗ : G∗ → G∗ correspond to each other by duality.

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Reductive Groups and Steinberg Maps

(This relation is symmetric.) With this notation, we can now state: Lemma 1.7.12 Let f : G → G and f ∗ : G∗ → G∗ correspond to each other by duality, as above. Assume that f : G → G  is an isomorphism with a closed subgroup of G. Then f ∗ : G∗ → G∗ is surjective and ker( f ∗ ) is a central torus. ∗ ∗ Furthermore, the restricted map f ∗ : (G∗ )F → (G∗ )F is surjective. Proof We follow [Bo06, 2.5], using the notation in 1.7.11. By restriction, f yields a closed embedding f : T0 → T0 . By 1.1.11(a), the induced map ϕ : X(T0 ) → X(T0 ) is surjective and ker(ϕ)  X(T0 / f (T0 )) is torsion-free. Since Hom(X(T0 ), Z)  Y (T0 )  X(T∗0 ), Hom(X(T0 ), Z)  Y (T0 )  X(T0∗ ), we obtain an exact sequence ϕˆ

{0} −→ X(T∗0 ) −→ X(T0∗ ) −→ Hom(ker(ϕ), Z) −→ {0} where, as above, ϕˆ is induced by f ∗ : T0∗ → T∗0 and the second map is given by identifying λ ∈ X(T0∗ ) with an element in Hom(X(T0 ), Z) and then restricting this to ker(ϕ) ⊆ X(T0 ). By 1.1.11(b), we deduce that f ∗ : T0∗ → T∗0 is surjective and that ∗ ))  Hom(ker(ϕ), Z). Hence, X(ker( f ∗ )) is torsionX(ker( f ∗ ))  X(T0∗ )/ϕ(X(T ˆ 0 ∗ free and so ker( f ) is a torus. Since G∗ = f ∗ (G∗ ).T∗0 and f ∗ (T0∗ ) = T∗0 , we also ∗

have f ∗ (G∗ ) = G∗ . Finally, since ker( f ∗ ) is connected, the fact that f ∗ (G∗ )F = ∗ (G∗ )F follows from Proposition 1.4.13(b).  The following result is cited, for example, in [Lu84a, 8.8], [Lu88, 8.1], [Lu92b, 0.1] in relation to certain reduction arguments; it appeared in an unpublished manuscript of Asai [As]. See [Ta19] for a further discussion of Asai’s reduction techniques. Proposition 1.7.13 (Cf. [As, §2.3]) Let G be connected reductive and F : G → G a Steinberg map. Then there exist a connected reductive group G• , a Steinberg map F • : G• → G• and a homomorphism f : G• → G, such that the following conditions hold: (a) G•der is semisimple of simply connected type; (b) f is a surjective homomorphism of algebraic groups and F ◦ f = f ◦ F • ; (c) ker( f ) is a central torus of G• . •

In particular, f induces a surjective homomorphism of finite groups G• F → GF . Furthermore, if G has a connected centre, then G• has a connected centre, too. Proof Asai [As] shows this by explicitly constructing the appropriate root datum for G• and then using Theorem 1.3.22 (extended isogeny theorem). Here is a more

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89

direct argument. Let πsc : (Gder )sc → Gder be a simply connected covering of the derived group of G, as in Remark 1.5.13. Assume first that πsc is bijective. Let Z := Z(G) and K := {z ∈ Z((Gder )sc ) | πsc (z) ∈ Z◦ }. We have an isogeny f1 : (Gder )sc × Z◦ → G,

(z, z ) → πsc (z)z ,

with ker( f1 ) = {(z, πsc (z)−1 ) | z ∈ K}; note that ker( f1 ) is finite. Let G• := ((Gder )sc × Z◦ )/ker( f1 ). Then f1 induces a bijective morphism of algebraic groups f : G• → G. Now (b), (c) are clear. To prove (a), note that G•der = G . ker( f1 )/ker( f1 )

where

G := (Gder )sc × {1};

here, G is a closed subgroup of (Gder )sc × Z◦ . We have a bijective homomorphism of algebraic groups (Gder )sc → G•der , sending g ∈ (Gder )sc to the image of (g, 1) in G• . On the other hand, since G ∩ ker( f1 ) = {1} and ker( f1 ) is finite, the product G . ker( f1 ) is semidirect and so the natural projection G . ker( f1 ) → G → (Gder )sc is a homomorphism of algebraic groups; see 1.1.8(c). Passing to the quotient by ker( f1 ), we obtain a homomorphism of algebraic groups G•der → (Gder )sc which is inverse to the above map (Gder )sc → G•der . Thus, (a) holds. Finally, since f is bijective, the centre of G• is connected if and only if Z is connected. Now consider the general case, where the map πsc may not be bijective. By Lemma 1.7.3, there exists a regular embedding i : G∗ → H. By duality, we obtain a homomorphism of algebraic groups i ∗ : H∗ → G; note that, as remarked in Definition 1.5.17, we can identify (G∗ )∗ with G. By Lemma 1.7.12, i ∗ is surjective and ker(i ∗ ) is a central torus of H∗ ; furthermore, by Lemma 1.5.22, the simply connected covering (H∗der )sc → H∗der is bijective. By the previous argument, there exists a bijective homomorphism of algebraic groups f1 : G• → H∗ such that (a), (b), (c) hold. Then (a), (b), (c) hold for the composition f = i ∗ ◦ f1 : G• → G. Finally, assume that Z is connected. Now the derived subgroup of H is isomorphic to that of G∗ . Hence, Lemma 1.5.22 implies that the centre of H∗ is connected as  well. Since f1 is bijective, it follows that G• also has a connected centre. ˜ be a regular Example 1.7.14 Assume that G is semisimple and let i : G → G ˜ embedding. Applying Proposition 1.7.13 to G, we obtain a homomorphism of ˜• → G ˜ satisfying the above three conditions. Furthermore, algebraic groups f : G ˜ is connected, we have that Z(G ˜ • ) is connected, too. Now f (G ˜• ) = since Z(G) der ˜ • → G which is a Gder = G and so, by restriction, we obtain an isogeny fˆ : G der

simply connected covering of G. We have a commutative diagram: G

i

−→

˜ G

→

↑ ˜• G

↑ ˜• G der

90

Reductive Groups and Steinberg Maps

Thus, a simply connected covering of G can always be chosen to be compatible with ˜ (This remark appears in [Lu88, 8.1(d)].) the given regular embedding i : G → G. The following result was first stated (for K of characteristic 0) by Lusztig [Lu88, Prop. 10] (see also [Lu84b]), with an outline of the strategy of the proof. The details, which are surprisingly complicated, were provided much later in [Lu08a]; in the meantime, Cabanes and Enguehard also gave a proof in [CE04, Chap. 16], along similar lines. First, one employs a reduction argument which reduces the proof to the case where G is simple of simply connected type. As far as such groups are concerned, one can then use Proposition 1.7.5, which shows that type Dn with n even is the only case which requires a special argument, but all the difficulty lies with this case. We state here an extension of Lusztig’s original result, valid for irreducible representations over any algebraically closed field. ˜ be a regular Theorem 1.7.15 (Multiplicity-Freeness Theorem) Let i : G → G embedding and K be any algebraically closed field. Then the restriction of every ˜ F˜ -module to GF (via i) is multiplicity-free. simple KG Proof We can only sketch the general strategy here, and highlight where the principal difficulty of the proof lies. First we note that the reduction argument described in the proof of [Lu88, Prop. 10] works for simple modules over any algebraically closed field K, not just for char(K) = 0. (Some adjustments of a different kind are required, since Lusztig considers Frobenius maps, not Steinberg maps in general; see [Ta19].) Hence, it suffices to prove the theorem in the case where G is simple of simply connected type. Furthermore, the reduction argument shows ˜ that it is sufficient to consider only one particular regular embedding i : G → G, namely, one satisfying the conditions in Proposition 1.7.5. So let us now assume that these conditions are satisfied. ˜ F˜ /i(GF ) is cyclic, then a standard result on representations of finite groups If G shows that the desired assertion holds; see, e.g., [Fei82, Theorem III.2.14]. (This uses that K is algebraically closed, but works without any assumption on char(K).) It remains to consider the case where G is of type Dn with n even, char(k)  2 and F is ‘untwisted’. Let us identify G with i(G) and use the notational conventions ˜ F˜ , H := G.Z ˜ F˜ , we have in Remark 1.7.6. Writing G = GF , G˜ = G H  G˜

˜ and G/H  (Z/Z◦ )F = Z  Z/2Z × Z/2Z.

˜ ˜ Let V be a simple KG-module and denote by VH its restriction to H. Since H = G.Z˜ F ˜ F˜ is contained in the centre of G, ˜ it is sufficient to show that VH is multiplicityand Z free. (To see this, one only needs to show that non-isomorphic simple H-submodules of VH remain simple and non-isomorphic upon restriction to G. And this easily follows, for example, by the argument in [Ge93a, p. 265].) Now, if char(K) = 2, then

1.7 Regular Embeddings

91

VH is multiplicity-free by some general results on representations of finite groups; see, e.g., [KlTi09, Lemma 3.14]. If char(K) = char(k) = p, then VH is even simple by [BrLu12, Lemma 3.4] (see also [Ca88, §B.11]). So, finally, assume that char(K)  char(k) and char(K)  2. In particular, char(K) ˜ By Clifford’s Theorem (see is either 0 or a prime not dividing the index of H in G. [HuBl82, Theorem VII.9.18]), VH is semisimple and there are two possibilities: either VH is multiplicity-free (with 1, 2 or 4 irreducible constituents) or the direct sum of 2 copies of a simple KH-module. In the case where char(K) = 0, it is shown by an elaborate counting argument (first published in [CE04]; see also [Lu08a]) that the second type does not occur. This argument involves: • knowing the action (by tensor product) of the four 1-dimensional representations ˜ ˜ of G/H on the simple KG-modules; • counting conjugacy classes and simple modules for Spin2n (q). (As noted in [Lu88, §13], this is ‘very long and unpleasant’.) Finally, it is shown in [Ge93a, §3], using the results on basic sets of Brauer characters in [GeHi91], that Lusztig’s argument can be adapted to work as well when char(K) > 0 (but still char(K)  char(k) and char(K)  2).  It would be highly desirable to find a more conceptual proof of this result which does not rely on a case-by-case analysis and the counting arguments for Spin2n (q). Remark 1.7.16 It is an intriguing challenge to try to formulate a general condition on a finite group Γ and a normal subgroup Γ   Γ such that the statement of Theorem 1.7.15 can be obtained as a special case of it. In [Bo06, §11.E], Bonnafé poses the following question. Suppose that Γ/Γ  is a p-group (for some prime p) and that Γ = CΓ (g)Γ  for every p-element g ∈ Γ. Is it true that the restriction of every irreducible character of Γ to Γ  is multiplicity-free? It is shown by Navarro [Na19] that this question has a negative answer, by finding a counter example using the library of groups in [GAP4].

2 Lusztig’s Classification of Irreducible Characters

Let G be a connected reductive algebraic group over k = F p and F : G → G be a Steinberg map. We will be interested in describing the complex irreducible characters of the finite group GF , where the term ‘describing’ is left deliberately vague: it can range from complete knowledge of all character values to rough information about the possible character degrees and their multiplicities, to bounds on character values on specific elements and so on. The Cambridge ATLAS [CCNPW] (see also [Bre18]) contains many character tables of individual finite groups of Lie type, even of some of the large groups of exceptional type (for example, 2E6 (2)). For small rank cases, complete character tables for a whole series of groups (for example, Sp4 (q) where q runs through all prime powers) have been determined, in many cases without using any of the machinery arising from the Deligne–Lusztig theory that we are going to introduce in this chapter. The most complete results are available for the groups GLn (q) by the pioneering work of Green [Gre55]. Section 2.1 introduces some notation and basic constructions for characters of finite groups. From Section 2.2 on, we consider finite groups of the form GF as above. We begin by recalling the fundamental construction of the virtual characters RTG (θ) using the -adic cohomology approach of Deligne–Lusztig [DeLu76]; in particular, we define the set Uch(GF ) of unipotent characters in Section 2.3. Since all this is already well covered and re-worked in existing textbooks (e.g., [Ca85], [DiMi20]), we will only introduce the required notation, state the main results (e.g., scalar product formulae and degree formulae) and illustrate them by examples, but without proofs. Only on a few occasions do we present a detailed argument when this appears to be a good illustration for the methods developed so far. Sections 2.4 and 2.5 form the technical core of this chapter. Here, we work out in some detail the basic formalism of Lusztig’s book [Lu84a] which yields an approach to the partition of Irr(GF ) into rational/geometric series of characters which is somewhat different from that developed in [Ca85], [DiMi20]. It also provides the technical language for the formulation of the main result, that is, ‘Main 92

2.1 Generalities about Character Tables

93

Theorem 4.23’ of [Lu84a]. Once this is established, the Jordan decomposition of characters can be stated in relatively smooth terms, both in the original case where the centre of G is connected, and in the general case; see Section 2.6. The importance and impact of this fundamental result can hardly be overstated. It is a tremendous achievement, both conceptually and in terms of technical complexity, which leads to an efficient classification of Irr(GF ) in terms of data in a group G∗ dual to G (as already introduced in Chapter 1). Much of this has been turned into explicit algorithms and computer programs; see [GHLMP], [MiChv], [Lue07]. In the final two Sections 2.7 and 2.8 we give a first introduction to the problem of computing the values of the irreducible characters of GF on all elements. This problem is not yet completely solved. We will mainly focus on uniform functions and the determinantion of the virtual characters RTG (θ).

2.1 Generalities about Character Tables We assume some basic familiarity with the ‘ordinary’ representation theory of finite groups over fields of characteristic 0 (in which case all of their finite-dimensional representations are semisimple); see, e.g., [FuHa91, Part I], [Hup67, Kap. V] or [Is76]. Our main interest will be in studying the characters of representations, where we work throughout over a fixed subfield K ⊆ C, which is algebraic over Q, invariant under complex conjugation and ‘large enough’, that is, K contains sufficiently many roots of unity and K is a splitting field for all finite groups under consideration. 2.1.1 Let Γ be a finite group. We note by CF(Γ) the vector space of all K-valued class functions on Γ, that is, functions f : Γ → K that are constant on the conjugacy classes of Γ. There is an inner product given by 1  f , f  := f (g) f (g) where f , f  ∈ CF(Γ). |Γ| g ∈Γ (Here, the bar denotes complex conjugation.) Let Irr(Γ) be the set of irreducible characters of Γ (afforded by irreducible representations of Γ over K). Let Cl(Γ) be the set of conjugacy classes of Γ. It is well known that |Irr(Γ)| = |Cl(Γ)| and that Irr(Γ) is an orthonormal basis of CF(Γ). The character table of Γ is the matrix

X(Γ) = χ(gC ) χ ∈Irr(Γ), C ∈Cl(Γ) where gC is a fixed representative of C ∈ Cl(Γ). We call f ∈ CF(Γ) a virtual character if f is an integral linear combination of Irr(Γ). We say that f is an actual character (or just a character) if f can be written as an integral combination of Irr(Γ)

94

Lusztig’s Classification of Irreducible Characters

with non-negative coefficients. It is then the character of an actual representation of Γ. Explicit examples certainly play an important and useful part in this theory. Table 2.1 displays the character tables of three finite groups of Lie type; they are printed in the output format of GAP [Scho97], [GAP4], which is modelled on the Cambridge ATLAS [CCNPW]. (A dot ‘.’ in the tables stands for the value 0; for the notation concerning irrationalities, see the GAP online help.) 2.1.2 A highly useful method for constructing characters of Γ is given by the process of induction, due to Frobenius. If Γ   Γ is a subgroup and f ∈ CF(Γ ), then we denote by IndΓΓ ( f ) ∈ CF(Γ) the induced class function. It is well known that, if f is a character of Γ , then IndΓΓ ( f ) is a character of Γ. For f ∈ CF(Γ ), the values of the induced class function are given by the following character formula:



1 f (xgx −1 ) (g ∈ Γ). IndΓΓ f (g) =  |Γ | −1  x ∈Γ : xgx

∈Γ

Thus, in order to work out these values, one needs to know the class fusion from Γ  to Γ, that is, the map η : Cl(Γ ) → Cl(Γ) such that C  ⊆ η(C ) for all C  ∈ Cl(Γ ). For a fixed g ∈ Γ, we denote g Γ  := gΓ  g −1 and define g f ∈ CF(g Γ ) by g f (gg  g −1 ) = f (g  ) for all g  ∈ Γ  . Then, clearly, we have IndΓg Γ (g f ) = IndΓΓ ( f )

for all f ∈ CF(Γ ).

(When we consider characters of finite groups of Lie type, we will see generalisations of the process of induction, defined by cohomological methods.) 2.1.3 A further useful construction is as follows. Let Γ   Γ be a normal subgroup and f ∈ CF(Γ/Γ ). Then we obtain a class function f˜ ∈ CF(Γ) by setting f˜(g) = f (gΓ ) for all g ∈ Γ. Clearly, if f is a character of Γ/Γ , then f˜ is a character of Γ, called the inflation of f . Conversely, if f ∈ CF(Γ), then we obtain a class function f  ∈ CF(Γ/Γ ) by setting 1 f (gg ) for all g ∈ Γ. f (gΓ ) =  |Γ | g ∈Γ Again, if f is a character of Γ, then f  is a character of Γ/Γ . Indeed, if V is a KΓ-module affording f , then the subspace of fixed points 

V Γ = {v ∈ V | g  .v = v for all g  ∈ Γ  } 

is still a K(Γ/Γ )-submodule of V, and f  is the character of V Γ . Remark 2.1.4 We shall need a few basic notions from Clifford theory (see, e.g., [Hup67, §V.17]). Let Γ   Γ be a normal subgroup. Then Γ acts by conjugation

2.1 Generalities about Character Tables

95

Table 2.1 Tables for Sp4 (2)  S6 , Suz(8) = 2B2 (8), SU4 (2)  PSp4 (3) Sp4(2) 2 4 4 4 4 1 1 1 3 3 . 1 3 2 1 . 1 2 1 2 . . . 1 5 1 . . . . . . . . 1 . 1a 2a 2b 2c 3a 6a 3b 4a 4b 5a 6b X.1 X.2 X.3 X.4 X.5 X.6 X.7 X.8 X.9 X.10 X.11

1 5 9 5 10 16 5 10 9 5 1

-1 1 -1 1 -1 1 -3 1 1 2 . -1 -3 1 -3 . . . -1 1 3 -1 -1 2 -2 -2 2 1 1 1 . . . -2 . -2 1 1 -3 -1 1 2 2 -2 -2 1 -1 1 3 1 3 . . . 3 1 -1 2 . -1 1 1 1 1 1 1

-1 -1 1 1 . . -1 . -1 1 1

1 1 -1 -1 . 1 1 -1 . -1 . . . . -1 . 1 . -1 . . . . 1 1 -1 . -1 . -1 1 1 1

Suz(8) 2 5 7 13

X.1 X.2 X.3 X.4 X.5 X.6 X.7 X.8 X.9 X.10 X.11 A C E F G

SU4(2) 2 3 5

X.1 X.2 X.3 X.4 X.5 X.6 X.7 X.8 X.9 X.10 X.11 X.12 X.13 X.14 X.15 X.16 X.17 X.18 X.19 X.20 A C E G I

= = = = =

= = = = =

6 6 4 4 . . . . . . . 1 . . . 1 . . . . . . 1 . . . . 1 1 1 . . . 1 . . . . . . . 1 1 1 1a 2a 4a 4b 5a 7a 7b 7c 13a 13b 13c 1 1 1 1 1 14 -2 A -A -1 14 -2 -A A -1 35 3 -1 -1 . 35 3 -1 -1 . 35 3 -1 -1 . 64 . . . -1 65 1 1 1 . 65 1 1 1 . 65 1 1 1 . 91 -5 -1 -1 1

1 -3 -3 -2 2 2 -1 7 4 8 10 6 6 -8 -8 -3 -3 -4 . 9

1 1 1 2 -2 -2 -1 3 4 . 2 2 2 . . -3 -3 4 . -3

1 A /A -3 B /B 6 -3 2 6 3 C /C D /D E /E 6 -8 .

1 /A A -3 /B B 6 -3 2 6 3 /C C /D D /E E 6 -8 .

1 . . . . . 1 D B C .

1 1 1 1 . 1 1 1 . 1 1 1 . E G F . F E G . G F E 1 -1 -1 -1 C . . . D . . . B . . . . . . .

2*E(4) = 2ER(-1), B = E(7)+E(7)ˆ6 E(7)ˆ3+E(7)ˆ4, D = E(7)ˆ2+E(7)ˆ5 -E(13)-E(13)ˆ5-E(13)ˆ8-E(13)ˆ12 -E(13)ˆ4-E(13)ˆ6-E(13)ˆ7-E(13)ˆ9 -E(13)ˆ2-E(13)ˆ3-E(13)ˆ10-E(13)ˆ11

6 6 5 3 3 2 1 4 3 . 3 3 2 2 1 2 4 2 1 4 4 3 3 1 . . 2 2 2 2 2 1 1 . . . . . . . . 1 . . . . . . 1a 2a 2b 3a 3b 3c 3d 4a 4b 5a 6a 6b 6c 6d 6e 6f 1 5 5 6 10 10 15 15 20 24 30 30 30 40 40 45 45 60 64 81

1 . . . . . 1 B C D .

1 1 1 1 1 1 1 1 1 -1 2 1 -1 . F /F I -I -1 2 1 -1 . /F F -I I 3 . 2 . 1 1 1 1 1 1 1 2 . . A /A -1 -1 1 1 2 . . /A A -1 -1 3 . 3 -1 . 2 2 -1 -1 . 3 -1 1 . 1 1 -2 -2 5 -1 . . . -2 -2 1 1 . 3 . . -1 2 2 2 2 3 3 -2 . . -1 -1 -1 -1 -3 . 2 . . /F F -I I -3 . 2 . . F /F I -I -2 1 . . . G /G G /G -2 1 . . . /G G /G G . . 1 1 . H /H . . . . 1 1 . /H H . . -3 -3 . . . 2 2 -1 -1 4 -2 . . -1 . . . . . . -3 -1 1 . . . .

1 . . -2 -1 -1 2 1 1 -1 -1 . . 1 1 . . -1 . .

. 2 . 9a

. 2 2 2 1 1 . . . 9b 12a 12b

1 1 1 1 1 1 J /J -J -/J 1 /J J -/J -J -1 . . -1 -1 1 -/J -J J /J 1 -J -/J /J J -1 . . . . . . . -1 -1 1 -1 -1 . . . . . . . -1 . . 1 1 -1 . . /J J -1 . . J /J . -/J -J . . . -J -/J . . . . . -J -/J . . . -/J -J 1 . . . . . 1 1 . . . . . . .

E(3)-2*E(3)ˆ2 = (1+3ER(-3))/2, B = 5*E(3)+2*E(3)ˆ2 = (-7+3ER(-3))/2 6*E(3)-3*E(3)ˆ2 = (-3+9ER(-3))/2, D = 2*E(3)+8*E(3)ˆ2 = -5-3ER(-3) -9*E(3) = (9-9ER(-3))/2, F = E(3)+2*E(3)ˆ2 = (-3-ER(-3))/2 -2*E(3) = 1-ER(-3) = 1-i3, H = 3*E(3) = (-3+3ER(-3))/2 E(3)-E(3)ˆ2 = ER(-3) = i3, J = -E(3) = (1-ER(-3))/2

96

Lusztig’s Classification of Irreducible Characters

on Γ  and, hence, on Irr(Γ ). Clearly, every g ∈ Γ  acts trivially on Irr(Γ ) and so we have an action of Γ/Γ  on Irr(Γ ). It is well known that, if χ ∈ Irr(Γ), then the irreducible constituents of χ|Γ form a single orbit under this action; in particular, all these constituents have the same multiplicity in χ|Γ . (The Multiplicity-Freeness Theorem 1.7.15 describes a situation where these multiplicities are equal to 1.) Now assume, furthermore, that Γ/Γ  is abelian. Let Θ be the group of all linear characters η ∈ Irr(Γ) (that is, group homomorphisms η : Γ → K× ) such that Γ  ⊆ ker(η). This group acts on Irr(Γ) by the usual pointwise product of class functions. Let χ1, χ2 ∈ Irr(Γ). Then the restrictions of χ1, χ2 to Γ  have an irreducible constituent in common if and only if χ1, χ2 are in the same Θ-orbit. More precisely, we have:   χ1 |Γ, χ2 |Γ = |{η ∈ Θ | η · χ1 = χ2 }|.

 Indeed, since Γ/Γ  is abelian, IndΓΓ χ1 |Γ = IndΓΓ (1Γ ) · χ1 = η ∈Θ η · χ1 , so the above equality immediately follows using Frobenius reciprocity. Remark 2.1.5 In the following section, we will construct representations of certain finite groups Γ over Q , an algebraic closure of the field of -adic numbers, where  is a prime number. These constructions will give rise to virtual characters of Γ over Q , that is, Z-linear combinations of characters afforded by finite-dimensional Q Γ-modules. Since K ⊆ C is algebraic over Q, we can find an embedding K ⊆ Q . Assume now that a virtual character f over Q has the property that f (g) ∈ K for all g ∈ Γ. Then one easily sees that f is also a virtual character over K, that is, a Z-linear combination of characters afforded by finite-dimensional KΓ-modules. (Note that K is assumed to be a splitting field for Γ.) 2.1.6 Assume now that we are also given a group automorphism σ : Γ → Γ. It will be convenient to introduce already at this stage some notation related to the presence of σ and its action on the elements and the characters of Γ. We say that two elements y, y  ∈ Γ are σ-conjugate if there exists some x ∈ Γ such that y  = xyσ(x)−1 . This defines an equivalence relation on Γ; the equivalence classes are called the σ-conjugacy classes of Γ. For g ∈ Γ, the σ-centraliser of g is defined to be the subgroup CΓ,σ (g) := {x ∈ Γ | xg = gσ(x)}. This is the stabiliser of g for the action of Γ on itself by σ-conjugation and so |Γ| = |CΓ,σ (g)||Cg |, where Cg is the σ-conjugacy class of g. As in [Bo06, 1B], [Leh78, §1], we shall denote by H 1 (σ, Γ) the set of σ-conjugacy classes of Γ. (Of course, if σ = idΓ is the identity, then H 1 (idΓ, Γ) = Cl(Γ) is just the set of ordinary conjugacy classes.) 2.1.7

In the setting of 2.1.6, a function f : Γ → K is called a σ-class function if

2.1 Generalities about Character Tables

97

f is constant on the σ-conjugacy classes of Γ. Let CFσ (Γ) be the K-vector space of all σ-class functions on Γ. There also is an inner product given by 1  f , f  σ := f (g) f (g) where f , f  ∈ CFσ (Γ). |Γ| g ∈Γ We obtain functions in CFσ (Γ) by the following construction. Given χ ∈ Irr(Γ), we define χ σ : Γ → K by χ σ (g) := χ(σ(g)) for all g ∈ Γ; then it is clear that we also have χ σ ∈ Irr(Γ). We set Irr(Γ)σ := { χ ∈ Irr(Γ) | χ σ = χ}. Now let χ ∈ Irr(Γ)σ . Let n = χ(1) and X : Γ → GLn (K) be a matrix representation affording χ. Since χ σ = χ, there exists an invertible matrix E ∈ GLn (K) such that the following invariance condition holds: X(σ(g)) = E · X(g) · E −1

for all g ∈ Γ.

(∗)

Since X is irreducible and K is a splitting field, E is unique up to multiplication by a non-zero scalar (Schur’s Lemma). In general, there is no canonical choice for E, but we can and will always assume that E is a matrix of finite order (see the argument in [Fei82, Theorem III.2.14]). Then we obtain a σ-class function χ˜ ∈ CFσ (Γ) by χ(g) ˜ := trace(X(g) · E) = trace(E · X(g))

for all g ∈ Γ.

Note that χ˜ is well defined up to multiplication by a root of unity; each such χ˜ will be called a σ-extension of χ. (See Remark 2.1.9 below for further explanations.) By (∗), we have χ(σ(g)) ˜ = χ(g) ˜ for all g ∈ Γ. Furthermore, since E has finite order, χ(g) ˜ is a cyclotomic integer for all g ∈ Γ. There are a few situations where there is a natural choice for E as above. Example 2.1.8 (a) Assume that σ is the identity. Then the invariance condition (∗) in 2.1.7 trivially holds with E = In (the identity matrix of size n); so we have χ˜ = χ with this choice of E. However, for greater flexibility, we do allow here to take E = δIn where δ is any root of unity; we then have χ˜ = δ χ and χ(1) ˜ = δ. (b) Assume that σ is an inner automorphism, that is, there exists some g0 ∈ Γ such that σ(g) = g0 gg0−1 for all g ∈ Γ. Then, clearly, we have χ = χ σ for all χ ∈ Irr(Γ). In this case, the above invariance condition simply holds for E = X(g0 ). Hence, a natural σ-extension χ˜ of χ is given by χ(g) ˜ := χ(gg0 ) = χ(g0 g)

for all g ∈ Γ.

(c) Let χ ∈ Irr(Γ)σ be a linear character. Then χ is a group homomorphism Γ → K×

98

Lusztig’s Classification of Irreducible Characters

and the invariance condition in 2.1.7 certainly holds for E = 1. Hence, in this case, a canonical σ-extension χ˜ of χ is given by χ(g) ˜ := χ(g)

for all g ∈ Γ.

This situation occurs, of course, when Γ is abelian. Note that we have |H 1 (σ, Γ)| = |Γσ | in this case, where Γσ := {g ∈ Γ | σ(g) = g}; see [Bo06, Chap. I, Exp. 1.1]. Remark 2.1.9 As in [DiMi85, §I.6], let Γ˜ be the semidirect product of Γ with the ˜ If σ has order cyclic group σ ⊆ Aut(Γ). We identify Γ with a subgroup of Γ. d  1, then Γ˜ is generated by Γ and an additional element σ ˜ ∈ Γ˜ (of order d) such ˜ we have the identity that, in Γ, σ·g· ˜ σ ˜ −1 = σ(g)

for all g ∈ Γ.

˜ for g ∈ Γ. Hence, two elements of Γ˜ are Note that σ·(g· ˜ σ)· ˜ σ ˜ −1 = g −1 ·(g·σ)·g ˜ conjugate in Γ if and only if they are already conjugate by an element of Γ. Consequently, for g ∈ Γ, we have CΓ,σ (g) = CΓ (g·σ) ˜ and the map g → g·σ ˜ induces a bijection between the σ-conjugacy classes of Γ and the usual conjugacy classes of ˜ Furthermore, Irr(Γ)σ consists precisely Γ˜ that are contained in the coset Γ·σ ˜ ⊆ Γ. ˜ Now let χ ∈ Irr(Γ)σ . of those irreducible characters of Γ that can be extended to Γ. Then we can find a matrix E satisfying the invariance condition in 2.1.7(∗) and such that E d = In (see [Fei82, Theorem III.2.14]). In this case, ( χ(g)) ˜ g∈Γ are the values of an extension of χ on the elements in the coset Γ σ. ˜ One can also define a process of σ-induction of characters from σ-invariant subgroups of Γ; see [Bo06, 1.C] for further details. 2.1.10 Let us now assume that a particular σ-extension χ˜ has been chosen for each χ ∈ Irr(Γ)σ . Then, by [DiMi85, Rem. I.6.3(iii)], we have the following orthogonality relations:  1 1 if χ = χ ,   χ(g) ˜ χ˜ (g) =  χ, ˜ χ˜ σ = 0 if χ  χ  . |Γ| g ∈Γ

Moreover, by the argument in [GKP00, 7.3], we have |Irr(Γ)σ | = |H 1 (σ, Γ)|. Hence, for any g, g  ∈ Γ, we also have 

if g, g  are σ-conjugate, |CΓ,σ (g)|  χ(g) ˜ χ(g ˜ )= 0 otherwise. σ χ ∈Irr(Γ)

In particular, the set { χ˜ | χ ∈ Irr(Γ)σ } is an orthonormal basis of CFσ (Γ). Definition 2.1.11

In the setting of 2.1.7, let us choose a representative gC ∈ C

2.1 Generalities about Character Tables

99

for each σ-conjugacy class C ∈ H 1 (σ, Γ). Furthermore, as above, we assume that a particular σ-extension χ˜ has been chosen for each χ ∈ Irr(Γ)σ . Then the matrix

Xσ (Γ) = χ(g ˜ C ) χ ∈Irr(Γ)σ , C ∈H 1 (σ,Γ) is called the σ-character table of Γ. This is a square matrix which does not depend on the choice of gC ∈ C. (But it does depend on the choice of the σ-extension χ˜ for each χ ∈ Irr(Γ)σ ; if a different choice of χ˜ is made, then this will result in multiplying the corresponding row of Xσ (Γ) by a root of unity.) By 2.1.7, the entries of Xσ (Γ) are cyclotomic integers; the orthogonality relations in 2.1.10 show that Xσ (Γ) is invertible. If we choose σ-extensions as in Remark 2.1.9, then the matrix Xσ (Γ) is part of the ATLAS ‘compound character table’ for Γ. If, moreover, σ is the identity, then Xσ (Γ) = X(Γ) is just the ordinary character table of Γ. We now give two examples. First of all, they illustrate that one cannot expect a natural choice of a σ-extension of a character. Secondly, we discuss this in some detail because a completely analogous problem – at a technically more elaborate level – is one of the key issues in completing the character tables of finite groups of Lie type (see Section 2.8, especially Remark 2.8.8). Example 2.1.12 Let Γ be a dihedral group of order 8. Thus, Γ is generated by two elements s1  s2 such that s1, s2 have order 2 and the product s1 s2 has order 4. We have Irr(W) = {1Γ, ε, ε , ε , φ1 }, where 1Γ is the trivial character, ε, ε , ε  are linear characters, and φ1 has degree 2. Here, the notation is such that ε(s1 ) = ε(s2 ) = −1; furthermore, ε (s1 ) = ε (s2 ) = −1

and

ε (s2 ) = ε (s1 ) = 1.

(See also Table 4.1, p. 274.) Now, there is an automorphism σ : Γ → Γ such that σ(s1 ) = s2 and σ(s2 ) = s1 . The σ-conjugacy classes of Γ are given by {1, s1 s2, s2 s1, s1 s2 s1 s2 }, {s1, s2 }, {s1 s2 s1, s2 s1 s2 }. In particular, we see that Irr(Γ)σ = {1Γ, ε, φ1 }. We find σ-extensions for 1Γ and ε by the construction in Example 2.1.8(c). The character φ1 is afforded by the matrix representation √     −1 0 2 1 , s2 → . X : s1 → √ 2 1 0 −1   0 1 The invariance condition in 2.1.7 is satisfied with E = ; note also that 1 0 E 2 = I2 . If we use E to define a σ-extension of φ1 , then the resulting σ-character

100

Lusztig’s Classification of Irreducible Characters

table of Γ is the first table printed in Table 2.2. If we replace E by −E, then this will change the signs of the values of φ˜1 .

Table 2.2 The σ-character tables of dihedral groups of order 8 and 12 Dih(8) 1˜ Γ ε˜ φ˜1

1 s1 s1 s2 s1 1 1 1 1 √ −1 −1 √ . 2 − 2

Dih(12) 1˜ Γ ε˜ φ˜1 φ˜2

1 s1 s1 s2 s1 s1 s2 s1 s2 s1 1 1 1 1 1 √ −1 −1 −1 √ . 3 . − 3 . 1 −2 1

Example 2.1.13 Let Γ be a dihedral group of order 12. Thus, Γ is generated by two elements s1  s2 such that s1, s2 have order 2 and the product s1 s2 has order 6. We have Irr(W) = {1Γ, ε, ε , ε , φ1, φ2 }, where 1Γ is the trivial character, ε, ε , ε  are linear characters, and φ1, φ2 have degree 2. We use the same notational convention for the linear characters as in the previous example; furthermore, the notation for φ1 and φ2 is such that φ1 ((s1 s2 )3 ) = −2 and φ2 ((s1 s2 )3 ) = 2. (See also Table 4.2, p. 276.) As in the previous example, there is an automorphism σ : Γ → Γ such that σ(s1 ) = s2 and σ(s2 ) = s1 . The σ-conjugacy classes of Γ are given by {1, s1 s2, s2 s1, s1 s2 s1 s2, s2 s1 s2 s1, (s1 s2 )3 }, {s1, s2 }, {s1 s2 s1, s2 s1 s2 }, {s1 s2 s1 s2 s1, s2 s1 s2 s1 s2 }. In particular, Irr(Γ)σ = {1Γ, ε, φ1, φ2 }. We find σ-extensions by similar arguments as in the previous example. The resulting σ-character table of Γ is the second table printed in Table 2.2 (see also [GKP00, Example 7.6]). Finally, let G be a connected reductive algebraic group over k = F p and F : G → G be a Steinberg map, as in Section 1.4. Even if we are primarily interested in the ordinary (untwisted) character theory of GF , we will naturally encounter σcharacter tables as well, where σ is typically induced by F in some way. As a first example, recall from 1.6.1 that F induces an automorphism σ : W → W, where W is the Weyl group of G; note that σ(S) = S where S is a suitable set of simple reflections in W. We can apply the above discussion to W, σ. Proposition 2.1.14 (Lusztig [Lu84a, 3.2, 14.2]) Let σ : W → W be as in 1.6.1. ˜ For each φ ∈ Irr(W)σ , there exists a σ-extension φ˜ such that φ(w) ∈ R for all w ∈ W. (Note that our σ-extensions are unique up to multiplication by a root of unity, and so there are exactly two real σ-extensions.) If, moreover, σ is ordinary ˜ in the sense of 1.6.2, then φ˜ can be chosen such that φ(w) ∈ Z for all w ∈ W. (This will play a role, for example, in Remark 2.4.17; see also Section 4.1.)

2.1 Generalities about Character Tables

101

Proof By exactly the same kind of reduction arguments as in the proof of [Lu84a, 3.2], it is sufficient to consider the case where W is irreducible. So let us now assume that this is the case. Now, since W is a finite Weyl group, it is well known that all irreducible characters of W itself are integer valued; see, e.g., [GePf00, Theorem 6.3.8]. So, if σ is the identity, then there is nothing to prove. Let us now assume that σ is not the identity. If W is of type An (n  2), D2n+1 (n  2) or E6 , then σ is given by conjugation by the longest element of W. So the assertion holds by Example 2.1.8(b). If W is of type D2n (n  2) and σ has order 2, then we use the interpretation of σ-extensions in Remark 2.1.9. In the present situation, the semidirect product ˜ = W  σ can be identified with a Weyl group of type B2n . So the assertion W holds again by the known result about the character values of the latter group. (See also Example 4.1.4 for further details.) If W is of type D4 and σ has order 3, then the assertion is proved in [Lu77b, 3.18] (for a slightly different argument see the proof of [Lu84a, 3.2]). Thus, it remains to consider the cases where σ is not ordinary and W is of type B2 , G2 or F4 . Then σ has order 2 and σ-extensions can be worked out by explicit computations, as in Examples 2.1.12 and 2.1.13. See Example 2.1.15 (below) for type F4 and [GKP00, §7] for further details.  Example 2.1.15 Let W be the Weyl group of type F4 , with generators S = {s1, s2, s3, s4 } labelled as in Table 1.1 (p. 20). The ordinary character table of W is printed in [Ca85, p. 413]; we have |Irr(W)| = 25 in this case. An explicit construction of the 25 irreducible characters can be found in [Lu84a, p. 97]. There is an automorphism σ : W → W such that σ(s1 ) = s4 , σ(s2 ) = s3 , σ(s3 ) = s2 and σ(s4 ) = s1 . We have |Irr(W)σ | = 11 and the table Xσ (W) has been determined in [GKP00, §7] (even for the corresponding generic Iwahori–Hecke algebra). This table, with some rescaling of the extensions, is printed in Table 2.3. (The reason for the rescaling will be explained in Remark 2.8.19 below.) In the table, we simply write 232 instead of s2 s3 s2 , for example. As in [Lu84a, 14.2] we specify, for each w j , ˜ in the natural reflection representation. ˜ ∈W the characteristic polynomial of w j σ (See also [Shi75, Table III].) The aim of the following sections is to explain how the problem of determining the character table of GF can be approached. In the context of Section 1.6, we would like to consider GF not just as an individual finite group but as a member of an infinite series of finite groups of Lie type, and obtain a uniform description of the irreducible characters of all finite groups in such a series. For certain groups of small rank (see Table 2.4), such a uniform description is explicitly available in the

102

Lusztig’s Classification of Irreducible Characters Table 2.3 The σ-character table of W of type F4 w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 w11 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 1 1 −1 1 1 1 1 1 2 . . −1 −1 . 2 −1 2 2 2 1 1 −1 . . 1 −1 . −3 3 3 1 −1 1 . . −1 −1 . −3 3 3 . . . 1 1 . 2 −1 −4 −2 −2 2 . . −1 −1 . . 1 −2 −4 −4 2 . . 1 1 . −2 −1 2 −2 −2 √ √ √ √ √ √ . −√2 . −√2 . . . −2√2 2√2 √2 √2 . − 2 . 2 . . . 2√2 −2√2 √2 −√2 . . . − 2 2 . . . . 4 2 −4 2 (Note: the scaling is different from that in [GKP00].) wj char. pol. of w j σ ˜ No. in [Shi75], [Ma90] w1 = () (q2 − 1)√2 1 w2 = 232 (q2 − 1)(q2 + 2q + 1) 4 4 2 w3 = 1 √ 3q −21 √ 4 w4 = 1213214321 q + √2q + q + √2q + 1 11 w5 = 12 q4 − 2q3 + q2 √− 2q + 1 10 w6 = 2 (q2 − 1)(q2 − 2q + 1) 3 q4 + 1 5 w7 = 12132132 q4 − q2 + 1 9 w8 = 1232 (q2√+ 1)2 8 w9 = 121321324321 w10 = 121321324321324321 (q2 + √2q + 1)2 7 w11 = 121321 (q2 − 2q + 1)2 6 φ˜1,0 = 1˜ 1 φ˜1,24 = 1˜ 4 φ˜4,8 = 4˜ 1 φ˜9,2 = 9˜ 1 φ˜9,10 = 9˜ 4  = 6˜ 1 φ˜6,6  φ˜6,6 = 6˜ 2 ˜ φ˜12,4 = 12 φ˜4,1 = 4˜ 2 φ˜4,13 = 4˜ 5 ˜ φ˜16,5 = 16

form of a generic character table, and these will serve as valuable examples. (The CHEVIE system [GHLMP] contains these in electronic form.) Example 2.1.16 Let us consider the series of groups GF = GL2 (q), where q is any prime power. The character table of GL2 (q) is explicitly described in [St51b]. This has found its way into textbooks on representation theory; see, for example, [FuHa91, §5.2], [Et11, §5.25]. The conjugacy classes of GL2 (q) are classified in terms of normal forms of matrices. Let σ be a generator of F×q and τ be a generator of F×q2 . There are four types of conjugacy classes with representatives as follows.   a 0 σ where 0  a  q − 2, A1 (a) := 0 σa   a 0 σ A2 (a) := where 0  a  q − 2, 1 σa

2.1 Generalities about Character Tables

103

Table 2.4 Known generic character tables of finite groups of Lie type Series

Author(s)

PSL2 (p) SL2 (q) GL2 (q) GL3 (q), GL4 (q) 2B (q2 ) 2 GU3 (q) 2G (q2 ) 2 Sp4 (q) CSp4 (q) SL3 (q), SU3 (q) G2 (q)

Frobenius [Fro96] (p prime) Jordan [Jor07], Schur [Schu07] (see also [Bo11]) Jordan [Jor07], Schur [Schu07], Steinberg [St51b] Steinberg [St51b] Suzuki [Suz62] Ennola [Enn63] Ward [War66] Srinivasan [Sr68] (q odd), Enomoto [Eno72] (q = 2m ) Shinoda [Shi82] (q odd) Simpson and Frame [SiFr73] Chang and Ree [ChRe74] (q = pm , p  2, 3), see also Hiss [Hi90b, Anhang B], Enomoto [Eno76] (q = 3m ), Enomoto and Yamada [EnYa86] (q = 2m ) Locker [Loc77] (q = 2m ), see also Lübeck [Lue93] Lübeck [Lue93] (q odd) Spaltenstein [Spa82b], Deriziotis and Michler [DeMi87] Malle [Ma90] (complete table in CHEVIE [GHLMP]) Geck and Pfeiffer [GePf92] (q odd), Geck [Ge95] (q = 2m ) Lübeck [GHLMP]

Sp6 (q) CSp6 (q) 3D (q) 4 2F (q2 ) 4 SO+ (q) (partial) 8 SO− (q) (partial) 8

(Here,‘partial’ means: unipotent characters only, as defined later in this chapter.)

 A3 (a, b) :=  B1 (a) :=

σa 0

0 σb



0 −τ a(q+1) 1 τ a +τ aq

where 0  a < b  q − 2,  where a ∈ Eq and (q + 1)  a.

Here, Eq ⊆ {0, 1, . . . , q2 − 2} is a set of representatives for the equivalence relation on Z defined by: a ∼ a  if a  ≡ a mod (q2 − 1) or a  ≡ qa mod (q2 − 1). Note that the matrix B1 (a) is diagonalisable over Fq2 (but not over Fq ), with eigenvalues τ a and τ aq . Class type Number of classes Size of class A1 (a) q−1 1 q−1 (q − 1)(q + 1) A2 (a) 1 A3 (a, b) q(q + 1) 2 (q − 1)(q − 2) 1 q(q − 1) q(q − 1) B1 (a) 2

104

Lusztig’s Classification of Irreducible Characters

The total number of conjugacy classes is q2 − 1. Hence, there are also q2 − 1 irreducible characters, which can be arranged into four families as follows. χ1(n)

of degree 1, where 0  n  q − 2,

χq(n)

of degree q, where 0  n  q − 2,

(m,n) χq+1 (n) χq−1

of degree q + 1, where 0  n < m  q − 2, of degree q − 1, where n ∈ Eq and (q + 1)  n.

The values are displayed in Table 2.5. Here, ε ∈ K and η ∈ K denote primitive roots of unity of order q − 1 and q2 − 1, respectively.

Table 2.5 The generic character table of GF = GL2 (q) A1 (a)

A2 (a)

A3 (a, b)

B1 (a)

ε 2na

ε 2na

ε n(a+b)

ε na

qε 2na (q+1)ε (m+n)a

ε (m+n)a

(q−1)η na(q+1)

−η na(q+1)

(n)

χ1

(n) χq (m,n) χq+1 (n) χq−1

.

ε n(a+b) + ε na+mb

ε ma+nb

.

−ε na . −(η na + η naq )

Example 2.1.17 Let us consider the series of groups GF = SL2 (q), where q is any prime power. The generic character table was determined in [Jor07], [Schu07]; see, for example, [FuHa91, §5.2], for a modern textbook reference. Again, the conjugacy classes are classified in terms of normal forms of matrices, but some additional care is required when q is odd. First, we set       1 0 1 1 1 ξ  I := , N := , N := , 0 1 0 1 0 1 where N  is only defined for q odd: in this case, {x 2 | x ∈ F×q }  F×q and we fix an element ξ ∈ F×q which is not a square. Let σ be a generator of F×q and τ0 := τ q−1 where τ is a generator of F×q2 . (So τ0 ∈ F×q2 has order q + 1.) Then we define:  S(a) := ! T(b) :=

σa 0 0 1

0 σ −a



−1 bq τ0b +τ0

where " where

1a 1a 1b 1b

q 2 −1 1 2 (q − 3) q 2 1 2 (q − 1)

if q is even, if q is odd, if q is even, if q is odd.

Note that T(b) is diagonalisable over Fq2 (but not over Fq ), with eigenvalues τ0b and

2.2 The Virtual Characters of Deligne and Lusztig

105

bq

τ0 . Representatives of the conjugacy classes of SL2 (q) are given by I, N, {S(a)}, {T(b)} I, −I, N, N , −N, −N , {S(a)}, {T(b)}

if q is even, if q is odd,

where a, b run over the index ranges specified above. The total number of conjugacy classes is q+1 if q is even, and q+4 if q is odd. We now use the known character table of GL2 (q) from Example 2.1.16. We define the following restrictions of characters of GL2 (q) to SL2 (q). ψ1 := restriction of χ1(0), ψq := restriction of χq(0), (i) (0,i) := restriction of χq+1 for 1  2i  q − 2, ψq+1 (j)

(j)

ψq−1 := restriction of χq−1 for 1  2 j  q. If q is even, then the restriction of every irreducible character of GL2 (q) to SL2 (q) is irreducible. If q is odd, then we have reducible restrictions as follows. Let i0 = 1 1 2 (q − 1) and j0 = 2 (q + 1). Then (i0 ) = ψ+ + ψ+ ψq+1

and

(j )

0 ψq−1 = ψ− + ψ−

where ψ± are distinct irreducible characters of SL2 (q) of degree 12 (q + 1) and ψ± are distinct irreducible characters of degree 12 (q − 1). The values of these characters involve the following algebraic numbers: √ √ ω = 12 (1 + δq) and ω∗ = 12 (1 − δq) where δ = (−1)(q−1)/2 . The generic character tables are displayed in Table 2.6. Here, ε ∈ K denotes again a primitive root of unity of order q − 1 and η0 ∈ K denotes a primitive root of unity of order q + 1. (We take η0 = η q−1 where η is the primitive root of unity of order q2 − 1 in the table of GL2 (q).) In Table 2.7, we print the tables for q = 2, 3, 4.

2.2 The Virtual Characters of Deligne and Lusztig In this section we explain the basic construction of Deligne and Lusztig [Lu75], [DeLu76], [Lu77b] in which the theory of -adic cohomology is used to obtain representations of finite groups acting on algebraic varieties. (See also [Lu14c, §6] for some historical comments about the origins of this construction.) 2.2.1 Let X be an algebraic variety over k = F p . Let  be a prime different from p and Q be the field of -adic numbers. It is a deep fact that one can attach to X a

106

Lusztig’s Classification of Irreducible Characters Table 2.6 The generic character table of GF = SL2 (q) q even ψ1 ψq (i) ψq+1

I 1 q q+1

N 1 . 1

S(a) 1 1 ε ai + ε −ai

ψq−1

q−1

−1

.

(j)

Here, 1  i 

q 2

− 1 and 1  j 

−I 1 q (−1)i (q+1)

ψq−1 q−1

(−1) j (q−1) −1

ψ+ ψ+ ψ− ψ−

1 2 (q+1) 1 (q+1) 2 1 (q−1) 2 1 (q−1) 2

1 δ(q+1) 2 1 δ(q+1) 2 − 12 δ(q−1) − 12 δ(q−1)

−b j

bj

−η0 − η0 q 2.

q odd I ψ1 1 ψq q (i) ψq+1 q+1 (j)

T(b) 1 −1 .

N 1 . 1

N 1 . 1

−N 1 . (−1)i

−N  S(a) 1 1 . 1 (−1)i ε ai + ε −ai

−1 (−1) j+1 (−1) j+1

ω ω∗ ω∗ ω −ω∗ −ω −ω −ω∗

δω δω∗ δω∗ δω

δω∗ δω δω δω∗

. (−1)a (−1)a . .

T(b) 1 −1 . bj

−b j

−η0 − η0 . . (−1)b+1 (−1)b+1

Here, 1  i  12 (q − 3) and 1  j  12 (q − 1).

Table 2.7 Character tables of GF = SL2 (q) for q = 2, 3, 4 q=2 ψ1 ψ2 (1) ψ1

I N T(1) 1 1 1 2 . −1 1 −1 1

q=4 ψ1 ψ4 (1) ψ5

I 1 4 5

(1)

ψ3

(2)

ψ3

N S(1) T(1) T(2) 1 1 1 1 . 1 −1 −1 1 −1 . .

3 −1

.

ζ

ζ∗

3 −1

.

ζ∗

ζ

q=3 ψ1 ψ3 (1) ψ2 ψ+ ψ+ ψ− ψ−

I 1 3 2 2 2 1 1

−I N N  −N −N  T(1) 1 1 1 1 1 1 3 . . . . −1 −2 −1 −1 1 1 . −2 ω ω∗ −ω −ω∗ . −2 ω∗ ω −ω∗ −ω . ∗ 1 −ω −ω −ω∗ −ω 1 1 −ω −ω∗ −ω −ω∗ 1

√ 1 Here, ω = √ 2 (1 + −3), √ 1 ζ = 2 (1 − 5) and ζ ∗ = 12 (1 + 5). (Note that SL2 (2)  S3 and SL2 (4)  A5 .)

family of Q -vector spaces Hic (X, Q ) (i ∈ Z), called the -adic cohomology groups with compact support; see [Ca85, Appendix], [Sr79, Chap. V] and the references there. These vector spaces are finite dimensional, zero for i < 0 and for large i. They are functorial, in the sense that a finite morphism f : X → X naturally induces a linear map f ∗ : Hic (X, Q ) → Hic (X, Q ) for each i. This can be used to construct representations of finite groups. In the following, we extend scalars from

2.2 The Virtual Characters of Deligne and Lusztig

107

Q to an algebraic closure Q and denote the corresponding cohomology spaces by Hic (X, Q ). Let Γ be a finite group acting as a group of algebraic automorphisms of X. Then i Hc (X, Q ) is a Q Γ-module for each i, where g ∈ Γ acts via (g ∗ )−1 . For g ∈ Γ, the alternating sum of traces



(−1)i Trace (g ∗ )−1, Hic (X, Q ) L(g, X) := i

is called the Lefschetz number of g on X. Note that the sum makes sense, since all Hic (X, Q ) are finite dimensional and zero for almost all i. Thus, the map g → L(g, X) is a virtual character of Γ over Q . It is known that L(g, X) = L(g −1, X) is a rational integer and does not depend on ; see [DeLu76, 3.3], [Lu77b, 1.2] for a proof. Hence, by Remark 2.1.5, the map g → L(g, X) is also a virtual character of Γ over K. 2.2.2 In the above setting, there is a way of defining Lefschetz numbers without reference to -adic cohomology at all and this can be used to compute L(g, X) in certain examples. For this purpose, let F : X → X be the Frobenius map relative to a rational structure on X over some finite subfield of k and assume that g ∈ Γ commutes with F. Then F n ◦ g also is a Frobenius map for n = 1, 2, . . . (see Remark 1.4.4(c)); in particular, F n ◦ g has only finitely many fixed points on X. So we can form the formal power series R(t, g) := −

∞

|XF

n ◦g

|t n ∈ Z[[t]].

n=1

Then R(t, g) is a rational function in t, which is independent of F and which only has simple poles and no pole at ∞; furthermore, the value of R(t, g) at ∞ is just L(g, X). (The deduction of these statements from properties of the -adic cohomology groups can be found, for example, in [Lu77b, 1.2], [Ca85, App. (h)], [DiMi20, §8.1].) Let us give two simple illustrations of this principle, which are taken from [Lu77b, §1]. Example 2.2.3 (a) Assume that X is a finite set; then L(g, X) = |Xg |

where

Xg = {x ∈ X | g.x = x}.

Indeed, consider a realisation of X as a closed subset of k d for some d  1. Since X is finite, there is a power q of p such that X ⊆ Fqd where Fq ⊆ k is the subfield with q elements. Then the standard Frobenius map Fq : k d → k d restricts to a Frobenius n map F on X such that F(x) = x for all x ∈ X. Thus, we have |XF ◦g | = |Xg | for all

108

Lusztig’s Classification of Irreducible Characters

n  1. This yields that R(t, g) = −

∞

|XF

n ◦g

|t n = −|Xg |

n=1

∞

t n = |Xg |

n=1

t t−1

|Xg |,

as required. and so the value at ∞ is L(g, X) = (b) Assume that f : X → X is a bijective morphism of affine varieties. If g : X → X and g  : X → X are automorphisms of finite order such that f ◦ g = g  ◦ f , then L(g, X) = L(g , X). Indeed, we can realise X and X as closed subsets of some affine spaces. Then X, X, g, g , f are given by finitely many polynomials all of whose coefficients will lie in some finite subfield of k. Thus, we can find Frobenius maps F : X → X and F  : X → X such that g commutes with F, g  commutes with F  and f ◦F = F  ◦ f . Then F n ◦ g and F n ◦ g  have the same number of fixed points for all n  1. Hence, the corresponding power series coincide and so L(g, X) = L(g , X). 2.2.4 Now let G be a linear algebraic group over k = F p and F : G → G be a Steinberg map. Let L : G → G, g → g −1 F(g), be the Lang–Steinberg map. (Recall from Theorem 1.4.8 that L is surjective if G is connected.) Let Y be a closed subset of G. Following [Lu77b, 2.1], we consider the variety L −1 (Y) = {x ∈ G | x −1 F(x) ∈ Y}. This is a closed subset of G stable under left multiplication by elements of GF . Assume furthermore that H is a subgroup of G such that |H| < ∞

and

h−1 YF(h) ⊆ Y

for all h ∈ H.

(∗)

(This is slightly more general than the situation described in [Lu77b, 2.1]; this generalisation will be useful in 2.3.18 below.) Then L −1 (Y) is stable under right multiplication by elements of H. So GF × H acts on L −1 (Y) by (g, h).x = gxh−1

(x ∈ L −1 (Y), g ∈ GF , h ∈ H).

Consequently, the vector spaces Hic (L −1 (Y), Q ) are Q (GF × H)-modules. Proposition 2.2.5 (Cf. [Lu77b, 2.1]) Assume that we are in the setting of 2.2.4, where H ⊆ G satisfies (∗). Let θ : H → K be a class function. Then

1 G (θ) : GF → K, g → L (g, h), L −1 (Y) θ(h) RH,Y |H| h ∈H

is a class function of GF . (Here, L (g, h), L −1 (Y) ∈ Z are Lefschetz numbers as G in 2.2.1.) If θ is a virtual character of H, then RH,Y (θ) is a virtual character of GF whose values are algebraic integers in Q(θ(h) | h ∈ H) ⊆ K.

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If H is of the form H = HF for a closed F-stable subgroup H ⊆ G, then we also G G write RH,Y (θ) instead of RH,Y (θ). Proof For each i, let ψi : GF ×H → Q be the character of the Q (GF ×H)-module

 Hic (L −1 (Y), Q ). Thus, L (g, h), L −1 (Y) = i (−1)i ψi (g, h) for all g ∈ GF and h ∈ H. Now assume first that θ : H → K is not just a class function but a character of H. Let V be a finite-dimensional KH-module affording θ. Choosing an embedding K ⊆ Q , we can also regard V as a Q (GF × H)-module, with the elements of GF acting trivially. We now use the construction in 2.1.3 (where we identify H with a normal subgroup of the direct product, with factor group GF ). Then the subspace of fixed points i

H Hc (L −1 (Y), Q ) ⊗ V is still a Q GF -module; let ψiθ : GF → Q be its character. By 2.1.3, we have 1 ψi (g, h)θ(h) for all g ∈ GF . ψiθ (g) = |H| h ∈H

 G (θ) = i (−1)i ψiθ is a virtual character of GF over Q . Hence, we see that RH,Y G Since Lefschetz numbers are integers, the values of RH,Y (θ) are algebraic integers G in Q(Trace(h, V) | h ∈ H) ⊆ K. So Remark 2.1.5 shows that RH,Y (θ) is also a G virtual character over K. Thus, we obtain a map RH,Y sending a character of H (over K) to a virtual character of GF (over K). Extending this map linearly, we obtain a map from virtual characters of H (over K) to virtual characters of GF (over K) and,  finally, a map CF(H) → CF(GF ) as desired. We shall assume from now on that G is connected and reductive 1 . It will also be useful at several places below to fix a setting as in 1.6.1, where T0 ⊆ G is an F-stable maximal torus that is maximally split, that is, contained in an F-stable Borel subgroup B0 ⊆ G. Let W = NG (T0 )/T0 be the Weyl group of G and q be the positive real number defined by F in Proposition 1.4.19. Definition 2.2.6 (Deligne–Lusztig [DeLu76], Lusztig [Lu77b]) Let T ⊆ G be an F-stable maximal torus and θ ∈ Irr(TF ). Choose a Borel subgroup B ⊆ G such that T ⊆ B, and let U = Ru (B) be the unipotent radical of B. (Note that B is not necessarily F-stable.) Then Y := U satisfies condition (∗) in 2.2.4 with respect to G the finite subgroup H := TF . Thus, we obtain a virtual character RT,U (θ) of GF , which is called a Deligne–Lusztig character. G As pointed out in [Lu75], the virtual characters RT,U (θ) provide a solution to a 1

There are also extensions of the following constructions to disconnected groups, see [Ma93a], [DiMi94], [Lu12b]; we will briefly discuss this in Section 4.8.

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series of conjectures made by Macdonald around 1968, on the basis of the character tables of finite groups of Lie type known at the time, most notably GLn (q) (see [Gre55]) and Sp4 (q) (see [Sr68]). We shall now state some of the basic properties G of the virtual characters RT,U (θ). The original proofs can be found in [DeLu76], G [Lu77b]; see also [Ca85], [DiMi20], [Sr79]. The construction of RT,U (θ) is a kind of ‘twisted induction’, generalising the usual induction of characters from subgroups, by the following result. (This is a special case of Harish-Chandra induction and will be studied in more detail in Section 3.2.) Proposition 2.2.7 Let T ⊆ G be an F-stable maximal torus that is contained in F G an F-stable Borel subgroup B ⊆ G. Then RT,U (θ) = IndG (θ) where θ ∈ Irr(TF ) is BF regarded by inflation as a character of BF . For the proof, see [Lu77b, 2.6] or [Ca85, 7.2.4]. Now the next major result is a scalar product formula which has a number of important consequences. Theorem 2.2.8 (Scalar product formula) Let T, T ⊆ G be F-stable maximal tori. Let θ ∈ Irr(TF ) and θ  ∈ Irr(TF ). Then G (θ), RTG,U (θ ) = |{g ∈ GF | gTg −1 = T and g θ = θ  }|/|TF |, RT,U

where U, U are the unipotent radicals of Borel subgroups containing T, T, respectively. The original proof in [DeLu76, §6] (see also [Lu75]) has the minor disadvantage that it does not cover certain extreme cases where q < 2 (with q as in Proposition 1.4.19). A simpler proof, which works in complete generality, is given in [Lu77b, 2.3]; an exposition of this latter proof can be found in [Ca85, §7.3]. See also [DiMi20, Cor. 9.3.1] where the proof is given via a Mackey formula. G (θ) is independent of U; it will henceCorollary 2.2.9 The virtual character RT,U G forth be denoted by RT (θ). Furthermore, the following hold.

(a) If θ ∈ Irr(TF ) is in general position (that is, we have g θ  θ for all g ∈ NG (T)F \ TF ), then either RTG (θ) or −RTG (θ) is in Irr(GF ). (b) For any ρ ∈ Irr(GF ), we have −|W| 1/2  RTG (θ), ρ  |W| 1/2 . The number of ρ ∈ Irr(GF ) that occur in RTG (θ) is bounded above by |W|. Proof Let U be the unipotent radical of another Borel subgroup containing T. Then Theorem 2.2.8 shows that we have G G G G G G (θ), RT,U (θ) = RT,U (θ), RT,U RT,U  (θ) = RT,U (θ), RT,U (θ) . G G G G (θ) − RT,U It follows that RT,U  (θ) has norm 0 and so RT,U (θ) = RT,U (θ). Thus,

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G (θ) is independent of U. For (a), just note that RTG (θ), RTG (θ) = 1 if θ is in RT,U general position. For (b), just note that

RTG (θ), ρ 2  RTG (θ), RTG (θ)  |NG (T)F |/|TF |  |NG (T)/T| = |W|.



By similar arguments, we obtain: Corollary 2.2.10 Two Deligne–Lusztig characters RTG (θ) and RTG (θ ) are either equal or orthogonal to each other. We have RTG (θ) = RTG (θ ) if and only if there exists some g ∈ GF such that gTg −1 = T and g θ = θ . In order to fix the sign in Corollary 2.2.9(a), we shall need the notion of the relative F-rank of G. Following [Ca85, p. 197], this is defined as follows. Definition 2.2.11 Let T be any F-stable maximal torus of G. Let ϕ be the endomorphism of X = X(T) induced by F. Denote by ϕR the canonical extension of ϕ to XR = R ⊗Z X. Then the relative F-rank of T is defined as the dimension of the q-eigenspace of ϕR on XR (with q as in Proposition 1.4.19). We also set εT := (−1)r

where

r := relative F-rank of T.

If T0 ⊆ G is our fixed maximally split torus, then we set εG := εT0

and

relative F-rank of G := relative F-rank of T0 .

(This does not depend on the particular choice of T0 since all maximally split tori are GF -conjugate.) The (absolute) rank of G is defined as dim T0 . By [Ca85, 6.5.7], the relative F-rank of T0 is larger than or equal to the relative F-rank of any F-stable maximal torus of G. Hence, if G is the complete root datum associated with G, F (see Example 1.6.11) and |G| ∈ R[q] is the corresponding order polynomial, then relative F-rank of G = max{i  0 | (q − 1)i divides |G| in R[q]}. In particular, if G is simple, then the relative F-rank of G and the sign εG can be easily read off the polynomials in Table 1.3 (p. 73). Theorem 2.2.12 (Degree formula) Let T be an F-stable maximal torus of G and θ ∈ Irr(TF ). Then

RTG θ (1) = εG εT |GF : TF | p, where, for any non-zero m ∈ Z, we denote by m p the p-part of m. For the proof (which uses properties of the Steinberg character of GF ), see [DeLu76, 7.1], [Lu77b, 2.9]; see also [Ca85, 7.5.1, 7.5.2]. Consequently, in Corollary 2.2.9(a), we have εG εT RTG (θ) ∈ Irr(GF ).

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Since the definition involves the cohomology spaces Hic (X, Q ) for certain subvarieties X ⊆ G, it may be expected that the problem of computing the values of RTG (θ) on arbitrary elements of GF is difficult, and this is indeed so. A first step consists of using the Jordan decomposition for an element g ∈ GF , which allows us to separate the problem according to the semisimple and the unipotent part of g. We begin by recalling some facts about centralisers of semisimple elements. 2.2.13 Let s ∈ G be semisimple. Then CG (s) is a closed subgroup which is, in general, not connected. Let us denote by H := CG◦ (s) the connected component of the identity. Now there always exists some maximal torus T ⊆ G such that s ∈ T (see, e.g., [Ge03a, Exc. 3.12]). Then T ⊆ CG (s) and, hence, s ∈ T ⊆ H. It is also known that every unipotent element u ∈ CG (s) already belongs to H (see [Hum95, 1.12]). Furthermore, by [Hum95, 2.2], the connected algebraic group H is reductive, of the same rank as G (since T ⊆ H). In fact, if s belongs to our reference torus T0 , then we have H = T0, Uα | α ∈ R such that α(s) = 1 and Rs := {α ∈ R | α(s) = 1} is the root system of H with respect to T0 . Note that, since all maximal tori are conjugate in G, every semisimple element of G is conjugate to an element in T0 . For later reference we also state the following result, which provides a basic criterion for when CG (s) is connected. Theorem 2.2.14 (Steinberg [St68, 8.5, 9.1]) Let πsc : Gsc → G be a simply connected covering of the derived subgroup of G, as in Remark 1.5.13. Let s ∈ G be semisimple. Then CG (s)/CG◦ (s) is isomorphic to a subgroup of ker(πsc ). In particular, if ker(πsc ) = {1}, then CG (s) is connected. See [Ca85, §3.5], [DiMi20, §11.2], [Hum95, §2.6], [Bo05], [MaTe11, §14.2], for a further discussion of this important result. Definition 2.2.15 Let Guni be the set of unipotent elements of G. Let T ⊆ G be F an F-stable maximal torus. Define a function QG T : Guni → K by G

F , for u ∈ Guni QG T (u) := RT 1T (u) where 1T stands for the trivial character of TF . This function is called a Green function (since it already occurred in Green’s work [Gre55] on GLn (q)); see [DeLu76, 4.1]. By Proposition 2.2.5, the values of QG T are integers; hence, we also have G −1 F . Furthermore, by Corollary 2.2.10, we have QG (u) = Q (u ) for all u ∈ G uni T T G F. = Q for all g ∈ G QG −1 T gTg

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Theorem 2.2.16 (Character formula) Let g ∈ GF and write g = su = us, where s is semisimple and u is unipotent. Let H := CG◦ (s). Then



1 QH (u) θ(x −1 sx). RTG θ (g) = F xTx −1 |H | F −1 x ∈G : x sx ∈T



Furthermore, if s = 1, then RTG θ (u) = QG T (u) ∈ Z does not depend on θ. For the proof, see [DeLu76, 4.2] or [Ca85, 7.2.8]. We observe that the expression QH (u) is meaningful: firstly, as already recalled above in 2.2.13, H is connected xTx −1 and reductive; secondly, if x −1 sx ∈ TF , then xTx −1 is an F-stable maximal torus contained in H; furthermore, u belongs to H (as noted in 2.2.13). Example 2.2.17 (a) The above character formula immediately shows that we have

RTG θ (g) = 0 if s is not conjugate in GF to any element of TF . (b) Let g ∈ GF and write g = us = su, where s is semisimple and u is unipotent. Taking the sum over all θ ∈ Irr(TF ) in the above character formula, and using the orthogonality relations for the characters of TF , we obtain the following identity:  F G

|T |QT (u) if s = 1, G

RT θ (g) = 0 otherwise. F θ ∈Irr(T )

(c) If u = 1, then the character formula reduces to the formula:

F

θ (s) = εT εH |HF | p RTG θ (s). IndG TF Hence, the difficulty of computing the values of RTG (θ) on semisimple elements is F

θ : for this purpose, one has to equivalent to that of computing the values of IndG TF determine the class fusion from TF to GF . We then also have the following formula for the values of the irreducible characters of GF on semisimple elements: Proposition 2.2.18 (Character values on semisimple elements, [DeLu76, 7.6] or [Ca85, 7.5.5]) Let ρ ∈ Irr(GF ) and s ∈ GF be semisimple. Then 1 εH εT RTG (θ), ρ θ(s), ρ(s) = F |H | p (T,θ) where H = CG◦ (s) and the sum runs over all pairs (T, θ) such that T ⊆ G is an F-stable maximal torus, s ∈ T and θ ∈ Irr(TF ). Corollary 2.2.19 For any irreducible character ρ ∈ Irr(GF ), there exists a pair (T, θ) such that RTG (θ), ρ  0. Proof Let s = 1. Then ρ(1)  0 and so some term in the sum in the formula in Proposition 2.2.18 must be non-zero. 

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The character formula immediately shows that, if z ∈ Z(G)F , then we have



θ (z) = θ(z)RTG θ (1). Actually, something stronger is true.

RTG

Proposition 2.2.20 (See [DeLu76, 1.22]) Let T ⊆ G be an F-stable maximal torus and θ ∈ Irr(TF ). Let ρ ∈ Irr(GF ) be such that RTG (θ), ρ  0. Then ρ(z) = θ(z)ρ(1) for all z ∈ Z(G)F . Proof Since it does not seem to be possible to prove this by purely charactertheoretic methods, it may be a good exercise to go through the details. Recall the construction of RTG (θ) from the proof of Proposition 2.2.5. Let U ⊆ G be the unipotent radical of a Borel subgroup containing T. Let ψi : GF × TF → Q be the character of the Q (GF × TF )-module Hic (L −1 (U), Q ), where the module structure is induced by the action of GF × TF on L −1 (U) via (g, t) : x → gxt −1 (see 2.2.4). Now assume that g = z ∈ Z(G)F . Then (z, t) acts on L −1 (U) in the same way as (1, z −1 t). An analogous statement holds for the induced linear action on Hic (L −1 (U), Q ) and so we have ψi (z, t) = ψi (1, z −1 t)

for all t ∈ TF .

Next, let Vθ be a one-dimensional Q TF -module with character θ. Then we have  RTG (θ) = i (−1)i ψiθ , where ψiθ : GF → Q is the character of the Q GF -module

Hic (L −1 (U), Q ) ⊗ Vθ

TF

.

Using the formula in 2.1.3, we obtain that 1 1 ψi (z, t)θ(t) = F ψi (1, z −1 t)θ(t) ψiθ (z) = F |T | |T | F F t ∈T t ∈T 1 = F ψi (1, t)θ(zt) = θ(z)ψiθ (1). |T | F t ∈T

ψiθ

Since is an actual character, we conclude that z ∈ Z(G)F acts by scalar multiplication with θ(z) on a module affording ψiθ . Since RTG (θ), ρ  0, there is some i such that ρ is a constituent of ψiθ . Hence, z will also act by scalar multiplication with θ(z) on a module affording ρ.  The determination of the values of the Green functions is a very hard problem (see Section 2.8 further below). Deligne and Lusztig [DeLu76] were able to determine these values on certain unipotent elements at least, which we now introduce. 2.2.21 The set Guni of unipotent elements is a closed irreducible subset of G, invariant under the conjugation action of G on itself (see [Hum95, §4.2]). In particular, Guni is a union of conjugacy classes of G, which are called the unipotent classes of G. It is known that the number of unipotent classes is finite (see [Lu76b]

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115

and also [DiMi20, §12.1]). Then a general argument shows that there is a unique unipotent class O0 such that Guni is the Zariski closure of O0 (see [Ca85, 5.1.2], [Hum95, §4.3]). This class O0 is called the regular unipotent class and its elements the regular unipotent elements of G. We have F(O0 ) = O0 and so O0F  . Furthermore, dim CG (u) = dim T0 for any regular unipotent u ∈ G. (If one does not want to use the finiteness of the number of unipotent classes, then see Steinberg’s existence proof in [Hum95, §4.5, §4.6].) Theorem 2.2.22 (See [DeLu76, Theorem 9.16]) Let T ⊆ G be an F-stable F maximal torus. Then QG T (u) = 1 for any regular unipotent u ∈ G . Note that, while most of the results in this section so far are fully covered with proofs in [Ca85], [DiMi20], [Sr79], the original article of Deligne and Lusztig appears to be the only source for a proof of the above result. Note also that, in [DeLu76], it is assumed that F is a Frobenius map but the proof applies verbatim for Steinberg maps, as pointed out by [DiMi20, Lemma 12.4.8]. (For the Suzuki and Ree groups, the result is also verified by explicit computations; see Example 2.8.18 below.) Even if the values of the Green functions are known, the character formula in Theorem 2.2.16 still requires some work as far as the sum over all x ∈ GF such that x −1 sx ∈ TF is concerned. The following result is certainly known and much used in the literature (see, e.g., [FoSr82, 1.10], [GePf92, 4.1], [Lue93, Chap. 6], [MaMa18, §II.5.1]) but in order to get all the technicalities right, it is useful to write down a detailed proof. Lemma 2.2.23 (Simplified character formula) Assume that we are in the setting of Theorem 2.2.16, where s is conjugate in GF to an element in TF . Let H := CG◦ (s) and T1, . . . , Tm be representatives of the HF -conjugacy classes of F-stable maximal tori of H that are conjugate in GF to T. For each i let xi ∈ GF be such that Ti = xi Txi−1 . Then θ i := xi θ ∈ Irr(TiF ) and



1

QH θ i (w −1 s w) RTG θ (su) = Ti (u) F |W(H, T ) | i F 1im w ∈W(G,Ti )

where W(G, Ti ) = NG (Ti )/Ti and W(H, Ti ) = NH (Ti )/Ti . Proof Let N := NG (T). Let D be a set of representatives for the (HF , NF )-double cosets in GF . Let x ∈ GF and write x = hdn where h ∈ HF , d ∈ D and n ∈ NF . We have x −1 sx ∈ T if and only if s ∈ xTx −1 if and only if xTx −1 ⊆ H. Using now x = hdn, we see that the latter condition is equivalent to dTd −1 ⊆ H and, hence, to d −1 sd ∈ T. So let D  be the set of all d ∈ D such that d −1 sd ∈ T. Then the formula

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in Theorem 2.2.16 can be re-written as follows:



1 QH (u)θ(x −1 sx). RTG θ (g) = F xTx −1 |H | d ∈D F F x ∈H dN

Now, if x = hdn as above (where d ∈ D ), then xTx −1 = hdTd −1 h−1 is an F-stable maximal torus of H that is conjugate in HF to Td := dTd −1 . Hence, −1 we have QH (u) = QH T d (u). Let Nd := NG (T d ) = dNd . Furthermore, let xTx −1 n1, . . . , nr ∈ NFd be a set of right coset representatives of HF ∩ NFd in NFd , that is, NFd is the disjoint union of (HF ∩ NFd )ni for 1  i  r. Then every x ∈ HF dNF has a unique expression as x = hni d where h ∈ HF and 1  i  r. So the above formula can be re-written as



RTG θ (g) = QH θ(d −1 ni−1 sni d) T d (u) d ∈D

=

d ∈D

1ir

QH T d (u)

1 d θ(n−1 sn). |HF ∩ NFd | F n∈N d

Now, HF ∩ NFd = (H ∩ NG (Td ))F = NH (Td )F ⊇ TFd ; we also have NFd ⊇ TFd . Since NH (Td )F /TFd  W(H, Td )F and NFd /TFd  W(G, Td )F , we obtain:

1 1 d −1 d

θ(n sn) = θ(w −1 s w). F F |W(H, T ) | |HF ∩ Nd | d F F w ∈W(G,T d )

n∈N d

Finally, it is easy to check that we can take D  = {x1, . . . , xm }.



Example 2.2.24 (a) Let T ⊆ G be an F-stable torus of type w. Let g ∈ G be such that T = gT0 g −1 and w is the image of g −1 F(g) ∈ NG (T0 ). Then the map

−1 T, is an isomorphism. (See [Ca85, 3.3.6].) CW,σ (w) → W(G, T)F , x → g xg F (b) Let g ∈ G with Jordan decomposition g = su = us. Assume that u is regular unipotent in CG◦ (s). Let T ⊆ G be an F-stable maximal torus such that s is conjugate in GF to an element in TF . Then Theorem 2.2.22 and Lemma 2.2.23 show that



|W(G, Ti )F : W(H, Ti )F |. RTG 1T (g) = 1im

We can further evaluate this as follows. Assume that w1 ∈ W is such that the F-stable maximal torus T1 ⊆ G is of type w1 . Now W(H, T1 ) is a subgroup of W1 := W(G, T1 ). Furthermore, F induces the automorphism σ1 : W1 → W1,

y → w1 σ(y)w1−1 .

(Indeed, let g ∈ G be such that T1 = gT0 g −1 and g −1 F(g) = w 1 . Then NG (T1 ) =

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117

−1 ) = g w 1 F(w)

w 1−1 g −1 = gNG (T0 )g −1 . Let w ∈ W and w  := σ1 (w). Then F(g wg  −1 g w g , as required.) Then, using (a), we obtain

RTG 1T (g) = |W(H, T1 )| −1 |CW1,σ1 (w)||C1 ∩ W(H, T1 )|, where C1 denotes the σ1 -conjugacy class of w1 in W1 . Definition 2.2.25 Following [Lu77b, 2.15], a class function f ∈ CF(GF ) is called a uniform function if f can be written as a K-linear combination of Deligne–Lusztig characters RTG (θ) for various T, θ. Example 2.2.26 (a) Let 1G be the trivial character of GF . Then RTG (1T ), 1G = 1 for all F-stable maximal tori T ⊆ G; see [Ca85, 7.4.1, 7.4.2], [Lu77b, 2.7]. Furthermore, 1G is uniform: 1 F G |T |RT (1T ) 1G = F |G | T where the sum runs over all F-stable maximal tori T ⊆ G. Evaluating the above identity at 1 and using the degree formula in Theorem 2.2.12, we obtain |GF | p =  T εG εT , where the sum runs over all F-stable maximal tori in G. (b) The character χreg of the regular representation of GF is uniform:

1 χreg = F εG εT RTG (θ), |G | p T F θ ∈Irr(T )

where, again, the first sum runs over all F-stable maximal tori T ⊆ G; see [Ca85, 7.5.6], [Lu77b, 2.11]. Example 2.2.27 Let StG be the Steinberg character of GF . An explicit model of StG is constructed in [St57]. In the present context, it is sufficient to know that StG is irreducible with the following properties: StG (1) = |GF | p

and

F

StG, IndG (1 )  0, BF B

where B is an F-stable Borel subgroup of G and 1B stands for the trivial character of BF . (See [Ca85, Chap. 6], [DiMi20, §7.4], and also Section 3.4.) This character is also uniform; we have 1 εG εT |TF |RTG (1T ), StG = F |G | T where the sum runs over all F-stable maximal tori T ⊆ G; see [Ca85, 7.6.6]. Remark 2.2.28 Having defined RTG : CF(TF ) → CF(GF ), we obtain by adjunction F F a unique linear map ∗RG T : CF(G ) → CF(T ) such that  G  ∗RG T ( f ), f =  f , RT ( f )

for all f ∈ CF(GF ) and f  ∈ CF(TF ).

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(Here, on the left-hand side of the equality, the inner product is taken in CF(TF ), while on the right-hand side it is taken in CF(GF ).) The values of ∗RG T ( f ) are given as follows (see [DiMi20, §10.1]). Let t ∈ TF and H := CG◦ (t). Then

∗ G

RT f (t) = |TF ||HF | −1 QH T (u) f (tu). F u ∈Huni

G ( f ) + f ⊥ where π G ( f ) is For any f ∈ CF(GF ), we can write uniquely f = πun un ⊥ G ( f ) the uniform and f is orthogonal to all uniform class functions. We call πun uniform projection of f . Explicitly, we have (see [DiMi20, §10.2]): 1 F G ∗ G

G (f) = F |T |RT RT ( f ) , πun |G | T

where the sum runs over all F-stable maximal tori T ⊆ G. Example 2.2.29 Let GF = GL2 (q). In Table 2.5, the most difficult characters to obtain are those of degree q − 1. Using the notation in Example 2.1.16, consider the cyclic subgroup H := B1 (1) of index q(q − 1). Inducing the linear characters of H to GL2 (q) yields (reducible) characters μ(n) (for 0  n  q2 − 2) with the q(q−1) following values. μ(n) q(q−1)

A1 (a) q(q − 1)η na(q+1)

A2 (a) 0

A3 (a, b) B1 (a) 0 η na + η naq

If n ∈ Eq and (q + 1)  n, then it turns out that (n) (0,n) (0,n) = χq(0) χq+1 − χq+1 − μ(n) ∈ Irr(GL2 (q)). χq−1 q(q−1)

(See [St51b, p. 227].) We can now re-interpret this as follows. A maximally split torus and the corresponding Weyl group are given by       0 1 ξ 0   × and W = s where s = . T0 =  ξ, ξ ∈ k 1 0 0 ξ Consider the element s ∈ W and let Ts ⊆ G be an F-stable maximal torus of type s (see 1.6.4, 1.6.21). By Example 1.6.5, we have     ξ 0  × q2 T0 [s] = ξ∈k ,ξ =ξ . 0 ξq For 0  a < q2 − 1 let B1 (a) be the diagonal matrix with diagonal entries τ a and τ aq , where τ is a generator of F×q2 . Thus, TsF  T0 [s] = {B1 (a) | 0  a < q2 − 1}.

For 0  n < q2 − 1 let θ n ∈ Irr(TsF ) be the linear character that, via TsF  T0 [s], corresponds to the linear character of T0 [s] sending B1 (a) to η na . Then we have (n) RTGs (θ n ) = − χq−1

for all n ∈ Eq such that (q + 1)  n.

2.2 The Virtual Characters of Deligne and Lusztig

119

A completely analogous situation occurs for GLn (q) and any n  2, where the most difficult characters to construct are the so-called ‘discrete series’ characters, of degree (q − 1)(q2 − 1) . . . (q n−1 − 1); see [Ja86, §3], [Gre99]. Example 2.2.30 Let us re-interpret the character table of GF = SL2 (q) in Table 2.6 (p. 106) in terms of the theory developed so far. In this case, a maximally split torus and the corresponding Weyl group are given by       0 1 ξ 0  ×

and W = s where s = . ξ ∈ k T0 =  −1 0 0 ξ −1 For a ∈ Z, let S(a) be the diagonal matrix with diagonal entries σ a and σ −a , where σ is a fixed generator of F×q . Then T0F = {S(a) | 0  a < q − 1} is cyclic of order q − 1. Now consider the element s ∈ W and let Ts ⊆ G be an F-stable maximal torus of type s (see 1.6.4, 1.6.21). As in Example 1.6.5, one sees that     ξ 0  × q2 q+1 ξ ∈ k , ξ = ξ and ξ = 1 T0 [s] =  0 ξq is cyclic of order q + 1. For b ∈ Z, let S (b) be the diagonal matrix with diagonal bq entries τ0b and τ0 , where τ0 = τ q−1 is an element of order q + 1 in F×q2 . (Note  that S (b) has the same eigenvalues as the matrix T(b) defined in Table 2.6.) Thus, TsF  T0 [s] = {S (b) | 0  b < q + 1}, where T(b) and S (b) correspond to each other under this isomorphism. For 0  i < q − 1 let θ i ∈ Irr(T0F ) be the character that sends S(a) to ε ai ; for 0  j < q + 1 let θ j ∈ Irr(TsF ) be the character that sends T(b) to η b j . Then we have (see also [Bo11, §5.3]): ⎧ ⎪ ψ +ψ ⎪ ⎪ ⎨ 1(i) q ⎪ G RT0 (θ i ) = ψq+1 ⎪ ⎪ ⎪ ⎪ ψ+ + ψ+ ⎩

if i = 0, if ε 2i  1, if i =

q−1 2 ,

⎧ ⎪ ψ −ψ ⎪ ⎪ ⎨ 1 (j) q ⎪ G  RTs (θ j ) = −ψq−1 ⎪ ⎪ ⎪ ⎪ −ψ− − ψ− ⎩

if j = 0, if η2j  1, if j =

q+1 2 .

(j)

(i) Thus, we see that ψ1 and ψq , as well as the characters ψq+1 and ψq−1 , are uniform, while ψ± , ψ± are not uniform. (The latter four characters take different values on the regular unipotent elements N, N , while all RTG (θ) have value 1 on N, N  by Theorem 2.2.22.) Assume now that q is odd and let i0 = (q − 1)/2, j0 = (q + 1)/2. Then we note that the values of the following rational linear combinations take a particularly simple form:

1  2 (ψ+ 1  2 (ψ+

− −

ψ+ ψ+

+ −

ψ− ψ−

− +

ψ−) ψ−)

I −I N N −N −N  S(a) T(b) √ √ . . δq − δq . . . . √ √ . . . . δ δq −δ δq . .

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Lusztig’s Classification of Irreducible Characters

(An interpretation for this special behaviour will be given in Example 2.7.27.) Note that RTG0 (θ i0 ) and RTGs (θ j0 ) do not have an irreducible constituent in common. However, the pairs (T0, θ i0 ) and (Ts, θ j0 ) are still related in a geometric way, which we will discuss in the following section (Example 2.3.6). It is a good exercise to re-interpret the character tables in Table 2.1 (p. 95) in a similar way. For the Suzuki groups 2B2 (q2 ), see [Ge03a, §4.6].

2.3 Unipotent Characters and Degree Polynomials We have seen in the previous section that every ρ ∈ Irr(GF ) occurs in some Deligne– Lusztig character RTG (θ). Since ±RTG (θ) is not irreducible in general, it makes sense to define a graph DL(GF ) as follows. It has vertices in bijection with Irr(GF ). Two characters ρ1  ρ2 in Irr(GF ) are joined by an edge if there exists some pair (T, θ) such that RTG (θ), ρi  0 for i = 1, 2. Thus, the connected components of DL(GF ) define a partition of Irr(GF ). In this section, we will concentrate on a particular connected component of DL(GF ): the one containing the trivial character of GF . We begin with a basic criterion for when two Deligne–Lusztig characters have an irreducible constituent in common, which may happen even if they are orthogonal to each other, since they are in general just virtual characters. This relies on the following auxiliary result. Lemma 2.3.1 Let T ⊆ G be an F-stable maximal torus. For any integer d  1, we define a norm map by NF d /F : T → T,

t → tF(t)F 2 (t) . . . F d−1 (t). d

Then NF d /F is an isogeny of algebraic groups, such that NF d /F (TF ) = TF . Proof Since T is abelian, it is clear that NF d /F is an abstract group homomorphism. d If t ∈ TF , then F(NF d /F (t)) = F(t)F 2 (t) . . . F d (t) = NF d /F (t) and so NF d /F (t) ∈ TF . Since F is a homomorphism of algebraic groups, so is NF d /F . Let us now consider the kernel of NF d /F . Let t ∈ T. Since T is F-stable and connected, we can apply the Lang–Steinberg Theorem and write t = s−1 F(s) for some s ∈ T. Then





NF d /F (t) = s−1 F(s) F s−1 F(s) . . . F d−1 s−1 F(s) = s−1 F d (s). d

Hence, NF d /F (t) = 1 if and only if s ∈ TF . We conclude that ker(NF d /F ) = {s−1 F(s) | s ∈ TF } ⊆ TF . d

d

So the kernel is finite and, hence, NF d /F is an isogeny. We have already seen

2.3 Unipotent Characters and Degree Polynomials

121

Fd

NF d /F (T ) ⊆ TF . It remains to show that equality holds. Let t ∈ TF . By Lang– Steinberg, we can write t = s−1 F d (s) where s ∈ T. Then the above computation shows that NF d /F (s−1 F(s)) = s−1 F d (s) = t. Finally,

F d (s−1 F(s)) = F d (s)−1 F F d (s) = (st)−1 F(st) = s−1 F(s) and so s−1 F(s) ∈ TF , as required. d



Using the norm map NF d /F , we obtain an induced map d

Irr(TF ) → Irr(TF ),

θ → θ ◦ NF d /F ,

d

which is injective (since NF d /F : TF → TF is surjective). Theorem 2.3.2 (Exclusion theorem) Suppose that two Deligne–Lusztig characters RTG1 (θ 1 ) and RTG2 (θ 2 ) have an irreducible constituent in common. Then (T1, θ 1 ) and (T2, θ 2 ) are ‘geometrically conjugate’, that is, there exist some d  1 and some d d g ∈ GF such that T2 = gT1 g −1 and g conjugates θ 1 ◦ NF d /F ∈ Irr(T1F ) to d d d θ 2 ◦ NF d /F ∈ Irr(T2F ) (via the isomorphism T1F → T2F , t → gtg −1 ). For the proof, see [DeLu76, 5.4, 6.3] or [Ca85, 4.1.1, 7.3.8] or [DiMi20, 11.1.3]; the term ‘exclusion theorem’ comes from [DeLu76, 7.10]. Remark 2.3.3 Let T ⊆ G be an F-stable maximal torus. Then the norm map is transitive, in the sense that NF d e /F = N(F d )e /F d ◦ NF d /F

for any integers d, e  1.

Consequently, if the pairs (T, θ) and (T, θ ) are geometrically conjugate via the norm map NF d /F , then they will also be geometrically conjugate via the norm map NF d e /F for any e  1. Definition 2.3.4 (Cf. [Ca85, §12.1]) Let ρ1, ρ2 ∈ Irr(GF ). We say that ρ1, ρ2 are geometrically conjugate if there are Deligne–Lusztig characters RTGi (θ i ) such that ρi, RTGi (θ i )  0 for i = 1, 2, and the pairs (T1, θ 1 ), (T2, θ 2 ) are geometrically conjugate (as in Theorem 2.3.2). This defines an equivalence relation on Irr(GF ); the equivalence classes are called geometric conjugacy classes of characters or geometric series of characters. Clearly, if ρ1, ρ2 belong to the same connected component of the graph DL(GF ), then ρ1, ρ2 belong to the same geometric series of characters. Lemma 2.3.5 Let ρ ∈ Irr(GF ) and T ⊆ G be an F-stable maximal torus. Then the number of θ ∈ Irr(TF ) such that RTG (θ), ρ  0 is bounded above by |W|. Proof Let us fix some θ such that RTG (θ), ρ  0. If RTG (θ ), ρ  0 for some θ  ∈ Irr(TF ), then Theorem 2.3.2 implies that there exists some d  1 and some

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g ∈ NG (T) such that g(θ ◦ NF d /F ) = θ  ◦ NF d /F . Hence, since W  NG (T)/T, there are at most |W| possibilities for θ  ◦ NF d /F . Finally, since the map θ  → θ  ◦ NF d /F is injective, there are at most |W| possibilities for θ  itself.  Example 2.3.6 Let GF = SL2 (q) as in Example 2.2.30. Assume that q is odd and consider the non-uniform characters ψ± , ψ±; we have RTG0 (θ i0 ) = ψ+ + ψ+

RTGs (θ j0 ) = −ψ− − ψ− .

and

Now, although RTG0 (θ i0 ) and RTGs (θ j0 ) do not have an irreducible constituent in common, the two pairs (T0, θ i0 ) and (Ts, θ j0 ) are geometrically conjugate. We leave it as an exercise to check this directly, using the norm map NF 2 /F . (Or see Example 2.4.7(b).) Lemma 2.3.7 ([Bo06, 9.11]) Let (T1, θ 1 ) and (T2, θ 2 ) be geometrically conjugate. Then θ 1 and θ 2 have the same restriction to Z◦ (G)F . Proof Let d  1 and g ∈ GF be such that T2 = gT1 g −1 , g(θ 1 ◦ NF d /F ) = θ 2 ◦ NF d /F . Let z ∈ Z◦ (G)F . As in the proof of Lemma 2.3.1, we have z = NF d /F (z1−1 F(z1 )) where z1 ∈ Z◦ (G) is such that z = z1−1 F d (z1 ). This yields



θ 2 (z) = θ 2 ◦ NF d /F (z1−1 F(z1 )) = θ 1 NF d /F (g −1 z1−1 F(z1 )g) d

= θ 1 (NF d /F (z1−1 F(z1 ))) = θ 1 (z), where the third equality holds since z1−1 F(z1 ) ∈ Z(G).



Irr(GF )

is called a unipotent Definition 2.3.8 ([DeLu76, 7.8]) A character ρ ∈ G character if RT (1T ), ρ  0 for some F-stable maximal torus T ⊆ G (where 1T stands for the trivial character of TF ). We also denote Uch(GF ) := set of all unipotent characters of GF . This set is a geometric series of characters. Indeed, by Example 2.2.26(a), every ρ ∈ Uch(GF ) is geometrically conjugate to 1G . Furthermore, if T ⊆ G is an Fstable maximal torus and θ ∈ Irr(TF ), then the Exclusion Theorem 2.3.2 shows that, for ρ ∈ Uch(GF ), we have RTG (θ), ρ = 0 unless θ = 1T . The same arguments also show that Uch(GF ) is precisely the connected component of the graph DL(GF ) (introduced in the beginning of this section) to which 1G belongs 2 . Example 2.3.9 Let B0 ⊆ G be an F-stable Borel subgroup such that T0 ⊆ B0 (as in 1.6.1). Then it is a classical part of the character theory of GF (well established before the work of Deligne and Lusztig) that there is a bijection '   ( F 1−1 Irr(Wσ ) ←→ ρ ∈ Irr(GF ) | IndG (1 ), ρ  0 , φ ↔ ρφ . B F B0 0

2

A characterisation of all connected components of DL(G F ) will only be achieved much later, in Remark 2.6.19.

2.3 Unipotent Characters and Degree Polynomials

123

See [CuRe87, §67, §68] and, for example, [GeJa11, §4.3] for a discussion taking into account more recent developments; we will also come back to this in Chapter 3. The above correspondence is canonical once a square root of q has been fixed. (The condition comes from the fact that the values of all characters ρφ are contained √ in Q( q) ⊆ K; see Remark 3.1.21 and Example 3.2.6 for further details.) By Proposition 2.2.7, we have ρφ ∈ Uch(GF ) for all φ ∈ Irr(Wσ ). In particular, the trivial character 1G and the Steinberg character StG of GF are unipotent. (See also Examples 2.2.26 and 2.2.27.) As we will see in later sections, the unipotent characters are of fundamental importance in the character theory of GF . It would certainly be interesting and desirable to find an elementary characterisation of these characters, without reference to the theory of -adic cohomology. For example, given the character table of GF (e.g., as in Table 2.1 or the Cambridge ATLAS), how can we identify the unipotent characters in this table? Assuming that q is large enough, such an elementary characterisation is mentioned in the introduction of [Lu77b]. To state it, we need some preparation. Remark 2.3.10

Let T be an F-stable maximal torus of G. Then Treg := {t ∈ T | CG◦ (t) = T}

is called the set of regular elements of G in T. (See, e.g., [Hum95, Chap. 4] for a further discussion.) Now let ρ ∈ Uch(GF ). As already noted, for θ ∈ Irr(TF ) we have RTG (θ), ρ = 0 unless θ = 1T . So the formula in Proposition 2.2.18 takes the following simple form (see [DeLu76, 7.9]): ρ(s) = RTG (1T ), ρ

F . for any s ∈ Treg

F , then T is the only maximal torus of G that contains s.) (Note that, if s ∈ Treg F   for all F-stable maximal tori T ⊆ G, then the knowledge of Thus, if Treg the multiplicities RTG (1T ), ρ is equivalent to the knowledge of the values of ρ on regular semisimple elements. F   if q is large Now, it is known that Treg is dense in T which implies that Treg enough. More precisely, we have:

Lemma 2.3.11 With T, Treg as above, there is a constant C > 0 (depending only F |/|TF |  1 − C/q. on the root datum of G) such that |Treg Proof We follow the argument in [Ca85, 8.4.2], with minor modifications. First note that the formulae in 1.6.21 show that |TF | = (q − ε1 ) · · · (q − εr ) where r = dim T and εi are roots of unity. Hence, (q − 1)r  |TF |  (q + 1)r .

(a)

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Lusztig’s Classification of Irreducible Characters

Let s ∈ TF be non-regular and set H := CG◦ (s). Then s ∈ T  H, and H is F-stable, closed, connected and reductive (see 2.2.13). Since s is non-regular, we have Z(H)  T and so dim Z(H)  r − 1. Now Z◦ (H) is an F-stable torus and so |Z◦ (H)F |  (q + 1)r−1 (by the same argument as above for TF ). Furthermore, by 1.3.10(c), Z(H)/Z◦ (H) is isomorphic to the centre of the semisimple group H/Z◦ (H). So, by Proposition 1.5.2 and Table 1.2 (p. 20), we have |Z(H)/Z◦ (H)|  r + 1. Thus, we obtain s ∈ Z(H)F

and

|Z(H)F |  (r + 1)(q + 1)r−1 .

(b)

Let R1 ⊆ X(T) be the set of roots of G with respect to T. Then H is generated by T and the root subgroups of G corresponding to those α ∈ R1 such that α(s) = 1 (see once more 2.2.13). Thus, H is completely determined by T and a subset of R1 . So, if we just use (b) and the fact that there are at most 2 |R1 | subsets of R1 at all, then the discussion so far shows that F |  2 |R1 | (r + 1)(q + 1)r−1 . |TF \ Treg

(c)

Using (a) and (c), we obtain the estimate F | |Treg

|TF |

=1−

F | |TF \ Treg

|TF |

 1−

C 2 |R1 | (r + 1)(q + 1)r−1  1− , (q − 1)r q

where C > 0 only depends on the root datum of G.



One can in fact give precise formulae for the number of regular elements in TF but we do not need this here; see, e.g., [Hum95, §8.9] and the references there. The above proof yields a very crude bound but it shows that, if q is large enough, then F  ; furthermore, the proportion of elements in TF that are regular tends to 1 Treg as q → ∞. Proposition 2.3.12 (Lusztig, see [Lu77b, Introduction]) Assume that every Fstable maximal torus of G contains sufficiently many regular elements fixed by F or, more precisely, that

F |Treg | > 1 − 2− |W | |TF | for any F-stable maximal torus T ⊆ G. (∗) F is Let ρ ∈ Irr(GF ). Then ρ is unipotent if and only if the restriction of ρ to Treg constant, for any F-stable maximal torus T ⊆ G.

Proof If ρ is unipotent, then we have seen in Remark 2.3.10 that ρ takes the constant value RTG (1T ), ρ on the regular semisimple elements inside a given TF . Conversely, let ρ ∈ Irr(GF ) be non-unipotent. By Corollary 2.2.19, there is an Fstable maximal torus T ⊆ G such that RTG (θ), ρ  0 for some θ ∈ Irr(TF ) which is not the trivial character. Using our assumption (∗), we will show that then there

2.3 Unipotent Characters and Degree Polynomials

125

exist regular semisimple elements s, s  ∈ TF such that ρ(s)  ρ(s ). Indeed, for regular semisimple elements, the formula in Proposition 2.2.18 reduces to:

F , ρ(s) = mi θ i (s) for all s ∈ Treg (†) 1in

Irr(TF )

are such that mi := RTG (θ i ), ρ  0. Note that none of the where θ i ∈ characters θ i is the trivial character of TF . Hence, θ 1, . . . , θ n together with the trivial character of TF are linearly independent functions TF → K, by a classical result due to Dedekind. By a slight generalisation of this result, these functions are F . Indeed, by [Lu90, 8.1], these n + 1 even linearly independent when restricted to Treg restrictions are linearly independent if

F |Treg | > 1 − 2−n |TF |. By Lemma 2.3.5, we have n  |W| and so this inequality does hold by our assumpF , the linear combination  tion (∗). Hence, on the set Treg 1in mi θ i is not a scalar F multiple of the trivial character of TF . Consequently, by (†), there exist s, s  ∈ Treg  such that ρ(s)  ρ(s ), as desired.  The following results show that unipotent characters are much better behaved with respect to normal subgroups of GF than arbitrary irreducible characters. Because of its importance for the classification of unipotent characters (see Chapter 4), we discuss this in some detail. This is also a good illustration of some of the methods developed so far. Lemma 2.3.13 Let η ∈ Irr(GF ) be a linear character such that η(u) = 1 for all unipotent elements u ∈ GF . Then η · RTG (θ) = RTG (η|T F · θ) for any F-stable maximal torus T ⊆ G and θ ∈ Irr(TF ). Proof This easily follows from the character formula in Theorem 2.2.16, applied to RTG (η|T F · θ). Just note that, if g = su = us is the Jordan decomposition of g ∈ GF

and if x ∈ GF is an element such that x −1 sx ∈ T, then we have η|T F · θ (x −1 sx) = η(s)θ(x −1 sx) = η(g)θ(x −1 sx).  F (see Remark 1.5.13). If Lemma 2.3.14 Let GuF = u ∈ GF | u unipotent ⊆ Gder F F ρ ∈ Uch(G ), then the restriction of ρ to Gu is irreducible. Furthermore, distinct unipotent characters of GF have distinct restrictions to GuF .

Proof First note that GuF is a normal subgroup of GF and GF /GuF is abelian (see Remark 1.5.13). Hence, by Remark 2.1.4, we must show that η · ρ  ρ for every non-trivial linear character η ∈ Irr(GF ) with GuF ⊆ ker(η). For this purpose, let T ⊆ G be an F-stable maximal torus such that RTG (1T ), ρ  0. Let η ∈ Irr(GF ) be a linear character such that GuF ⊆ ker(η). Using Lemma 2.3.13, we deduce that

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Lusztig’s Classification of Irreducible Characters

RTG (η|T F ), η · ρ  0. Hence, if ρ = η · ρ, then the Exclusion Theorem 2.3.2 implies that η|T F must be trivial. But, the two statements (b) and (c) in Remark 1.5.13 imply  that GF = GuF .TF and so η itself must be trivial, as desired. Proposition 2.3.15 (See [DeLu76, 7.10]) Let G be another connected reductive algebraic group over k and F  : G → G be a Steinberg map. Let π : G → G be a surjective homomorphism of algebraic groups such that F  ◦ π = π ◦ F and ker(π) ⊆ Z(G). Then the following hold. (a) If T ⊆ G is an F-stable maximal torus, then T := π(T) ⊆ G is an F -stable  maximal torus and RTG (1T ) = RTG (1T ) ◦ π|G F .  F (b) We have a bijection Uch(G ) → Uch(GF ), ρ → ρ ◦ π|G F . Proof (a) This is contained in [DeLu76, 7.10]; see also [DiMi20, 11.3.8]. Here, one has to use once more the definition of RTG (1T ) as in Proposition 2.2.5.  (b) By Proposition 1.4.13, π(GF ) is a normal subgroup of GF , with abelian  factor group of order prime to p. Thus, all unipotent elements of GF are contained  in π(GF ). By Lemma 2.3.14, the restriction of any ρ ∈ Uch(GF ) to π(GF ) is   F  F irreducible. Hence, ρ := ρ ◦ π|G F ∈ Irr(G ) for any ρ ∈ Uch(G ); furthermore, the map ρ → ρ is injective. Now let T ⊆ G be an F -stable maximal torus and    write RTG (1T ) = i ni ρi where ρi ∈ Uch(GF ) and ni ∈ Z. Then (a) shows that    RTG (1T ) = i ni ρi . Thus, we conclude that Uch(GF ) = {ρ | ρ ∈ Uch(GF )}. Remark 2.3.16 Let G be another connected reductive algebraic group over k and F  : G → G be a Steinberg map. Let f : G → G be an isotypy (see 1.3.21), that is, f is a homomorphism of algebraic groups such that ker( f ) ⊆ Z(G) and  ⊆ f (G); also assume that f ◦ F = F  ◦ f . Let T  ⊆ G  be an F  -stable maximal Gder  torus and consider the F-stable maximal torus T := f −1 (T) ⊆ G. Let θ  ∈ Irr(TF ) and θ = θ  ◦ f |T F ∈ Irr(TF ). If ker( f ) is connected, then 

RTG (θ) = RTG (θ ) ◦ f |G F . See [Bo00, Cor. 2.1.3] and [DiMi20, Prop. 11.3.10]. If one is only interested in θ = 1T and θ  = 1T , then the assumption on ker( f ) can be dropped, as in Proposition 2.3.15(a) above. Example 2.3.17 Let α : G → G be a bijective homomorphism of algebraic groups such that α ◦ F = F ◦ α. Then α restricts to an automorphism of GF which we denote by the same symbol. Let T ⊆ G be an F-stable maximal torus. Then the formula in Remark 2.3.16 shows that G RTG (1T ) = Rα(T) (1α(T) ) ◦ α.

It follows that, if ρ ∈ Uch(GF ), then ρ ◦ α ∈ Uch(GF ). This will be used, for

2.3 Unipotent Characters and Degree Polynomials

127

example, in the situation where G is a closed subgroup of some larger group and α is given by conjugation with an element in the normaliser of G. As noted in Definition 2.3.8, the unipotent characters form a geometric series of characters of GF . As preparation for the discussion of the remaining geometric series, we introduce an alternative model for RTG (θ), which appears already in [DeLu76] (see [Lu77b, 3.3]) and which is interesting and useful in its own right. One advantage of this model is that we do not have to deal with arbitrary maximal tori in G, but that everything is done in terms of a fixed maximally split torus T0 ⊆ G as in 1.6.1. We will treat this in some detail, as this is also a good illustration for working with Lefschetz numbers. 2.3.18 Recall the setting of 1.6.1. Thus, T0 ⊆ G is an F-stable maximal torus that is maximally split, that is, it is contained in an F-stable Borel subgroup B0 ⊆ G. Let W = NG (T0 )/T0 , identified with the Weyl group of the corresponding root datum R = (X, R, Y, R∨ ) (relative to T0 ). For any w ∈ W, we consider the finite subgroup

⊆ T0 T0 [w] := {t ∈ T0 | F(t) = w −1 t w}

(see 1.6.4),

where w is a representative of w in NG (T0 ). Then, as discussed in 1.6.21, every subgroup of GF of the form TF (where T ⊆ G is an F-stable maximal torus) can be realised as T0 [w] for some w ∈ W. More precisely, given T, there exists some g ∈ G such that T = gT0 g −1 and conjugation with g defines an isomorphism TF  T0 [w], where w is the image of g −1 F(g) in W; thus, T is a torus of type w.

0 satisfies condition (∗) Let U0 = Ru (B0 ) be the unipotent radical of B0 . Then wU in 2.2.4 with respect to the finite subgroup T0 [w]. Hence, by Proposition 2.2.5, we obtain virtual characters of GF by setting θ := RTG0 [w], wU Rw

0 (θ)

for any θ ∈ Irr(T0 [w]).

We now identify these characters with the RTG (θ) introduced earlier. θ does not depend on the choice of the Lemma 2.3.19 The virtual character Rw

Furthermore, let g ∈ G be such that g−1 F(g) = w and consider representative w. the corresponding torus T = gT0 g −1 of type w, as in 2.3.18. Then we have θ = RTG (g θ) Rw

where

g

θ(t) := θ(g −1 tg) for t ∈ TF .

θ only depends on w. An arbitrary representative of Proof First we show that Rw w in NG (T0 ) is of the form h0 w where h0 ∈ T0 . Let F  : G → G be defined by

F (x) = wF(x) w −1 for all x ∈ G. By Lemma 1.4.14, F  also is a Steinberg map on G, and we have F (T0 ) ⊆ T0 . Hence, by Theorem 1.4.8 (Lang–Steinberg), we can write h0 = h−1 F (h) for some h ∈ T0 . We claim that we have a bijection

0 ) → L −1 (h0 wU

0 ), f : L −1 (wU

x → xh.

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0 . Since F(h) ∈ T0 normalises Indeed, let x ∈ G be such that x −1 F(x) ∈ wU

U0 , we have (xh)−1 F(xh) ∈ h−1 wF(h)U . We have h−1 wF(h) = h0 w and so 0

0 . Thus, f is a bijection. Furthermore, f commutes with the (xh)−1 F(xh) ∈ h0 wU

0 ) and on L −1 (h0 wU

0 ). action of an element (g , t) ∈ GF × T0 [w] on L −1 (wU Hence, the property of Lefschetz numbers in Example 2.2.3(b) implies that, for all g  ∈ GF and t ∈ T0 [w], we have





0 ) = L (g , t), L −1 (h0 wU

0) . L (g , t), L −1 (wU θ only depends on w but not on the choice of the representative w

∈ NG (T0 ). Thus, Rw θ = RG (g θ). Let T = gT g −1 as above. Then we Next we establish the identity Rw 0 T also have T = F(g)T0 F(g)−1 and so F(g)B0 F(g)−1 is a Borel subgroup containing T. Consequently, by Corollary 2.2.9, we have

1  L (g , t), L −1 (F(g)U0 F(g)−1 ) θ(g −1 tg) RTG (g θ)(g ) = F |T | t ∈T F



1 = L (g , gtg −1 ), L −1 (F(g)U0 F(g)−1 ) θ(t) |T0 [w]| t ∈T0 [w]

for all g  ∈ GF . So it remains to show that the Lefschetz number appearing in the

0 )). For this purpose, note that the morphism above sum equals L((g , t), L −1 (wU

0 ), f : L −1 (F(g)U0 F(g)−1 ) → L −1 (wU

y → yg,

0 w −1 g −1 if is bijective. (Indeed, for y ∈ G we have L (y) ∈ F(g)U0 F(g)−1 = g wU −1 −1

0 .) It is also straightforward and only if L (yg) = g L (y)F(g) ∈ g F(g)U0 = wU to check that f commutes with the action of GF (by left multiplication on both sides) and transforms the action of TF on L −1 (F(g)U0 F(g)−1 ) into the action of

0 ). Hence, a further application of Example 2.2.3(b) yields the T0 [w] on L −1 (wU desired equality.  2.3.20 We define X(G, F) to be the set consisting of all pairs (T, θ), where T is an F-stable maximal torus of G and θ ∈ Irr(TF ). If (T, θ) ∈ X(G, F) and g ∈ GF , then T := gTg −1 is an F-stable maximal torus of G and we obtain an irreducible character g θ ∈ Irr(TF ) by setting g θ(t ) := θ(g −1 t  g) for all t  ∈ TF . Thus, GF acts on X(G, F) via

g, (T, θ) → (gTg −1, g θ). (a) GF × X(G, F) → X(G, F), We denote by X(G, F) the set of GF -orbits for this action. On the other hand, we define X(W, σ) := {(w, θ) | w ∈ W, θ ∈ Irr(T0 [w])}. If w, w , x ∈ W are such that w  = xwσ(x)−1 , then a straightforward computation shows that we have a group

x −1 , where x is a fixed representative of x isomorphism T0 [w] → T0 [w ], t → xt in NG (T0 ); furthermore, if θ ∈ Irr(T0 [w]), then we obtain an irreducible character

2.3 Unipotent Characters and Degree Polynomials

129

for all t ∈ T0 [w ]. Thus, W acts on ∈ Irr(T0 [w ]) by setting xθ(t) := θ( x −1 t x) X(W, σ) via

(b) W × X(W, σ) → X(W, σ), x, (w, θ) → (xwσ(x)−1, xθ). xθ

We denote by X(W, σ) the set of W-orbits for this action. Then it is straightforward to verify that we have a natural bijection X(W, σ) ←→ X(G, F),

(c)

defined as follows: The W-orbit of a pair (w, θ) ∈ X(W, σ) corresponds to the GF orbit of a pair (T, θ ) ∈ X(G, F) if there exists some g ∈ G such that T = gT0 g −1 , g −1 F(g) ∈ NG (T0 ) is a representative of w, and θ (gtg −1 ) = θ(t) for all t ∈ T0 [w]. Remark 2.3.21 Via 2.3.20(c), the W-orbits of pairs (w, 1) correspond to the GF orbits of pairs (T, 1) (where, in both cases, 1 stands for the trivial character). Hence, the correspondence in 2.3.20(c) is a refinement of the well-known correspondence between σ-conjugacy classes of W and GF -conjugacy classes of F-stable maximal tori of G. For further details see, e.g., [Ca85, 3.3.3], [Ge03a, 4.3.5]. θ instead of RG (θ), certain identities involving Deligne–Lusztig characUsing Rw T ters simplify considerably. We give a few examples.

Example 2.3.22 (a) The scalar product formula in Theorem 2.2.8 can be reexpressed as follows. If (wi, θ i ) ∈ X(W, σ) for i = 1, 2, then θ1 θ2 , Rw = |{x ∈ W | x.(w1, θ 1 ) = (w2, θ 2 )}|. Rw 1 2

In particular, the scalar product is zero unless w1, w2 are σ-conjugate in W. (See [Ge18, 2.6] for further details about the translation of the previous formula into the new one.) (b) Let (w, θ) ∈ X(W, σ) correspond to the pair (T, θ ) ∈ X(G, F) under the θ = RG (θ  ); furthermore, ε ε = (−1)l(w) where bijection in 2.3.20(c). Then Rw G T T l(w) denotes the usual length of w ∈ W; see [Ca85, 7.5.2]. Consequently, the degree formula in Theorem 2.2.12 translates to the formula θ Rw (1) = (−1)l(w)

|GF | p |GF | p = (−1)l(w) , |T0 [w]| det(q idXR − ϕ0 ◦ w −1 )

where the second equality holds by Lemma 1.6.6. Example 2.3.23 If θ = 1 is the trivial character of T0 [w], then we shall simply 1 . Then the trivial character 1 and the Steinberg character write Rw instead of Rw G F StG of G (see Example 2.2.27) can be expressed as follows: 1 1 Rw and StG = (−1)l(w) Rw . 1G = |W| w ∈W |W| w ∈W

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Lusztig’s Classification of Irreducible Characters

We have Rw, 1G = 1 and Rw, StG = (−1)l(w) for all w ∈ W. The character of the regular representation of GF is given by

1 θ θ Rw (1) Rw . χreg = |W| (w,θ)∈X(W,σ)

Compare with the formulae in Examples 2.2.26 and 2.2.27! The above formulations are taken from [DiMi20, 10.2.5 and 10.2.6]. Finally, let ρ ∈ Irr(GF ). Taking the scalar product of ρ with the above expression for χreg , we obtain

1 θ θ Rw , ρ Rw (1), ρ(1) = |W| which simplifies to ρ(1) =

1 |W |



(w,θ)∈X(W,σ)

w ∈W Rw ,

ρ Rw (1) if ρ is unipotent.

Proposition 2.3.24 (See [Lu77b, 3.12]) We have

1 ρ(1) ρ = Rw (1) Rw . |W| w ∈W F ρ∈Uch(G )

Proof

This follows from the identity

1 ρ(1) ρ = χreg = |W| F ρ∈Irr(G )

θ θ Rw (1) Rw

(w,θ)∈X(W,σ)

and the Exclusion Theorem 2.3.2. (See also [Lu77a, Lemma 7.7].)



Combining the formula for ρ(1) in Example 2.3.23 with the degree formula for θ in Example 2.3.22(b), we are led to the following definition which, in this form, Rw appeared in [Ge18, 3.3]. Definition 2.3.25 Let G = (R, ϕ0 W) be the complete root datum associated with G, F (see Example 1.6.11). For any ρ ∈ Irr(GF ), the corresponding degree polynomial is defined as

1 |G| θ (−1)l(w) Rw , ρ q−N ∈ R[q], Dρ := |W| |Tw | (w,θ)∈X(W,σ)

where N is the number of positive roots, |G| ∈ R[q] is the order polynomial of GF and |Tw | ∈ R[q] is the order polynomial of T0 [w]; note that q N |Tw | divides |G| in R[q] (see 1.6.21) and, hence, Dρ is indeed a polynomial (and not just a rational function). Using the formula in Example 2.3.23, we see that the actual character degree ρ(1) is obtained by evaluating the polynomial Dρ at q; in particular, Dρ  0. Analogous to the notation |GF | p (for the p-part of |GF |), we will also write |G|q instead of q−N |G|.

2.3 Unipotent Characters and Degree Polynomials

131

Remark 2.3.26 Let ρ ∈ Irr(GF ). Using the degree polynomial Dρ ∈ R[q], we can define numerical invariants of ρ as follows. Aρ := degree of Dρ, aρ := largest non-negative integer such that qaρ divides Dρ . We obtain a further invariant nρ ∈ R>0 by the condition that Dρ = ±

1 aρ q + combination of powers qi where i > aρ . nρ

All we can say at this stage is that 0  aρ  Aρ  N (since q−N |G|/|Tw | ∈ R[q] has degree N). The above invariants first appeared in [Lu79a, §8]; see also [Lu84a, 4.26]. They are an important tool for organising the irreducible characters into ‘series’ and ‘families’ (see Section 4.2 for a further discussion). Remark 2.3.27 One can show that the degree polynomials behave in many ways like true character degrees. For example, by analogy with results known for character degrees, the following are true: (a) Dρ divides |G| in R[q]. θ , ρ  0, then D divides |G|/|T | in R[q] (cf. [Ge92, 2.5]). (b) If Rw ρ w See [Ge18, Remark 3.11] for some hints concerning the proofs. Furthermore, let G ⊆ ˜ be a regular embedding (see Section 1.7). Let ρ˜ ∈ Irr(G ˜ F˜ ). By Theorem 1.7.15, G we have ρ| ˜ G F = ρ1 + · · · + ρr with distinct irreducible characters ρi of GF . Since the ˜ F , we have ρi (1) = 1 ρ(1) for all i. The analogous relation ρi are conjugate under G r ˜ also holds for the degree polynomials: (c) Dρi = r1 Dρ˜ for i = 1, . . . , r (see [Ge18, Lemma 6.5]). Remark 2.3.28 Once the degree polynomials of all unipotent characters are determined, and the Jordan decomposition of characters is establised, one observes that Dρ , for any ρ ∈ Irr(GF ), is of the form Dρ =

1 aρ ±q + · · · + q Aρ , nρ

where

nρ Dρ ∈ Z[q].

This will follow by a combination of Remark 2.3.27(c) (reduction to the case where Z(G) is connected), Proposition 2.5.11 (reduction to the case where ρ is unipotent), Remark 4.2.1 (further reduction to the case where G is simple) and, finally, by inspection of the tables of unipotent character degrees in the appendix of [Lu84a] (see also [Ca85, §13.8, §13.9]). Remark 2.3.29 Let G be the complete root datum associated with G, F, as above. We have a corresponding (infinite) series of finite groups of Lie type {G(q ) | q  ∈

132

Lusztig’s Classification of Irreducible Characters

PG } as in Remark 1.6.12, and our given group GF is isomorphic to the member G(q) of this series. Then it easily follows from Remark 2.3.28 that the set of all possible degree polynomials {Dρ | ρ ∈ Irr(G(q )) for some q  ∈ PG } is finite. By [Ge12a, 4.2], this can also be proved directly by a general argument.

Table 2.8 The unipotent characters of GF = Sp4 (q) T1 T2 T3 T4 T5 W-class {s1 s2, s2 s1 } {s2, s1 s2 s1 } {1} {s1 s2 s1 s2 } {s1, s2 s1 s2 } |TiF | q2 + 1 q2 − 1 (q − 1)2 (q + 1)2 q2 − 1 Dθ0 = 1 Rs1 s2 = θ 0 − θ 9 + θ 10 + θ 13, Rs2 = θ 0 + θ 11 − θ 12 − θ 13, Dθ9 = 12 q(q + 1)2 R1 = θ 0 + 2θ 9 + θ 11 + θ 12 + θ 13, Dθ10 = 12 q(q − 1)2 Rs1 s2 s1 s2 = θ 0 − 2θ 10 − θ 11 − θ 12 + θ 13, Dθ11 = Dθ12 = 12 q(q2 + 1) Dθ13 = q4 Rs1 = θ 0 − θ 11 + θ 12 − θ 13 .

Example 2.3.30 Let GF = Sp4 (q) where |GF | = q4 (q2 −1)2 (q2 +1). The complete character table (for q odd) was determined in [Sr68] (see [Pr82] for the correction of some minor errors). The unipotent characters are explicitly described in [Sr91, A.1] in terms of the Deligne–Lusztig characters RTG (1T ); see [Eno72] for q a power of 2. The Weyl group W = s1, s2 is dihedral of order 8 (with F acting trivially on W), where we fix the notation so that s1 corresponds to a long root and s2 to a short root. Corresponding to the five conjugacy classes of W, there are five GF -conjugacy classes of F-stable maximal tori in G. As in [Sr91], we denote representatives by Ti (1  i  5). There are six unipotent characters, denoted by θ 0 = 1G , θ 9 , θ 10 , θ 11 , θ 12 and θ 13 = StG in [Sr68]. This information, plus the decomposition of the RTGi (1Ti ) into unipotent characters, is contained in Table 2.8. Under the correspondence in Example 2.3.9, and using the notation for the characters of W in Example 2.1.12, we have: θ 0 ↔ 1W,

θ 9 ↔ φ1,

θ 11 ↔ ε ,

θ 12 ↔ ε ,

θ 13 ↔ ε.

Finally, the notation is different in [Eno72], but the information in Table 2.8 remains valid for q even. See also [Shi82] where the character table of the conformal symplectic group CSp4 (q) (for q odd) is determined. √ 2m+1 Example 2.3.31 Let GF = 2B2 (q2 ) be a Suzuki group, where q = 2 for F 4 2 4 some m ∈ Z0 ; see Example 1.4.22. We have |G | = q (q − 1)(q + 1). The known

2.3 Unipotent Characters and Degree Polynomials

133

Table 2.9 Values of unipotent characters of GF = 2B2 (q2 ) g 1 u u0 F F |CG (g) | |G | q4 2q2 1G 1 1 1 √ √ √ 1 2 q(q2 −1) − 1 2 q 1 −2 q  2√ 2√ 2 √ 1 2 q(q2 −1) − 1 2 q − 1 −2 q  2 2 2 StG q4 . . T0 is of type 1 ∈ W and |T0F | = q2 −1; √ T1 is of type s1 s2 s1 and |T1F | = q2 + 2q+1; √ T2 is of type s1 and |T2F | = q2 − 2q+1;

π1b π2c u0−1 π0a 2 F F 2q |T0 | |T1 | |T2F | 1 1 1 1 √ − 12 −2 q . 1 −1 √ 1 −2 q . 1 −1 2 . 1 −1 −1 R1 = 1G + StG . Rs1 s2 s1 = 1G +  +   − StG . Rs1 = 1G −  −   − StG .

character table in [Suz62] has been re-interpreted in [Ge03a, §4.6] in terms of the Deligne–Lusztig characters RTG (θ). The Weyl group W = s1, s2 is again dihedral of order 8 but now F induces an automorphism σ : W → W such that σ(s1 ) = s2 and σ(s2 ) = s1 . Corresponding to the three σ-conjugacy classes of W in Example 2.1.12, there are three GF -conjugacy classes of F-stable maximal tori in G; representatives will be denoted by T0, T1, T2 . We have Uch(GF ) = {1G, StG, ,   }, where ,   are complex conjugate to each other. The values of the unipotent characters are given in Table 2.9, where u , u0 are unipotent elements; furthermore, π0a , π1b , π2c are regular semisimple and a, b, c run over certain index ranges that we do not specify here. (There are q2 + 3 conjugacy classes.) The degree polynomials are given by the polynomial expressions in Table 2.9. We have √ a = a  = 1, A = A  = 3 and n = n  = 2. Thus, the invariant nρ is not always an integer. √ Finally, as in Example 2.2.30, we note that the values of the linear combination 12 2( −  ) take a particularly simple form: √ 1 2

1 u 2( −  ) . .

√

u0 −1 q

u−1 √0 − −1 q

{π0a } .

{π1b } .

{π2c } .

√ 3 If we consider the example where q = 2 , then the complete character table is printed in Table 2.1 (p. 95). By just looking at character degrees, we see that {X.2, X.3} = {,   }. In order to match these characters exactly, one would need to specify actual representatives of the classes labelled by 4a, 4b. √ 2m+1 Example 2.3.32 Let GF = 2F4 (q2 ) be a Ree group of type F4 , where q = 2 for some m ∈ Z0 ; see Example 1.4.22. By [Lu84a, §14.2], there is a unique

134

Lusztig’s Classification of Irreducible Characters

unipotent character ρ ∈ Uch(GF ) such that Dρ = 13 q4 (q2 −1)2 (q4 −q2 +1)(q8 −q4 +1) and

RTG (1T ), ρ) ∈ 2Z

for any F-stable maximal torus T ⊆ G. In particular, we have aρ = 4, Aρ = 20 and nρ = 3. Somewhat related to this particular behaviour is the fact that ρ is rational valued but its Frobenius–Schur indicator is −1 and, hence, it can not be realised by a representation over Q (see [Ge03b, §7]).

2.4 Towards Lusztig’s Main Theorem 4.23 In the previous section, we introduced the partition of Irr(GF ) into geometric series, where the unipotent characters Uch(GF ) appear to play a special role. The determination of that partition essentially relies on knowing the scalar products θ , ρ for all ρ ∈ Irr(GF ), all w ∈ W and all θ ∈ Irr(T [w]). One may hope that Rw 0 this information—once available—would then lead to: • a parametrisation of the characters inside a geometric series and • explicit formulae for character degrees and character values, at least on semisimple elements (via Proposition 2.2.18). This hope has been fully realised in Lusztig’s work, where the book [Lu84a] develops the main results assuming that Z(G) is connected and [Lu88] extends this to the general case (via regular embeddings and the Multiplicity-Freeness Theorem 1.7.15). It is the purpose of this section to introduce the basic formalism of [Lu84a] (with complete proofs for a number of intermediate results), and to explain the ‘Main Theorem 4.23’ in [Lu84a, p. 131]. (See also the survey [Ca95].) In non-technical terms, that main result yields explicitly computable formulae for θ , ρ and a Jordan decomposition of characters by which the the multiplicities Rw characters in an arbitrary series of Irr(GF ) are put in bijection with the unipotent characters of a (usually) smaller group. As far as the unipotent characters themselves are concerned, there is a purely combinatorial classification which is ‘independent of q’, as follows. ¯ Theorem 2.4.1 (Lusztig [Lu84a, 4.23]) There exist a finite set X(W, σ) and a ¯ collection of integers {m(w, x) ¯ | w ∈ W, x¯ ∈ X(W, σ)} ⊆ Z (both depending only on W and σ) such that the following holds. There is a bijection 1−1 ¯ σ), Uch(GF ) ←→ X(W,

ρ ↔ x¯ρ,

such that Rw, ρ = m(w, x¯ρ ) for all w ∈ W and ρ ∈ Uch(GF ). In particular, for

2.4 Towards Lusztig’s Main Theorem 4.23

135

each ρ ∈ Uch(GF ), the corresponding degree polynomial is given by |G|q 1 . (−1)l(w) m(w, x¯ρ ) Dρ = |W| w ∈W |Tw | Remark 2.4.2 (a) Using Proposition 2.3.15 and the arguments in [Lu76c, 1.18] (see also Remark 4.2.1 in Chapter 4), the proof of Theorem 2.4.1 can be reduced to the case where G is simple of adjoint type. In this case, and assuming that q is sufficiently large, the above result had already been obtained in earlier papers by Lusztig; see [Lu80a], [Lu81c], [Lu82b] (and [Lu84a, Appendix] for the Suzuki and Ree groups). Then [Lu84a, 4.23] shows that these results remain valid for any q. A more conceptual explanation for the fact that Uch(GF ) is classified ‘independently of q’ is provided by [Lu14b]. It even makes sense to define ‘exotic’ parameter sets and corresponding degree polynomials for any (not necessarily crystallographic) finite Coxeter group (see [Lu93], [Lu94]). This is the starting point for the ‘spetses’ philosophy in [BMM93], [BMM99], [BMM14]. ¯ (b) Let x¯1 ∈ X(W, σ) correspond to the trivial character of GF . Then, by Example 2.2.26, we have m(w, x¯1 ) = 1 for all w ∈ W and one easily sees that x¯1 is uniquely determined by this condition (see, e.g., [Ge18, 4.9]). In general, the correspondence ρ ↔ x¯ρ is not uniquely determined by the above properties. For example, the two unipotent characters ,   of GF = 2B2 (q2 ) in Example 2.3.31 cannot be distinguished by their multiplicities in the various Rw . (A similar thing necessarily happens whenever there exist unipotent characters that are not rational valued, which is the case in almost all groups of exceptional type.) These uniqueness issues will also be discussed in further detail in Chapter 4. ¯ Remark 2.4.3 The explicit description of the sets X(W, σ) occupies almost all of [Lu84a, Chap. 4]. This is a formidable piece of technical and combinatorial machinery. It proceeds in various stages starting with the case where W is irreducible and σ = idW , then going on to the case where σ is non-trivial but ordinary, and finally presenting reduction arguments for the general case. (For the case where σ is not ordinary, see [Lu84a, §14.2].) For a somewhat different parametrisation, see [Lu14a, §3], [Lu14c, §18]. (This is also described in [Ge18, §4].) The integers m(w, x) ¯ are not directly described in [Lu84a] but, assuming that σ is ordinary, in terms of the σ-character table of W and a certain ‘non-abelian Fourier matrix’, ¯ with rows indexed by X(W, σ) and columns indexed by a second set X(W, σ); see Section 4.2 for further details. (In the case where W is irreducible and σ is not ordinary, the integers m(w, x) ¯ are printed in [Lu84a, Appendix].) ¯ Thus, given any G, F with corresponding W, σ, the set X(W, σ) and the integers m(w, x) ¯ can be worked out explicitly, by an entirely combinatorial procedure. This is electronically available in Michel’s version of CHEVIE [MiChv], through the functions

136

Lusztig’s Classification of Irreducible Characters and

UnipotentCharacters

DeligneLusztigCharacter

which take as input an arbitrary finite Weyl group and an automorphism. The first ¯ function displays the parameter set X(W, σ) (including the corresponding degree polynomials) and the second function returns, for any w ∈ W, the decomposition of the Deligne–Lusztig character Rw into unipotent characters; see the online help for further information and examples. (E.g., one can immediately recover the formulae in Examples 2.3.30, 2.3.31, 2.3.32 in this way. One can even use as input for the function UnipotentCharacters a finite Coxeter group of non-crystallographic type I2 (m), H3 or H4 , in which case one obtains the ‘exotic’ parameter sets and degree polynomials in [Lu93].) Given the amount of complicated mathematics involved in them, the above two functions may be regarded as a highlight of modern computer algebra techniques! Our next task is to explain, following [Lu84a], how the characters in an arbitrary geometric series of Irr(GF ) can be described in terms of the information available for unipotent characters in Theorem 2.4.1 (for G itself and further, usually smaller groups). 2.4.4 As in [DeLu76, §5], we fix some choices of a purely number-theoretic nature that allow us to connect the multiplicative group of k = F p with roots of unity in C. It is known that k × is (non-canonically) isomorphic to (Q/Z) p , the group of all elements of Q/Z of order prime to p; see, e.g., [Ca85, 3.1.3], [Hum95, §2.7]. We fix once and for all a group isomorphism ∼

ι : k × −→ (Q/Z) p . Furthermore, the exponential map induces a group isomorphism ∼

exp : (Q/Z) p −→ μ p,

x + Z → exp(2πix),

where μ p := {z ∈ C | z n = 1 for some n ∈ Z1 such that p  n} is the group of all roots of unity in C of order prime to p. Composing these two isomorphisms, we obtain a group isomorphism ∼

ψ : k × −→ μ p ⊆ C×

(ψ = exp ◦ι).

The following constructions will depend on these choices 3 . 2.4.5 We now introduce the basic set-up of [Lu84a, 2.1] 4 . Let us consider a pair (λ, n) where λ ∈ X = X(T0 ) and n  1 is an integer prime to p. (Here, T0 is our 3

4

Isomorphisms ι, ψ as above can be obtained from a realisation of k as k = A/p where A is the ring of algebraic integers in C and p is a maximal ideal of A such that p ∈ p. Definite choices would arise from a construction of k via Conway polynomials; see [LuPa10, §4.2]. Lusztig actually works with line bundles L over G/B0 instead of characters λ ∈ X(T0 ), but one can pass from one to the other as described in [Lu84a, 1.3.2].

2.4 Towards Lusztig’s Main Theorem 4.23

137

fixed maximally split torus of G, as in 2.3.18.) Let Zλ,n be the set of all w ∈ W for which there exists some λw ∈ X such that

(♠) λw (t n ) = λ F(t)w −1 t −1 w for all t ∈ T0 . Note that the character λw , if it exists, is uniquely determined by w (since k × = {ξ n | ξ ∈ k × } and, hence, also T0 = {t n | t ∈ T0 }). Also note that, using additive notation in X, we can re-phrase (♠) as nλw = λ ◦ F − w.λ. Assume now that Zλ,n  . Then, for any w ∈ Zλ,n , the restriction of λw to T0 [w] is a group homomorphism λ¯w : T0 [w] → k ×

such that

λ¯w (t)n = 1 (t ∈ T0 [w]).

∼

Using our chosen isomorphism ψ : k × −→ μ p ⊆ C× from 2.4.4, we obtain a linear character θ w := ψ ◦ λ¯w ∈ Irr(T0 [w]), such that θ w (t)n = 1 for all t ∈ T0 [w]. If w ∈ Zλ,n and θ = θ w ∈ Irr(T0 [w]), then we will simply write (λ, n)  (w, θ) in the following. We shall see that this relation provides an efficient tool for dealing with geometric conjugacy in a way that does not involve the norm map. Definition 2.4.6 (Lusztig) Let (λ, n) be a pair as in 2.4.5, with Zλ,n  . We define a corresponding subset of Irr(GF ) by θw , ρ  0 for some w ∈ Zλ,n }. Eλ,n := {ρ ∈ Irr(GF ) | Rw

(Note that the Eλ,n are originally introduced in [Lu84a, 2.19], assuming that Z(G) is θw . connected, and using individual cohomology spaces instead of the characters Rw Then it is shown in [Lu84a, 6.5] that the original definition is equivalent to the one above. Here, the above definition in terms of virtual characters will work for all our purposes 5 , without any assumption on Z(G).) Example 2.4.7 (a) Let n = 1. Let λ0 ∈ X be the neutral element, that is, λ0 (t) = 1 for all t ∈ T0 . For w ∈ W, we can just set λw := λ0 and then (♠) in 2.4.5 holds. Hence, Zλ0,1 = W. For any w ∈ W, the restriction of λw to T0 [w] is trivial and so θ w = 1 is the trivial character of T0 [w]. Hence, we have (λ0, 1)  (w, 1) for all w ∈ W. This shows that Eλ0,1 = Uch(GF ) is the set of unipotent characters. (b) Let GF = SL2 (q) where q is odd. As in Example 2.2.30, we write W = {1, s} and T0 = {S(ξ) | ξ ∈ k × } where, for any ξ ∈ k × , we denote by S(ξ) the diagonal matrix with diagonal entries ξ, ξ −1 . Let n = 2 and define λ ∈ X by λ(S(ξ)) = ξ for all ξ ∈ k × . Now note that, if t ∈ T0 , then t q−1 = F(t)t −1 and t q+1 = F(t)s −1 t −1 s . Thus, if we define λ1, λs ∈ X by λ1 (S(ξ)) = ξ (q−1)/2

and

λs (S(ξ)) = ξ (q+1)/2

for all ξ ∈ k ×,

then (♠) holds for w = 1 and w = s, respectively. So Zλ,2 = W. We have (λ, 2)  5

We will see in Corollary 2.4.29 below that the sets Eλ, n are precisely the geometric series of characters as introduced in the previous section.

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Lusztig’s Classification of Irreducible Characters

(1, θ 1 ) and (λ, 2)  (s, θ s ) where θ 1 ∈ Irr(T0 [1]) and θ s ∈ Irr(T0 [s]) are the unique non-trivial characters of order 2. Thus, we obtain Eλ,2 = {ψ+ , ψ+, ψ− , ψ− }. (We will see in Example 2.4.25 that (1, θ 1 ) and (s, θ s ) are geometrically conjugate and, hence, ψ+ , ψ+, ψ− , ψ− are geometrically conjugate.) Lemma 2.4.8 (Cf. [Lu84a, 6.2(iii)]) Let w ∈ W and θ ∈ Irr(T0 [w]). Let n  1 be any integer prime to p such that θ(t)n = 1 for all t ∈ T0 [w]. Then there exists some λ ∈ X such that w ∈ Zλ,n and (λ, n)  (w, θ). If μ ∈ X also satisfies these conditions, then μ = λ + nν for some ν ∈ X.

Proof Define F  : T0 → T0 by F (t) = wF(t) w −1 for all t ∈ T0 . Then T0 [w] = T0F and, as in the proof of Lemma 1.6.6, we have a surjective map 

X(T0 ) → Hom(T0F , k × ),

λ → λ|T F  ,



(∗)

0

with kernel {λ ◦ F  − λ | λ ∈ X }. Now, the values of θ are nth roots of unity in C× . ∼ Hence, using the isomorphism ψ : k × −→ μ p , we can write uniquely θ = ψ◦θ¯ where  θ¯ : T0F → k × is a group homomorphism. By (∗), we know that θ¯ is the restriction of some λ1 ∈ X. Now θ n is the trivial character and so nλ1 is in the kernel of the map in (∗). Hence, there exists some λ  ∈ X such that λ  ◦ F  − λ  = nλ1 . Setting λ := w −1 .λ  ∈ X, we obtain

λ1 (t n ) = (nλ1 )(t) = λ (F (t)t −1 ) = λ(F(t)w −1 t −1 w)

for all t ∈ T0,

that is, we have w ∈ Zλ,n and (λ, n)  (w, θ), as desired. Now assume that μ ∈ X also satisfies these conditions, that is, we have

μ1 (t n ) = μ(F(t)w −1 t −1 w)

for all t ∈ T0,

where μ1 ∈ X is such that θ¯ is the restriction of μ1 . Then the restriction of λ1 − μ1 to  T0F is trivial and so μ1 − λ1 = ν  ◦ F  − ν  for some ν  ∈ X. Setting ν := w −1 .ν  ∈ X, it follows that



μ F(t)w −1 t −1 w = μ1 (t n ) = λ1 (t n )ν  F (t n )(t n )−1

n





n = λ F(t)w −1 t −1 w ν  F (t)t −1 ) = λ F(t)w −1 t −1 w ν F(t)w −1 t −1 w

is surjective by the Lang– for t ∈ T0 . But the map T0 → T0 , t → F(t)w −1 t −1 w, Steinberg theorem. Hence, we conclude that μ(t) = λ(t)ν(t)n for all t ∈ T0 , that is, μ = λ + nν.  Remark 2.4.9 Let (λ, n) be a pair as in 2.4.5, with Zλ,n  . Let n   1 also be an integer prime to p. If w ∈ Zλ,n and t ∈ T0 , then 





n    λw (t nn ) = λw (t n )n = λ F(t n )w −1 (t n )−1 w = λ F(t)w −1 t −1 w .

2.4 Towards Lusztig’s Main Theorem 4.23

139

So w ∈ Zn λ,nn and the two pairs (λ, n) and (n  λ, nn ) give rise to the same λw ∈ X and, hence, to the same character θ w ∈ Irr(T0 [w]). Thus, (λ, n)  (w, θ)

⇒

(n  λ, nn )  (w, θ).

One checks that the reverse implication also holds and so Zλ,n = Zn λ,nn . As in [Lu84a, 6.1], we say that (λ, n) is indivisible if it is impossible to write λ = n1 λ1 where n = n1 n1 with n1  1 and n1  2. Thus, in the definition of Eλ,n we may assume without loss of generality that (λ, n) is indivisible. Remark 2.4.10 Assume that (λ, n)  (w, θ). Then (λ, n) is indivisible if and only if n  1 is the smallest integer such that θ(t)n = 1 for all t ∈ T0 [w]. Indeed, the assumption (λ, n)  (w, θ) implies, in particular, that θ(t)n = 1 for all t ∈ T0 [w]. So, if m  1 is the smallest integer such that θ(t)m = 1 for all t ∈ T0 [w], then n = mn  for some integer n   1. By Lemma 2.4.8, there exists some μ ∈ X such that (μ, m)  (w, θ). Then we also have (n  μ, n)  (w, θ); see Remark 2.4.9. Again by Lemma 2.4.8, we have n  μ − λ = mn  ν for some ν ∈ X. Hence, (λ, n) is indivisible if and only if n  = 1. Corollary 2.4.11 If ρ ∈ Irr(GF ), then there exists an indivisible pair (λ, n) as in 2.4.5 such that Zλ,n   and ρ ∈ Eλ,n . Proof By Corollary 2.2.19 and Lemma 2.3.19, there exists some pair (w, θ) ∈ θ , ρ  0. Let n  1 be the smallest integer such that θ(t)n = 1 X(W, σ) such that Rw for all t ∈ T0 [w]; then n is prime to p. By Lemma 2.4.8, there exists a pair (λ, n) as in 2.4.5 such that (λ, n)  (w, θ) and so ρ ∈ Eλ,n . By Remark 2.4.10, the pair (λ, n) is indivisible.  Lemma 2.4.12 Let (λ, n) be a pair as in 2.4.5, with Zλ,n  . Then Zλ,n is a coset with respect to the subgroup ˆ λ,n := {w ∈ W | w.λ − λ ∈ nX } ⊆ W. W Proof Let w, w  ∈ Zλ,n . Using additive notation in X, we have nλw = λ ◦ F − w.λ and nλw = λ ◦ F − w  .λ. Setting x := w −1 w , it follows that

x.λ − λ = w −1 . w  .λ − w.λ = n(w −1 .λw − w −1 .λw ) ∈ nX. ˆ λ,n and so Zλ,n is contained in the coset w W ˆ . Now consider any Thus, x ∈ W

−1 λ,n −1 −1 −1 ˆ λ,n . Then λ(F(t) y w t w y ) = λ(F(t)) (wy).λ (t ). Writing y.λ = λ + nν y∈W where ν ∈ X, we also have

(wy).λ = w.(y.λ) = w.(λ + nν) = w.λ + nw.ν = λ ◦ F − nλw + nw.ν and so λ ◦ F − (wy).λ = n(λw − w, ν) ∈ nX. Thus, wy ∈ Zλ,n .



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Lusztig’s Classification of Irreducible Characters

ˆ λ,n is not 2.4.13 Let (λ, n) be a pair as in 2.4.5, with Zλ,n  . In general, W a reflection subgroup of W. But one can always find a reflection subgroup inside ˆ λ,n , as follows. As in [Lu84a, 1.8], we define a subgroup by W ˆ λ,n, Wλ,n := {w ∈ W | w.λ − λ ∈ nZR} ⊆ W where R ⊆ X is the set of roots of G with respect to T0 . We also set Rλ,n := {α ∈ R | λ, α∨ ∈ Z is divisible by n}. Note that, for α ∈ R, we have wα .λ − λ = −λ, α∨ α. Hence, one easily sees that wα ∈ Wλ,n if and only if α ∈ Rλ,n . Thus, {wα | α ∈ Rλ,n } is precisely the set of all reflections of W that are contained in Wλ,n . Now, it is known that Wλ,n is generated by {wα | α ∈ Rλ,n }; see [Bou68, Chap. VI, Exc. 1 of §2] and [Hum95, 2.10]. So Wλ,n is a Weyl group with root system Rλ,n ; furthermore, R+ ∩ Rλ,n is a positive system in Rλ,n and so there is a unique base Πλ,n for Rλ,n such that Πλ,n ⊆ R+ ∩ Rλ,n . Thus,

Wλ,n, Sλ,n is a Coxeter system, where Sλ,n := {wα | α ∈ Πλ,n }. (a) We also note that, by [Lu84a, 1.9], every coset wWλ,n (w ∈ W) contains a unique element w1 of minimal length (for the usual length function of W) and we have l(w1 y) > l(w1 )

for all y ∈ Wλ,n , y  1.

(b)

When Z(G) is connected, we obtain a more precise statement, as follows. Lemma 2.4.14 (Cf. [Lu84a, 2.15, 2.19]) In the above setting, assume that Z(G) ˆ λ,n and the following hold. is connected. Then Wλ,n = W (a) There is a unique element w1 ∈ Zλ,n of minimal length (for the usual length function of W). We have Zλ,n = w1 Wλ,n . (b) There is a well-defined group automorphism γ : Wλ,n → Wλ,n such that γ(y) = σ(w1 yw1−1 ) for all y ∈ Wλ,n . We have γ(Sλ,n ) = Sλ,n . ˆ λ,n = {w ∈ W | w.λ − λ ∈ nX } follows rather Proof The equality Wλ,n = W directly from the fact that Z(G) is connected, as shown in [DiMi20, 11.2.1]. Combining this with Lemma 2.4.12, we see that Zλ,n is a coset of Wλ,n . As already mentioned above, such a coset contains a unique element w1 of minimal length; by [Lu84a, 1.9], this element is characterised by the property that w1 .α ∈ R+ for any α ∈ R+ ∩ Rλ,n . Thus, (a) holds. For (b), let y ∈ Wλ,n and t ∈ T0 . Setting x := σ(w1 yw1−1 ) ∈ W, we must show that x.λ − λ ∈ nX. Let t ∈ T0 and consider t1 := w 1 y −1 w 1−1 t w 1 y w 1−1 ∈ T0 . Since

2.4 Towards Lusztig’s Main Theorem 4.23

141

x ≡ F(w 1 )F( y )F(w 1 )−1 (modulo T0 ), we obtain



(x.λ) F(t) = λ x −1 F(t) x = (λ ◦ F)(t1 ) = (w1 .λ)(t1 ) · λw1 (t1 )n

= λ( y −1 w 1−1 t w 1 y ) · λw1 (t1 )n = w1 y.λ (t) · λw1 (t1 )n = (λ ◦ F)(t) · λw1 y (t −1 )n · λw1 (t1 )n

n = λ(F(t)) · λw1 y (t −1 ) · λw1 (t1 ) , where we used that w1 ∈ Zλ,n (third equality) and w1 y ∈ Zλ,n (sixth equality). Now, the map t → λw1 y (t −1 ) · λw1 (t1 ) is a character of T0 (where t1 is defined in terms of t as above). Thus, we see that (x.λ − λ) ◦ F ∈ nX. We must show that this implies that x.λ − λ ∈ nX. Now, by Proposition 1.4.18, there exist integers d, m  1 such that the map induced by F d on X is given by scalar multiplication with pm . Since (x.λ − λ) ◦ F ∈ nX, we also have pm (x.λ − λ) = (x.λ − λ) ◦ F d ∈ nX. As n is prime to p, this easily implies that x.λ − λ ∈ nX. It remains to show that γ(Sλ,n ) = Sλ,n . Recall from 1.6.1 that there is a permutation α → α† of R such that α† ◦ F = qα α; we have α† ∈ R+ for all α ∈ R+ . We claim that this induces a permutation R+ ∩ Rλ,n → R+ ∩ Rλ,n,

α → (w1 .α)† .

Indeed, let α ∈ R+ ∩ Rλ,n . By Remark 1.2.10, we have σ(wα ) = wα† and so w(w1 .α)† = σ(ww1 .α ) = σ(w1 wα w1−1 ) = γ(wα ) ∈ Wλ,n . Hence, (w1 .α)† ∈ Rλ,n . On the other hand, we also have w1 .α ∈ R+ (as noted in the proof of (a)) and so (w1 .α)† ∈ R+ , as required. But then the above permutation also preserves the unique base Πλ,n contained in R+ ∩ Rλ,n .  We obtain the following extension of Theorem 2.4.1 to non-unipotent characters ) of GF . Recall from Corollary 2.4.11 that Irr(GF ) = (λ,n) Eλ,n where the union runs over all pairs (λ, n) as in 2.4.5, with Zλ,n  . Theorem 2.4.15 (Lusztig [Lu84a, 4.23]) Assume that Z(G) is connected. Let (λ, n) be a pair as in 2.4.5, with Zλ,n  . Let Wλ,n , w1 , γ be as in Lemma 2.4.14, such that Zλ,n = w1 Wλ,n . Then there is a bijection 1−1 ¯ λ,n, γ), Eλ,n ←→ X(W θw

ρ ↔ x¯ρ,

such that Rw1 y1 , ρ = (−1)l(w1 ) m(y, x¯ρ ) for all y ∈ Wλ,n and ρ ∈ Eλ,n . y

¯ λ,n, γ) and the integers Remark 2.4.16 (a) In the above statement, the set X(W m(y, x) ¯ are determined as in Remark 2.4.3, by an entirely combinatorial procedure. We shall see in the next section that there exists a connected reductive algebraic ˆ over k and a Steinberg map Fˆ : H ˆ →H ˆ such that Wλ,n can be identified group H

142

Lusztig’s Classification of Irreducible Characters

ˆ (relative to a maximally split torus of H) ˆ and γ is the with the Weyl group of H automorphism induced by the action of Fˆ (as in 1.6.1). Thus, in combination with Theorem 2.4.1, we obtain bijections 1−1 1−1 ˆ Fˆ ) ¯ λ,n, γ) ←→ Uch(H Eλ,n ←→ X(W

which are almost what is meant by the Jordan decomposition of characters. (b) The above result is established in [Lu84a] assuming that F : G → G is a Frobenius map. By the arguments in [Lu84a, 8.8], the proof can be reduced to the case where the derived subgroup Gder is simple of simply connected type. As indicated in [Lu84a, 14.2], it can be shown that the above theorem also holds for the Suzuki and Ree groups and, hence, for any Steinberg map F : G → G. (For an extension of the reduction arguments to the more general situation involving Steinberg maps, see also [Ta19].) There is a certain amount of case-by-case arguments involved in the proof of Theorem 2.4.15. More conceptual arguments are nowadays available via Lusztig’s work on ‘categorical centres’; see [Lu14b], [Lu16], [LuYu19]. Remark 2.4.17 In the setting of Theorem 2.4.15, let y, y  ∈ Wλ,n . By [Lu84a, Prop. 3.9] (or, rather, its proof), we have θw

θw

y

Rw1 y1 , Rw1 y1  = |{x ∈ Wλ,n | y  x = γ −1 (x)y}| y

= |{x  ∈ Wλ,n | y  γ(x ) = x  y}|. (Compare with the formula in Example 2.3.22(a).) Let us now consider the γcharacter table of Wλ,n (as in Section 2.1), where we fix a γ-extension φ˜ for each φ ∈ Irr(Wλ,n )γ as in Proposition 2.1.14. Following [Lu77b, 3.17], [Lu84a, 3.7], we define the corresponding uniform almost character 6 by

1 θw1 y F ˜ Rφ˜ := φ(y)R w1 y ∈ CF(G ). |Wλ,n | y ∈W λ, n

Note that this is, in general, an R-linear combination of irreducible characters of GF . ˜ then this will result in replacing Rφ˜ If we replace the chosen σ-extension φ˜ by −φ, by −Rφ˜ . By [Lu84a, Prop. 3.9], we have  1 if φ = φ , Rφ˜ , Rφ˜  = 0 if φ  φ  (see also [Lu77b, 3.19]). Consequently, we also have

θw y ˜ Rφ˜ Rw1 y1 = for all y ∈ Wλ,n . φ(y) φ ∈Irr(Wλ, n )γ

6

This is a special case of the more general definitions in [Lu84a, 4.24], [Lu19].

2.4 Towards Lusztig’s Main Theorem 4.23

143

Thus, a knowledge of the integers m(y, x) ¯ is equivalent to a knowledge of the scalar products Rφ˜ , ρ for φ ∈ Irr(Wλ,n )γ and ρ ∈ Eλ,n . Furthermore, by [Lu84a, 4.26.1], we have

ρ(1) = Rφ˜ , ρ Rφ˜ (1) for ρ ∈ Eλ,n . φ ∈Irr(Wλ, n )γ

In fact, it turns out that the matrix of scalar products Rφ˜ , ρ has a much simpler shape than the matrix of integers m(y, x); ¯ this will be further discussed in Section 4.2. Example 2.4.18 Assume that Wλ,n is a direct product of Weyl groups of type Ami (for various mi  0). Then one can extract the following information from [Lu84a, 4.4, 4.19] and the general reduction arguments in [Lu84a, 4.21]. First, we can identify ¯ λ,n, γ) and Irr(Wλ,n )γ . If x¯ ∈ X(W ¯ λ,n, γ) corresponds to φ ∈ Irr(Wλ,n )γ under X(W ˜ this identification, then m(y, x) ¯ = ±φ(y) for all y ∈ Wλ,n , where the sign only depends on the choice of the (real-valued) γ-extension φ˜ of φ. Furthermore, we have Eλ,n = {ρ ∈ Irr(GF ) | ρ = ±Rφ˜ for some φ ∈ Irr(Wλ,n )γ }. Indeed, by Theorem 2.4.15, the subspace of CF(GF ) spanned by Eλ,n has dimension ¯ λ,n, γ)| = |Irr(Wλ,n )γ |. So the orthogonality relations in Remark 2.4.17 imply | X(W

that this subspace is also spanned by {Rφ˜ | φ ∈ Irr(Wλ,n )γ }. Consequently, every θw

ρ ∈ Eλ,n can be written as a linear combination of the virtual characters Rw1 y1 for ˜ for y ∈ Wλ,n (where x¯ and φ correspond to each y ∈ Wλ,n . Since m(y, x) ¯ = ±φ(y) as above), we conclude that ρ = ±Rφ˜ if x¯ = x¯ρ , as required. In order to fix the sign, one has to choose the γ-extension φ˜ so that Rφ˜ (1) > 0 (and there is a unique such choice). We can now state the following result that shows why the groups GLn (q) and GUn (q) are so much easier to deal with than groups of other types. (The situation is already more complicated for SLn (q); see Example 2.2.30 and also [Leh73], [Leh78], [Bo06].) y

Corollary 2.4.19 (Cf. [LuSr77]) Assume that G = GLn (k) or, more generally, that Z(G) is connected and W is a direct product of Weyl groups of type Ami (for various mi ). Then every ρ ∈ Irr(GF ) is a uniform almost character, that is, ρ = ±Rφ˜ where φ ∈ Irr(Wλ,n ) for some pair (λ, n) as in 2.4.5 such that Zλ,n  . Proof Since W is a direct product of Weyl groups of type Ami (for various mi ), an analogous statement holds for the reflection subgroups Wλ,n . So the assertion is an immediate consequence of Corollary 2.4.11 and the discussion in Example 2.4.18. Note that [LuSr77, 2.2] provides a direct argument showing that Uch(GF ) = {±Rφ˜ |  φ ∈ Irr(W)σ }. See also [Lu77a, 7.14], [DiMi20, §11.7].

144

Lusztig’s Classification of Irreducible Characters

Example 2.4.20 Assume that G = GLn (k) and F : G → G is the standard Frobenius map (see Example 1.4.21) such that GF = GLn (q). We have W  Sn (the symmetric group of degree n) and σ : W → W is the identity. There is a natural labelling of the irreducible characters of W by the partitions ν n; see Example 4.1.2. We shall write Irr(W) = {φν | ν

n}. (1 n )

is the sign character. For each For example, φ(n) is the trivial character and φ ν n, we let φ˜ν := φν be the trivial extension as in Example 2.1.8(a) and form the corresponding unipotent uniform almost character 1 ν 1 Rν := φ (w)Rw (cf. Remark 2.4.17). |W| w ∈W Then Uch(GF ) = {Rν | ν n}; see Example 2.4.18 (and [DiMi20, 11.7.2] where it is shown how the signs in Example 2.4.18 are fixed). On the other hand, we can apply the discussion in Example 2.3.9. To each φν ∈ Irr(W) corresponds a character ρν ∈ Uch(GF ), such that ρν occurs in the permutation character of GF on the cosets of B0F . Then ρν = Rν for all ν n. (This is a very special case of [Lu84a, 12.6, 1 to GF are indeed the Green 12.14.2].) Hence, the restrictions of the functions Rw uni functions first investigated in [Gre55]. (The discussion of this case will be continued in Example 2.8.7.) Our next aim is to establish, following [Lu84a, Chap. 6], an important fact related to Theorem 2.4.15: the various sets Eλ,n are precisely the geometric series of characters. This will work without any assumptions on the centre Z(G). We begin θ. with a version of Theorem 2.3.2 for the characters Rw 

2.4.21 Let w ∈ W and recall from 1.6.4 that T0 [w] = T0F where F  : T0 → T0

is defined by F (t) = wF(t) w −1 for t ∈ T0 . To indicate the dependence on w, we

thus, will denote F  by wF (short-hand for: F followed by conjugation with w); T0 [w] = T0wF . Now, for any integer d  1 and t ∈ T0 , we have

w)

. . . F d−1 (w)F

d (t)F d−1 (w)

−1 . . . F(w)

−1 w −1 . (wF)d (t) = F d (t) = wF( Hence, setting y := wσ(w) . . . σ d−1 (w) ∈ W, we have (wF)d (t) = y F d (t) y −1 for t ∈ T0 . Let d0  1 be the order of σ. If d is a multiple of d0 , then

d/d0 y = wσ(w) . . . σ d0 −1 (w) . Hence, if d is a sufficiently large multiple of d0 , then y = 1 and so (wF)d (t) = F d (t) for all t ∈ T0 . Since W is finite, we can even find d  1 such that d0 | d

and

(wF)d (t) = F d (t) for all t ∈ T0 and all w ∈ W.

2.4 Towards Lusztig’s Main Theorem 4.23

145 (wF) d

d

Any such d will be called admissible. If d is admissible, then T0 = T0F does not depend on w any more; let Nd(w) := N(wF) d /wF : T0 → T0 be the corresponding norm map. By Lemma 2.3.1, we have Nd(w) (T0F ) = T0wF = T0 [w] d

for all w ∈ W.

As in Remark 2.3.3, one sees that Nd(w) is transitive. Definition 2.4.22 (Cf. [BoRo93, §4.4]) Recall from 2.3.20 the definition of the set X(W, σ). We say that two pairs (w1, θ 1 ) and (w2, θ 2 ) in X(W, σ) are geometrically conjugate if, in the above setting, there exists an admissible integer d  1 such that θ 2 ◦ Nd(w2 ) = y (θ 1 ◦ Nd(w1 ) ) ∈ Irr(T0F ) d

Here, we tacitly assume that y ∈ NG (T0 ) Example 1.4.11(b)).

Fd

for some y ∈ W.

, which we may since σ d = idW (see

Lemma 2.4.23 The correspondence X(G, F) ←→ X(W, σ) in 2.3.20 induces a bijection between geometric conjugacy classes of pairs (T, θ) (see Theorem 2.3.2) and geometric conjugacy classes of pairs (w, θ) as defined above. Proof For i = 1, 2 let wi ∈ W and θ i ∈ Irr(T0 [wi ]). Let Ti := gi T0 gi−1 where gi ∈ G is such that w i = gi−1 F(gi ). Assume first that (T1, g1 θ 1 ) and (T2, g2 θ 2 ) are d geometrically conjugate. So there exist an integer d  1 and an element x ∈ GF such that T2 = xT1 x −1 and x θˆ1 = θˆ2 , where

d for i = 1, 2. θˆi := gi θ i ◦ NF d /F ∈ Irr(TiF ) Replacing d by a multiple if necessary we may assume, by Remark 2.3.3 and the discussion in 2.4.21, that d is admissible and also that F d (gi ) = gi for i = 1, 2. d Now define θ i ∈ Irr(T0F ) by θˆi = gi θ i for i = 1, 2. We note that y := g2−1 xg1 ∈ d NG (T0 )F ; furthermore, θ 2 = y θ 1 . Hence, it remains to show that θ i = θ i ◦ Nd(wi ) for i = 1, 2. For this purpose, let t ∈ T. Then gi−1 tgi ∈ T0 and we compute: (wi F)(gi−1 tgi ) = w i F(gi )−1 F(t)F(gi )w i−1 = g −1 F(t)gi ; hence, (wi F)m (gi−1 tgi ) = gi−1 F m (t)gi for all m  1. We obtain that Nd(wi ) (gi−1 tgi ) = (gi−1 tgi )(wi F)(gi−1 tgi ) · · · (wi F)d−1 (gi−1 tgi ) = gi−1 tF(t) · · · F d−1 (t)gi = gi−1 NF d /F (t)gi . Now let s ∈ T0F . Then t := gi sgi−1 ∈ TF and so



θ i(s) = θˆi (t) = θ i gi−1 NF d /F (t)gi = θ i Nd(wi ) (gi−1 tgi ) = θ i (Nd(wi ) (s)). d

d

Thus, θ i = θ i ◦ Nd(wi ) for i = 1, 2. So, (w1, θ 1 ) and (w2, θ 2 ) are geometrically

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conjugate, as required. One can also run this argument backwards, which proves the reverse implication.  Corollary 2.4.24 (Exclusion Theorem; 2nd version) Assume that two Deligne– θ1 θ2 and Rw (as in 2.3.18) have an irreducible constituent in Lusztig characters Rw 1 2 common. Then the pairs (w1, θ 1 ) and (w2, θ 2 ) are geometrically conjugate. Proof

Immediate by Theorem 2.3.2, Lemma 2.3.19 and Lemma 2.4.23.



Example 2.4.25 Let (λ, n) be a pair as in 2.4.5, with Zλ,n  . For each w ∈ Zλ,n , we have a corresponding character θ w ∈ Irr(T0 [w]). We claim that all pairs {(w, θ w ) | w ∈ Zλ,n } are geometrically conjugate. Indeed, let w ∈ Zλ,n . Then θ w = ψ ◦ λ¯w and θ w (t)n = 1 for all t ∈ T0 [w]. Recall from 2.4.5 that λw ∈ X is defined by the condition:



= (w.λ) F (t)t −1 for all t ∈ T0, λw (t n ) = λ(F(t)w −1 t −1 w)

w −1 for t ∈ T0 . Now consider the norm map where we set again F (t) := wF(t) d d (w) F Nd : T0 → T0 [w] and the linear character θ w ◦ Nd(w) ∈ Irr(T0F ), where d  1 is an admissible integer as in 2.4.21. Let t ∈ T0 and set t  := Nd(w) (t). Then



F (t ) = F  Nd(w) (t) =F  tF (t) . . . F d−1 (t) =t −1 Nd(w) (t)F d (t) = t −1 t  F d (t) where the last equality holds since d is admissible. Hence, we obtain:





λw (t n ) = (w.λ) F (t )t −1 = (w.λ) t −1 t  F d (t)t −1 = (w.λ) F d (t)t −1 . On the other hand, the left-hand side equals

n

= λw Nd(w) (t n ) = λw ◦ Nd(w) (t n ). λw (t n ) = λw Nd(w) (t)



Thus, we obtain that λw ◦ Nd(w) (t n ) = (w.λ) F d (t)t −1 for all t ∈ T0 . This means (d) (where the super-script (d) indicates that F d is used instead of F) that 1 ∈ Zw.λ,n

(w) and that (♠) holds with (w.λ)(d) 1 := λw ◦ Nd ∈ X. Hence

(d) Fd θ w ◦ Nd(w) = (ψ ◦ λ¯w ) ◦ Nd(w) = ψ ◦ (w.λ)(d) 1 = θ 1 ∈ Irr(T0 )

and so (w.λ, n)  (1, θ 1(d) ) = (1, θ ◦ Nd(w) ) (where  is defined with respect to F d ). Now let also w  ∈ Zλ,n ; then θ w = ψ ◦ λ¯w . Setting x := w  w −1 ∈ W, we have



n = (w.λ) x −1 t −1 xF

= (λw ◦ Nd(w) ) ( x −1 t x)

d ( x −1 t x) (λw ◦ Nd(w) )( x −1 t n x)

= x and xw = w , the right-hand side equals for all t ∈ T0 . Since F d ( x)





(w.λ) x −1 t −1 F d (t) x = (w  .λ) t −1 F d (t) = λw ◦ Nd(w ) (t n ).

2.4 Towards Lusztig’s Main Theorem 4.23 147



 Thus, λw ◦ Nd = x λw ◦ Nd(w) and so θ w ◦ Nd(w ) = x θ w ◦ Nd(w) . Hence, the pairs (w, θ w ) and (w , θ w ) are geometrically conjugate, as claimed. (w )

Lemma 2.4.26 (Cf. [Lu84a, 6.2(i)]) Let (λ, n) be a pair as in 2.4.5. Let w ∈ Zλ,n and assume that (λ, n)  (w, θ) where θ ∈ Irr(T0 [w]). (a) Let μ := λ + nν for some ν ∈ X. Then w ∈ Zμ,n and (μ, n)  (w, θ). (b) For any y ∈ W, we have y.w ∈ Zσ(y).λ,n and (σ(y).λ, n)  y.(w, θ) (where the action of y on X(W, σ) is defined in 2.3.20(b)). Proof By assumption, we have θ = ψ ◦ λ¯w and θ(t)n = 1 for all t ∈ T0 [w]. Recall from 2.4.5 that λw ∈ X is defined by the condition:

λw (t n ) = λ(F(t)w −1 t −1 w)

for all t ∈ T0 .

for t ∈ T0 . Since T0 is (a) Define νw : T0 → k × by νw (t) := ν(F(t)w −1 t −1 w) abelian and w ∈ NG (T0 ), we have νw ∈ X. Now set μw := λw + νw ∈ X. Then we obtain

n μw (t)n = λw (t)n νw (t)n = λ(F(t)w −1 t −1 w)ν(F(t) w −1 t −1 w)

= μ(F(t)w −1 t −1 w)

= λ + nν (F(t)w −1 t −1 w) for all t ∈ T0 and so w ∈ Zμ,n . Since the restriction of νw to T0 [w] is trivial, both λw and μw have the same restriction and so (μ, n)  (w, θ), as desired. (b) A straightforward computation shows that, for any t ∈ T0 , we have



(y.λw )(t n ) = λw ( y −1 t y )n = · · · = (σ(y).λ) F(t)w −1 t −1 w  where w  := y.w = ywσ(y)−1 . Thus, w  ∈ Zσ(y).λ,n and (σ(y).λ)w = y.λw . Now let θ  ∈ Irr(T0 [w ]) be the linear character such that (σ(y).λ, n)  (w , θ ). For t ∈ T0 , we have y t y −1 ∈ T0 [w ] and



θ ( y t y −1 ) = ψ (y.λw )( y t y −1 ) = ψ λw (t) = θ(t), which shows that θ  = y θ, as required.



Corollary 2.4.27 Let (λ, n) be a pair as in 2.4.5, with Zλ,n  . Let x ∈ W and ν ∈ X. Then the following hold. (a) We have Zx.λ+nν,n = Zx.λ,n   and Eλ,n = Ex.λ+nν,n . θ  θw = Rww , where y ∈ W is (b) If w ∈ Zλ,n , then w  := ywσ(y)−1 ∈ Zx.λ,n and Rw such that x = σ(y). (These statements appear in [Lu84a, 6.5(ii)] but the proof there is more complicated, because Lusztig’s original definition of Eλ,n is different; see the comments in Definition 2.4.6.)

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Proof Firstly, Lemma 2.4.26(a) immediately shows that Zλ+nν,n = Zλ,n and Eλ+nν,n = Eλ,n for all ν ∈ X. So we may assume now that ν = 0. Let x ∈ W and set y := σ −1 (x) ∈ W. Then Lemma 2.4.26(b) shows that, for any w ∈ Zλ,n , we have w  := y.w = ywσ(y)−1 ∈ Zx.λ,n and (x.λ, n)  y.(w, θ w ) = (w , θ w ). θw = Furthermore, the scalar product formula in Example 2.3.22(a) implies that Rw θw  y.θw Ry.w = Rw . In particular, we conclude that {y.w | w ∈ Zλ,n } ⊆ Zx.λ,n and  Eλ,n ⊆ Ex.λ,n . By symmetry, we also get the reverse inclusions. We can now establish the following strengthening of Example 2.4.25. First, some notation. Let Λ(G, F) be the set of all indivisible pairs (λ, n) as in 2.4.5, with Zλ,n  . For i = 1, 2 let (λi, ni ) ∈ Λ(G, F). Then write (λ1, n1 ) ∼ (λ2, n2 ) if n1 = n2 and there exists some x ∈ W and ν ∈ X such that λ2 = x.λ1 + n1 ν. This defines an equivalence relation on Λ(G, F). Proposition 2.4.28 (Cf. [DeLu76, 5.7], [Lu84a, 6.5]) The relation (λ, n)  (w, θ) induces a bijection between equivalence classes of pairs (λ, n) ∈ Λ(G, F) and geometric conjugacy classes of pairs (w, θ) ∈ X(W, σ) (as in Definition 2.4.22), which in turn are in bijection with geometric conjugacy classes of pairs (T, θ) ∈ X(G, F) (see Lemma 2.4.23). Proof By Lemma 2.4.8 and Remark 2.4.10, every pair in X(W, σ) arises from a pair in Λ(G, F) via the relation . Now let (λi, ni ) ∈ Λ(G, F) for i = 1, 2. Let wi ∈ Zλi ,ni and set θ i := θ wi . We must show that (λ1, n1 ) ∼ (λ2, n2 ) if and only if (w1, θ 1 ) and (w2, θ 2 ) are geometrically conjugate. First note that, by Remark 2.4.10, ni  1 is the smallest integer such that θ i (t)ni = 1 for all t ∈ T0 [wi ]. Now consider d the norm map Nd(wi ) : T0F → T0 [wi ] where d  1 is an admissible integer. We have seen in Example 2.4.25 that (wi .λi, ni )  (1, θ i ◦ Nd(wi ) )

for i = 1, 2,

(∗)

where  is defined using F d . Assume now that (w1, θ 1 ) and (w2, θ 2 ) are geometrically conjugate, that is, there exists some x ∈ W such that x

(θ 1 ◦ Nd(w1 ) ) = θ 2 ◦ Nd(w2 ) ∈ Irr(T0F ) d

where

d

x ∈ NG (T0 )F .

First we note that this implies that n1 = n2 . We now apply Lemma 2.4.26 to (w1 .λ1, n1 )  (1, θ 1 ◦ Nd(w1 ) ). Since, here,  is defined with respect to F d and since σ d = idW , we obtain



(xw1 .λ1, n1 )  x. 1, θ 1 ◦ Nd(w1 ) = 1, x (θ 1 ◦ Nd(w1 ) ) = (1, θ 2 ◦ Nd(w2 ) ). Finally, since both (w2 .λ2, n2 )  (1, θ 2 ◦ Nd(w2 ) ) and (xw1 .λ1, n1 )  (1, θ 2 ◦ Nd(w2 ) ), we can apply Lemma 2.4.8 which yields that w2 .λ2 = xw1 .λ1 + n1 ν for some ν ∈ X. Hence, (λ1, n1 ) ∼ (λ2, n2 ). Conversely, assume that (λ1, n1 ) ∼ (λ2, n2 ).

2.4 Towards Lusztig’s Main Theorem 4.23

149

Hence, n1 = n2 and there exists some x ∈ W such that λ2 = x.λ1 + n1 ν for some ν ∈ X. We set y := w2 xw1−1 ∈ W. Then w2 .λ2 = w2 x.λ1 + n1 w2 .ν = yw1 .λ1 + n1 w2 .ν. By (∗), Lemmas 2.4.8 and 2.4.26 (applied with F d ), we have

(yw1 .λ1 + n1 w2 .ν, n1 )  1, y (θ 1 ◦ Nd(w1 ) ) . Since also (w2 .λ2, n2 )  (1, θ 2 ◦ Nd(w2 ) ), we conclude that θ 2 ◦ Nd(w2 ) = y (θ 1 ◦ Nd(w1 ) ),  that is, (w1, θ 1 ) and (w2, θ 2 ) are geometrically conjugate. Corollary 2.4.29 (Cf. [DeLu76, 10.1], [Lu84a, 6.5]) We have a partition * Irr(GF ) = Eλ,n (Eλ,n as in Definition 2.4.6) (λ,n)

where (λ, n) runs over a set of representatives for the classes of Λ(G, F) under the equivalence relation ∼. Two characters ρ1, ρ2 ∈ Irr(GF ) belong to the same piece in the above partition if and only if ρ1, ρ2 belong to the same geometric series of characters (as in Definition 2.3.4). ) Proof First, by Corollary 2.4.11, we have Irr(GF ) = (λ,n) Eλ,n where the union runs over all (λ, n) ∈ Λ(G, F). Now let (λ1, n1 ) and (λ2, n2 ) be pairs in Λ(G, F) and assume that there exists some ρ ∈ Eλ1,n1 ∩ Eλ2,n2 . For i = 1, 2 let wi ∈ Zλi ,ni be such θi that ρ, Rw  0 where we set θ i := θ wi . Then, by Corollary 2.4.24, the pairs (w1, θ 1 ) i and (w2, θ 2 ) are geometrically conjugate. Consequently, by Proposition 2.4.28, we have (λ1, n1 ) ∼ (λ2, n2 ). So Corollary 2.4.27 shows that Eλ1,n1 = Eλ2,n2 . Thus, we have a disjoint union as stated above. By Proposition 2.4.28, it is now sufficient to show that, if ρ1, ρ2 ∈ Eλ,n , then ρ1, ρ2 belong to the same geometric series of characters. There exist wi ∈ Zλ,n θi such that Rw , ρi  0 where we set θ i := θ wi for i = 1, 2. By Example 2.4.25, i (w1, θ 1 ) and (w2, θ 2 ) are geometrically conjugate. Hence, ρ1, ρ2 belong to the same geometric series of characters by Definition 2.3.4 and Lemma 2.4.23.  Lemma 2.4.30 Let (λ, n) be a pair as in 2.4.5, with Zλ,n  . Let ρ ∈ Eλ,n and θ  , ρ  0. Then there exists some w ∈ Z (w , θ ) ∈ X(W, σ) be such that Rw  λ,n such  θ w θ that Rw = Rw . Proof Let m  1 be the smallest integer such that θ (t)m = 1 for all t ∈ T0 [w ]. By Lemma 2.4.8, there exists some μ ∈ X such that (μ, m)  (w , θ ); furthermore, (μ, m) is indivisible by Remark 2.4.10. Now ρ ∈ Eλ,n ∩ Eμ,m and so (λ, n) ∼ (μ, m) by Corollary 2.4.29. Write μ = x.λ + nν where x ∈ W and ν ∈ X. If we set θ  θw θ  where w ∈ Z y := σ −1 (x), then Corollary 2.4.27 shows that Rw = Rww = Rw  λ,n is defined by the condition that w  = y.w = ywσ(y)−1 . 

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Finally, we re-interpret the degree polynomials in Definition 2.3.25 in terms of the formalism of Theorem 2.4.15. First, some preparations. Remark 2.4.31 In the setting of Remark 2.4.17, we have Rφ˜ (1) = Dφ˜ (q), where the polynomial Dφ˜ ∈ R[q] is defined by Dφ˜ :=

|G|q 1 ˜ . (−1)l(w1 y) φ(y) |Wλ,n | y ∈W |Tw1 y | λ, n

(As in Definition 2.3.25, one sees that Dφ˜ indeed is a polynomial, and not just a rational function in q.) These polynomials first appeared in [St51a], [Lu77a, §2], [Lu77b, 3.16], [BeLu78, §1]; following [AlLu82, §3], Dφ˜ is called the (twisted) fake degree of φ. Note that Dφ˜ only depends on the complete root datum associated with G, F and the choice of the γ-extension φ˜ of φ. Now assume that n = 1 and λ = λ0 is the trivial character. Then Wλ0,1 = W and so w1 = 1; furthermore, the automorphism γ is just the automorphism σ : W → W induced by F, and Eλ0,1 = Uch(GF ) is the set of unipotent characters of GF ; see Example 2.4.7. In this case, the class functions Rφ˜ (for φ ∈ Irr(W)σ ) are the unipotent uniform almost characters; the above formula now reads: |G|q 1 ˜ Dφ˜ = (−1)l(w) φ(w) (φ ∈ Irr(W)σ ). |W| w ∈W |Tw | We will see alternative interpretations of Dφ˜ in 4.1.6, 4.1.26, and Proposition 4.2.5. See also Proposition 2.4.32 at the end of this section. Proposition 2.4.32 (Simplified degree polynomials) Assume that we are in the setting of Theorem 2.4.15, and let ρ ∈ Eλ,n . Then the degree polynomial Dρ ∈ R[q] (see Definition 2.3.25) can be re-written as follows.

|G|q 1 Dρ = (−1)l(y) m(y, x¯ρ ) |Wλ,n | y ∈W |Tw1 y | λ, n

= Rφ˜ , ρ Dφ˜ , φ ∈Irr(Wλ, n )γ

where the fake degree polynomials Dφ˜ are defined as in Remark 2.4.31. Proof By Corollary 2.4.24, the sum over all pairs (w, θ) ∈ X(W, σ) in the definition of Dρ can be restricted to those pairs that lie in a fixed geometric conjugacy class of X(W, σ). Furthermore, that geometric conjugacy class is a union of orbits under the action of W on X(W, σ); see 2.3.20. Let {(wa, θ a ) | a ∈ I} be a complete set of representatives for those orbits (where I is some finite index set). By the scalar θb θa product formula in Example 2.3.22(a), we have Rw , Rw = 0 for a, b ∈ I with a b θa θ a  b; furthermore, we have Rw = Rwa if (w, θ) ∈ X(W, σ) is in the same W-orbit as

2.4 Towards Lusztig’s Main Theorem 4.23

151

θa θa , Rw is precisely the size of the stabiliser of (wa, θ a ) under (wa, θ a ). Finally, Rw a a the action of W. Thus, the defining formula for Dρ can be re-written as follows.

Dρ =

θa

Rw , ρ |G|q . (−1)l(wa ) θ a θ a Rwa , Rwaa |Twa | a ∈I

By Lemma 2.4.30, we may assume that wa ∈ Zλ,n and θ a = θ wa for a ∈ I. θ  = Rθw as in Lemma 2.4.30, then Example 2.3.22(a) also shows (Note that, if Rw  w that the pairs (w , θ ) and (w, θ w ) are in the same W-orbit of X(W, σ).) So we can write wa = w1 ya , where ya ∈ Wλ,n for a ∈ I. The scalar product formula in Remark 2.4.17 shows that θw

θw

θa θa , Rw = Rw1 y1 aa , Rw1 y1 aa = |CWλ, n,γ (ya )| Rw a a y

y

(where the γ-centraliser is defined as in 2.1.6). Hence, we obtain θw y

Rw1 y1 aa , ρ |G|q l(w1 ya ) . (−1) Dρ = |CWλ, n,γ (ya )| |Tw1 ya | a ∈I

We now claim that {ya | a ∈ I} is a complete set of representatives of the γconjugacy classes of Wλ,n . This is seen as follows. Let a, b ∈ I, a  b. Then θb θa , Rw = 0 and so the formula in Remark 2.4.17 shows that ya, yb are not Rw a b γ-conjugate in Wλ,n . On the other hand, by Example 2.4.25, all pairs (w1 y, θ w1 y ) (for y ∈ Wλ,n ) are in the same geometric conjugacy class of X(W, σ). Hence, each such pair (w1 y, θ w1 y ) must be in the same W-orbit as (wa, θ a ) = (w1 ya, θ w1 ya ), for some a ∈ I. But then θw

θw

|{x  ∈ Wλ,n | ya γ(x ) = x  y}| = Rw1 y1 , Rw1 y1 aa  0 y

y

(see again Remark 2.4.17 and Example 2.3.22(a)), and we conclude that y, ya are γ-conjugate. Thus, the above claim is proved. Hence, the formula for Dρ can be re-written as

|G|q 1 θw y Dρ = . (−1)l(w1 y) Rw1 y1 , ρ |Wλ,n | y ∈W |Tw1 y | λ, n

θw

It remains to use the formula for Rw1 y1 , ρ in Theorem 2.4.15 and the identities  concerning Rφ˜ in Remark 2.4.17. y

Remark 2.4.33 Finally, we remark that the proofs of the main results of [Lu84a] involve a further ingredient that we did not discuss here at all: the theory of ‘cells’ of finite Coxeter groups, as introduced by Kazhdan and Lusztig [KaLu79], and further developed by Lusztig; see [Lu03b] and the survey [Cu88]. At the time of writing [Lu84a], some crucial properties of cells (see, e.g., [Lu81a]) were established using the theory of primitive ideals in enveloping algebras of semisimple Lie algebras.

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Lusztig’s Classification of Irreducible Characters

As pointed out in [Lu87b, p. 253] (see also [Lu18b]), the further development of the cell theory made it possible to avoid that dependence on the theory of primitive ideals.

2.5 Geometric Conjugacy and the Dual Group The basic assumptions of the previous section remain in force; in particular, we ∼ assume that an isomorphism ι : k × → (Q/Z) p as in 2.4.4 has been fixed. We will now reformulate the partition of Irr(GF ) into geometric series in terms of ‘dual groups’, already introduced in Section 1.5. Following [Lu84a, 8.4], this can be done in a rather straightforward way by using the formalism of pairs (λ, n) and the sets Eλ,n from the previous section. Since this set-up is somewhat different from that in [Ca85], [DiMi20], we will give detailed proofs whenever appropriate. 2.5.1 Let G∗ be a connected reductive algebraic group over k. Let F ∗ : G∗ → G∗ be a Steinberg map such that the pairs (G, F) and (G∗, F ∗ ) are in duality; see Definition 1.5.17 and Example 1.6.19. Such a duality is defined with respect to maximally split tori T0 ⊆ G and T∗0 ⊆ G∗ . Thus, we are given an isomorphism ∼ δ : X(T0 ) → Y (T∗0 ) satisfying the conditions in Definition 1.5.17. Now, by [Ca85, 3.1.2], the map k × × Y (T∗0 ) → T∗0 , (ξ, ν) → ν(ξ), induces an isomorphism of ∼ abelian groups k × ⊗Z Y (T∗0 ) → T∗0 . (We have already used this in Example 1.5.6.) ∼ Combining this with our chosen isomorphism ι : k × → (Q/Z) p and the map ∼ δ : X(T0 ) → Y(T∗0 ), we obtain an isomorphism of abelian groups

∼ (x + Z) ⊗ λ → δ(λ) ι−1 (x + Z) . (a) (Q/Z) p ⊗Z X(T0 ) −→ T∗0, Next, recall that W = NG (T0 )/T0 is the Weyl group of G and σ : W → W is the automorphism induced by F; see 1.6.1. Similarly, W∗ = NG∗ (T∗0 )/T∗0 is the Weyl group of G∗ and σ ∗ : W∗ → W∗ is the automorphism induced by F ∗ . By [Ca85, ∼ 4.2.3, 4.3.2], there is a group isomorphism W → W∗ , w → w ∗ , such that

∗ (b) σ(w) = (σ ∗ )−1 (w ∗ ) and δ(w.λ) = w ∗ .δ(λ) for all λ ∈ X(T0 ). Furthermore, l(w) = l ∗ (w ∗ ) for all w ∈ W, where l : W → Z0 and l ∗ : W∗ → Z0 are the length functions as in 1.6.1. (See also Remark 1.5.19.) Using the isomorphism ∼ W → W∗ , w → w ∗ , we obtain a natural bijection    ∗F∗  F G -classes of F-stable G -classes of F ∗ -stable . (c) ←→ maximal tori T ⊆ G maximal tori T∗ ⊆ G∗ ∗

Here, the GF -conjugacy class of T ⊆ G corresponds to the G∗ F -conjugacy class

2.5 Geometric Conjugacy and the Dual Group

153

of T∗ ⊆ G∗ if there exists some w ∈ W such that T is of type w and T∗ is of type (w ∗ )−1 ; see [Ca85, 4.3.4]. Lemma 2.5.2 In the above setting, assume that T ⊆ G and T∗ ⊆ G∗ correspond ∗ to each other via 2.5.1(c). Then we have |TF | = |T∗ F |; furthermore, the order polynomials of (T, F) and (T∗, F ∗ ) are equal. Proof By [DeLu76, 5.2] or [Ca85, 4.3.4], the two pairs (T, F) and (T∗, F ∗ ) are in duality, that is, there exists an isomorphism δ : X(T) → Y (T∗ ) satisfying the conditions in Definition 1.5.17. Using the formula in 1.6.21, it follows that (T, F) and (T∗, F ∗ ) have the same order polynomials. Alternatively, one can argue as follows. ∗ By [Ca85, 4.4.2], we have |TF | = |T∗ F |. By [DeLu76, 5.3], this equality remains valid when we replace F,F ∗ by F n , (F ∗ )n , for any n  1. But then not only the individual orders are the same, but also the corresponding order polynomials.  Remark 2.5.3

Let X = X(T0 ). For working with (Q/Z) p ⊗Z X, we note:

(a) Every element of (Q/Z) p ⊗Z X is a tensor of the form ( n1 + Z) ⊗ λ where λ ∈ X and n  1 is an integer prime to p. (b) Let λ, μ ∈ X and n, m  1 be integers prime to p. Then ( n1 +Z) ⊗ λ = ( m1 +Z) ⊗ μ if and only if there exist integers d, d   1, both prime to p, such that dn = d  m and d  μ = dλ + dnν for some ν ∈ X. (c) Let m  1 be an integer prime to p. Then every element of (Q/Z) p ⊗Z X of order m can be written as ( m1 + Z) ⊗ μ where μ ∈ X. This immediately follows from standard properties of tensor products and the fact that X is a free Z-module of finite rank. (We omit further details.) The following results contain a basic feature of a duality as above: it replaces characters of a torus by elements in a dual torus. We set

for any pair (λ, n) as in 2.4.5. tλ,n := δ(λ) ι−1 ( n1 + Z) ∈ T∗0 Thus, tλ,n is the image of ( n1 + Z) ⊗ λ under the isomorphism in 2.5.1(a). Lemma 2.5.4 Let λ ∈ X and n  1 be an integer prime to p, as in 2.4.5. (a) We have F ∗ (tλ,n ) = tλ◦F,n and tw.λ,n = w ∗ tλ,n (w ∗ )−1 for any w ∈ W. (b) The G∗ -conjugacy class of tλ,n is F ∗ -stable if and only if Zλ,n  . If Zλ,n  , then F ∗ (tλ,n ) = w ∗ tλ,n (w ∗ )−1 for all w ∈ Zλ,n . (c) (λ, n) is indivisible if and only if n is the order of the element tλ,n .

Proof Let ξ := ι−1 ( n1 + Z) ∈ k × ; then tλ,n = δ(λ) ξ . (a) By Definition 1.5.17(b), we have F ∗ ◦ δ(λ) = δ(λ ◦ F). This yields that





F ∗ (tλ,n ) = F ∗ δ(λ)(ξ) = F ∗ ◦ δ(λ) (ξ) = δ λ ◦ F (ξ) = tλ◦F,n .

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Now let w ∈ W. Then



tw.λ,n = δ(w.λ) ξ = w ∗ .δ(λ) (ξ) = w ∗ δ(λ) ξ (w ∗ )−1 = w ∗ tλ,n (w ∗ )−1 where the second equality holds by 2.5.1(b) and the third equality just expresses the natural action of W∗ on Y (T∗0 ); see 1.3.1. (b) First assume that F ∗ (tλ,n ) is G∗ -conjugate to tλ,n . It is well known (see, e,g., [Ca85, 3.7.1]) that then we can already find some w ∈ W such that F ∗ (tλ,n ) = w ∗ tλ,n (w ∗ )−1 . Now, by (a), the left-hand side equals tλ◦F,n and the right-hand side equals tw.λ,n . So we have ( n1 +Z) ⊗ (λ ◦ F) = ( n1 +Z) ⊗ (w.λ). Using Remark 2.5.3(b), it follows that λ ◦ F = w.λ + nμ for some μ ∈ X and so Zλ,n  . Conversely, assume that w ∈ Zλ,n , that is, λ ◦ F = w.λ + nλw . Using (a), we obtain





F ∗ (tλ,n ) = tλ◦F,n = δ(w.λ + nλw ) ξ = δ(w.λ) ξ · δ(λw ) ξ n . We certainly have δ(λw )(ξ n ) = 1 since ξ n = ι−1 (1 + Z) = 1 ∈ k × . Hence, using (a) again, the right-hand side of the above equation equals w ∗ tλ,n (w ∗ )−1 and so the G∗ -conjugacy class of tλ,n is F ∗ -stable. n = 1 and so (c) Let m  1 be the order of tλ,n . Since ξ n = 1, we have tλ,n m | n. Under the isomorphism in 2.5.1(a), tλ,n corresponds to an element of order m in (Q/Z) p ⊗Z X. By Remark 2.5.3(c), there exists some μ ∈ X such that tλ,n =

δ(μ) ι−1 ( m1 + Z) and so ( n1 + Z) ⊗ λ = ( m1 + Z) ⊗ μ. Using Remark 2.5.3(b), it follows that (λ, n) is indivisible if and only if n = m.  Proposition 2.5.5 (Cf. [DeLu76, 5.7, 5.22]) The map (λ, n) → tλ,n induces a bijection between equivalence classes of pairs (λ, n) ∈ Λ(G, F) (see Proposition 2.4.28) and F ∗ -stable conjugacy classes of semisimple elements in G∗ . Proof Let (λi, ni ) ∈ Λ(G, F) for i = 1, 2. By Lemma 2.5.4(b), the G∗ -conjugacy class of tλi ,ni is F ∗ -stable. We must show that (λ1, n1 ) ∼ (λ2, n2 ) if and only if tλ1,n1 and tλ2,n2 are conjugate in G∗ . Assume first that (λ1, n1 ) ∼ (λ2, n2 ), that is, n1 = n2 and λ2 = w.λ1 + n1 ν for some w ∈ W and ν ∈ X. Let ξ := ι−1 ( n11 + Z) ∈ k × . Then

tn1,λ1 = δ(λ1 ) ξ and







tλ2,n2 = tλ2,n1 = δ(λ2 ) ξ = δ(w.λ1 + n1 ν) ξ = δ(w.λ1 ) ξ · δ(ν) ξ n1 .



Now, δ(ν) ξ n1 = 1 since ξ n1 = ι−1 (1 + Z) = 1 ∈ k × . Furthermore, δ(w.λ1 ) ξ = w ∗ tλ1,n1 (w ∗ )−1 by Lemma 2.5.4(a). Hence, tλ1,n1 and tλ2,n2 are conjugate in G∗ . Conversely, assume that tλ1,n1 and tλ2,n2 are conjugate in G∗ . Then tλ1,n1 and tλ2,n2 have the same order and so n := n1 = n2 ; see Lemma 2.5.4(c). As in the proof of Lemma 2.5.4(b), there exists some w ∈ W such that tλ2,n2 = w ∗ tλ1,n1 (w ∗ )−1 . But tw.λ1,n1 = w ∗ tλ1,n1 (w ∗ )−1 and so ( n1 + Z) ⊗ (w.λ1 ) = ( n1 + Z) ⊗ λ2 . Using Remark 2.5.3(b), it follows that (λ1, n1 ) ∼ (λ2, n2 ), as required. Finally, we show that every F ∗ -stable conjugacy class of semisimple elements in G∗ corresponds to

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155

an equivalence class of pairs in Λ(G, F) by the above procedure. So let C be such a conjugacy class in G∗ . By [Ca85, 3.7.1], we can find an element s ∈ C ∩ T∗0 . Furthermore, since C is F ∗ -stable, there exists some w ∈ W such that F ∗ (s) = w ∗ s (w ∗ )−1 . Using the isomorphism in 2.5.1(a) and Remark 2.5.3, we can write s = tλ,n where (λ, n) is a pair as in 2.4.5 and n is the order of s. By Lemma 2.5.4, we have Zλ,n   and so (λ, n) ∈ Λ(G, F).  Combining the above result with Proposition 2.4.28, we obtain a bijection {geometric conjugacy classes of pairs (T, θ) ∈ X(G, F)} 1−1

{F ∗ -stable conjugacy classes of semisimple elements in G∗ }. + By Corollary 2.4.29, we have a partition Irr(GF ) = (λ,n) Eλ,n where the union runs over a set of representatives for the equivalence classes of Λ(G, F) (as above), and each piece Eλ,n is a geometric series of characters. ←→

Proposition 2.5.6 The number of geometric conjugacy classes of pairs in X(G, F) is |Z◦ (G)F |ql where l = dim T0 − dim Z◦ (G) is the semisimple rank of G. Proof By [St68, 14.8] (or [Ca85, 3.7.6]), it is known that the number of F-stable conjugacy classes of semisimple elements of G is given by |Z◦ (G)F |ql . By [Ca85, ∗ 4.4.5], we have |Z◦ (G∗ )F | = |Z◦ (G)F |. Furthermore, l also is the semisimple rank of G∗ . Hence, we conclude that |Z0 (G)F |ql is the number of F ∗ -stable conjugacy classes of semisimple elements of G∗ . By Proposition 2.5.5, the latter number is the number of geometric conjugacy classes of pairs in X(G, F).  The map (λ, n) → tλ,n gives rise to a number of further constructions. Lemma 2.5.7 (Cf. [DeLu76, 5.2], [Lu84a, 8.4]) For each w ∈ W, there is a unique isomorphism of abelian groups ∼

Irr(T0 [w]) −→ T∗0 [(w ∗ )−1 ],

θ → sθ ,

satisfying the following condition. If θ ∈ Irr(T0 [w]) and (λ, n) is any pair as in Lemma 2.4.8 such that (λ, n)  (w, θ), then sθ = tλ,n ∈ T∗0 . Proof Let X = X(T0 ). Let θ ∈ Irr(T0 [w]) and n  1 be the smallest integer such that θ(t)n = 1 for all t ∈ T0 [w]. By Lemma 2.4.8, there exists some λ ∈ X such that w ∈ Zλ,n and θ = ψ ◦ λ¯w (see 2.4.5). Then we set sθ := tλ,n . We must show that this is well defined. So let (μ, m) be any pair as in 2.4.5 such that (μ, m)  (w, θ). Then n | m. Writing m = dn where d  1, we also have (dλ, dn)  (w, θ); see Remark 2.4.9. Now Lemma 2.4.8 implies that μ = dλ + dnν for some ν ∈ X. Hence, by Remark 2.5.3(b), the pairs (μ, m) and (λ, n) define the same element of (Q/Z) p ⊗Z X and so sθ = tλ,n = tμ,m , as desired.

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Next, since (λ, n)  (w, θ), we have w ∈ Zλ,n and so we obtain F ∗ (tλ,n ) =

∗ )−1 ; see Lemma 2.5.4(b). Thus, sθ = tλ,n ∈ T∗0 [(w ∗ )−1 ], as desired. So we λ,n (w have a well-defined map θ → sθ . To see that this is a group homomorphism, we set m := |Irr(T0 [w])|; note that this is prime to p. For each θ ∈ Irr(T0 [w]), we have sθ = tμ,m where μ ∈ X is such that (μ, m)  (w, θ). Working with this fixed m, it is then easy to see that θ → sθ is a group homomorphism. Finally, we show that the map θ → sθ is bijective. First, let θ ∈ Irr(T0 [w]) and write again sθ = tλ,n where, as above, n  1 is the smallest integer such that θ(t)n = 1 for all t ∈ T0 [w]. So (λ, n) is indivisible by Remark 2.4.10; consequently, n is the order of sθ = tλ,n by Lemma 2.5.4(c). Hence, if θ  1, then n > 1 and so sθ  1. Thus, θ → sθ is injective. On the other hand, by Lemma 2.5.2, we have w ∗ t

|Irr(T0 [w])| = |T0 [w]| = |T∗0 [(w ∗ )−1 ]|. Hence, the map θ → sθ is also surjective.



Since we systematically work in the set-up of [Lu84a] (as in 2.4.5), the above construction of an isomorphism Irr(T0 [w])  T∗0 [(w ∗ )−1 ] is somewhat different from that in [Ca85, 4.4.1] or [DiMi20, 11.1.7, 11.1.14] (although, of course, the two constructions can be seen to be equivalent by following the exact sequences in [DeLu76, 5.2]; see also [DiMi20, 11.1.7]). We can now state the following version of the dictionary between geometric conjugacy classes and the dual group. Corollary 2.5.8 For i = 1, 2 let (wi, θ i ) ∈ X(W, σ) and consider the corresponding semisimple elements sθi ∈ T∗0 [(wi∗ )−1 ] ⊆ T∗0 , obtained via the isomorphisms in Lemma 2.5.7. Then (w1, θ 1 ) and (w2, θ 2 ) are geometrically conjugate if and only if sθ2 = y sθ1 y −1 for some y ∈ W∗ . Proof For i = 1, 2 let (λi, ni ) ∈ Λ(G, F) be such that (λi, ni )  (wi, θ i ). By Proposition 2.4.28, the pairs (λ1, n1 ) and (λ2, n2 ) are equivalent if and only if (w1, θ 1 ), (w2, θ 2 ) are geometrically conjugate. Now sθi = tλi ,ni for i = 1, 2. It remains to use Proposition 2.5.5 and to recall that two elements in T∗0 are conjugate in G∗ if and only if they are conjugate in NG∗ (T∗0 ).  Remark 2.5.9 For future reference, we state a compatibility property (see (∗) below) of the isomorphism in Lemma 2.5.7. Assume that (G, F ) and (G∗, F ∗ ) also is a pair of connected reductive groups in duality, with respect to maximally split tori T0 ⊆ G and T0∗ ⊆ G∗ . Let f : G → G  and f ∗ : G∗ → G∗ be central isotypies that correspond to each other by duality, exactly as in 1.7.11; in particular, we have f (T0 ) ⊆ T0 and f ∗ (T0∗ ) ⊆ T∗0 . Using f and f ∗ , we may identify W with the Weyl group of G (with respect to T0 ) and W∗ with the Weyl group of G∗ (with

2.5 Geometric Conjugacy and the Dual Group

157

respect to T0∗ ). Thus, for each w ∈ W, we also have an isomorphism ∼

Irr(T0 [w]) −→ T0∗ [(w ∗ )−1 ],

θ  → sθ  .

Using f ◦ F = F  ◦ f and f ∗ ◦ F ∗ = F ∗ ◦ f ∗ , one immediately checks that



and f ∗ T0∗ [(w ∗ )−1 ] ⊆ T∗0 [(w ∗ )−1 ]. f T0 [w] ⊆ T0 [w] Now let θ  ∈ Irr(T0 [w]). Then we claim that sθ = f ∗ (sθ  )

where

θ := θ  ◦ f |T0 [w] ∈ Irr(T0 [w]).

(∗)

To see this, let (λ , n) ∈ Λ(G, F ) be such that (λ , n)  (w, θ ). Then sθ  = δ (λ ) ξ where ξ := ι−1 ( n1 + Z) ∈ k × and δ  : X(T0 ) → Y (T0∗ ) is the isomorphism that defines the duality between (G, F ) and (G∗, F ∗ ). Now set λ := λ  ◦ f ∈ X(T0 ) and  ◦ f ∈ X(T ). A straightforward computation shows that nλ = λ◦ F −w.λ λw := λw 0 w and so w ∈ Zλ,n ; furthermore, we have (λ, n)  (w, θ) and





sθ = δ(λ) ξ = δ(λ  ◦ f ) (ξ) = f ∗ ◦ (δ (λ )) (ξ) = f ∗ δ (λ )(ξ) = f ∗ (sθ  ), where the third equality holds by the compatibility relation in 1.7.11. 2.5.10 Let us fix a pair (λ, n) ∈ Λ(G, F) and set s := tλ,n ∈ T∗0 . Since Zλ,n  , the G∗ -conjugacy class of s is F ∗ -stable (see Lemma 2.5.4). Let R∗ ⊆ X(T∗0 ) be the set of roots of G∗ with respect to T∗0 . Recall from Definition 1.5.17 that R∗ = {α∗ | α ∈ R}, where α∗ is determined by the condition that δ(α) = α∗ ∨ ∈ Y (T∗0 ). Then, by 2.2.13, we have CG◦ ∗ (s) = T∗0, Uα∗ | α∗ ∈ Rs∗ where Rs∗ := {α∗ ∈ R∗ | α∗ (s) = 1} is the root system of CG◦ ∗ (s) with respect to T∗0 . Let ξ = ι−1 ( n1 + Z) ∈ k × . Then

∗ ∨ α∗ (s) = α∗ (tλ,n ) = α∗ δ(λ)(ξ) = ξ α ,δ(λ) = ξ λ,α , where we used the relations in 1.1.11 and Definition 1.5.17(a). Thus, since ξ has order n, we have α∗ ∈ Rs∗ if and only if λ, α∨ ∈ Z is divisible by n. Hence, we obtain that Rs∗ = {α∗ | α ∈ Rλ,n }, So the Weyl group of

CG◦ ∗ (s)

with Rλ,n ⊆ R as in 2.4.13.

(with respect to T∗0 ) is given by the subgroup

∗ := {w ∗ | w ∈ Wλ,n } ⊆ W∗ . Wλ,n

Assume now that Z(G) is connected. By Lemma 1.5.22, this implies that the fundamental group of G∗ is trivial. Hence, Theorem 2.2.14 shows that CG∗ (s) = CG◦ ∗ (s) is connected. Let w1 be the unique element of minimal length in Zλ,n (as in

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Lemma 2.4.14) and define F  : G∗ → G∗ by F (g) = (w 1∗ )−1 F ∗ (g)w 1∗ for g ∈ G∗ . Then F  is a Steinberg map by Lemma 1.4.14. We have seen in Lemma 2.5.4(b) that F ∗ (tλ,n ) = w 1∗ tλ,n (w 1∗ )−1 . Hence, we obtain that F (s) = s and so CG∗ (s) is F -stable. ∗ is given by The map induced by F  on Wλ,n γ (w ∗ ) = (w1∗ )−1 σ ∗ (w ∗ )w1∗

for all w ∈ Wλ,n .

Recall from Lemma 2.4.14 that we have an automorphism γ : Wλ,n → Wλ,n such

∗ that γ(w) = σ(w1 ww1−1 ) for all w ∈ Wλ,n . By 2.5.1(b), we have σ ∗ (w ∗ ) = σ −1 (w) for all w ∈ W. This yields

∗

∗

∗ γ (w ∗ ) = (w1∗ )−1 σ −1 (w) w1∗ = w1−1 σ −1 (w)w1 = γ −1 (w) ∼

∗ , w → w ∗ , for all w ∈ Wλ,n . Thus, under the isomorphism Wλ,n → Wλ,n ∗ ) corresponds to γ −1 ∈ Aut(W ). (a) γ  ∈ Aut(Wλ,n λ,n ∗ ∗ is = {w ∗ | w ∈ Sλ,n } of Wλ,n It follows that the set of simple reflections Sλ,n  ∗ preserved by γ . Finally, this also shows that the torus T0 ⊆ CG∗ (s) is maximally split with respect to F . Note that, by [Ca85, 3.5.5],

B∗s := T∗0, Uα∗ | α ∈ R+ ∩ Rλ,n is a Borel subgroup of CG∗ (s), and this Borel subgroup is F -stable (see the proof of Lemma 2.4.14(b)). We conclude that ∗ and the automorphism γ  arise from the connected reduc(b) the Weyl group Wλ,n tive algebraic group CG∗ (s) and the Steinberg map F  by the same procedure by which W, σ arise from G, F as in 1.6.1.

Now let H := CG∗ (s). We can apply the construction of dual groups to the pair (H, F ). Thus, we obtain a connected reductive algebraic group H∗ and a Steinberg map F ∗ : H∗ → H∗ such that (H∗, F ∗ ) is in duality with (H, F ) with respect to suitable maximally split tori in H and in H∗ . Then we can identify the Weyl group of H∗ with Wλ,n ; by (a) and 2.5.1(b), the automorphism induced by F ∗ on Wλ,n will then be γ. Thus, as already announced in Remark 2.4.16(a), we can now state: ∗ 1−1 ¯ λ,n, γ), ψ ↔ x¯ψ , satisfying the (c) There is a bijection Uch(H∗ F ) ←→ X(W conditions in Theorem 2.4.1.

Now, it is even known that (H∗, F ∗ ) can be replaced by (H, F ) in (c). This will be further discussed in the next section (see Remark 2.6.5). Proposition 2.5.11 In the setting of Theorem 2.4.15, let ρ ∈ Eλ,n . As in 2.5.10(c),

2.5 Geometric Conjugacy and the Dual Group let ψ ∈ Uch(H∗ Then

F ∗

159

¯ λ,n, γ). ) be the unipotent character such that x¯ψ = x¯ρ ∈ X(W Dρ =

|G|q Dψ, |H∗ |q

where G and H∗ are the complete root data associated with G, F and with H∗, F ∗ , respectively; furthermore, Dρ ∈ R[q] and Dψ ∈ R[q] are the degree polynomials of ρ and ψ, respectively (see Definition 2.3.25). Proof

By Theorems 2.4.1 and 2.4.15, and Proposition 2.4.32, we have

|G|q 1 Dρ = , (−1)l(y) m(y, x¯ρ ) |Wλ,n | y ∈W |Tw1 y | λ, n

|H∗ |q 1 (−1)l(y) m(y, x¯ψ ) ∗ . Dψ = |Wλ,n | y ∈W |Ty | λ, n

Here, |Tw1 y | is the order polynomial of an F-stable maximal torus of G of type w1 y, while |Ty∗ | is the order polynomial of an F ∗ -stable maximal torus of H∗ of type y. ¯ λ,n, γ), it only remains to show that Since x¯ψ = x¯ρ ∈ X(W |Tw1 y | = |Ty∗ |

for all y ∈ Wλ,n .

This is seen as follows. By duality (see 2.5.1), an F-stable maximal torus T of G of type w1 y corresponds to an F ∗ -stable maximal torus T∗ of G∗ of type (w1∗ y ∗ )−1 ; furthermore, (T, F) and (T∗, F ∗ ) have the same order polynomial (see Lemma 2.5.2). Similarly, an F ∗ -stable maximal torus of H∗ of type y corresponds to an F stable maximal torus of H of type (y ∗ )−1 , and these two tori have the same order polynomial. Now we use the order formula in 1.6.21. Let X ∗ = X(T∗0 ) and ϕ∗ : X ∗ → X ∗ be the map induced by F ∗ . Write ϕ∗ = qϕ0∗ where ϕ0∗ : XR∗ → XR∗ has finite order. Then, by 1.6.21 and the above discussion, we have

|Tw1 y | = det q idXR∗ − ϕ0∗ ◦ (w1∗ y ∗ ) . Similarly, let ϕ  : X ∗ → X ∗ be the map induced by F . Then |Ty∗ | = det q idXR∗ − ϕ0 ◦ y ∗ ), where ϕ  = qϕ0 and ϕ0 has finite order. By a computation as in 1.6.4, one checks that ϕ  = ϕ∗ ◦ w1∗ and, hence, also ϕ0 = ϕ0∗ ◦ w1∗ . Comparing the above two formulae, we conclude that |Tw1 y | = |Ty∗ |, as desired.  We introduce the following notation, dual to that in 2.3.20. (We first formulate this in terms of G, in order to avoid cumbersome notation.)

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2.5.12 Define Y(G, F) as the set of all pairs (T, s), where T is an F-stable maximal torus of G and s ∈ TF . Then GF acts on Y(G, F) via

GF × Y(G, F) → Y(G, F), g, (T, s) → (gTg −1, gsg −1 ). (a) We denote by Y(G, F) the set of GF -orbits for this action. On the other hand, the group W does not only act on itself by σ-conjugation but also on the set Y(W, σ) := {(w, t) | w ∈ W, t ∈ T0 [w]}, via

x −1 ) (b) W × Y(W, σ) → Y(W, σ), x, (w, t) → (xwσ(x)−1, xt (where x denotes a representative of x in NG (T0 )). We denote by Y(W, σ) the set of W-orbits for this action. Then it is straightforward to verify that there is a natural bijection Y(W, σ) ←→ Y(G, F)

(c)

defined as follows: The W-orbit of a pair (w, t) ∈ Y(W, σ) corresponds to the GF orbit of a pair (T, s) ∈ Y(G, F) if there exists some g ∈ G such that T = gT0 g −1 , g −1 F(g) ∈ NG (T0 ) is a representative of w, and s = gtg −1 . Remark 2.5.13 The isomorphism in Lemma 2.5.7 has the following compatibility property. Let (w1, θ 1 ) and (w2, θ 2 ) be two pairs in X(W, σ) that are in the same W-orbit, that is, there exists some x ∈ W such that w2 = xw1 σ(x)−1 and θ 2 = xθ 1 . Then 2.5.1(b) shows that (w2∗ )−1 = y ∗ (w1∗ )−1 σ ∗ (y ∗ )−1

where

y := σ(x) ∈ W∗ .

Thus, (w1∗ )−1 and (w2∗ )−1 are σ ∗ -conjugate in W∗ . Now let (λ, n) be a pair as in 2.4.5 such that (λ, n)  (w1, θ 1 ). Then sθ1 = tλ,n ∈ T∗0 [(w1∗ )−1 ]. By Lemma 2.4.26, we have (y.λ, n)  (w2, θ 2 ). So Lemma 2.5.4 shows that y ∗ sθ1 ( y ∗ )−1 = y ∗ tλ,n ( y ∗ )−1 = ty.λ,n = sθ2 ∈ T∗0 [(w2∗ )−1 ].



In particular, the pairs (w1∗ )−1, sθ1 and (w2∗ )−1, sθ2 in Y(W∗, σ ∗ ) are in the same W∗ -orbit. Corollary 2.5.14 (Cf. [DeLu76, 5.21]) 1−1

1−1

There are natural bijections 1−1

X(G, F) ←→ X(W, σ) ←→ Y(W∗, σ ∗ ) ←→ Y(G∗, F ∗ ), where the first one is given by 2.3.20, the third one by 2.5.12, and the middle one is ∼ induced by the isomorphisms Irr(T0 [w]) → T∗0 [(w ∗ )−1 ] (w ∈ W) in Lemma 2.5.7. Proof Let (w1, θ 1 ) ∈ X(W, σ). Using Lemma 2.5.7, we obtain an element sθ1 ∈

T∗0 [(w1∗ )−1 ] and, hence, a pair (w1∗ )−1, sθ1 ∈ Y(W∗, σ ∗ ). Let also (w2, θ 2 ) ∈

X(W, σ), with corresponding (w2∗ )−1, sθ2 ∈ Y(W∗, σ ∗ ). Now, if (w1, θ 1 ) and

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161

∗ −1 (w2, θ 2 ) are in the same W-orbit, then Remark 2.5.13 shows that (w1 ) , sθ1 ∗ −1



and (w2 ) , sθ2 are in the same W∗ -orbit. Conversely, assume that (w1∗ )−1, sθ1

and (w2∗ )−1, sθ2 are in the same W∗ -orbit. So there exists some y ∈ W such that (w2∗ )−1 = y ∗ (w1∗ )−1 (y ∗ )−1 and sθ2 = y ∗ sθ1 (y ∗ )−1 . We write y = σ(x) where x ∈ W. Then w2 = xw1 σ(x)−1 . Let θ  := xθ 1 ∈ Irr(T0 [w2 ]). Then Remark 2.5.13 shows that sθ2 = y ∗ sθ1 (y ∗ )−1 = y ∗ tλ,n (y ∗ )−1 = ty.λ,n = sθ  . Since the map ∼ Irr(T0 [w2 ]) −→ T∗0 [(w2∗ )−1 ] in Lemma 2.5.7 is an isomorphism, it follows that θ 2 = θ  = xθ 1 . Thus, we obtain a bijection between W-orbits of X(W, σ) and  W∗ -orbits of Y(W∗, σ ∗ ), giving rise to the middle correspondence. Remark 2.5.15 Let (T, θ) ∈ X(G, F) and (T∗, s) ∈ Y(G∗, F ∗ ) be such that the ∗ GF -orbit of (T, θ) corresponds to the G∗ F -orbit of (T∗, s) via the bijections in Corollary 2.5.14. Then, for any integer r ∈ Z, the GF -orbit of (T, θ r ) also corres∗ ponds to the G∗ F -orbit of (T∗, sr ) via those bijections. To see this, one first passes from (T, θ) to a pair (w, θ ) ∈ X(G, F) as in 2.3.20; then (T, θ r ) passes to the pair (w, θ r ). Similarly, one passes from (T∗, s) to a pair ((w ∗ )−1, t) ∈ Y(G∗, F ∗ ) as in 2.5.12; then (T∗, sr ) passes to ((w ∗ )−1, t r ). Now θ  ∈ Irr(T0 [w]) and t ∈ T∗0 [(w ∗ )−1 ] correspond to each other under the above group ∼ isomorphism Irr(T0 [w]) → T∗0 [(w ∗ )−1 ]. Hence, θ r and t r also correspond to each other in this way. (See also [Hi90a, §2].) Remark 2.5.16 We note that |X(G, F)| = |Y(G∗, F ∗ )|. Indeed, by [St67, 14.14] (see also [Ca85, 3.4.1]), the number of F-stable maximal tori in G is given by ∗ |GF | 2p . Furthermore, we have |GF | = |G∗ F |; see Example 1.6.19. So the number of F ∗ -stable maximal tori in G∗ is given by the same number. It remains to group the F-stable maximal tori of G and the F ∗ -stable maximal tori of G∗ into GF -classes ∗ and G∗ F -classes, respectively, and to use the bijection in 2.5.1(c). Definition 2.5.17 (See [Lu77a, 7.5]) Let T∗ ⊆ G∗ be an F ∗ -stable maximal torus ∗ ∗ and s ∈ T∗ F . Via Corollary 2.5.14, the G∗ F -orbit of (T∗, s) corresponds to the GF -orbit of some (T, θ) ∈ X(G, F). Then we denote 7 RTG∗ (s) := RTG (θ). By Corollary 2.2.10, this is well defined, that is, RTG∗ (s) does not depend on the choice of the pair (T, θ) ∈ X(G, F) in its GF -orbit. Following [Lu77a, §7], we will now re-express some properties of the virtual characters RTG (θ) in terms of the new notation RTG∗ (s). 7

We note explicitly that there is no known direct way of constructing a virtual character of G F from a pair (T∗, s) ∈ Y(G∗, F ∗ ). Whenever we want to prove something about RTG∗ (s), we have to translate θ . the desired statement back to the original definition of RTG (θ) or R w

162

Lusztig’s Classification of Irreducible Characters

Proposition 2.5.18 ([Lu77a, 7.5.1]) For i = 1, 2 let (T∗i , si ) ∈ Y(G∗, F ∗ ). Then we have the following scalar product formula: ∗

∗

RTG∗ (s1 ), RTG∗ (s2 ) = |{x ∈ G∗ F | xT∗1 x −1 = T∗2, xs1 x −1 = s2 }|/|T∗1 F |. 1

2

∗

In particular, if (T∗1, s1 ) and (T∗2, s2 ) are not in the same G∗ F -orbit, then RTG∗ (s1 ) and RTG∗ (s2 ) are orthogonal.

1

2

Proof For i = 1, 2 let (T∗i , si ) correspond to (Ti, θ i ) ∈ X(G, F) via the bijections in Corollary 2.5.14. If (T1, θ 1 ) and (T2, θ 2 ) are not in the same GF -orbit, then we have RTG1 (θ 1 ), RTG2 (θ 2 ) = 0 by Corollary 2.2.10. But then (T∗1, s1 ) and (T∗2, s2 ) are not ∗

in the same G∗ F -orbit either, so the right-hand side of the desired scalar product formula is zero. Now assume that (T1, θ 1 ), (T2, θ 2 ) are in the same GF -orbit, and, ∗ hence, that (T∗1, s1 ), (T∗2, s2 ) are in the same G∗ F -orbit. For i = 1, 2 let (Ti, θ i ) correspond to (wi, θ i) ∈ X(W, σ) (as in 2.3.20). Then RTG1 (θ 1 ), RTG2 (θ 2 ) equals θ

θ

Rw11 , Rw22 = |{x ∈ W | x.(w1, θ 1 ) = (w2, θ 2 )}|;

(1)

see Lemma 2.3.19 and Example 2.3.22(a). Similarly, for i = 1, 2 let (T∗i , si ) correspond to (wi, si) ∈ Y(W∗, σ ∗ ) (as in 2.5.12). Then one easily sees that the right-hand side of the desired formula RTG1 (θ 1 ), RTG2 (θ 2 ) equals |{y ∈ W∗ | y.(w1, s1 ) = (w2, s2 )}|.

(2)

But, we have si = sθi for i = 1, 2 (as in Lemma 2.5.7) and so the compatibility of the isomorphisms in Lemma 2.5.7 with the actions of W and of W∗ (see Remark 2.5.13) shows that the right-hand side of (1) equals (2).  Example 2.5.19 We give a few examples where a relation in Section 2.2 involving the virtual characters RTG (θ) is reformulated in terms of RTG∗ (s). (a) For i = 1, 2 let (T∗i , si ) ∈ Y(G∗, F ∗ ). If RTG∗ (s1 ) and RTG∗ (s2 ) have an irreducible 1 2 constituent in common, then s1, s2 are conjugate in G∗ . Indeed, for i = 1, 2 let (T∗i , si ) correspond to (Ti, θ i ) via Corollary 2.5.14. So, if RTG∗ (s1 ) and RTG∗ (s2 ) have an irreducible constituent in common, then (T1, θ 1 ) 1 2 and (T2, θ 2 ) are geometrically conjugate by Theorem 2.3.2. Via Propositions 2.4.28 and 2.5.5, this translates into the condition that s1, s2 are conjugate in G∗ . (b) Using Remark 2.5.16 (see also [Lu77a, 7.5.3]), the character of the regular representation of GF in Example 2.2.26(b) can be re-written as follows:

1 εG εT∗ RTG∗ (s). χreg = F |G | p (T∗,s)∈Y(G∗,F ∗ ) If s1, . . . , sn is a set of representatives of the conjugacy classes of semisimple

2.5 Geometric Conjugacy and the Dual Group elements of G∗

F∗

163

and Hi := CG∗ (si ) for 1  i  n, then this also equals: χreg =

n

|GF | p ∗

i=1

|HiF |

T∗

εG εT∗ RTG∗ (si );

F ∗ -stable

where the second sum runs over all maximal tori T∗ ⊆ H; see [Lu77a, ∗ F 7.7]. Note that |GF | = |G∗ |, εG = εG∗ and εT = εT∗ , if T and T∗ correspond to each other as in 2.5.1(c). Indeed, as already mentioned in Example 1.6.19, (G, F) ∗ and (G∗, F ∗ ) have the same order polynomial and so |GF | = |G∗ F |. For the other two identities, one can then use the characterisation of the relative rank in terms of order polynomials (see Definition 2.2.11). The linear characters of GF have an elegant description via the above dictionary between G and G∗ . As in Remark 1.5.13, let F . GuF = u ∈ GF | u unipotent ⊆ Gder

We denote by Θu ⊆ Irr(GF ) the group of all linear characters η : GF → K× such that GuF ⊆ ker(η). We can construct a canonical group homomorphism ∗

Θu → T∗0 F ,

η → zη, ∼

∗

as follows. If η ∈ Θu , then we apply the isomorphism Irr(T0F ) −→ T∗0 F in ∗ Lemma 2.5.7 to η|T F ∈ Irr(T0F ) and obtain an element zη ∈ T∗0 F . 0

∗

Proposition 2.5.20 ([Lu77a, 7.4.2]) We have zη ∈ Z(G∗ )F for any η ∈ Θu . The ∗ resulting map Θu → Z(G∗ )F , η → zη , is an isomorphism. ∗

If z ∈ Z(G∗ )F and η ∈ Θu is such that zη = z, then it will also be convenient to denote η by zˆ. Proof We follow the hint given in [Lu77a, 7.4.2]. By Remark 1.5.13(b), we have F ) where π : G → G is a simply connected covering of the derived GuF = πsc (Gsc sc sc −1 (T ) ⊆ G . Then, by Remark 1.5.13(c) the inclusion ˜ 0 := πsc subgroup of G. Let T 0 sc F ) = GF /GF . Let ˜ F )  GF /πsc (Gsc T0 ⊆ G induces an isomorphism T0F /πsc (T u 0 ˜ F ) ⊆ ker(θ)}. Irr (T0F ) := {θ ∈ Irr(T0F ) | πsc (T 0 Then we have an isomorphism Θu → Irr (T0F ), η → η|T F , and zη is obtained by ∼

∗

0

applying the isomorphism Irr(T0F ) → T∗0 F , θ → sθ , in Lemma 2.5.7 to η|T F ∈ 0 Irr (T0F ). Hence, we must show that the isomorphism θ → sθ in Lemma 2.5.7 maps ∗ ∗ the subset Irr (T0F ) ⊆ Irr(T0F ) onto Z(G∗ )F ⊆ T∗0 F . ∗ : G∗ → G∗ , For this purpose, as in 1.7.11, we consider a dual central isotypy πsc sc ∗ ∗ ∗ ∗ with respect to maximally split tori T0 ⊆ G and T˜ 0 ⊆ Gsc . There are Steinberg ∗ ◦ F ∗ = F˜ ∗ ◦ π ∗ . Furthermore, maps F ∗ : G∗ → G∗ and F˜ ∗ : G∗sc → G∗sc such that πsc sc

164

Lusztig’s Classification of Irreducible Characters

∼ ∼ ˜ ∗ ) satisfying the there are isomorphisms δ : X(T0 ) → Y (T∗0 ) and δ˜ : X(T˜ 0 ) → Y (T 0 appropriate conditions. As in Remark 2.5.9, we also have an isomorphism ∼

˜∗

˜ ∗ )F , Irr(T˜ 0F ) −→ (T 0 ∼

θ˜ → sθ˜,

∗

which is compatible with Irr(T0F ) → T∗0 F , θ → sθ , in the following sense. If ∗ (s ). ˜ F˜ ), then sθ˜ = πsc θ ∈ Irr(T0F ) and θ˜ := θ ◦ πsc |T˜ F˜ ∈ Irr(T θ 0 0  F Now we can argue as follows. Let θ ∈ Irr (T0 ). By definition, this means that θ˜ := ∗ (s ) = s = 1 and ˜ F and so sθ˜ = 1. But then πsc θ ◦ πsc |T˜ F is the trivial character of T θ θ˜ 0

∗

0

∗

∗

∗ ) ⊆ Z(G∗ )F . Conversely, let z ∈ Z(G∗ )F ⊆ T∗ F and θ ∈ Irr(TF ) so sθ ∈ ker(πsc 0 0 be such that z = sθ . Since G∗sc is semisimple of adjoint type (see Example 1.5.20(a)), ∗ (s ) = π ∗ (z) = 1. Hence, if we have Z(G∗sc ) = {1} (see Example 1.5.3) and so πsc θ sc F ∗ (s ) = 1 and so θ˜ is the ˜ ˜ we set θ := θ ◦ πsc |T˜ F ∈ Irr(T0 ) as above, then sθ˜ = πsc θ 0  trivial character, that is, θ ∈ Irr (T0F ).

Proposition 2.5.21 ([Lu77a, 7.5.5]) We have η · RTG∗ (s) = RTG∗ (zη s) for any η ∈ Θu and any pair (T∗, s) ∈ Y(G∗, F ∗ ). Proof Let (T, θ) ∈ X(G, F) correspond to (T∗, s) via Corollary 2.5.14. Then we already know from Lemma 2.3.13 that η · RTG (θ) = RTG (η|T F · θ). So we must show that the pairs (T, η|T F · θ) ∈ X(G, F) and (T∗, zη s) ∈ Y(G∗, F ∗ ) also correspond to each other via Corollary 2.5.14. For this purpose, we go explicitly through the correspondences in Corollary 2.5.14. Let w ∈ W be such that T is of type w. Let g ∈ G be such that w = g−1 F(g) and T = gT0 g −1 . Define θ  ∈ Irr(T0 [w]) by θ (t) = θ(gtg −1 ) for all t ∈ T0 [w] (see 2.3.20). Now T∗ is of type (w ∗ )−1 . Let h ∈ G∗ be such that (w ∗ )−1 = h−1 F ∗ (h) and T∗ = hT∗0 h−1 . Define s  ∈ T∗0 [(w ∗ )−1 ] by s = hs  h−1 (see 2.5.12). Thus, we have

(T, θ) ←→ (w, θ ) ←→ (w ∗ )−1, s  ←→ (T∗, s), ∼

where s  = sθ  by the isomorphism Irr(T0 [w]) → T∗0 [(w ∗ )−1 ] in Lemma 2.5.7. Since ∗ zη ∈ Z(G∗ )F , we have zη s = zη hs  h−1 = hzη s  h−1 and so (T∗, zη s) ∈ Y(G∗, F ∗ )

corresponds to (w ∗ )−1, zη s  ∈ Y(W∗, σ ∗ ) as in 2.5.12. Define η  ∈ Irr(T0 [w]) by η (t) = η(gtg −1 ) for all t ∈ T0 [w]. Then (T, η|T F · θ) ∈ X(G, F) corresponds to (w, η  · θ ) ∈ X(W, σ) via 2.3.20. Hence, it remains to show that sη = zη via the isomorphism in Lemma 2.5.7. Now, by Example 2.2.25 and Lemma 2.3.13, we have RTG (η|T F ), η = η · RTG (1T ), η = RTG (1T ), 1G = 1 and, similarly, RTG0 (η|T F ), η = 1. Thus, by Theorem 2.3.2, the two pairs (T, η|T F ) 0 and (T0, η|T F ) in X(G, F) are geometrically conjugate and, hence, the correspond0 ing two pairs (w, η ) and (1, η|T F ) in X(W, σ) are geometrically conjugate (see 0 Lemma 2.4.23). Via Lemma 2.5.7, the first of these pairs gives rise to the element

2.5 Geometric Conjugacy and the Dual Group

165

F∗

sη ∈ T∗0 [(w ∗ )−1 ], while the second pair gives rise to zη ∈ T∗0 (by the definition of the isomorphism in Proposition 2.5.20). These two elements are conjugate in G∗ by Corollary 2.5.8. Since zη ∈ Z(G∗ ), we conclude that sη = zη , as desired.  Next, it will be useful to know how the virtual characters RTG∗ (s) behave with ˜ as in Section 1.7. We use the convention respect to a regular embedding i : G → G ˜ Let (G ˜ ∗, F˜ ∗ ) be in duality in Remark 1.7.6 where we identify G with i(G) ⊆ G. ∗ ˜ ˜ ˜ with (G, F). We have a corresponding central isotypy i : G∗ → G∗ such that ˜ is a closed embedding, we have: i ∗ ◦ F˜ ∗ = F ∗ ◦ i ∗ ; see 1.7.11. Since i : G → G ˜ ∗ ), • K := ker(i ∗ ) is an F˜ ∗ -stable torus contained in Z(G ∗ F˜ ∗ ∗ F ∗ ∗ ∗ ∗ ∗ ˜ ) )=G ; ˜ ) = G and i (G • i (G see Lemma 1.7.12. In this setting, we can now state: Proposition 2.5.22 (Cf. [Lu88, p. 164]) Let T∗ ⊆ G∗ be an F ∗ -stable maximal ∗ ˜ ∗ . Then there exists a semisimple torus and s ∈ T∗ F . Let T˜ ∗ := i ∗ −1 (T∗ ) ⊆ G  ∗ ˜ ˜ ∗ )F such that i ∗ (s˜) = s. For any such s˜ we have RG∗ (s) = RG˜ (s˜) F . element s˜ ∈ (T ∗ ˜ T G T ˜ ∗ . Hence, Proof First note that T˜ ∗ := i ∗ −1 (T∗ ) is an F˜ ∗ -stable maximal torus of G −1 −1 ∗ ∗ ∗ ˜ consists of semisimple elements. Now note that i (s) is an F˜ ∗ -stable i (s) ⊆ T coset of K = ker(i ∗ ). The connected group K acts transitively on this coset by ˜∗ multiplication and so Proposition 1.4.9 shows that i ∗ −1 (s)F  . Thus, there exists ∗ ˜ ∗ )F˜ such that i ∗ (s˜) = s. a semisimple element s˜ ∈ (G ˜ be the maximally Next note that we are in a setting as in Remark 2.5.9. Let T˜ 0 ⊆ G ˜ Let T˜ ∗ := i ∗ −1 (T∗ ) ⊆ G ˜∗ ˜ 0 . (We have T0 = T0 .Z(G).) split torus such that T0 = G∩T 0

0

be the corresponding dual maximally split torus. Using i and i ∗ , we can identify W ˜∗ ˜ (with respect to T ˜ 0 ) and W∗ with the Weyl group of G with the Weyl group of G ∗ ˜ ). Let w ∈ W be such that T is of type w. Then T ˜ also is of type (with respect to T ˜ 0 [w]. Let θ˜ ∈ Irr(T˜ 0 [w]) w. Furthermore, one immediately checks that T0 [w] ⊆ T be such that s˜ = sθ˜ via the isomorphism in Lemma 2.5.7. Let ˜ T0 [w] = θ˜ ◦ i|T0 [w] ∈ Irr(T0 [w]). θ := θ|

Let g ∈ G be such that w = g −1 F(g) and T = gT0 g −1 . Then we also have ˜ F˜ ) T˜ = g T˜ 0 g −1 . Define θ  ∈ Irr(TF ) by θ (t) = θ(g −1 tg) for all t ∈ TF and θ˜ ∈ Irr(T ˜ ˜ −1 t˜g) for all t˜ ∈ T˜ F (see 2.3.20). Then by θ˜(t˜) = θ(g ˜ ˜ ˜ RTG˜ ∗ (s˜) = RTG˜ ∗ (sθ˜ ) = RTG˜ (θ˜)

RTG∗ (sθ ) = RTG (θ ); ˜  furthermore, it is clear that θ  = θ˜ |T F . Hence, we have RTG (θ ) = RTG˜ θ˜ G F . (This is just a very special case of the formula in Remark 2.3.16.) Combining this with ˜ the previous equalities, we deduce that RTG∗ (sθ ) = RTG˜ ∗ (s˜)|G F . It remains to use that sθ = i ∗ (s˜) = s, which holds by Remark 2.5.9(∗).  and

166

Lusztig’s Classification of Irreducible Characters

˜ F˜ . So we have an isomorphism Now, Proposition 2.5.20 also holds for G ∼ ˜ ∗ )F˜ ∗ , ˜ u −→ Z(G Θ

η˜ → zη˜ ,

˜ F → K× such that G ˜ uF˜ ⊆ ker(η); ˜ u is the group of all linear characters η˜ : G ˜ where Θ ˜ ˜ uF . also note that GuF = G ˜ u such that GF ⊆ ker(η)}. ˜ Lemma 2.5.23 We have ker(i ∗ ) = {zη˜ | η˜ ∈ Θ ˜ u such that GF ⊆ ker(η). ˜ F ) be ˜ Let Irr (T Proof Let Θ be the group of all η˜ ∈ Θ 0 ˜ ˜ Now note that TF = GF ∩ T˜ F˜ the group of all θ˜ ∈ Irr(T˜ 0F ) such that T0F ⊆ ker(θ). 0 0 ˜ F˜ = GF .T ˜ F˜ ; see Lemma 1.7.7. Hence, we have an isomorphism and G ˜

0

˜ F˜ ), Θ → Irr (T 0

η˜ → η| ˜ T˜ F˜ . 0

We can now argue exactly as in the proof of Proposition 2.5.22, where the role of ˜ and, hence, the role of Z(G∗sc ) = πsc : Gsc → G is replaced by that of i : G → G ∗ ∗  ker(πsc ) is replaced by that of K = ker(i ). We have the following refined version of the Exclusion Theorem 2.3.2, which shows an advantage of the new notation RTG∗ (s): it is not at all obvious how to even state the result in terms of the original notation RTG (θ) ! Theorem 2.5.24 (Refined Exclusion Theorem; see [Lu77a, 7.5.2]) Assume that RTG∗ (s1 ) and RTG∗ (s2 ) (as in Definition 2.5.17) have an irreducible constituent in 1

2

∗

common. Then s1, s2 are conjugate not only in G∗ (see Example 2.5.19) but in G∗ F . Proof

Assume that ρ ∈ Irr(GF ) is a common irreducible constituent of RTG∗ (s1 ) 1

and RTG∗ (s2 ). Following the hint in [Lu77a, 7.5.2], we consider a regular embed2 ˜ as above, and let i ∗ : G ˜ ∗ → G∗ be a corresponding dual central ding i : G → G −1 ∗ ∗ ∗ ˜ ∗ . By Frobenius reciprocity and ˜ isotypy. Let i ∈ {1, 2} and T := i (T ) ⊆ G i

i

Proposition 2.5.22, we have  G˜   G  ˜ F˜ G RT˜ ∗ (s˜i ), IndG F (ρ) = RT∗ (si ), ρ  0, i

i

˜ ∗ )F˜ ∗ (T i

i ∗ (s˜i )

˜ F˜ ) such is such that = si . Hence, there is some ρ˜i ∈ Irr(G where s˜i ∈ ˜ that RG˜ ∗ (s˜i ), ρ˜i  0 and ρ occurs in the restriction of ρ˜i to GF . Since this holds for T i

i = 1 and i = 2, we see by Remark 2.1.4 that ρ˜2 = η˜ · ρ˜1 for some linear character ˜ F˜ ) such that GF ⊆ ker(η). η˜ ∈ Irr(G ˜ Using Proposition 2.5.21, we obtain     ˜   G˜ ˜ RT˜ ∗ (zη˜ s˜1 ), ρ˜2 = η˜ · RTG˜ ∗ (s˜1 ), η˜ · ρ˜1 = RTG˜ ∗ (s˜1 ), ρ˜1  0. 1

1

Hence, ρ˜2 is a common irreducible constituent of

1

˜ RTG˜ ∗ (zη˜ s˜1 ) 1

˜

and RTG˜ ∗ (s˜2 ). By Ex2

˜ ∗ . But, as already ample 2.5.19(a), the elements zη˜ s˜1 and s˜2 are conjugate in G

2.6 The Jordan Decomposition of Characters

167

˜ is connected, the centraliser C ˜ ∗ (s˜2 ) is connected and so noted in 2.5.10, since Z(G) G ˜ ∗ )F˜ ∗ (see Example 1.4.10). Hence, i ∗ (zη˜ )s1 zη˜ s˜1 and s˜2 are already conjugate in (G ∗ and s2 are conjugate in G∗ F . Finally, since GF ⊆ ker(η), ˜ we have i ∗ (zη˜ ) = 1 by F∗ ∗ Lemma 2.5.23. Thus, s1 and s2 are conjugate in G .  For somewhat different proofs, see [DiMi20, Prop. 12.4.4] and [Bo06, 11.8].

2.6 The Jordan Decomposition of Characters Let G be a connected reductive algebraic group over k and F : G → G be a Steinberg map. Based on the preparations from the previous sections, we are now ready to formulate and to discuss the fundamental Jordan decomposition of characters of the finite group GF , both in the original case where Z(G) is connected (see [Lu84a]), and in the general case (see [Lu88], [Lu08a]). Let G∗ be a connected reductive algebraic group over k and F ∗ : G∗ → G∗ be a Steinberg map such that (G, F) and (G∗, F ∗ ) are in duality as in Definition 1.5.17; we also assume that the choices in 2.4.4 have been made. Recall from the previous section that we have then a canonical bijection between GF -orbits of pairs (T, θ) (where T ⊆ G is an F-stable maximal torus and θ ∈ ∗ Irr(TF )) and G∗ F -orbits of pairs (T∗, s) (where T∗ ⊆ G∗ is an F ∗ -stable maximal ∗ torus and s ∈ T∗ F ). If the pairs (T, θ) and (T∗, s) correspond to each other in this way, then we write RTG∗ (s) = RTG (θ). ∗

Definition 2.6.1 (Cf. [Lu77a, 7.6]) Let s ∈ G∗ F be semisimple. We define E (GF , s) to be the set of all ρ ∈ Irr(GF ) such that RTG∗ (s), ρ  0 for some F ∗ stable maximal torus T∗ ⊆ G∗ with s ∈ T∗ . This set is called a rational series of characters of GF , or Lusztig series of characters. For example, we have E (GF , 1) = Uch(GF ); see Definition 2.3.8. Indeed, just note that the pairs (T∗, 1) ∈ Y(G∗, F ∗ ) correspond to the pairs (T, 1T ) ∈ X(G, F). Thus, the notions of ‘rational series of characters’ and ‘geometric series of characters’ coincide as far as Uch(GF ) is concerned. (We will see that this is not the case in general; see Remark 2.6.19.) ∗

Theorem 2.6.2 ([Lu77a, 7.6]) If s1, s2 ∈ G∗ F are semisimple and conjugate in ∗ G∗ F , then E (GF , s1 ) = E (GF , s2 ). We have a partition * E (GF , s) Irr(GF ) = s

where s runs over a set of representatives of the conjugacy classes of semisimple

168

Lusztig’s Classification of Irreducible Characters F∗

elements in G∗ . If ρ ∈ E (GF , s), then ρ(1) =

|GF | p εG εT∗ RTG∗ (s), ρ |HF ∗ | T∗

where H := CG∗ (s) and the sum runs over all F ∗ -stable maximal tori T∗ ⊆ H◦ . (Note that H need not be connected here!) Proof Since every ρ ∈ Irr(GF ) occurs in some Deligne–Lusztig character RTG (θ), ) we have Irr(GF ) = s E (GF , s) where s runs over all semisimple elements of ∗ ∗ G∗ F . Now let s1, s2 be semisimple elements in G∗ F . We claim that E (GF , s1 ) = ∗ ∗ E (GF , s2 ) if s1, s2 are conjugate in G∗ F . Indeed, let x ∈ G∗ F be such that s2 = xs1 x −1 . Let T∗1 ⊆ G∗ be an F ∗ -stable maximal torus in G∗ such that s1 ∈ T∗1 . Setting ∗ T∗2 := xT∗1 x −1 , we have s2 ∈ T∗2 and (T∗2, s2 ) ∈ Y(G∗, F ∗ ) is in the same G∗ F -orbit as (T∗1, s1 ). So RTG∗ (s2 ) = RTG∗ (s1 ). It follows that E (GF , s1 ) ⊆ E (GF , s2 ). The reverse 2

1

inclusion holds by symmetry and so E (GF , s1 ) = E (GF , s2 ), as claimed. On the other hand, Theorem 2.5.24 shows that, if E (GF , s1 ) ∩ E (GF , s2 )  , ∗ then s1, s2 are conjugate in G∗ F and so E (GF , s1 ) = E (GF , s2 ). Consequently, we obtain a partition as stated above. The formula for ρ(1) follows by using the formula for χreg in Example 2.5.19(b) and the fact that we have a partition of Irr(GF ) as above.  ∗

Remark 2.6.3 Let us fix a semisimple element s ∈ G∗ F . Let C be the G∗ conjugacy class of s, which is F-stable. By Proposition 2.5.5, C corresponds to the equivalence class of a pair (λ, n) ∈ Λ(G, F). Corresponding to (λ, n), we have a subset Eλ,n ⊆ Irr(GF ) that is a geometric series of characters; see Corollary 2.4.29. Then the definitions immediately imply that  E (GF , s ). (a) Eλ,n = s ∈C ,F ∗ (s )=s

(Just follow once more the correspondences in Propositions 2.4.28 and 2.5.5, and Corollary 2.5.14.) In particular, this shows that every geometric series of characters is a union of rational series of characters. Now assume that Z(G) is connected. Then CG∗ (s) is connected (see Theorem 2.2.14) and so the set {s  ∈ C | F ∗ (s ) = s  } is a ∗ single G∗ F -conjugacy class (see Example 1.4.10). Thus, we conclude that Eλ,n = E (GF , s)

(if Z(G) is connected);

(b)

that is, every geometric series of characters is just a rational series of characters in this case. We can now state the Jordan decomposition of characters for the connected centre case: Theorem 2.6.4 ([Lu84a, 4.23], [Lu88, Cor. 6.1]) Assume that Z(G) is connected.

2.6 The Jordan Decomposition of Characters

169

Let s ∈ G∗ be a semisimple element such that F ∗ (s) = s. Let H := CG∗ (s); note that H is connected. There is a bijection ∗

1−1

E (GF , s) ←→ Uch(HF ),

ρ ↔ ρu,

such that, for any F ∗ -stable maximal torus T∗ ⊆ H, we have RTG∗ (s), ρ = εG εH RTH∗ (1T∗ ), ρu . ∗

(Here RTH∗ (1T∗ ) is for the Deligne–Lusztig character of HF corresponding to the ∗ trivial character of T∗ F , as in the original set-up of Section 2.2.) A bijection as above will not in general be unique; we will discuss the uniqueness question under additional assumptions in Section 4.7. If we speak about Jordan decomposition later, we will mean any bijection satisfying the conditions in Theorem 2.6.4. Theorem 2.6.4 is obtained from Theorem 2.4.15 and the discussion in 2.5.10, via the various correspondences established in the previous sections, by which one can pass from equivalence classes of pairs (λ, n) as in 2.4.5 to F ∗ -stable conjugacy classes of semisimple elements in G∗ and, hence, to conjugacy classes of semisimple ∗ elements in G∗ F . It also involves the following fact (already mentioned in 2.5.10): Remark 2.6.5

If we take s = 1 in Theorem 2.6.4, then we obtain a bijection ∗

1−1

Uch(GF ) ←→ Uch(G∗ F ),

ρ ↔ ρ∗,

∗

such that RTG (1T ), ρ = RTG∗ (1T∗ ), ρ∗ for corresponding maximal tori T ⊆ G and T∗ ⊆ G∗ as in 2.5.1(c). This was first stated explicitly in [Lu88, 6.1]. Note, however, that this bijection is not really a consequence of Theorem 2.6.4 but one needs to prove it beforehand, in order to be able at all to reformulate Theorem 2.4.15 as above. A proof of the above bijection can be obtained as follows. Using Proposition 2.3.15 and the reduction arguments in [Lu76c, 1.18] (see also Remark 4.2.1 in Chapter 4), it is sufficient to consider the case where G is simple. Then the only case which requires a special argument is when G is of type Bn or Cn , where the desired statement is known by the results in [Lu81c], [Lu84a, Chap. 9] (see also Remark 4.2.18 below). A more conceptual proof can nowadays be obtained by [Lu14b, §7]. While Theorem 2.4.15 already contains all the essential information, the above formulation is particularly useful in many applications; it leads to elegant formulations of certain identities that would be hard to state otherwise. Corollary 2.6.6 In the setting of Theorem 2.6.4, we have ∗

∗

ρ(1) = |G∗ F : HF | p ρu (1) ∗

for all ρ ∈ E (GF , s). ∗

In particular, ρ(1) is divisible by |G∗ F : HF | p for any ρ ∈ E (GF , s).

170 Proof

Lusztig’s Classification of Irreducible Characters Using the degree formula in Theorem 2.6.2, we have ρ(1) =

|GF | p εG εT∗ RTG∗ (s), ρ |HF ∗ | T∗

where the sum runs over all F ∗ -stable maximal tori in H. On the other hand, by Proposition 2.2.18 (applied to H and s = 1) and Theorem 2.3.2, we obtain 1 ρu (1) = F ∗ εH εT∗ RTH∗ (1T∗ ), ρu |H | p T∗ where, again, the sum runs over all F ∗ -stable maximal tori in H. So Theorem 2.6.4 immediately yields the desired formula.  Example 2.6.7 In the setting of Theorem 2.6.4, assume that s is a regular element as in Remark 2.3.10, that is, H = CG∗ (s) ⊆ G∗ is a maximal torus. Then E (GF , s) = G {εG εH RH (s)}. Now, we have seen in Lemma 2.3.11 that, if q is large, then most elements in a torus are regular. Hence, as already noted in the introduction of [Lu77b], ‘almost all’ irreducible characters of GF are of the form ±RTG∗ (s) where ∗ s ∈ T∗ F is regular. 2.6.8 It had already been shown in [DeLu76, §10] (that is, long before the full version of Theorem 2.6.4 became available) that every geometric series E (GF , s) (where Z(G) is connected) contains certain distinguished characters which – in the language of Theorem 2.6.4 – correspond to the trivial and the Steinberg character ∗ of HF ; thus, their degrees are given by ∗

∗

|G∗ F : HF | p

and

∗

∗

∗

|G∗ F : HF | p |HF | p,

respectively (see Example 2.2.27 for the degree of the Steinberg character; also note that, in the situation of Example 2.6.7, there is only one character in E (GF , s)). We will now explain how the character corresponding to the trivial character of ∗ HF is obtained, following (and streamlining) the exposition in [Ca85, Chap. 8]. In the beginning of the following discussion, we make no assumption on Z(G). Let O0 ⊆ G be the conjugacy class of regular unipotent elements; see 2.2.21. Let l := dim T0 − dim Z◦ (G) be the semisimple rank of G. By [Ca85, 5.1.9], we have |O0F ||Z◦ (G)F |ql = |GF |. Following [DeLu76, 10.2], we define a class function ΔG ∈ CF(GF ) by  ◦ if g ∈ O0F , |Z (G)F |ql ΔG (g) = 0 otherwise. (In [Ca85, §8.3], this class function is denoted by Ξ.) Then, for f ∈ CF(GF ), we have  f , ΔG  0 if and only if the average value of f on O0F is non-zero.

2.6 The Jordan Decomposition of Characters

171

Definition 2.6.9 (Cf. [Ca85, p. 280]) Let ρ ∈ Irr(GF ). Using the above notation, we say that ρ is a semisimple character if ρ, ΔG  0. Thus, ρ is semisimple if and only if the average value of ρ on O0F is non-zero. Let S0 (GF ) denote the set of all semisimple characters of GF . Clearly, the trivial character of GF is semisimple, and so are all those linear F in their kernel. characters with Guni The reason why the values of ΔG are taken to be |Z◦ (G)F |ql on elements in O0F (and not just to be the constant value 1) is the following: (♣)

ΔG is a virtual character of GF if Z(G) = Z◦ (G) is connected.

See [Ca85, 8.1.7] or [DeLu76, 10.3] for a proof, which uses the Gelfand–Graev character of GF . (The proof is elementary in the sense that it does not use the theory of -adic cohomology; note that, for the following discussion, we do not require the multiplicity-freeness of Gelfand–Graev characters in [Ca85, Thm. 8.1.3].) 2.6.10 By Theorem 2.2.22, any Deligne–Lusztig character takes value 1 on O0F . Hence, for any pair (T∗, s) ∈ Y(G∗, F ∗ ), we obtain that 1 RTG∗ (s), ΔG = F |Z◦ (G)F |ql = 1. (a) |G | F u ∈ O0

It follows that there exists at least one ρ ∈ S0 (GF ) such that RTG∗ (s), ρ  0. In particular, the rational series of characters E (GF , s) contains at least one semisimple character. Thus, ∗

E (GF , s) ∩ S0 (GF )   for every semisimple element s ∈ G∗ F .

(b)

Now assume that Z(G) = Z◦ (G) is connected. Then the rational series of characters are precisely the geometric series of characters; see Remark 2.6.3. Furthermore, by (♣), the class function ΔG is an integral linear combination of the semisimple characters of GF . Thus, |S0 (GF )|  ΔG, ΔG = |Z(G)F |ql , where the last equality is clear by the definition of ΔG . On the other hand, by Proposition 2.5.6, |Z(G)F |ql also is the number of geometric conjugacy classes of characters. Taking into account (b), we conclude that ∗

for every semisimple element s ∈ G∗ F .

|E (GF , s) ∩ S0 (GF )| = 1

(c)

Thus, if Z(G) is connected, then E (GF , s) contains exactly one semisimple character, which we will denote by ρs ∈ S0 (GF ). Hence ΔG has at least |Z(G)F |ql irreducible constituents. Since ΔG, ΔG = |Z(G)F |ql , we conclude that

s ρs (s ∈ {±1}) (d) ΔG = s

172

Lusztig’s Classification of Irreducible Characters

where s runs over a set of representatives of the conjugacy classes of semisimple ∗ elements of G∗ F . Now we can state the following result, which summarises a big part of [DeLu76, §10] and interprets it in terms of the Jordan decomposition. ∗

Theorem 2.6.11 Assume that Z(G) is connected and let s ∈ G∗ F be a semisimple element. As above, let ρs ∈ S0 (GF ) be the unique semisimple character that belongs to E (GF , s). Then the following hold. (a) For any F ∗ -stable maximal torus T∗ ⊆ G∗ with s ∈ T∗ , we have RTG∗ (s), ρs = s , where s = ±1 is as in 2.6.10(d). ∗ ∗ (b) Let H := CG∗ (s). Then ρs (1) = |G∗ F : HF | p and s = εG εH . (c) Via the Jordan decomposition of characters in Theorem 2.6.4, the character ∗ ρs ∈ E (GF , s) corresponds to the trivial character of HF .  Proof (a) We have 1 = RTG∗ (s), ΔG = s s RTG∗ (s), ρs where s  runs over a ∗ set of representatives of the conjugacy classes of semisimple elements of G∗ F . Using the partition in Theorem 2.6.2, we conclude that all terms in the sum are zero, except for one term corresponding to the unique s  that is conjugate to s. Thus, 1 = s RTG∗ (s), ρs , as desired. (b) Using the degree formula in Theorem 2.6.2, we have ρs (1) =

|GF | p |GF | p G ∗ R ∗ (s), ρs = s ε ε εG εT∗ , G T ∗ T |HF | T∗ |HF ∗ | T∗

where the sums run over all F ∗ -stable maximal tori T∗ ⊆ H; the second equality  ∗ holds by (a). Now, by Example 2.2.26(a), we have T∗ εG εT∗ = εG εH |HF | p . This yields the formula for ρs (1) and the identity s = εG εH . ∗ (c) Let ψ ∈ Uch(HF ) correspond to ρs via a bijection as in Theorem 2.6.4. Let T∗ ⊆ H be an F ∗ -stable maximal torus. Using (a) and (b), we obtain RTH∗ (1T∗ ), ψ = εG εH RTG∗ (s), ρs = εG εH s = 1. ∗

So Remark 2.4.2(b) shows that ψ must be the trivial character of HF .



Remark 2.6.12 Assume that Z(G) is connected. By (♣), the class function ΔG is a virtual character of GF . We have seen in 2.6.10 that ΔG, ΔG = |Z(G)F |ql and that the scalar product of ΔG with any Deligne–Lusztig character equals 1. Then there is a general argument which implies that ΔG and all irreducible constituents of ΔG are uniform; see [Ca85, 8.4.4] or [DeLu76, 10.6]. In particular, each ρs is uniform. An explicit expression of ΔG as a linear combination of Deligne–Lusztig characters

2.6 The Jordan Decomposition of Characters is given by ΔG =

1 |GF |

(T∗,s)∈Y(G∗,F ∗ )

173

∗

|T∗ F |RTG∗ (s).

This immediately follows by re-writing [DeLu76, 10.7.4] in terms of RTG∗ (s). Remark 2.6.13 Assume that Z(G) is connected. (a) If (λ, n) ∈ Λ(G, F) corresponds to s as in Remark 2.6.3, then one easily sees that ±ρs is equal to the uniform almost character Rφ˜ (see Remark 2.4.17), where φ is the trivial character of Wλ,n and φ˜ is its canonical γ-extension as in Example 2.1.8(c). (b) We have mentioned above that there is a further distinguished member of ∗ ∗ E (GF , s), corresponding to the Steinberg character of HF = CG∗ (s)F . The most natural and elegant way to obtain this character is by using a duality operation on CF(GF ) which we will discuss in Section 3.4. Our final aim in this section is to present Lusztig’s extension of Theorem 2.6.4 to the case where Z(G) is not necessarily connected. This passes by an intermediate version involving regular embeddings as in Section 1.7. The following discussion is a worked-out version of [Lu84a, §14.1]. ˜ and use the notational convention in Let us fix a regular embedding i : G → G ˜ Let (G ˜ ∗, F˜ ∗ ) be in duality with Remark 1.7.6, where we identify G with i(G) ⊆ G. ˜ ˜ ∗ → G∗ such ˜ (G, F). As in 1.7.11, we have a corresponding central isotypy i ∗ : G ∗ ∗ ∗ ∗ that i ◦ F˜ = F ◦ i . Recall from Lemma 1.7.12 that ˜ ∗ ), • K := ker(i ∗ ) is an F˜ ∗ -stable torus contained in Z(G ∗ F˜ ∗ ∗ F ∗ ∗ ∗ ∗ ∗ ˜ ) )=G . ˜ ) = G and i (G • i (G ˜

˜ F /GF is abelian of order prime to p. In particular, every unipotent Also recall that G ˜ F ˜ already belongs to GF . element of G 2.6.14 In order to simplify the notation, let us write G := GF ,

˜ F˜ , G˜ := G

∗

G∗ := G∗ F ,

˜ ∗ )F˜ ∗ . G˜ ∗ := (G

We denote the restriction of i ∗ to G˜ ∗ again by the same symbol. Now G˜ acts by conjugation on G and, hence, on Irr(G). Every g ∈ G acts trivially on Irr(G) and ˜ so we have an action of G/G on Irr(G). Let Θ be the group of all linear characters ˜ ˜ via the usual tensor product η ∈ Irr(G) such that G ⊆ ker(η). Then Θ acts on Irr(G) ˜ of class functions. Given ρ˜ ∈ Irr(G), let Θ( ρ) ˜ := {η˜ ∈ Θ | η˜ · ρ˜ = ρ} ˜ be the stabiliser of ρ˜ in Θ. By Theorem 1.7.15, we have ρ| ˜ G = ρ1 + · · · + ρr , where

174

Lusztig’s Classification of Irreducible Characters

ρ1, . . . , ρr are distinct irreducible characters of G. We have r =  ρ| ˜ G, ρ| ˜ G = |Θ( ρ)|; ˜

see Remark 2.1.4.

˜ F˜ . Then Furthermore, let Z = Z(G), Z := ZF and Z˜ := Z(G) Θ( ρ) ˜ ⊆ {η˜ ∈ Θ | Z˜ ⊆ ker(η)}, ˜ since any element of Z˜ acts as a scalar in a representation affording ρ. ˜ Consequently, ◦ ˜ ˜ r = |Θ( ρ)| ˜ divides | G : G. Z | and, hence, the order of (Z/Z )F , where we use the isomorphism in Remark 1.7.6(b). In particular, if G itself has a connected centre, ˜ to G is irreducible. then the restriction of ρ˜ ∈ Irr(G) ˜ u ⊆ Irr(G) ˜ be the group of all Remark 2.6.15 As in the previous section, let Θ × ˜ = 1 for u ∈ G˜ unipotent. Since all linear characters η˜ : G˜ → K such that η(u) unipotent elements of G˜ already belong to G, we conclude that Θ (as defined above) ˜ u . So, by Lemma 2.5.23, we have is contained in Θ ˜∗

KF = {zη˜ | η˜ ∈ Θ}.

(a)

Using Proposition 2.5.21, one immediately sees that ˜ s˜) := {η˜ · ρ˜ | ρ˜ ∈ E (G, ˜ s˜)} = E (G, ˜ zη˜ s˜) η˜ · E (G,

(b)

˜ permutes the for any semisimple element s˜ ∈ G˜ ∗ . Thus, the action of Θ on Irr(G) ˜ s˜). By Theorem 2.6.2, the stabiliser of E (G, ˜ s˜) under this geometric series E (G, action is the subgroup Θ(s˜) := {η˜ ∈ Θ | zη˜ and zη˜ s˜ are G˜ ∗ -conjugate}.

(c)

˜ s˜). Clearly, if ρ˜ ∈ Thus, the subgroup Θ(s˜) ⊆ Θ permutes the characters in E (G, ˜ E (G, s˜), then Θ( ρ) ˜ ⊆ Θ(s˜). Proposition 2.6.16 Let s ∈ G∗ be semisimple and s˜ ∈ G˜ ∗ be any semisimple element such that i ∗ (s˜) = s. Then ' ( ˜ s˜) . E (G, s) = ρ ∈ Irr(G) |  ρ| ˜ G, ρ  0 for some ρ˜ ∈ E (G, Proof First we show the inclusion ‘⊆’. Let ρ ∈ E (G, s). There exists an F ∗ -stable ˜ ∗ := maximal torus T∗ ⊆ G∗ such that s ∈ T∗ and RTG∗ (s), ρ  0. Then s˜ ∈ T −1 ∗ ∗ ∗ ˜ i (T ) ⊆ G . Using Proposition 2.5.22 and Frobenius reciprocity, we obtain   G   G˜ ˜ RT˜ ∗ (s˜), IndG G (ρ) = RT∗ (s), ρ  0. ˜ ˜ such that RG˜ ∗ (s˜), ρ So there exists some ρ˜ ∈ Irr(G) ˜  0 and ρ˜ occurs in IndG G (ρ). T˜ ˜ s˜) and ρ occurs in ρ| But then ρ˜ ∈ E (G, ˜ G (by Frobenius reciprocity), as required. To prove the reverse inclusion we use an indirect argument, as follows. Let s1, . . . , sn ∈

2.6 The Jordan Decomposition of Characters

175

G∗ be a set of representatives of the conjugacy classes of semisimple elements of G∗ . For each i, we choose a semisimple element s˜i ∈ G˜ ∗ such that i ∗ (s˜i ) = si . By the previous part of the proof, we have for all i: ' ( ˜ s˜i ) . E (G, si ) ⊆ E˜i := ρ ∈ Irr(G) |  ρ| ˜ G, ρ  0 for some ρ˜ ∈ E (G, Assume that there exists some ρ ∈ E˜i ∩ E˜j . Thus, we have  ρ| ˜ G, ρ  0 for some ˜ s˜i ) and  ρ˜  |G, ρ  0 for some ρ˜  ∈ E (G, ˜ s˜ j ). By Remark 2.1.4, there ρ˜ ∈ E (G,

˜ By Remark 2.6.15, we have exists some η˜ ∈ Θ such that ρ˜  = η˜ · ρ. ˜∗

zη˜ ∈ KF

and

˜ s˜i ) = E (G, ˜ zη˜ s˜i ). ρ˜  ∈ η˜ · E (G,

By Theorem 2.6.2, s˜ j and zη˜ s˜i are conjugate in G˜ ∗ . Hence s j and i ∗ (zη˜ )si are ˜∗ conjugate in G∗ . But i ∗ (zη˜ ) = 1 since zη˜ ∈ KF and so i = j. Hence, {E˜i | 1  i  n} ) is a family of disjoint subsets of Irr(G). But we also have Irr(G) = 1in E (G, si ) and so we must have E (G, si ) = E˜i for all i.  For a somewhat different proof of the above proposition, see [Bo06, 11.7]. Proposition 2.6.17 ([Lu84a, §14.1], [Lu88, §11]) ˜ s˜). By Theorem 1.7.15, we can write ρ˜ ∈ E (G, ρ| ˜ G F = ρ1 + · · · + ρr

where

Let s˜ ∈ G˜ ∗ be semisimple and

ρ1, . . . , ρr ∈ Irr(GF )

and ρi  ρ j for i  j. Let s := i ∗ (s˜) ∈ G∗ and T∗ ⊆ G∗ be an F ∗ -stable maximal ˜ ∗ , we have ˜ ∗ = i ∗ −1 (T∗ ) ⊆ G torus with s ∈ T∗ . Then, setting T

˜ ρi ∈ E (G, s) and RTG∗ (s), ρi = RTG˜ ∗ (s˜), ρ˜  for 1  i  r, ρ˜  ∈O

˜ s˜) is the orbit of ρ˜ under the action of Θ(s˜). where O ⊆ E (G, Since every ρ ∈ Irr(GF ) occurs as some ρi as above, this result in combination with Theorem 2.6.2 reduces the problem of computing the multiplicities RTG∗ (s), ρ ˜ for all (T∗, s) ∈ Y(G∗, F ∗ ) to an analogous problem for Irr(G). Proof The fact that ρi ∈ E (G, s) is clear by Proposition 2.6.16. Now, by Remark 2.1.4 and Theorem 1.7.15, we have

˜ IndG ρ˜  G (ρi ) = ρ˜  ∈O

where O  is the full orbit of ρ˜ under the action of Θ. Hence, by Proposition 2.5.22 and Frobenius reciprocity, we obtain:

 ˜ ˜ ˜ F˜

˜ ρi = RTG˜ ∗ (s˜), ρ˜  . RTG∗ (s), ρi = RTG˜ ∗ (s˜)|G F , ρi = RTG˜ ∗ (s˜), IndG GF ρ˜  ∈O

Let

ρ˜ 

∈

O

be such that the corresponding term in the above sum is non-zero. Then

176

Lusztig’s Classification of Irreducible Characters

˜ s˜). Writing ρ˜  = η˜ · ρ˜ and using Remark 2.6.15, we see that η˜ preserves ρ˜  ∈ E (G, ˜ s˜) and, hence, η˜ ∈ Θ(s˜). So ρ˜  ∈ O as required. E (G,  The following result completely describes the semisimple characters of G. (See [DiMi20, §12.4] for a slightly different discussion.) Corollary 2.6.18 Let s ∈ G∗ be semisimple. Let s˜ ∈ G˜ ∗ be semisimple such that ˜ be the unique semisimple character that belongs to E (G, ˜ s˜); i ∗ (s˜) = s. Let ρ˜ ∈ Irr(G) see 2.6.10(c). (a) The semisimple characters in E (G, s) are precisely the irreducible constituents of ρ| ˜ G. (b) Let ρ ∈ E (G, s) be a semisimple character. Then RTG∗ (s), ρ = ±1 for any F ∗ -stable maximal torus T∗ ⊆ G∗ such that s ∈ T∗ . ˜ s˜) and let ρ ∈ Irr(G) be an irreducible constituent Proof (a) Take any ψ˜ ∈ E (G, ˜G ˜ G ; by Proposition 2.6.16, we have ρ ∈ E (G, s). By Clifford’s theorem, ψ| of ψ| g ˜ ˜ is a sum of conjugates of ρ, that is, characters of the form ρ where g˜ ∈ G. Since ˜ all conjugates of ρ have the same O0F ⊆ G is invariant under conjugation in G, F average value on O0 . So ψ˜ has a non-zero average value on O0F if and only if ρ ˜ then ρ is semisimple and all has a non-zero average value on O0F . Thus, if ψ˜ = ρ, semisimple characters in E (G, s) are constituents of ρ| ˜ G. (b) Note that η˜ · ρ˜ is also semisimple, for every η˜ ∈ Θ. Hence, since ρ˜ is the ˜ s˜), the orbit of ρ˜ under the action of Θ(s˜) is unique semisimple character in E (G, O = { ρ}. ˜ So Proposition 2.6.17 yields that ˜

˜ = ±1 RTG∗ (s), ρ = RTG˜ ∗ (s˜), ρ where the last equality holds by Theorem 2.6.11(a).



Remark 2.6.19 Recall from the beginning of Section 2.3 the definition of the graph DL(G). As discussed in [Ge18, 6.14], it now also follows that the partition of Irr(G) defined by the connected components of this graph is precisely the partition into rational series of characters as in Theorem 2.6.2. Indeed, by Corollary 2.6.18, each character in E (G, s) is directly linked to a fixed semisimple character in E (G, s). Hence, all characters in E (G, s) belong to the same connected component of DL(G). Assume, if possible, that E (G, s) is strictly contained in a connected component of DL(G). Then there exists some ρ1 ∈ E (G, s) and some ρ2 ∈ Irr(G) such that ρ2  E (G, s) and ρ1, ρ2 are directly linked in DL(G), that is, there is some pair (T∗, s ) ∈ Y(G∗, F ∗ ) such that RTG∗ (s ), ρi  0 for i = 1, 2. But then s, s  are conjugate in G∗ by Theorem 2.5.24. So we have E (G, s) = E (G, s ) by Theorem 2.6.2 and, hence, ρ2 ∈ E (G, s), contradiction. (For a slightly different argument, see [DiMi20, Thm. 12.4.13].)

2.6 The Jordan Decomposition of Characters

177

To illustrate this, consider again the example where G = SL2 (q), with q odd. By Example 2.2.30, {ψ+ , ψ+ } and {ψ− , ψ− } are connected components of DL(G) and so these are also rational series of characters; in fact, we have E (GF , s) = {ψ+ , ψ+ }

and

E (GF , s ) = {ψ− , ψ− }

where s is the unique element of order 2 in a maximally split torus of G∗ = PGL2 (k) and s  is the unique element of order 2 in an F ∗ -stable maximal torus of G∗ that is not maximally split. On the other hand, by Example 2.4.7(b), the union E (GF , s) ∪ E (GF , s ) = {ψ+ , ψ+, ψ− , ψ− } is a geometric series of characters of GF . In particular, this shows that rational series of characters and geometric series of characters are not the same in general — and this already happens in the smallest possible case where Z(G) is not connected! In order to make an effective use of Proposition 2.6.17, one needs to know the ˜ Ideally, it should be possible to describe this action via the action of Θ on Irr(G). bijections in Theorem 2.6.4 and, hence, reduce the problem to a question about unipotent characters. This reduction can indeed be done, but requires more work. Lemma 2.6.20 ([Lu88, §8]) Let s˜ ∈ G˜ ∗ be semisimple and s := i ∗ (s˜) ∈ G∗ . Let A(s) := CG∗ (s)/CG◦ ∗ (s). Then F ∗ induces an automorphism of A(s), which we denote by the same symbol. There is a canonical isomorphism ∗

∼

A(s)F −→ Θ(s˜) ∗

= x; then the defined as follows. Let x ∈ CG∗ (s)F and x ∈ G˜ ∗ be such that i ∗ ( x) −1 coset of x in A(s) is sent to the unique η˜ ∈ Θ(s˜) such that zη˜ = x s˜−1 x s˜. Hence, ˜∗ ˜ s˜). via this isomorphism, we obtain an action of A(s)F on E (G,

˜ ∗ , B = G∗ , f = i ∗ , σ( x) ˜ = s˜−1 x˜ s˜ Proof We apply Lemma 1.1.9 with A = G ∗ −1 ∗ σ τ ˜ ( x˜ ∈ G ), τ(x) = s xs (x ∈ G ). Then A = CG˜ ∗ (s˜) and B = CG∗ (s). Since CG˜ ∗ (s˜) is connected, we have i ∗ (Aσ ) = CG◦ ∗ (s) by 1.3.10(e); furthermore, C = {a−1 σ(a) | a ∈ ker( f )} = {1} and we obtain a canonical isomorphism ∼

˜ ∗ } ∩ K. δ : CG∗ (s)/CG◦ ∗ (s) −→ { x˜ −1 s˜−1 x˜ s˜ | x˜ ∈ G ˜ ∗ }. Now The group on the right-hand side equals { z˜ ∈ K | s˜, z˜ s˜ are conjugate in G ∗ F is compatible with δ and we obtain an isomorphism ∗

∼

∗

˜ ∗ }. δ : A(s)F −→ { z˜ ∈ KF | s˜, z˜ s˜ are conjugate in G Note also that, by Example 1.4.11(b), the inclusion CG ∗ (s) ⊆ CG∗ (s) induces an ∗ ∗ ∗ isomorphism CG∗ (s)F /CG◦ ∗ (s)F  A(s)F . Finally, since CG˜ ∗ (s˜) is connected, the group on the right side of the above isomorphism equals ∗

{ z˜ ∈ KF | s˜, z˜ s˜ are conjugate in G˜ ∗ }

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Lusztig’s Classification of Irreducible Characters

and the latter group is isomorphic to Θ(s˜) via Remark 2.6.15(c). The above descrip∗ tion of the map A(s)F → Θ(s˜) follows from the description of δ in Lemma 1.1.9.  ˜ := C ˜ ∗ (s˜) and H := CG∗ (s); we also set In the setting of Lemma 2.6.20, let H G ˜ F˜ ∗ , H := HF ∗ and H ◦ := H◦ F ∗ . Then we obtain a natural action of A(s)F˜ ∗ H˜ := H ˜ = H◦ . Hence, ˜ as follows. In the above proof we already saw that i ∗ (H) on Uch(H), by Proposition 2.3.15, we obtain a canonical bijection 1−1

˜ Uch(H ◦ ) −→ Uch(H),

ψ → ψ ◦ i ∗ | H˜ .

Now the conjugation action of H on H ◦ induces an action of H on Uch(H ◦ ) (see Example 2.3.17). Clearly, H ◦ is in the kernel of this action and so we obtain a natural ∗ action of A(s)F on Uch(H ◦ ). Via the above bijection, this becomes an action of ∗ ˜ Now we can state: A(s)F on Uch(H). Theorem 2.6.21 ([Lu88, 8.1]) Let s˜ ∈ G˜ ∗ be semisimple and s := i ∗ (s˜) ∈ G∗ . Let ˜ := C ˜ ∗ (s˜) and H˜ := H ˜ F˜ ∗ . Then we can choose the bijection H G 1−1 ˜ ˜ s˜) ←→ Uch(H), E (G,

ρ˜ ↔ ρ˜u,

so that ρ˜ ↔ ρ˜u commutes with the action of A(s) ∗ ˜ just defined. and the action of A(s)F on Uch(H)

(see Theorem 2.6.4) F∗

˜ s˜) (see Lemma 2.6.20) on E (G,

The proof of this result is quite a tour de force. It proceeds by a reduction to the case where G is simple of simply connected type. Then it uses the full power of Theorem 2.4.15 (that is, the Main Theorem 4.23 of [Lu84a]), including the explicit ¯ λ,n ) and the multiplicities m(y, x). ¯ Various description of the parameter sets X(W special situations have to be considered case-by-case, including one where G is of type E7 ; see also [Lu84a, §14.1]. We can now state the general version of the Jordan decomposition of characters, where the formulation does not refer to a regular embedding of G. Theorem 2.6.22 ([Lu88, 5.1], [Lu08a, 5.2]) Let s ∈ G∗ be semisimple such that F ∗ (s) = s. Let H := CG∗ (s) and UF ∗ (H) be the set of all pairs (O, a) where O is ∗ ∗ ∗ an orbit of A(s)F on Uch(H◦ F ) and a ∈ A(s)F fixes some (or, equivalently, any) ∗ character in O. (Since A(s)F is abelian, all characters in an orbit O have the same stabiliser.) Then there is a bijection 1−1

E (GF , s) ←→ UF ∗ (H),

ρ ↔ (Oρ, aρ ),

such that, for any F ∗ -stable maximal torus T∗ ⊆ H◦ , we have

◦ RTH∗ (1), ρi where Oρ = {ρ1, . . . , ρr }. RTG∗ (s), ρ = εG εH 1ir

2.6 The Jordan Decomposition of Characters

179

This is now a formal consequence of Theorems 2.6.4, 2.6.21, Proposition 2.6.17 and general Clifford theory; see [Lu88, §11] and [CE04, 15.14] for further details. ∗

Remark 2.6.23 If A(s)F = {1}, then Theorem 2.6.22 essentially reduces to the statement of Theorem 2.6.4, even if H is not connected. Indeed, in this case, every ∗ orbit O consists of one unipotent character of H◦ F and so ' ∗( ∗ 1−1 (ψ, 1) ↔ ψ. UF ∗ (H) = (ψ, 1) | ψ ∈ H◦ F ←→ Uch(H◦ F ), 1−1

∗

Hence, we obtain a bijection E (GF , s) ←→ Uch(H◦ F ) satisfying the scalar product formula in Theorem 2.6.4. In general, the following facts are known. ∗

(a) By Lemma 2.6.20 and the discussion in 2.6.14, the order of A(s)F divides the order of (Z/Z◦ )F . ∗ (b) The exponent of A(s)F divides the order of the element s or, more precisely, the order of the image of s in Gad where πad : G → Gad denotes an adjoint quotient as in Remark 1.5.12 (see [Bo05, Cor. 2.9], [BrMi89, Lemma 2.1] and [Bor70, Part E, II, 4.4, 4.6]). ∗

So, if the order of s is prime to the order of (Z/Z◦ )F , then A(s)F = {1}. For example, if G is a simple algebraic group of classical type Bn , Cn or Dn and s has ∗ odd order, then A(s)F = {1}. ∗

Remark 2.6.24 In the setting of Theorem 2.6.22, let us fix an orbit O of A(s)F on ∗ Uch(H◦ F ) and consider the set of characters {ρ ∈ E (GF , s) | Oρ = O}. In [Lu88, §3], it is explained how one can define a natural action of the group (Z/Z◦ )F on E (GF , s). Then each set of characters as above is precisely an orbit under this action of (Z/Z◦ )F ; see [Lu88, 5.1]. Example 2.6.25 Let G = GF = SL4 (q) where q is odd; then Z = Z(G)F has order 2 (if 4 | q + 1) or order 4 (if 4 | q − 1). Since G is simple of simply connected type, we can take G∗ = PGL4 (k) and G∗ = PGL4 (q) (see 1.5.18). Consider the following two matrices in GL4 (k): 0 0   0 0 0 1   1 0  0 0 1 0   0 1  0 0  . and a =  s =    0   0 1 0 0   0 0 −1 0 −1   1 0 0 0   0 0 Let s¯ be the image of s in G∗ . Let H := CG∗ (s¯). Then H◦ has type A1 × A1 , the centre of H◦ has dimension 1, and H/H◦ is cyclic of order 2, generated by a¯ (the image of a). We have Uch(H ◦ ) = {ψ1, ψ2, ψ2, ψ3 } where ψ1 is the trivial character,

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Lusztig’s Classification of Irreducible Characters

ψ3 is the Steinberg character (of degree q2 ) and ψ2, ψ2 have degree q. There are four H ◦ -conjugacy classes of F ∗ -stable maximal tori in H◦ ; we denote representatives by T∗1, T∗2, T2∗, T∗3 where ∗

∗

∗

∗

|T∗1 F | = (q − 1)3, |T∗2 F | = |T2∗ F | = (q − 1)2 (q + 1), |T∗3 F | = (q − 1)(q + 1)2 . The Deligne–Lusztig characters have the following decompositions: ◦

RTH∗ (1T∗1 ) = ψ1 + ψ2 + ψ2 + ψ3, 1 ◦

RTH∗ (1T∗2 ) = ψ1 + ψ2 − ψ2 − ψ3, 2

◦ RTH∗ (1T∗2 ) 2

= ψ1 − ψ2 + ψ2 − ψ3,

◦

RTH∗ (1T∗3 ) = ψ1 − ψ2 − ψ2 + ψ3 . 3

Now, the action of a¯ on H◦ exchanges the two components of type A1 . This action fixes T∗1 and T∗3 but exchanges T∗2 , T2∗ (up to conjugation in H◦ ). Similarly, it fixes ∗ ψ1 and ψ3 but exchanges ψ2 , ψ2 . Thus, the orbits of A(s¯)F on Uch(H ◦ ) are O2 = {ψ2, ψ2 },

O1 = {ψ1 },

O3 = {ψ3 }.

F∗

The stabilisers of ψ1 , ψ3 in A(s¯) have order 2, while the stabilisers of ψ2 , ψ2 in ∗ A(s¯)F are trivial. Hence, we obtain ¯ (O2, 1), (O3, 1), (O3, a)}. ¯ UF ∗ (H) = {(O1, 1), (O1, a), Thus, E (G, s¯) contains exactly five irreducible characters. Let ρ ∈ E (G, s¯) correspond to (O2, 1) under a bijection as in Theorem 2.6.22. Then we obtain: ◦

◦

1 H◦ T∗2 H◦ T∗ 2 H◦ T∗3

1 H◦ T∗2 H◦ T∗ 2 H◦ T∗3

RTH∗ (s¯), ρ = RTH∗ (1T∗1 ), ψ2 + RTH∗ (1T∗1 ), ψ2 =

1+1 =

2,

RTH∗ (s¯), ρ = R (1T∗2 ), ψ2 + R (1T∗2 ), ψ2 =

1−1 =

0,

RTH∗ (s¯), ρ = R (1T∗2 ), ψ2 + R (1T∗2 ), ψ2 = −1 + 1 =

0,

1

2

2

RTH∗ (s¯), 3

ρ = R (1 ), ψ2 + R (1 T∗1

T∗3

), ψ2

= −1 − 1 = −2.

(Note that εG εH = 1 since a maximally split torus of G is contained in H◦ .) See [Ge18, 6.11] for the discussion of a similar example in G = Sp4 (q). Remark 2.6.26 Let s ∈ G∗ be semisimple; as above, we write H = CG∗ (s), ∗ ∗ H := HF and H ◦ := H◦ F . As in [Lu88, §12], we define Uch(H) to be the set of all ρ ∈ Irr(H) such that the restriction of ρ to H ◦ has an irreducible constituent that is unipotent. (See Proposition 4.8.19 for a different characterisation.) Note that, if ρ ∈ Uch(H), then all irreducible constituents of the restriction of ρ to H ◦ are unipotent. (Indeed, the irreducible constituents of ρ| H ◦ form an orbit under the natural action of H on Irr(H ◦ ) by conjugation; so, if one of them is unipotent, then all are unipotent by Example 2.3.17.) Now, it is known that the restriction of any

2.7 Average Values and Unipotent Support

181

ρ ∈ Uch(H) to H ◦ is multiplicity-free (see [DiMi20, Prop. 11.5.3]). Hence, as in 1−1

[Lu88, §12], one can conclude that there is a bijection E (G, s) ←→ Uch(H); see [DiMi20, Chap. 11] for a further discussion.

2.7 Average Values and Unipotent Support Let G be a connected reductive algebraic group over k = F p and F : G → G be a Steinberg map. The results exposited in the previous sections show that there is an efficient classification of the irreducible characters of GF . This classification also yields all character values on semisimple elements of GF (see Proposition 2.2.18), which includes the character degrees, but does not tell us much about the remaining values. In order to attack this problem in general, Lusztig developed the theory of character sheaves [LuCS], which tries to produce some geometric objects over the algebraic group G from which the irreducible characters of GF , for any F as above, could be deduced in a uniform manner (as exemplified by the generic character tables in Section 2.1). In this and the following section, we give but a brief introduction into this theory. Our focus will be on uniform functions and the determination of the values of the virtual characters RTG (θ), which already is a highly complex and technically intricate story. We begin with some elementary remarks relating conjugacy classes in the finite group GF to F-stable conjugacy classes of G. 2.7.1 Let C be an F-stable conjugacy class of G. Then C F is non-empty (see Example 1.4.10) and we pick an element g ∈ C F . Let A(g) := CG (g)/CG◦ (g) be the finite group of components of the centraliser of g. (If it is necessary to indicate the underlying algebraic group, then we write AG (g) instead of just A(g).) Since F(g) = g, the groups CG (g) and CG◦ (g) are F-stable and so F induces an automorphism of A(g) which we denote by the same symbol. By Example 1.4.10, we have: C F is a single GF -conjugacy class if A(g) = {1}.

(a)

In general, the following happens. Let a ∈ A(g) and a ∈ CG (g) be a representative of a. By the Lang–Steinberg theorem, we can write a = x −1 F(x) for some x ∈ G. One immediately checks that ga := xgx −1 ∈ C F . Thus, we have associated with any element a ∈ A(g) an element ga ∈ C F ; this depends on the choices of a and x as above, but the GF -conjugacy class of ga does not depend on these choices. If we

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Lusztig’s Classification of Irreducible Characters

denote this GF -conjugacy class by Ca , then we obtain a canonical bijection 1−1

H 1 (F, A(g)) ←→ {GF -conjugacy classes contained in C F },

(b)

where the F-conjugacy class of the element a ∈ A(g) corresponds to the GF conjugacy class Ca . (See, e.g., [Ge03a, 4.3.5] for a detailed proof; also recall from 2.1.6 that H 1 (F, A(g)) denotes the set of F-conjugacy classes of A(g).) Now let us consider what happens if we replace the chosen element g ∈ C F by another element g  ∈ C F . Then g  ∈ Ca for some a ∈ A(g). We can reverse the above procedure and find some x ∈ G such that g  = xgx −1 and a = x −1 F(x). Then CG (g ) = xCG (g)x −1 and CG◦ (g ) = xCG◦ (g)x −1 . So conjugation by x induces an ∼ isomorphism A(g) → A(g ) and it is straightforward to check that this restricts to an isomorphism ∼

C A(g),F (a) −→ A(g )F ;

(c)

here, C A(g),F (a) denotes the F-centraliser of a (as in 2.1.6). 2.7.2 In the setting of 2.7.1, consider the irreducible characters of the finite group A(g). For ς ∈ Irr(A(g))F , we fix an F-extension ς˜ ∈ CFF (A(g)) of ς (as in 2.1.7). F Then define a class function Y(g, ς) ˜ ∈ CF(G ) by  ς(a) ˜ if g  = ga ∈ C F for some a ∈ A(g),  Y(g, ς) (g ) = (a) ˜ 0 if g   C F . As noted in the remarks following Definition 2.1.11, the F-character table of A(g) is invertible. In particular, it follows that F the functions {Y(g, ς) ˜ | ς ∈ Irr(A(g)) } are linearly independent.

(b)

Now let Cl(G)F be the set of all F-stable conjugacy classes of G. For each C ∈ Cl(G)F , let us pick an element gC ∈ C F ; for each ς ∈ Irr(A(gC ))F , we choose a particular F-extension ς˜ of ς. Then (a), (b) show that the functions F F B := {Y(gC , ς) ˜ | C ∈ Cl(G) , ς ∈ Irr(A(gC )) }

(c)

form a vector space basis of CF(GF ). Thus, the problem of computing the character table of GF can be reformulated as the problem of determining the base change from the basis Irr(GF ) of CF(GF ) to the basis B. Note that this involves the issue of making choices of the representatives gC ∈ C F and of F-extensions of the F-invariant irreducible characters of A(gC ). Remark 2.7.3 The advantage of using the functions in 2.7.2(c) (rather than just the indicator functions of the conjugacy classes of GF ) is that these have a topological interpretation: each Y(gC , ς) ˜ is a ‘characteristic function’ of an F-stable, irreducible, G-equivariant Q -local system on C . (This is a special case of a more general

2.7 Average Values and Unipotent Support

183

geometric principle; see [Lu04b, 19.7] and also Example 2.7.27 below.) It opens the possibility of using the powerful machinery of ‘intersection cohomology’, which is the underlying theme in the theory of character sheaves; see [Lu84c], [Lu06] and the references there. Beyond its application to the character theory of GF , this machinery has turned out to be extremely helpful in attacking a broad range of problems in representation theory; see [Spr82] and the surveys [Lu91], [Lu14c]. Remark 2.7.4 Let C be an F-stable conjugacy class of G. A priori, we may choose any g ∈ C F and then perform the constructions in 2.7.2. So the question arises whether there are natural choices for g. We certainly have a favourable situation when we can find some g ∈ C F such that F acts trivially on A(g). For example, this clearly happens when | A(g)|  2 for g ∈ C . Further conditions are described in [Sho86, §5], [Ta13, §2]. However, one should keep in mind that this is not always the case. We give two examples. (a) Let GF = SL3 (q) where q ≡ −1 mod 3. Let u ∈ GF be a regular unipotent element (the Jordan normal form of u consists of one block with 1 on the diagonal). Now, we have Z(G) = {ζ I3 | ζ 3 = 1} and one easily checks that A(u) is cyclic of order 3, equal to the image of Z(G) in A(u). Since q ≡ −1 mod 3, the elements of Z(G) are inverted by F. Hence, F acts non-trivially on A(u) and so A(u)F = {1}. (This example can be easily generalised to other types of groups; see [Ta13, §2].) (b) Consider the Ree group GF = 2F 4 (q2 ). There exists an F-stable unipotent class C of G (denoted by F4 (a2 ) in [Spa85, p. 330]) such that A(g) is dihedral of order 8 for any g ∈ C , but there is no g ∈ C F such that F acts trivially on A(g). (By [Shi75, Table II], the class C splits into only three classes in GF , with representatives u10, u11, u12 .) Definition 2.7.5 Let C be an F-stable conjugacy class of G. For any class function f ∈ CF(GF ) we define the average value of f on C F by

| A(g j ) : A(g j )F | f (g j ), AV( f , C ) := 1 jr

where g1, . . . , gr ∈ C F are representatives of the GF -conjugacy classes contained in C F and, as above, we denote by A(g) := CG (g)/CG◦ (g) the finite group of components of the centraliser of an element g ∈ C F . (Note that AV( f , C ) does not depend on the choice of g1, . . . , gr ; also note that | A(g j )| does not depend on j, but | A(g j )F | may well depend on j.) It will turn out that the above definition of an average value is more efficient than just averaging the values of f over the whole set C F . Remark 2.7.6 Let C be an F-stable conjugacy class of G. Let us fix an element g ∈ C F . Let 1 = a1, a2, . . . , ar ∈ A(g) be a set of representatives of the F-conjugacy

184

Lusztig’s Classification of Irreducible Characters

classes of A(g). For 1  j  r, let g j ∈ Ca j (where Ca j is defined as in 2.7.1). Then A(g j )  C A(g),F (a j ) and so | A(g j ) : A(g j )F | = size of the F-conjugacy class of a j in A(g).  Hence, we may also write AV( f , C ) = a ∈ A(g) f (ga ), where ga ∈ C F is associated with a ∈ A(g) as in 2.7.1. F Example 2.7.7 Let C be an F-stable conjugacy class of G and Y(gC , ς) ˜ ∈ CF(G ) F be one of the basis functions in 2.7.2, where gC ∈ C and ς˜ is an F-extension of ς ∈ Irr(A(gC )). Then we have  ς(1)| ˜ A(gC )| if ς is the trivial character of A(gC ), AV(Y(gC , ς) ˜ ,C) = 0 otherwise.

Indeed, using the notation in Remark 2.7.6, we have

AV(Y(gC , ς) Y(gC , ς) ˜ ,C) = ˜ (ga ) = a ∈ A(gC )

ς(a) ˜

a ∈ A(gC )

˜ F where the inner product is defined in and the right-hand side equals | A(gC )| ς, ˜ 1 ˜ 2.1.7 and 1 denotes the trivial extension of the trivial character of A(gC ). So the orthogonality relations in 2.1.10 yield the desired identity. Example 2.7.8 Let C be an F-stable conjugacy class of G. (a) Let g ∈ C F and assume that CG (g) is connected. Then C F is a single conjugacy class of GF (see Proposition 1.4.9) and so AV( f , C ) = f (g). (b) Let g ∈ C F and assume that A(g) is abelian and F acts trivially on A(g). By Example 2.1.8(c), we then have |H 1 (F, A(g))| = | A(g)| and so A(g )F = A(g ) for any g  ∈ C F . Hence, in this case, we have

AV( f , C ) = f (g j ), 1 jr

where r = | A(g)| and g1, . . . , gr ∈ are as in Definition 2.7.5. (c) Let g ∈ C F and assume that A(g) is isomorphic to the symmetric group S3 . Then every (abstract) group automorphism of A(g) is inner and we can choose g so that F acts trivially on A(g) (see, e.g., [Ta13, Lemma 2.3]). Let 1 = a1, a2, a3 ∈ A(g) be representatives of the conjugacy classes of A(g), where a2 corresponds to a 2cycle and a3 corresponds to a 3-cycle in S3  A(g). Thus, we have CF

AV( f , C ) = f (g1 ) + 3 f (g2 ) + 2 f (g3 ), with g j ∈ Ca j for j = 1, 2, 3; note that | A(g j )F | = |CS3 (a j )|.

2.7 Average Values and Unipotent Support

185

Remark 2.7.9 Let C ⊆ G be as in Definition 2.7.5. As in [Ge96, §1], we have AV( f , C ) =  f , αC where the function αC : GF → K is defined by  if g ∈ C F , | A(g)||CG◦ (g)F | αC (g) := 0 otherwise. One easily sees that αC is a class function. The formula AV( f , C ) =  f , αC immediately follows from the fact that |CG (g)F | = | A(g)F ||CG◦ (g)F | for g ∈ GF (see Example 1.4.11(b)). The following result (mentioned in the proof of [GeMa00, Theorem 3.7]) shows that AV(ρ, C ) behaves well with respect to regular embeddings. ˜ be a regular embedding (see Section 1.7). Let C be Lemma 2.7.10 Let G ⊆ G ˜ an F-stable conjugacy class of G. Then C is also an F-stable conjugacy class in ˜ Let ρ˜ ∈ Irr(G ˜ F˜ ) and ρ ∈ Irr(GF ) be a constituent of the restriction of ρ˜ to GF . G. Then ˜ | A(g)|AV( ρ, ˜ C ) = r | A(g)|AV(ρ, C ), ˜ where A(g) = CG˜ (g)/C ◦˜ (g) (for g ∈ C ) and r  1 is the number of irreducible G constituents of the restriction of ρ˜ to GF . ˜ Let x ∈ G ˜ F˜ . Since G ˜ = G.Z, ˜ := Z(G). ˜ Proof Let us denote Z := Z(G) and Z −1 −1 we can write x = yz where y ∈ G and z ∈ Z˜ and so xC x = yC y = C . ˜ By Remark 2.7.9, we have ˜ Thus, C is also an F-stable conjugacy class in G. ˜ F ˜ AV( ρ, ˜ C ) =  ρ, ˜ α˜ C where α˜ C : G → K is defined by α˜ C (g) :=

◦ (g)F˜ | ˜ | A(g)||C ˜ G 0

if g ∈ C F , otherwise.

˜ Let g ∈ C F . Let a = | A(g)| and a˜ = | A(g)|. (Note that these two numbers do ˜ = G.Z, ˜ it is clear that C ˜ (g) = CG (g).Z. ˜ Since Z ˜ is not depend on g.) Since G G ◦ ◦ ◦ ◦ ˜ connected, this implies that C ˜ (g) = CG (g).Z. Since Z ⊆ CG (g), we can use G Lemma 1.7.8 and obtain that ˜ ˜ F˜ |/|Z◦ F |. |CG◦˜ (g)F | = |CG◦ (g)F || Z

˜ F˜ | = |GF || Z ˜ F˜ |/|Z◦ F | and so Now Lemma 1.7.8 also shows that | G ˜ F˜ |αC (g) a|GF | α˜ C (g) = a| ˜G

for all g ∈ C F .

Thus, the restriction of α˜ C to GF is a scalar multiple of αC , and we obtain aAV( ρ, ˜ C ) = a ρ, ˜ α˜ C = a ˜ ρ| ˜ G F , αC = aAV( ˜ ρ| ˜ GF , C ) ˜ F˜ and the right(where the left-hand average value is taken with respect to G F ˜ GF , C ) = hand average value with respect to G ). It remains to show that AV( ρ|

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Lusztig’s Classification of Irreducible Characters

rAV(ρ, C ). To see this, we write ρ| ˜ G F = ρ1 + · · · + ρr where ρi ∈ Irr(GF ) and ˜ F˜ . ρ1 = ρ. By Clifford’s theorem, each ρi is conjugate to ρ via some element xi ∈ G ˜ ˜ F to GF implies Now, the fact that αC is the restriction of a class function on G ˜ F F ˜ that αC is invariant under the action of G on CF(G ) by conjugation. Hence, we ˜ G F , C ) =  ρ| ˜ G F , αC = r ρ, αC = have ρi, αC = ρ, αC for all i and so AV( ρ| rAV(ρ, C ), as required.  The following result is obtained as a combination of [Ge96, Prop. 1.3] and [DiMi15, Cor. 6.8]. An analogous result concerning the indicator function on C F is established in [Ge18, §8], confirming a conjecture in [Lu77b, 2.16]. Theorem 2.7.11 Let G be connected reductive and F be a Frobenius map. Let C be an F-stable conjugacy class of G. Then the function αC ∈ CF(GF ) is uniform (see Definition 2.2.25). Thus, we have G

AV(ρ, C ) = AV πun (ρ), C for any ρ ∈ Irr(GF ), G (ρ) denotes the uniform projection as in Remark 2.2.28. where πun

Proof By [Ge96, Prop. 1.3], the statement holds in the special case where C is unipotent. The proof uses the results on Green functions in the following section, most notably the full power of Theorem 2.8.3 and 2.8.4 below. In order to deal with the general case, we follow the reduction argument in [DiMi15, §6] and introduce the following notation. Let s ∈ GF be semisimple and Hs := CG (s). For any f ∈ CF(GF ), we define ds ( f ) ∈ CF(H◦s F ) by  f (sx) if x ∈ H◦s F is unipotent, ds ( f )(x) := 0 otherwise. Now let C be an arbitrary F-stable conjugacy class of G and consider the class function f = αC ∈ CF(GF ). Let C  := {x ∈ Hs | x unipotent and sx ∈ C }. If C  = , then ds ( f ) is identically zero. Now assume that C   . Since |Hs : H◦s | < ∞ and since every unipotent element of Hs belongs to H◦s (see 2.2.13), we conclude that C  is a finite union of conjugacy classes of H◦s ; we denote these classes by C1, . . . , Cm where m  1. For 1  i  m we set ei := |CHs (x) : CH◦s (x)| (where x ∈ Ci ). Let us now work out the class function ds (αC ) ∈ CF(H◦s F ). Let x ∈ H◦s F be unipotent and set g := sx = xs. If g  C , then ds (αC )(x) = αC (g) = 0. Now assume that g ∈ C . Then x ∈ C  and so x ∈ Ci for a unique i ∈ {1, . . . , m}; in particular, Ci is F-stable. We claim that, in this case, we have ds (αC )(x) = ei αCi (x)

(where x ∈ Ci ).

2.7 Average Values and Unipotent Support

187

Indeed, since g = sx = xs, were s is semisimple and x is unipotent, we certainly have CG (g) = CHs (x). This also implies that CG◦ (g) = CH◦ s (x). Since CH◦ ◦ (x) is s connected and contained in CG (g), we have CH◦ ◦ (x) ⊆ CG◦ (g). On the other hand, s CH◦ ◦ (x) has finite index in CHs (x) and so CH◦ ◦ (x) ⊇ CH◦ s (x) = CG◦ (g). Thus, we have s s CG◦ (g) = CH◦ s (x) = CH◦ ◦ (x). It follows that s

AH◦s (x) = CH◦s (x)/CH◦ ◦s (x) ⊆ CHs (x)/CH◦ ◦s (x) = CG (g)/CG◦ (g) = AG (g) and | AG (g)| = ei | AH◦s (x)|. Thus, we conclude that αC (g) = | AG (g)||CG◦ (g)F | = ei | AH◦s (x)||CH◦ ◦s (x)F | = ei αCi (x). Since the left-hand side equals ds (αC )(x) by definition, the above claim is proved. It follows that

ds (αC ) = ei αCi ∈ CF(H◦s F ), i

where the sum runs over all i ∈ {1, . . . , m} such that Ci is F-stable. Now each αCi in the above sum is a uniform function by the special case dealt with in [Ge96, Prop. 1.3]. Hence, ds (αC ) is a uniform function. We have seen that this holds for every semisimple element s ∈ GF . So [DiMi15, Cor. 6.3] implies that αC ∈ CF(GF ) is uniform.  Remark 2.7.12 In the above result, we assumed that F is a Frobenius map. This was used when we referred to Theorems 2.8.3 and 2.8.4 in the proof. We will verify in Example 2.8.18 below by an explicit computation that the statements of these two theorems continue to hold for the Suzuki and Ree groups. Hence, Theorem 2.7.11 will also hold for these groups. Corollary 2.7.13 ([Lu84a, Introduction, p. xx]) Assume that F is a Frobenius map. Then, for any ρ ∈ Irr(GF ) and any F-stable conjugacy class C of G, the average value AV(ρ, C ) can be explicitly determined (in the form of an algorithm). In fact, at the time of writing [Lu84a], Lusztig had to assume that the characteristic p is large enough, but this condition can now be removed. Sketch of proof By Lemma 2.7.10, it is sufficient to determine AV(ρ, C ) in the case where Z(G) is connected. Let us now assume that this is the case. Then we G (ρ) using Theorem 2.4.15 (the ‘Main Theorem 4.23’ of [Lu84a]). can determine πun So the next step will be to work out the average values AV(RTG (θ), C ), for any Fstable maximal torus T ⊆ G and any θ ∈ Irr(TF ). Using the character formula in Theorem 2.2.16 (or Lemma 2.2.23), we can reduce to the case where C is unipotent. G Thus, finally, we need to determine the average values AV(QG T , C ), where Q T is the Green function corresponding to an F-stable maximal torus T ⊆ G. For this purpose,

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Lusztig’s Classification of Irreducible Characters

F we express QG T in the basis B of CF(G ) defined in 2.7.2. By Example 2.7.7, G AV(QT , C ) only depends on the coefficient of the basis function Y(gC , ς) ˜ ∈ B where ς is the trivial character of A(gC ). An algorithm for the determination of that coefficient will be explained in the following section; see 2.8.11. 

We now focus on unipotent classes of G; these play a special role in the theory of reductive algebraic groups. It is shown in [Lu76b] (by a general argument) that the number of unipotent classes of G is always finite. An exposition of this argument can be found in [DiMi20, §12.1]; it uses Lusztig induction, to be discussed in Section 3.3. For G simple, the unipotent classes have been explicitly determined in all cases; see, e.g., [Miz80] where this is done for G of type E7 , E8 , which is a truly amazing achievement! In the study of unipotent classes, one typically encounters exceptional situations when the characteristic p is ‘small’. More precisely, the distinction is between ‘good’ and ‘bad’ primes for G. Before we continue with average values, this is now a good place to introduce these notions, which we could completely avoid so far. 2.7.14 Good and bad primes. Recall from [SpSt70, I, §4], [Ca85, §1.14] that p is a good prime for G, if p is good for each simple factor of G, and that the conditions for the various simple types are as follows. An Bn, Cn, Dn G2, F4, E6, E7 E8

: : : :

no condition, p  2, p  2, 3, p  2, 3, 5.

It turns out that the classification of unipotent classes of G is independent of p, as long as p is a good prime for G. (If p is a bad prime, then usually there are more unipotent classes than in good characteristic.) See [Spa82a], [Hum95] for further details, and [Lu05], [ClPr13] for more recent developments regarding the bad prime case. The following result (or, rather, a slightly different version where the average  values AV(ρ, C ) are replaced by the global sums g ∈C F ρ(g)) was first conjectured by Lusztig [Lu80b, §1]. Quite remarkably, it is uniformly true in all characteristics although, as mentioned above, unipotent classes behave differently in bad or good characteristic. Theorem 2.7.15 ([Lu92a], [GeMa00]) Let G be connected reductive and F be a Frobenius map. Let ρ ∈ Irr(GF ). Then there exists a unique F-stable unipotent class O of G such that AV(ρ, O)  0, and such that AV(ρ, O ) = 0 for any F-stable unipotent class O  of G, unless O  = O or dim O  < dim O.

2.7 Average Values and Unipotent Support

189

Given ρ ∈ Irr(GF ), the unique unipotent class O attached to ρ as above is called the unipotent support of ρ and denoted by Oρ . For example, the unipotent support of the trivial character is the class of regular unipotent elements; at the other extreme, the unipotent support of the Steinberg character is the class of the identity element. The proof of the above result uses the full power of the whole theory developed so far (e.g., Theorem 2.7.11 and the main results of the book [Lu84a]), as well as some further ingredients that we did not discuss here at all, most notably the theory of ‘generalised Gelfand–Graev representations’ (see [Kaw86], [Kaw87] and also [Lu92a], [GeHe08], [Ta16] for more recent developments concerning these representations). In [Lu92a, §11], Theorem 2.7.15 is established assuming that p, q are sufficiently large; subsequently, it is shown in [GeMa00] that these assumptions can be removed. The following complementary result shows that the invariants aρ and nρ , which were defined in Remark 2.3.26 using the degree polynomial of the character ρ, can be recovered from Oρ and the average value of ρ on Oρ . (As far as aρ is concerned, this was also conjectured in [Lu80b, §1]; see also [Lu09a] for further interpretations of these invariants.) Proposition 2.7.16 ([Lu92a], [GeMa00]) Assume that F is a Frobenius map. Let ρ ∈ Irr(GF ). Then aρ = dim G − rank(G) − dim Oρ and aρ AV(ρ, Oρ ) = ±n−1 ρ q | A(u)|

(u ∈ Oρ ).

Remark 2.7.17 If GF is a Suzuki or Ree group (where F is not a Frobenius map), then the complete character table of GF is known by [Suz62], [War66], [Ma90]. As mentioned in [GeMa00, §5], the statements of Theorem 2.7.15 and Proposition 2.7.16 continue to hold in these cases as well. (In [GeMa00, §5], this was stated somewhat incorrectly, because we did not use the correct definition of nρ .) ˜ be a regular embedding (see Section 1.7). Let Example 2.7.18 Let G ⊆ G ˜ F ˜ ) and write ρ| ˜ G F = ρ1 + · · · + ρr where the ρi are distinct irreducible ρ˜ ∈ Irr(G characters of GF . Then all ρi have the same unipotent support, and this is the unipotent support of ρ. ˜ This immediately follows from Lemma 2.7.10. Example 2.7.19 Let O0 be the class of regular unipotent elements of G (see 2.2.21). Let ρ ∈ Irr(GF ). Then we claim that Oρ = O0

⇐⇒

ρ is semisimple (see Definition 2.6.9).

To see this, assume first that Z(G) is connected. Then it easily follows from the discussion in [Ge18, Remark 3.7] that αO0 = | A(u)|ΔG , where u ∈ O0 and the function

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Lusztig’s Classification of Irreducible Characters

ΔG is defined in 2.6.8. Hence, we have AV(ρ, O0 ) = ρ, αO0 = | A(u)|ρ, ΔG . Now ρ is semisimple if and only if ρ, ΔG  0, in which case we have ρ, ΔG = ±1; see 2.6.10(d). Hence, the desired equivalence holds; furthermore, we obtain that AV(ρ, O0 ) = ±| A(u)|

if ρ is semisimple (and Z(G) is connected).

˜ Now assume that Z(G) is not connected. Then consider a regular embedding G ⊆ G ˜ F ) be such that ρ occurs in the restriction of ρ˜ to as in Section 1.7. Let ρ˜ ∈ Irr(G GF . By Example 2.7.18, the characters ρ and ρ˜ have the same unipotent support. Furthermore, by Corollary 2.6.18, ρ is semisimple if and only if ρ˜ is semisimple. Hence, the desired equivalence holds in general. As far as the interaction of the unipotent support and individual character values are concerned, we have the following result. Theorem 2.7.20 ([Lu92a, §11]) Assume that F is a Frobenius map and p, q are sufficiently large. Let ρ ∈ Irr(GF ) and Oρ be the unipotent support of ρ. Let g ∈ GF be such that ρ(g)  0. Let u be the unipotent part of g and O be the conjugacy class of u in G. Then either dim O < dim Oρ or O = Oρ . Remark 2.7.21 Here, the assumption that p, q are sufficiently large means that one can operate with the Lie algebra of G as if we were in characteristic 0. (In particular, the variety of nilpotent elements in the Lie algebra may be identified with the variety of unipotent elements in G, via an exponential map; see [Lu92a, 1.3].) It is shown in [Ta19, §9] that the conclusion of Theorem 2.7.20 continues to hold if we only assume that p is a good prime for G and Z(G) is connected. It is likely that the assumption on Z(G) is unnecessary. However, the following examples will show that some assumptions on the characteristic p are necessary. Example 2.7.22 Let GF = PCSp4 (q) where q = p f ( f  1). The unipotent characters of GF are denoted as in Table 2.8 (p. 132); the classification and the degree polynomials do not depend on p or q. However, we will see a difference when we look at the values of the unipotent characters; here, we just focus on unipotent elements. The values are printed in Table 2.10; they can be extracted from the tables in [Sr68] and [Eno72]. (We work with the simple group G = PCSp4 (k) of adjoint type, because the unipotent classes are easier to describe in this case.) If p  2, then there are four unipotent classes which we denote by Oμ , where the subscript μ specifies the Jordan type of the elements in the class. (There is an isogeny Sp4 (k) → G which induces a bijection between the unipotent classes of Sp4 (k) and of G; thus, we may also speak of the Jordan type of unipotent elements in G.) For example, the elements in O(211) have one Jordan block of size 2 and two F splits into two classes in GF which we denote by blocks of size 1. The set O(22)

2.7 Average Values and Unipotent Support

191

Table 2.10 Unipotent characters of PCSp4 (q) on unipotent elements p2 (u)F |

|CG θ0 θ9 θ 10 θ 11 θ 12 θ 13

p=2 (u)F |

|CG θ0 θ9 θ 10 θ 11 θ 12 θ 13

O(211) 4 q (q2 − 1)

O(22) 3 2q (q − 1)

 O(22) 3 2q (q + 1)

1

1

1 2 2 q(q + 1) 1 q(q − 1)2 2 1 2 2 q(q + 1) 1 q(q2 + 1) 2 q4

1 2 q(q + 1) − 12 q(q − 1) − 12 q(q − 1) 1 q(q + 1) 2

.

1 q . q . .

1 . q . q .

O(211) 4 q (q2 − 1)

∗ O(22) 4 q (q2 − 1)

O(1111) |GF |

O(1111) |GF |

O(22)

O(4)

q4

2q2

1

1

1

1

1

1 q(q + 1)2 2 1 2 2 q(q − 1) 1 q(q2 + 1) 2 1 2 2 q(q + 1) q4

1 2 q(q + 1) − 12 q(q − 1) − 12 q(q − 1) 1 q(q + 1) 2

1 q(q + 1) 2 − 12 q(q − 1) 1 q(q + 1) 2 − 12 q(q − 1)

q 2 q 2 q 2 q 2

q 2 q 2 q −2 q −2

.

.

.

O(4)

.

q2 1 . . . . .  O(4)

2q2 1 q −2 q −2 q 2 q 2

.

 ; each of the remaining classes O gives rise to exactly one class in O(22) and O(22) μ F G , which we denote by Oμ . If p = 2, then we use a similar convention for denoting unipotent classes as above; just note that, now, there are two unipotent classes of G with elements of Jordan ∗ . type (22), which we denote by O(22) and O(22) By inspection of Table 2.10, we obtain the unipotent support of all ρ ∈ Uch(GF ):

ρ θ0 θ 9, θ 10, θ 11, θ 12 θ 13

aρ 0 1 4

nρ 1 2 1

Oρ O(4) O(22) O(1111)

AV(ρ, Oρ ) 1 (p2), 2 (p=2) q (p2), q2 (p=2) q4 (p2), q4 (p=2)

We also see in Table 2.10 that, for p = 2, the conclusion of Theorem 2.7.20 fails: the four characters θ 9, θ 10, θ 11, θ 12 have non-zero values on O(4) but the unipotent support is a class of strictly smaller dimension. (It may also be instructive to work with the individual character tables for p = q = 3 and p = q = 2 printed in Table 2.1, p. 95.) Finally, by considering the whole character table of GF for p = 2, one sees that ∗ O(22) is not the unipotent support of any irreducible character of GF . (This is not an isolated phenomenon, see [GeMa00, Remark 3.9].)

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Lusztig’s Classification of Irreducible Characters

√ 2m+1 Example 2.7.23 Let GF = 2B2 (q2 ) be a Suzuki group, where q = 2 for F some m  0. The values of the unipotent characters of G are printed in Table 2.9 (p. 133). Let O  be the unipotent class of the element denoted by u  in that table. We see that O  is the unipotent support of the characters ,  ; furthermore, we have √ AV(, O ) = AV( , O ) = − 12 2q, √ which is consistent with the formula in Proposition 2.7.16 since n = n  = 2 and aρ = 1. Again, we observe that ,   take non-zero values on a class of strictly bigger dimension than that of O . Finally, we briefly indicate how Lusztig’s theory of character sheaves enters the picture. By the basic construction of [DeLu76] (as explained in the previous sections), we obtain representations of GF on cohomology spaces Hic (X, Q ). In the theory of character sheaves, a completely different approach is used. One obtains class functions on GF by taking the trace of the action of the Frobenius map F on certain cohomology spaces associated with G. A priori, it is not at all clear that these class functions have anything to do with actual (or even just virtual) representations of GF . But, by this theory, Lusztig obtains a new basis of CF(GF ) which is, at least in principle, computable (see also the next section). Hence, in this picture, the whole problem of computing the character table of GF amounts to finding the base change from this new basis of CF(GF ) to the basis Irr(GF ). The new basis comes about by working with perverse sheaves in the bounded derived category DG of constructible Q -sheaves on G, in the sense of Beilinson, Bernstein, Deligne [BBD82]. The objects of this category are extremely complicated, and we will not even try to attempt to explain this. (See [Lu87a], [Lau89], [MaSp89], [Sho88] for further expositions and introductions.) 2.7.24 The ‘character sheaves’ on G, introduced and studied by Lusztig [LuCS], are certain irreducible perverse sheaves in DG that are equivariant for the action of G on itself by conjugation. What is remarkable is that in many situations we do not need to know exactly how these objects are defined, but only how to manipulate them (assuming some general familiarity with sheaf theory). Let K ∈ DG. Then K is represented by a complex of Q -sheaves K:

. . . → Ki−1 → Ki → Ki+1 → . . . ,

such that Ki = {0} if |i| is large. For any i ∈ Z, we have the ith cohomology sheaf, denoted by H i (K). Given g ∈ G, the stalks Hgi (K) are finite-dimensional Q -vector spaces. Using the Frobenius map F : G → G, we can form the inverse

2.7 Average Values and Unipotent Support

193

image F ∗K ∈ DG, which has the property that i Hgi (F ∗K) = HF(g) (K)

for i ∈ Z, g ∈ G. ∼

Let us assume now that K is isomorphic to F ∗K in DG. An isomorphism φ : F ∗K → K induces linear maps φi,g : Hgi (F ∗K) → Hgi (K) for each i ∈ Z and g ∈ G. If i (K) = H i (K) and so we obtain endomorphisms g ∈ GF , then Hgi (F ∗K) = HF(g) g i

φi,g ∈ End Hg (K) for any i ∈ Z. Following [LuCS, II, 8.4.1], the function

g → (−1)i Trace(φi,g, Hgi (K)), χK,φ : GF → Q , i

is called the characteristic function of K (with respect to φ). If K is irreducible, then, by a version of Schur’s Lemma, φ is unique up to a non-zero scalar; hence, χK,φ is unique up to a non-zero scalar. Theorem 2.7.25 ([LuCS, V, §25], [Lu12a]) (1) Let A ∈ DG be a character sheaf with F ∗A  A. Then there is an isomorphism ∼ φ : F ∗A → A such that the values of χA,φ are cyclotomic numbers (so we can assume χA,φ (g) ∈ K for all g ∈ GF ) and we have  χA,φ, χA,φ = 1. (2) If A, φ are as in (1), then the values of χA,φ on all elements of GF can be computed ‘in principle’. (3) The characteristic functions { χA,φ | A, φ as in (1), up to isomorphism} form an orthonormal basis of the space of class functions on GF . The existence of such a basis was already conjectured in [Lu84a, 13.7]. In [LuCS, Part V], the above theorem is proved under some mild conditions on p. These conditions were subsequently removed in [Lu12a]. We just mention that a part of the proof in [Lu12a] relies on explicit computations using data and programs in CHEVIE, concerning a canonical map {conjugacy classes of W }  {unipotent classes of G}

(see [Lu11]).

See [Ge11], [MiChv] for some further explanations about the computations. Remark 2.7.26 The problem of computing the character table of GF is now reduced to the problem of finding the base change from Irr(GF ) to the basis of class functions in Theorem 2.7.25(c). That is, we need to express each characteristic function χA,φ explicitly as a linear combination of the irreducible characters of GF . Conjecturally (see [LuCS, II, p. 226]), these linear combinations should be given by the almost characters defined by [Lu84a, 4.24.1] (in the case where Z(G) is connected) and by [Lu18a] (in general); see also Remark 2.4.17 where the uniform almost characters were defined. The conjecture is known in many cases, but not in

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Lusztig’s Classification of Irreducible Characters

general; solving this problem involves, in particular, the tricky issue of specifying a ∼ particular isomorphism φ : F ∗ A → A for any F-stable character sheaf A on G. The first successful realization of this whole program was carried out in [Lu86a], where character values on unipotent elements are determined. For further cases see [Lu92b], [Sho95], [Sho97], [Sho06a], [Sho09], [Bo06], [Wal04] and the references there. (See also the surveys [Sho98], [Ge18].) Example 2.7.27 There is a notion of ‘cuspidal’ character sheaves; see [Lu84c, Def. 2.4], [LuCS, I, Def. 30.10; II, §7], or [Lu92b, 1.1]. Assume now that G is simple. Then there are only finitely many cuspidal character sheaves on G (up to isomorphism), and these have been classified in all cases; see [Lu84c, 2.10], [Lu84c, §10–§15], [LuSp85]; see also [Sho95, I, §7; II, §5]. Assume that A ∈ DG ∼ is a cuspidal character sheaf such that F ∗ A  A; let φ : F ∗ A → A be an isomorphism as in Theorem 2.7.25. Then, by [Lu04b, 19.7] and the ‘cleanness’ result of [Lu12a], there is an F-stable conjugacy class C of G (which has some very specific properties) such that χA,φ = q(dim G−dim C )/2Y(g, ς) ˜,

(♦)

where g ∈ C F and ς˜ is a suitable F-extension of some ς ∈ Irr(A(g))F , as in 2.7.2. Here, the choice of φ determines ς˜ and vice versa. In [Lu92b, Theorem 0.8], the decomposition of χA,φ as a linear combination of Irr(GF ) is determined in each case, assuming that p is sufficiently large, and up to specifying the exact choice of ς. ˜ We have in fact already encountered some instances of this problem. For example, let GF = SL2 (q) where q is odd. Then the two class functions 1  2 (ψ+

− ψ+ + ψ− − ψ−)

and

1  2 (ψ+

− ψ+ − ψ− + ψ−)

shown in Example 2.2.30 are characteristic functions of F-stable cuspidal char√ acter sheaves. Similarly, the class function 12 2( −  ) of GF = 2B2 (q2 ) in Example 2.3.31 is such a characteristic function. Finally, if GF = PCSp4 (q), then the class function Γ1 := 12 (θ 9 + θ 10 − θ 11 − θ 12 )

(notation as in Table 2.10)

is the characteristic function of an F-stable cuspidal character sheaf. If p  2, then Γ1 takes values ±q on the conjugacy class of elements su ∈ GF where s is a certain involution and u is regular unipotent in CG◦ (s); see [Sr91, A.1, p. 192]. If p = 2, then we can read off Table 2.10 that Γ1 takes values ±q on the two unipotent classes  (and the whole character table in [Eno72] shows that Γ is 0 everywhere O(4), O(4) 1 else). It is an on-going project to extend the above-mentioned [Lu92b, Theorem 0.8]

2.8 On the Values of Green Functions

195

to the cases where p is small, and to determine the appropriate choice of ς˜ in each case; see [Sho06a], [Sho09], [Ta14b], [Ge19a], [He19].

2.8 On the Values of Green Functions In this section, we consider in more detail the problem of computing the values of a Deligne–Lusztig character RTG (θ) at an arbitrary element g ∈ GF . The character formula in Theorem 2.2.16 essentially reduces this problem to the case where g = u is unipotent and, hence, to Green functions. All this is part of the more general program sketched at the end of the previous section. 2.8.1 Let T ⊆ G be an F-stable maximal torus. Recall from Definition 2.2.15 that the Green function QG T is defined by G

F . QG for u ∈ Guni T (u) := RT 1T (u) G −1 F Also recall that QG T (u) = Q T (u ) ∈ Z for all unipotent u ∈ G . It will be convenient G F F . Let g ∈ GF to regard QT as a function on all of G , which takes value 0 outside Guni F F and write g = us = su where s ∈ G is semisimple and u ∈ G is unipotent. By Example 2.2.17(b), we have  F G

|T |QT (u) if s = 1, G

RT θ (g) = (a) 0 otherwise. F θ ∈Irr(T )

Combining (a) with the orthogonality relations for RTG (θ) in Theorem 2.2.8, we obtain the following orthogonality relations for Green functions: G QG T , Q T =

|{g ∈ GF | T = gTg −1 }| |TF ||TF |

(b)

where T ⊆ G is another F-stable maximal torus (see also [Ca85, 7.6.2]). 2.8.2 Let T ⊆ G be an F-stable maximal torus and let w ∈ W be such that T 1 (where the is of type w (see 2.3.18). By Lemma 2.3.19, we have RTG (1T ) = Rw 1 (u) superscript 1 stands for the trivial character of T0 [w]). We set Q w (u) := Rw F F for u ∈ Guni (and also regard Q w as a function on all of G , which takes value 0 F ). Then Q G = Q . As in Example 2.3.22, we may re-write the above outside Guni w T orthogonality relations as follows: Q w, Q w =

|{x ∈ W | w  = xwσ(x)−1 }| |Tw |(q)

(w, w  ∈ W)

where σ : W → W is the automorphism induced by F and |Tw | = det(q idXR − ϕ0 ◦ w −1 )

for all w ∈ W (see 1.6.21).

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Note that Q w = Q w if w, w  ∈ W are σ-conjugate; moreover, Q w, Q w = 0 if w, w  are not σ-conjugate in W. We now perform a transformation in analogy to that in Remark 2.4.17. For each φ ∈ Irr(W)σ , let us fix a real-valued σ-extension ˜ which exists by Proposition 2.1.14. Then we set φ, 1 ˜ φ(w)Q w . Q φ˜ := |W| w ∈W  ˜ We can invert these relations and also write Q w = φ ∈Irr(W)σ φ(w)Q φ˜ for all w ∈ W. Thus, the problem of computing the values of the Green functions Q w is equivalent to that of computing the values of the functions Q φ˜ . For each unipotent class O of G, we fix once and for all a representative uO ∈ O; here, we tacitly assume that F(uO ) = uO if O is F-stable. Let IGF be the set of all pairs (O, ς) where O is an F-stable unipotent class of G and ς ∈ Irr(A(uO ))F . If (O, ς) ∈ IGF and ς˜ is an F-extension of ς, then we have a corresponding function F Y(uO, ς) ˜ ∈ CF(G ) as in 2.7.2. Theorem 2.8.3 (Springer, Kazhdan, Lusztig, Shoji, . . .) Assume that F : G → G is a Frobenius map. For each φ ∈ Irr(W)σ , there exists a unique pair (O, ς) ∈ IGF F | Q (u)  0} ⊆ O and such that {u ∈ Guni φ˜ Q φ˜ |O F = q(dim G−dim T0 −dim O )/2 Y(uO, ς) ˜ where ς˜ is a suitable F-extension of ς. (See Remark 2.8.8 below for further comments about ς.) ˜ The resulting map Irr(W)σ → IGF is injective. The map Irr(W)σ → IGF is called the Springer correspondence. The pairs in the image of this map will be called the uniform pairs in IGF . These pairs and the correspondence itself are explicitly known for all G; see the summaries in [Ca85, §13.3] (with some restrictions on p) and [Lu84c], [LuSp85], [Spa85]. See also [Hum95, Chap. 9] for further comments and references. Theorem 2.8.4 (Springer, Kazhdan, Lusztig, Shoji, . . .) With the assumptions of Theorem 2.8.3, let Υ ⊆ CF(GF ) be the subspace spanned by {Q φ˜ | φ ∈ Irr(W)σ }. F (a) Υ is equal to the subspace spanned by the functions Y(uO, ς) ˜ where (O, ς) ∈ IG is a uniform pair and ς˜ is any F-extension of ς. In particular, each such Y(uO, ς) ˜ is a uniform function (cf. Definition 2.2.25). (b) If (O, ς) ∈ IGF is not a uniform pair, then Y(uO, ς) ˜ is orthogonal to the subspace Υ, for any F-extension ς˜ of ς.

It is almost impossible to trace here the exact history of the proof of these important results, which stretches from [DeLu76] up until quite recently. Based on the fundamental papers of Springer [Spr76], [Spr78] and Kazhdan [Kaz77],

2.8 On the Values of Green Functions

197

first instances of the above results were established by Shoji [Sho82], [Sho83] and Beynon–Spaltenstein [BeSp84], assuming that p, q are sufficiently large. It turned out to be a formidable task to remove the assumptions on p and q. The state of knowledge up until around 1986, incorporating further results by Borho– MacPherson, Lusztig and others, is summarised in Shoji’s survey articles [Sho86], [Sho88]. At that stage the general plan was laid out, with Lusztig’s theory of character sheaves [LuCS] being the new essential ingredient. In this picture one had to show that the Green functions Q w coincide with another type of Green functions defined in terms of character sheaves, for which the properties in Theorem 2.8.3 are known to hold by [LuCS, 24.1, 24.2] and those in Theorem 2.8.4 would hold by [LuCS, 24.4], assuming certain ‘cleanness’ assumptions. In [Spa82b], [Ma93b], Green functions were explicitly computed for some groups of exceptional type and the ‘bad’ prime p = 2; thus establishing Theorems 2.8.3, 2.8.4 in these cases. A decisive step in the general direction was taken by [Lu90], which was then completed by [Sho95, Part II, 5.5]. As a result, the two types of Green functions are indeed known to coincide without any assumption on p, q; this immediately yields Theorem 2.8.3. The final step is provided by [Lu12a] which establishes the cleanness assumption of [LuCS, §24] in complete generality and immediately implies Theorem 2.8.4. (Actually, the full version of cleanness is not needed to get Theorem 2.8.4; see [Ge96, §3] for a further discussion. On the other hand, there are also ‘generalised’ Green functions in [LuCS] for which the full version of cleanness is required.) Remark 2.8.5 The interpretation in terms of character sheaves (as mentioned above) yields the following compatibility property of the Springer correspondence Irr(W)σ → IGF . Let n  1. Then F n : G → G also is a Frobenius map; it n induces the automorphism σ n : W → W. Clearly, we have Irr(W)σ ⊆ Irr(W)σ n and IGF ⊆ IGF . Now [LuCS, 24.2] yields a commutative diagram n  - I Fn Irr(W)σ G ∪ Irr(W)σ



∪ - IF G

where the top arrow is the correspondence in Theorem 2.8.3 with respect to G, F n . Example 2.8.6 Let us consider the trivial character 1W and the sign character ε of W. We choose the trivial σ-extensions 1˜ W = 1W and ε˜ = ε (see Example 2.1.8(c)). F . By Example 2.3.23, we have Now let u ∈ Guni  |GF | p if u = 1, Q1˜ W (u) = 1G (u) = 1 and Q ε˜ (u) = StG (u) = 0 if u  1. (Note that an irreducible character of a finite group with p-defect 0 has value 0 on all non-trivial p-elements.) Hence, the defining conditions in Theorem 2.8.3

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immediately show that 1W corresponds to the uniform pair (O0, 1) ∈ IGF where O0 is the regular unipotent class and 1 stands for the trivial character of A(uO0 ). Similarly, ε corresponds to the uniform pair ({1}, 1) ∈ IGF where, again, 1 stands for the trivial character of A(1). Example 2.8.7 Let G = GLn (k) and F : G → G be the standard Frobenius map (see Example 1.4.21) such that GF = GLn (q). We have W  Sn (the symmetric group of degree n) and σ : W → W is the identity. As above, let Q w (for w ∈ 1 to GF . We have noted in Example 2.4.20 that W) denote the restriction of Rw uni the functions Q w are the same as the Green functions originally introduced and investigated in [Gre55]. Let us write again Irr(W) = {φν | ν n} and set 1 ν φ (w)Q w for any ν n. Qν := |W| w ∈W Now the unipotent classes of G are also naturally parametrised by the partitions of n. For μ n, let uμ ∈ G be the block-diagonal matrix formed by the Jordan blocks (with eigenvalue 1) of sizes given by the non-zero parts of μ. Let Oμ be the G-conjugacy class of uμ . Then {Oμ | μ n} is the set of unipotent classes of G and, clearly, we have F(uμ ) = uμ for all μ n. Further note that CG (g) is connected for any g ∈ G, and so A(uμ ) = {1} for all μ n. Finally, if we set

(i − 1)μi where μ = (μ1  μ2  · · ·  μr ) n, n(μ) := 1ir

then (dim G − dim T0 − dim Oμ )/2 = n(μ); see, e.g., [Ge03a, 2.6.1]. Based on the fundamental results of [Gre55], and extensions of these results in [Mor63], it is shown in [Ohm77, 2.14] that, for any μ, ν n, we have: Qν (uν ) = q n(ν)

and

Qν (uμ ) = 0 unless μ  ν.

Here, we write μ  ν if μ is less than or equal to ν in the lexicographical ordering. n, it is well known that Oμ ⊆ O μ ⇔ μ  μ, where  Now, given μ, μ denotes the dominance order on partitions; see, e.g., [Ge03a, 2.6.5]. It was noticed in [Kaw85, 3.2.19] that, in the above assertion about Qν (uμ ), we can replace the lexicographical ordering by the dominance order. Hence, we conclude that F {u ∈ Guni | Qν (u)  0} ⊆ O ν

and

Qν |OνF = q n(ν)Y(uν , ς) ˜

F where Y(uν , ς) ˜ (u) = 1 for u ∈ Oν (that is, ς˜ = ς is the trivial id-extension of the trivial character of A(uν ) = {1}). Thus, we have explicitly verified Theorems 2.8.3 and 2.8.4 for GF = GLn (q), where the correspondence Irr(W)σ → IGF is a bijection: it sends the character φν of W  Sn to the pair (Oν, 1) (where 1 stands for the trivial character of A(uν ) = {1}). For a further detailed discussion of the Green

2.8 On the Values of Green Functions

199

functions in this case, including both combinatorial and cohomological aspects, see [HoSh79] and [Mac95]. Remark 2.8.8 Let φ ∈ Irr(W)σ and (O, ς) ∈ IGF as in Theorem 2.8.3. Let ς˜ be the F-extension of ς such that Q φ˜ |O F = q(dim G−dim T0 −dim O )/2 Y(uO, ς) ˜ . In general, what can we say about the F-extension ς? ˜ This is quite a delicate matter, which is solved in almost all cases, but there are still open cases for type E8 and p = 2, 3, 5. (See the references below.) To begin with, the values of the Green functions Q w are rational integers and so are the values of the σ-extensions of the characters in Irr(W)σ (since F is a Frobenius map and so σ is ordinary; see Proposition 2.1.14). It follows that the values of the function ς˜ are in Q. On the other hand, by [LuCS, 24.2.4], [Lu04b, 19.7], the F-extension ς˜ is indeed formed using a matrix E of finite order (cf. 2.1.9) and so the values of ς˜ are cyclotomic integers in K. Hence, ς(a) ˜ ∈Z

for all a ∈ A(uO ).

Finally, assume that F acts trivially on A(uO ), or that ς is a linear character. (This will cover most cases in practice.) Then the above discussion shows that there is a root of unity δ such that ς(a) ˜ = δς(a) for a ∈ A(uO ). Since the values of ς˜ are in Q, it follows that δ = ±1 and ς(a) ∈ Z for all a ∈ A(uO ). Furthermore, in this case, δ = ±1 is determined by the equality Y(uO, ς) ˜ (uO ) = δς(1).

(♥)

See the remarks in [Sho86, 5.1], and also [BeSp84], [Sho06b], [Sho07], [Ge19b] for further results and details. It is important to keep in mind that, even if F acts trivially on A(uO ) and if ς is a linear character, then there are examples where δ = −1; see [BeSp84, §3, Case V], [Ge19c, §9]. 2.8.9 The Lusztig–Shoji algorithm. We can now describe the fundamental algorithm in [LuCS, 24.4] which yields explicit expressions of the functions Q φ˜ as linear combinations of the functions Y(uO, ς) ˜ . (It modifies and simplifies an algorithm described earlier by Shoji; see [Sho86, §5].) First, we need to prepare some notation. Let φ ∈ Irr(W)σ and (O, ς) ∈ IGF be the corresponding uniform pair as in Theorem 2.8.3, such that Q φ˜ |O F = q(dim G−dim T0 −dim O )/2 Y(uO, ς) ˜ . Then we denote the function Y(uO, ς) ˜ by Yφ˜ . Furthermore, we set Xφ˜ := q−dO Q φ˜

where

dO := (dim G − dim T0 − dim O)/2;

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Lusztig’s Classification of Irreducible Characters

it will also be convenient to set dφ := dO in this situation. The functions Xφ˜ will be called modified Green functions. By 2.7.2(b) and Theorem 2.8.4, the functions {Yφ˜ | φ ∈ Irr(W)σ } form a basis of the vector space Υ; in particular, we obtain a system of equations

Xφ˜ = pφ,φYφ˜  φ ∈Irr(W) σ

where the coefficients pφ,φ ∈ K are uniquely determined. Using the relations in 2.1.10 and 2.8.2, we obtain |GF |Xφ˜ , Xφ˜  = ω˜ φ,φ (q) where ω˜ φ,φ :=

1 −dφ −dφ |G| ˜ q φ(w)φ˜(w) ∈ R[q] |W| |T | w w ∈W

for φ, φ  ∈ Irr(W)σ .

Thus, the scalar products Xφ˜ , Xφ˜  are obtained from a symmetric matrix

˜ := ω˜ φ,φ Ω φ,φ ∈Irr(W) σ with entries in R[q], which can be computed explicitly using the σ-character table of W and the order polynomials |Tw | ∈ R[q]. Now we obtain:

pψ,φ pψ,φ |GF |Yψ˜ , Yψ˜  . ω˜ φ,φ (q) = |GF |Xφ˜ , Xφ˜  = ψ,ψ ∈Irr(W) σ

Let us set λ˜φ,φ := |GF |Yφ˜ , Yφ˜  ∈ K for all φ, φ  ∈ Irr(W)σ . Since the values of each Q φ˜ and of each Yφ˜ are in Q, it follows that pφ,φ ∈ Q

λ˜φ,φ = λ˜φ,φ ∈ Q

and

for all φ, φ  ∈ Irr(W)σ .

We can now write the above equations as a single matrix equation

P := pψ,φ ψ,φ ∈Irr(W)σ , ˜ · P = Ω(q) ˜

where Ptr · Λ ˜ := λ˜φ,φ Λ φ,φ ∈Irr(W) σ . ˜ But if In general, this system of equations will not have a unique solution for P, Λ. we take into account the additional information in Theorem 2.8.3, then it does have a unique solution. Indeed, let O1, . . . , Om be the F-stable unipotent classes of G; let us also denote ui := uOi ∈ OiF . Here, we choose the labelling such that dim O1  dim O2  · · ·  dim Om . This gives rise to a partition Irr(W)σ = I1  · · ·  Im, where each subset I j consists of those φ ∈ Irr(W)σ that correspond (via Theorem 2.8.3) to a uniform pair of the form (O j , ς) ∈ IGF for some ς ∈ Irr(A(u j )). We now enumerate the characters in Irr(W)σ in a way that is compatible with the

2.8 On the Values of Green Functions

201

˜ has a block diagonal shape, where above partition of Irr(W)σ . Then it is clear that Λ the blocks correspond to the sets I1, . . . , Im . Furthermore, Theorem 2.8.3 implies that P has an upper block triangular shape with identity matrices on the diagonal. More precisely, we can write: ⎡ ⎢ ⎢ ⎢ P = ⎢⎢ ⎢ ⎢ ⎢ ⎣

Ie1 P1,2 · · ·

P1,m .. .

0 Ie2 .. .. . Pm−1,m . 0 · · · 0 Ie m

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

and

⎡ ⎢ ⎢ ⎢ ˜ Λ = ⎢⎢ ⎢ ⎢ ⎢ ⎣

˜1 0 ··· 0 ⎤ Λ ⎥ .. ⎥ ˜2 . ⎥⎥ 0 Λ ⎥; .. .. . 0 ⎥⎥ . ˜ m ⎥⎦ 0 ··· 0 Λ

here, e j = |I j | and Ie j denotes the identity matrix of size e j . For 1  i < j  m, the block Pi, j has size ei × e j and entries pφ,φ ∈ Q for ˜ j has size e j × e j and entries λ˜φ,φ ∈ Q φ ∈ Ii and φ  ∈ I j ; similarly, the block Λ for φ, φ  ∈ I j . ˜ from the above It is then a standard linear algebra problem to compute P and Λ matrix equation; see [LuCS, 24.4] for further details, and [LaSr90], [Ge03d, §5] for ˜ and the Springter correspondence the discussion of some examples. Thus, given Ω σ F Irr(W) → IG , we have a purely automatic procedure for the computation of P ˜ (See also [GeMa99] for a variation of that algorithm.) This is implemented and Λ. in Michel’s version of CHEVIE [MiChv], through the functions UnipotentClasses

and

ICCTable

which take as input a finite Weyl group (or, more generally, a root datum) and a prime p (the characteristic of the field over which G is defined). The functions use extensive databases within CHEVIE which hold the explicitly known information about the Springer correspondence Irr(W)σ → IGF . ˜ has the property that every principal minor is nonRemark 2.8.10 The matrix Ω ˜ zero. (This follows from the fact that Ω(q) is the matrix of mutual scalar products of linearly independent class functions on GF .) Hence, the system of equations ˜ · P = Ω(q) ˜ can be solved at the level of rational functions in q. We will not Ptr · Λ ˜ can be expressed as formalise this here, but it means that the entries of P and Λ values at q of well-defined rational functions in q. By letting q vary over a suitable infinite set of prime powers, and using [LuCS, 24.5.2], one can even show that those rational functions are polynomials in q. (See the argument in the first part of the proof of [LuCS, 24.8].) Using the output of the above algorithm, we can now also solve the issue concerning the F-extensions ς˜ in Remark 2.8.8, at least in some special cases. 2.8.11 Let Irr(W)1σ be the set of all φ ∈ Irr(W)σ such that φ corresponds to a pair (O, 1) ∈ IGF (as in Theorem 2.8.3) where 1 stands for the trivial character of

202

Lusztig’s Classification of Irreducible Characters

A(uO ). Thus, given φ ∈ Irr(W)1σ with corresponding pair (O, 1) ∈ IGF , there is a sign δ = ±1 such that Yφ˜ (u) = Y(uO, 1) ˜ (u) = δ

for all u ∈ O F

(see Remark 2.8.8). Let 1˜ W = 1W be the trivial σ-extension of the trivial character of W. Then, as in [BeSp84, p. 587], we claim that δ = pφ,1W

(where pφ,1W is an entry of the matrix P in 2.8.9).

F ; the uniform pair Indeed, by Example 2.8.6, we have Q1˜ W (u) = 1 for all u ∈ Guni F corresponding to 1W is given by (O0, 1) ∈ IG where O0 is the regular unipotent class. By 2.2.21, we have dO0 = 0. As in 2.8.9, we have

pψ,1W Yψ˜ (u) 1 = Q1˜ W (u) = q dO0 X1˜ W (u) = ψ ∈Irr(W) σ

F . By 2.7.2(b), the functions {Y | ψ ∈ Irr(W)σ } are linearly for all u ∈ Guni ψ˜ independent. Hence, we deduce that pψ,1W = 0 if ψ  Irr(W)1σ ; furthermore, if ψ = φ ∈ Irr(W)1σ , then the non-zero values of pφ,1W Yφ˜ must be equal to 1, and so pφ,1W δ = 1, as desired. This also shows that, if φ ∈ Irr(W)1σ , then the corresponding pair (O, 1) ∈ IGF is a uniform pair.

Remark 2.8.12 Explicit descriptions of the subsets Irr(W)1σ ⊆ Irr(W)σ can be found in [Lu05, 1.3] (completing earlier work, assuming that p is large enough, in [Lu79b]). In particular, we obtain an explicit bijection ∼

Irr(W)1σ −→ {F-stable unipotent classes of G},

φ → Oφ,

where Oφ is the F-stable unipotent class such that φ corresponds to the uniform pair (Oφ, 1) ∈ IGF . Thus, the number of F-stable unipotent classes of G is seen to be bounded above by the cardinality of Irr(W)σ and, hence, by the number of σ-conjugacy classes of W. Example 2.8.13 Let GF = GLn (q) as in Example 2.8.7. In this case, the characters of W  Sn and the F-stable unipotent classes of G are both parametrised by the partitions of n. Furthermore, each piece I j in the partition of Irr(W)σ in 2.8.9 is just a singleton set. The requirement to order the unipotent classes by increasing dimension means that we have to order the partitions ν n according to decreasing ˜ is value of n(ν). Then P is an upper triangular matrix with 1 on the diagonal and Λ a diagonal matrix. Let us write P = (pμν )μ,ν

n

where

pμν := pφμ ,φν .

We already saw in Example 2.8.7 that each function Y(uμ , ς) ˜ takes the constant value 1

2.8 On the Values of Green Functions

203

Table 2.11 Modified Green functions for GLn (q), n = 2, 3, 4 n=2 n(ν) (11) 1 (2) 0

(11) 1 1

(2) . 1

n=3 n(ν) (111) 3 (21) 1 (3) 0

(1111) 1 q2 + q + 1 q2 + 1 2 q +q+1 1

n=4 n(ν) (1111) 6 (211) 3 (22) 2 (31) 1 (4) 0

(211) . 1 1 q+1 1

(111) 1 q+1 1 (22) . . 1 1 1

(21) . 1 1

(31) . . . 1 1

(3) . . 1

(4) . . . . 1

on OμF . Thus, we conclude that Qν (uμ ) = q n(ν) pμν

for all μ, ν

n.

Finally, by [Gre55] (see also [LuCS, 24.8] for a more general argument), it is known that there exist polynomials πμν ∈ Q[q] (depending only on n, and defined in terms of ‘Hall polynomials’) such that pμν = πμν (q) for all prime powers q. By the algorithm in 2.8.9, we obtain the example matrices Ptr in Table 2.11. Note that the ˜ are given by entries of the diagonal matrix Λ λ˜μμ = |GF : CG (uμ )F |

for all μ

n

in this case, and these numbers are also obtained by evaluating certain well-defined polynomials at q. Thus, if we denote these polynomials by cμ ∈ Q[q] (μ n), then we actually have a system of polynomial equations

ω˜ μν = πν μ cν πνν for μ, ν n, ν n

where the matrix (πμν ) is upper triangular with 1 on the diagonal. Example 2.8.14 Let GF = G2 (q) where |GF | = q6 (q2 −1)(q6 −1) and q is a power of a prime p  2, 3. The Green functions in this case were first determined in [Spr76, 7.16], before the above machinery was available. The Weyl group W = s1, s2 is a dihedral group of order 12, with F acting trivially on W. We fix the notation so that s1 corresponds to a simple long root (which we denote by β) and s2 corresponds to a simple short root (which we denote by α). There are five unipotent classes in G, which are all F-stable. The relevant information about these classes, as well as the Springer correspondence Irr(W)σ → IGF , are given in Table 2.12; here, the characters of W are denoted as in Example 2.1.13. (For the information about the classes Oi , see [Cha68] and Remark 2.8.15 below;

204

Lusztig’s Classification of Irreducible Characters Table 2.12 The Springer correspondence for G2 , p  2, 3 Oi O1 O2 O3 O4

dOi ui |CG (ui )F | A(ui ) ς φ ∈ Irr(W) = 6 1 |GF | {1} 1 ε = A1 3 x3α+2β (1) q6 (q2 −1) {1} 1 ε  = A˜ 1 2 x2α+β (1) q4 (q2 −1) {1} 1 φ2 = G2 (a1 ) 1 xβ (1)x2α+β (−3) 6q4 S3 ς(3) φ1 ς(21) ε ς(111) non-uniform O5 = G2 0 xα (1)xβ (1) q2 {1} 1 1W Ptr ε ε  φ2 φ1 ε 1W

 1 1 q2 + 1 q4 + 1 q2 1

A1 . 1 1 1 . 1

A˜ 1 . . 1 1 1 1

G2 (a1 ) . . . . . . 1 . . 1 1 .

G2 . . . . . 1

for the Springer correspondence, see [Ca85, p. 427] or [Spa85, p. 329].) By the Lusztig–Shoji algorithm in 2.8.9, we obtain the matrix Ptr printed in Table 2.12. Now let φ ∈ Irr(W)σ and (O, ς) ∈ IGF be the corresponding uniform pair. It remains to determine the F-extension ς˜ that enters in the definition of Yφ˜ = Y(uO, ς) ˜ ; see Remark 2.8.8. First of all, since σ is the identity, we can take φ˜ = φ for each φ ∈ Irr(W)σ . Since F acts trivially on A(uO ), we can apply Remark 2.8.8(♥). So there exists a sign δ = ±1 such that ς˜ = δς and Yφ˜ (uO ) = Y(uO, ς) ˜ (uO ) = δς(1). We claim that δ = 1 in all cases. If ς = 1 is the trivial character of A(uO ), then this immediately follows from the entries in the last row of the above matrix Ptr and 2.8.11. It remains to consider the case where O = O4 and ς = ς(21) is the character of degree 2 of A(u4 )  S3 . Now one either has to enter a study of the geometry of the variety of Borel subgroups containing u4 (e.g., as in [BeSp84, §3]), or one has to use ad hoc information that is available in some other way. In the present situation, we can argue as follows. Let B0 ⊆ G be an F-stable Borel subgroup containing our reference torus T0 such that u4 ∈ B0F . Since A(u4 )  S3 is non-abelian and of order prime to p, we certainly have CG (u4 )F  B0F and so F

(1 )(u4 ) > 1. IndG B F B0 0

On the other hand, by Proposition 2.2.7, the above induced character equals R11 =

2.8 On the Values of Green Functions RTG0 (1T0 ). We now compute:

R11 (u4 ) = Q1 (u4 ) = φ ∈Irr(W) σ

φ(1)Q φ˜ (u4 ) =

ψ,φ ∈Irr(W) σ

205

φ(1)pψ,φ q dφ Yψ˜ (u4 )

= φ1 (1)qYφ˜1 (u4 ) + ε (1)qYε˜  (u4 ) + 1W (1)Yφ˜1 (u4 ) = (2q + 1)Yφ˜1 (u4 ) + qYε˜  (u4 ) = (2q + 1) + 2qδ. Since R11 (u4 ) > 1, we must have δ = 1 in this case as well. Hence, for every uniform pair (O, ς) ∈ IGF , the function Y(uO, ς) ˜ is formed using ς˜ = ς. (Using the character tables in [Eno76], [EnYa86], one also recovers easily the Green functions for G2 (q) where q is a power of 2 or 3.) Remark 2.8.15 Consider the unipotent class O4 = G2 (a1 ) in Table 2.12. According to [Cha68] (see also [Hum95, 8.16]), we have A(u)  S3 for u ∈ O4 , and O4F splits into three classes in the finite group GF , with centraliser orders 6q4, 2q4, 3q4 . Hence, these three classes into which O4F splits can be distinguished just by looking at centraliser orders (or class sizes). So it is natural to fix a representative u4 ∈ O4F such that |CG (u4 )F | = 6q4 , in which case F acts trivially on A(u4 ). By [Cha68, 3.18], such a representative is given by  xβ (1)x2α+β (1) if q ≡ 1 mod 3, u4 = if q ≡ −1 mod 3, xβ (1)x2α+β (ξ) where 0  ξ ∈ Fq is a non-square. In fact, the formulae in [Cha68, §2] for the action of elements of T0F on unipotent elements show that we can just take u4 = xβ (1)x2α+β (ξ ) where 0  ξ  ∈ Fq is a square if q ≡ 1 mod 3, and a non-square if q ≡ −1 mod 3. Finally, it would be desirable to fix ξ  in some way, in order to obtain a uniform description of u4 , valid for all q (assuming that p  2, 3). It is an easy application of quadratic reciprocity to see that we can always take ξ  = −3. Example 2.8.16 Let again G = GLn (k) and F : G → G be the standard Frobenius map (see Example 1.4.21) such that GF = GLn (q). Let us now consider the automorphism γ : G → G defined by γ(g) = Jn (g tr )−1 Jn

with Jn ∈ G as in Example 1.3.19. 

Then F  := F ◦ γ is a Frobenius map such that GF = GUn (q) is the finite general unitary group (see Example 1.4.21). The automorphism σ  : W → W induced by F  is given by conjugation with the longest element w0 ∈ W; note that Jn is the permutation matrix associated with w0 . Write again Irr(W) = {φν | ν n}. Each φν is invariant under σ  and we could define a corresponding σ -extension as in

206

Lusztig’s Classification of Irreducible Characters Table 2.13 Modified Green functions for GUn (q), n = 2, 3, 4 n=2 n(ν) (11) 1 (2) 0

(11) 1 1

n=4 n(ν) (1111) 6 (211) 3 (22) 2 (31) 1 (4) 0

(2) . 1

n=3 n(ν) (111) 3 (21) 1 (3) 0

(1111) 1 q2 − q + 1 q2 + 1 2 q −q+1 1

(211) . 1 1 1−q 1

(111) 1 1−q 1 (22) . . 1 1 1

(21) . 1 1

(31) . . . 1 1

(3) . . 1

(4) . . . . 1

Example 2.1.8(c). However, following [LuCS, 17.2] (see also Remark 4.1.30), it will be more convenient to define a σ -extension by φ˜ν (w) := (−1)n(ν) φν (ww0 )

for all w ∈ W.

As in Example 2.8.7, we denote the unipotent classes of G by {Oμ | μ n}. Since these are characterised by the Jordan normal form of matrices, it is clear that  F (Oμ ) = Oμ for all μ n; let us fix a representative uμ ∈ OμF for each μ n.  (It will not matter which representative we choose since CG (uμ ) is connected and   so OμF is a single conjugacy class in GF .) Using the above σ -extensions of the characters of W, we obtain

1  (−1)n(ν) φν (w0 w)Q w and Xν = q−n(ν) Qν , Qν = |W| w ∈W 

 are the Green functions of GF . By the algorithm in 2.8.9, we obtain the where Q w matrices in Table 2.13. We notice that these matrices are obtained from those in Table 2.11 by simply ‘changing q to −q’. The general theory developed so far now yields a rather straightforward proof of this fact.

Theorem 2.8.17 (Ennola duality; see [Enn63], [HoSp77, 3.1], [Kaw85, 4.1]) the setting of Example 2.8.16, we have Qν (uμ ) = q n(ν) πμν (−q)

for all μ, ν

In

n,

where πμν ∈ Q[q] are the polynomials in Example 2.8.13. Proof We consider the algorithm in 2.8.9 for G, F  and compare it with the analogous algorithm for G, F. Thus, we have to consider two systems of matrix equations ˜  · P = Ω ˜ (q) ˜ · P = Ω(q), ˜ P tr · Λ and Ptr · Λ

2.8 On the Values of Green Functions

207

where the first one refers to G, F  and the second to G, F; all of the above matrices have rows and columns indexed by the partitions of n. In order to obtain unique solutions to the above systems of equations, we have to consider the unipotent classes {Oμ | μ n} and order them by increasing dimension. Via Theorem 2.8.3, this gives rise to two partitions * *  Iμ and Irr(W)σ = Iμ Irr(W)σ = μ n

μ n

G, F 

where, again, the first one refers to and the second one to G, F. (These partitions define the required block-triangular shape of the matrices P, P .) Now, 2 2 since γ has order 2 and commutes with F, we have F 2 = F 2 and so GF = GF =  GLn (q2 ). Since Irr(W) = Irr(W)σ = Irr(W)σ , we conclude using Remark 2.8.5 that   the correspondence Irr(W)σ → IGF is actually the same as the correspondence Irr(W)σ → IGF . Thus, Iμ = Iμ = {φμ } for all μ n and the matrices P, P  have the same triangular shape with 1 on the diagonal. In particular, the rows and ˜ and Ω ˜  are indexed and ordered in the same way. Now let us compare columns of Ω these two matrices. Let |G | ∈ R[q] be the order polynomial associated to G, F  and |G| ∈ R[q] be the order polynomial associated to G, F. Let T0 ⊆ G be the maximal torus consisting of the diagonal matrices; note that T0 is maximally split both for  | ∈ R[q] be the order polynomial of an F  -stable F and for F . For w ∈ W, let |Tw maximal torus in G of type w. Similarly, let |Tw | ∈ R[q] be the order polynomial  ) ˜  = (ω˜ μν of an F-stable maximal torus in G of type w. Then the entries of Ω μ,ν n ˜ = (ω˜ μν )μ,ν n are the polynomials given by and Ω 1 −n(μ)−n(ν) |G | ˜μ  ˜ν q = ω˜ μν  | φ (w) φ (w) ∈ R[q], |W| |T w w ∈W 1 −n(μ)−n(ν) |G| μ ω˜ μν = q φ (w)φν (w) ∈ R[q]. |W| |T | w w ∈W By Example 1.6.18(a), we have |G | = (−1)n |G|(−q). Next we show that  | = (−1)n |Tww0 |(−q) |Tw

for any w ∈ W.

(∗)

This is seen as follows. Let X = X(T0 ) be the character group of T0 . Let ϕ : X → X be the map induced by F; then ϕ is just given by multiplication with q and so |Tww0 | = det(q idX − (ww0 )−1 ); see 1.6.21. Let ϕ  : X → X be the map induced by F ; for t ∈ T0 , we have F (t) = γ(F(t)) = γ(t)q = Jn t −1 Jn )q = (w 0−1 t w 0 )−q . Hence, ϕ  is given by −qw0 and so  |Tw | = det(q idX − (−w0 ) ◦ w −1 ) = det(q idX + (ww0 )−1 ),

208

Lusztig’s Classification of Irreducible Characters

where the right-hand side equals (−1)n |Tww0 |(−q), as desired. Thus, (∗) is proved. Combining this with the definition of φ˜ν in Example 2.8.16, we conclude that  ω˜ μν = ω˜ μν (−q)

for all μ, ν

n.

˜ are obtained by evaluating certain As noted in Example 2.8.13, the entries of P, Λ ˜ · P = Ω(q) ˜ well-defined polynomials at q and the equation Ptr · Λ can actually ˜ ˜  = Ω(−q), the polynomial equation in be solved at the polynomial level. Since Ω Example 2.8.13 yields that  ω˜ μν (q) = ω˜ μν (−q) =

ν n

πν μ (−q)dν (−q)πνν (−q)

for μ, ν

n.

˜  are determined by the system of equations On the other hand, P  and Λ

 (q) = pν  μ λ˜ν ν pν ν for μ, ν n. ω˜ μν ν n

 ) have the same upper triangular shape with 1 on Since the matrices (πμν ) and (pμν the diagonal for the given ordering of the partitions of n, the above two systems of equations have the same solution. So we conclude that  pμν = πμν (−q)

and

 λ˜μμ = dμ (−q)

for all μ, ν

n.

 OνF .

(From the results for Using 2.8.11, we also see that Yφ˜ν is equal to 1 on F G = GLn (q), we know that πν,(n) = 1 for all ν n.) Hence, we conclude that  for all μ, ν n.  Qν (uμ ) = q n(ν) pμν 

An analogous argument also works for the Green functions of the groups GF = 2E (q) and 2D 6 2n+1 (q) where, again, the automorphism induced on W is given by conjugation with the longest element; see [Sho86, §6(B)]. Example 2.8.18 Let G be simple of type B2 , G2 or F4 , and F : G → G be a Steinberg map such that GF is Suzuki or Ree group. Although F is not a Frobenius map, we can simply run the algorithm in 2.8.9 in these cases as well and see what we get; the output is displayed in Table 2.14. By comparison with the known tables of unipotent characters in [Suz62], [War66], [Ma90], one verifies that Theorems 2.8.3 and 2.8.4 still hold. For 2B2 , we use the σ-extensions specified in Example 2.1.12. There are three F-stable unipotent classes of G, specified by indicating the Jordan normal form; the Springer correspondence in this case can be found in [LuSp85, 6.1]. The Green functions can be extracted from the complete character table in [Suz62, Thm. 13] and the multiplicity formulae in [Lu84a, p. 373]; see also [Ge03a, Prop. 4.6.8]. For 2G2 , we use the σ-extensions specified in Example 2.1.13. There are four Fstable unipotent classes of G, denoted as in [Spa85, p. 329] where one can also find

2.8 On the Values of Green Functions

209

Table 2.14 Values of modified Green functions for 2B2 , 2G2 , 2F 4 2B

2

φ ε˜ φ˜1 1˜ 2F

4

dφ 4 1 0

φ φ˜1,24 φ˜4,13 φ˜9,10 φ˜4,8  φ˜6,6 ˜ φ16,5 φ˜12,4  φ˜6,6 φ˜9,2 φ˜4,1 φ˜1,0

 O(22) O(4) 1 σ ρ ρ−1 1 . . . −q2 +1 1 . . 1 1 1 1

2G

φ ε˜ φ˜2 φ˜1 1˜

2

dφ 6 2 1 0

 1 1 −q2 +1 −q4 +1 1

 A˜ 1 dφ u0 = 1 u1 24 1 . 13 −q10 +q6 −q4 +1 1 10 q12 −q10 +q6 −q2 +1 −q2 +1 8 8 q +1 1 6 q12 −q10 −q2 +1 −q2 +1 5 q14 −2q10 +q8 +q6 −2q4 +1 q6 −2q4 +1 4 q16 +1 1 4 q14 −q12 +2q8 −q4 +q2 q8 −q4 +q2 2 q12 −q10 +q6 −q2 +1 q6 −q2 +1 1 −q10 +q6 −q4 +1 q6 −q4 +1 0 1 1

cont’d F4 (a3 ) A˜ 2 +A1 C3 (a1 ) φ dφ u5 u6 u7 u8 u9 . . . . . φ˜1,24 24 . . . . . φ˜4,13 13 . . . . . φ˜9,10 10 . . . . . φ˜4,8 8  6 1 . . . . φ˜6,6 φ˜16,5 5 1 1 . . . 1 1 1 1 1 φ˜12,4 4  4 . 1 2 . −1 φ˜6,6 2 2 ˜ 1 1 1 φ9,2 2 −q +1 −q +1 φ˜4,1 1 1 −q2 +1 −2q2 +1 1 q2 +1 1 1 1 1 1 φ˜1,0 0

A˜ 1 G2 (a1 ) X T T −1 . . . 1 . . 1 1 1 1 1 1

G2 Y YT YT −1 . . . . . . . . . 1 1 1

A1 + A˜ 1 (B2 )2 u2 u3 u4 . . . . . . 1 . . 1 1 1 −q2 +1 . . 4 2 2 −q +1 −q +1 −q +1 1 1 1 q2 1 1 −q2 +1 q4 −q2 +1 q4 −q2 +1 −q4 +1 −q2 +1 −q2 +1 1 1 1

F4 (a2 ) u10 u11 u12 . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 1 1 1 1

F4 (a1 ) F4 u13 u14 u15 u16 u17 u18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . 1 1 1 1 1 1

the Springer correspondence. Representatives of the unipotent classes are denoted as in [War66, Table III]. The Green functions can be extracted from the complete character table in [War66, p. 87] and the multiplicity formulae in [Lu84a, p. 376]; see also [Ge03a, Example 4.5.12]. For 2F4 , we use the σ-extensions specified in Example 2.1.15. There are ten Fstable unipotent classes of G, denoted as in [Spa85, p. 330] where one can also find the Springer correspondence. Representatives of the unipotent classes are denoted

210

Lusztig’s Classification of Irreducible Characters

as in [Shi75, Table II]. The Green functions have been first determined in [Ma90, §5]. Remark 2.8.19 Let G, F be as in Example 2.8.18 above. Let φ ∈ Irr(W)σ and consider the corresponding pair (O, ς) ∈ IGF as in Theorem 2.8.3. Then we observe that, in all cases, ς is the trivial character or there exists a representative uO ∈ O F such that F acts trivially on A(uO ). Hence, as in Remark 2.8.8, there is a sign δ = ±1 such that Y(uO, ς) ˜ (uO ) = δς(1). Clearly, the sign δ depends on the choice of the σ-extension of φ that is used to define Q φ˜ . Hence, replacing φ˜ by −φ˜ if necessary, we can achieve that δ = 1. This defines a particular set of σ-extensions of the characters in Irr(W)σ , and these are exactly those chosen in Examples 2.1.12, 2.1.13, 2.1.15.

3 Harish-Chandra Theories

In this chapter we introduce the concepts and main results of Harish-Chandra theory for the finite groups of Lie type. Harish-Chandra theories provide an inductive approach to the classification of irreducible characters or modules in whole families of groups of Lie type. The first of these, ordinary Harish-Chandra theory, can be developed solely on the basis of the fact that a group of Lie type G is in a natural way a group with a BN-pair: here B and N are the group of F-fixed points of an F-stable Borel subgroup and of the normaliser of a maximally split maximal torus, respectively, in an underlying algebraic group G in characteristic p with a Steinberg endomorphism F. Attached to such a BN-pair is its collection of standard Levi subgroups. It induces a subdivision of the set IrrK (G) of irreducible characters of G over a field K of characteristic different from p into (ordinary) Harish-Chandra series. These are indexed by socalled cuspidal characters of standard Levi subgroups (up to G-conjugation), and the Harish-Chandra series above a given cuspidal character is parametrised by the set of irreducible characters of a suitable endomorphism algebra. The latter turns out to be closely related to an Iwahori–Hecke algebra associated to a finite Coxeter group. This construction was first proposed by Harish-Chandra [HaCh] and then studied in more detail by Springer [Spr70] and Howlett–Lehrer [HoLe80]. The main drawback of this inductive approach to the classification of irreducible characters is the fact that it does not yield any information about the cuspidal characters of G itself, which have to be determined in some other way. We discuss the concepts and main results in this area in Section 3.1. We then specialise this approach to the case of finite groups of Lie type in Section 3.2 where stronger results in particular on the attached Iwahori–Hecke algebras and on character degrees are available. The second instance of Harish-Chandra theories we present here is the so-called d-Harish-Chandra theories introduced by Fong–Srinivasan [FoSr86] and then in full generality by Broué–Malle–Michel [BMM93] which have become a funda211

212

Harish-Chandra Theories

mental tool in the block theory of finite reductive groups. Their definition and the investigation of their properties is in terms of Lusztig’s induction functor which is based on -adic cohomology theory and thus on the Weil conjectures; no elementary construction for the corresponding generalised induction and restriction is known. Lusztig induction and its basic properties are discussed in Section 3.3. We then present the Alvis–Curtis–Kawanaka–Lusztig duality functor and use it to define the important Steinberg character of a finite reductive group in Section 3.4. Finally, in Section 3.5 we introduce d-tori and d-split Levi subgroups for arbitrary d  1 and lay out the foundations of d-Harish-Chandra theory.

3.1 Harish-Chandra Theory for BN-Pairs The purpose of this section is to introduce a setting in which the usual HarishChandra theory for finite groups with a BN-pair can be formulated. For a more general approach allowing for different collections of subquotients, so-called Mackey systems, see for example [CE04, §1] and [DD93]. 3.1.1 (Finite BN-pairs) We fix a prime p. Let G be a finite group with an algebraic BN-pair in characteristic p satisfying the commutator relations (see [Ca85, §2]) with associated Weyl group W generated by the set of simple reflections S. The cardinality |S| is called the rank of G. We write T := B ∩ N, and for any w ∈ W = N/T we choose once and for all a representative w ∈ N. For I ⊆ S let PI := B, s | s ∈ I be the associated standard parabolic subgroup of G and UI := O p (PI ) the largest normal p-subgroup of PI . It can be shown that the extension PI of UI splits and there exists a natural complement to UI in PI , the standard Levi subgroup LI , uniquely determined by the requirement that T  LI (see [Ca85, §2.6]). The semidirect product PI = UI  LI is called the (standard) Levi decomposition of PI . Here, LI is again a group with an algebraic BN-pair (BI , NI ) in characteristic p, where BI := B ∩ LI and NI = T, s | s ∈ I , with Weyl group WI = I and set of simple reflections I (see [Ca85, Prop. 2.6.3]). If PI = UI .LI , for I ⊆ S, is a standard parabolic subgroup of G, then any g g conjugate P = PI (with g ∈ G) is called a parabolic subgroup of G, and L = LI is called a Levi complement of P, or Levi subgroup of G. We will mainly be interested in the following important special case: Example 3.1.2 Let G be a connected reductive linear algebraic group in characteristic p > 0 and F : G → G a Steinberg endomorphism. We have seen in Theorem 1.3.2 that G has an algebraic BN-pair (B0, N0 ), where B0 is a Borel subgroup of G and N0 is the normaliser in G of a maximal torus T0 of B0 .

3.1 Harish-Chandra Theory for BN-Pairs

213

Now assume that T0 and B0 are chosen to be F-stable. Then the finite group of fixed points G = GF has an algebraic BN-pair (B, N), where B := B0F and N := NG (T0 )F are the groups of F-fixed points of B0 , N0 , respectively, see [MaTe11, Thm. 24.10]. Let W be the Weyl group of G with respect to T0 with set of simple ˜ Then W = WF is again a Coxeter group, with set of simple reflections reflections S. S in bijection with the set of F-orbits on S˜ ([MaTe11, Thm. C.5]). Let I˜ ⊆ S˜ be an F-stable subset and denote by I ⊆ S the subset corresponding to the F-orbits ˜ Then for a parabolic subgroup PI˜ of G with F-stable Levi decomposition in I. PI˜ = UI˜ .LI˜ we have PI = PFI˜ = UFI˜ .LFI˜ . The Levi subgroups of the BN-pair G are thus exactly the G-conjugates of F-fixed points L I = LFI˜ , hence of F-stable Levi subgroups of F-stable parabolic subgroups of G. We will call such F-stable Levi subgroups of F-stable parabolic subgroups of G split or 1-split Levi subgroups for short. These can be characterised as follows (this will be the motivation for our definition of d-split Levi subgroups in 3.5.1): Lemma 3.1.3 In the situation of Example 3.1.2 the split Levi subgroups of G are up to G-conjugation exactly the centralisers of split subtori of T0 . Here, an F-stable torus S  G is called (1-)split if F(t) = t q for all t ∈ S. In particular, SF  F×q × · · · ×F×q (dim S factors), and the order polynomial of S is given by (q−1)dim S . Observe that any F-stable torus T has a uniquely determined maximal

F-stable split subtorus T1 . Indeed, if T has complete root datum (X, , Y, ), ϕ ,

 then the root datum of T1 is given by (X , , Y , ), ϕ  where X  is the largest quotient of X on which the characteristic polynomial of ϕ is a power of q − 1, Y  is the kernel of ϕ − 1 on Y , and ϕ  the map on X  induced by ϕ (see also 3.5.1). Proof Let S  T0 be a split subtorus. Then its centraliser L := CG (S) is the F-stable Levi subgroup generated by T0 together with the root subgroups {Uα | α ∈ R, α|S = 0}, where R is the root system of G with respect to T0 (see [DiMi20, Thms. 1.3.3(iii) and 2.3.1(iv)]). In particular the Weyl group WL of L is generated by the reflections sα with α|S = 1, that is, the reflections that centralise Y (S). Thus WL is a parabolic subgroup of W (see e.g. [MaTe11, Cor. A.29]), F-stable as L is. Then a set of simple roots for WL is WF -conjugate to a set of simple roots corresponding to an F-stable subset I˜ ⊆ S˜ (see [MaTe11, Thm. C.5]) and hence WL is WF -conjugate to WI˜ . But then L is G-conjugate to LI˜ , so contained in the F-stable parabolic subgroup PI˜ of G as claimed. Conversely, if L is 1-split, then after conjugation in G we may assume that L is a standard Levi subgroup. In particular it contains T0 and so Z◦ (L)  T0 . By construction of the BN-pair for G = GF (see e.g. [MaTe11, Thm. C.5]) the Weyl group W = WF of G acts faithfully on Y (T1 ) for the maximal split subtorus T1

214

Harish-Chandra Theories

of T0 . Reversing the above arguments we then see that L = CG (S), where S is the maximal split subtorus of Z◦ (L).  Example 3.1.4 (BN-pair in GLn (q) and GUn (q)) In G = GLn by Example 1.3.7 the subgroup B0 of all upper triangular matrices of GLn together with the group N0 of monomial matrices form an algebraic BN-pair in G. Both of these are Fstable with respect to the standard Frobenius map F raising matrix entries to their qth power. So, by Example 3.1.2 the subgroup B of upper triangular matrices in GF = GLn (q) together with the group N = N0F of monomial matrices over Fq form a finite algebraic BN-pair for GLn (q), with Weyl group W = N/(B ∩ N)  Sn the symmetric group of degree n. The standard parabolic subgroups are the subgroups consisting of block upper triangular matrices of a fixed shape, and the corresponding Levi subgroups are the block diagonal matrices in GLn (q) of the same fixed shape. Now consider GLn with the Frobenius map F  from Example 1.3.19 with fixed   point group G  := GF = GUn (q). Again, both B0 and N0 are F -stable, so B  = B0F  and N  = N0F form a finite algebraic BN-pair in GUn (q). But here F acts as the non-trivial graph automorphism on the Dynkin diagram of the Weyl group W of G, and the Weyl group W  = N /(B ∩ N ) is a proper subgroup of the Weyl group of G, namely a Weyl group of type Bm with m = $n/2%. The F-orbits on S˜ are the subsets si := { s˜i, s˜n−i }, for 1  i  m. The standard parabolic subgroup of GUn (q) corresponding to a subset S \ {si1 , . . . , sir } of the orbit set S, with i1 < · · · < ir , consists of the block upper triangular matrices in G  with fixed symmetric vector of block lengths i1, i2 − i1, . . . , ir − ir−1, ir − ir−1, . . . , i1 . Its standard Levi subgroup consists of the block diagonal matrices of the same block shape. We will specialise to this setting of finite reductive groups from Section 3.2 onwards. Let us for the moment return to an arbitrary finite algebraic BN-pair G in characteristic p. Throughout let K be a field of characteristic different from p. Definition 3.1.5 Attached to any parabolic subgroup P  G with Levi decomposition P = U.L there are two natural functors G RLP : KL-mod −→ KG-mod, ∗ G R LP :

KG-mod −→ KL-mod,

called Harish-Chandra induction and Harish-Chandra restriction respectively, defined as follows: via the identification L  P/U, any KL-module X can be considered as a KP-module with U acting trivially, and then G P RLP (X) := IndG P (Infl L (X))

is the induction to G of this inflated module; conversely, if Y is a KG-module, then L acts on the U-fixed points UY and the resulting KL-module is denoted by ∗RG LP (Y ).

3.1 Harish-Chandra Theory for BN-Pairs

215

G (X) = KG ⊗KP Infl LP (X). It is immediate from this definition that Thus, RLP G dim(RLP (X)) = |G : P| dim X = |G : L| p dim X.

There’s another interpretation of Harish-Chandra induction and restriction due to Broué which is sometimes more useful. Consider the (KG, KL)-bimodule K[G/U], where the actions are given by left and right multiplication respectively. Then the , decompositions gP = l ∈L gUl for g in G induce a canonical isomorphism ∼

G (X) −→ K[G/U] ⊗KL X, RLP

g ⊗ x → gU ⊗ x,

and similarly we have ∗ G R LP (Y )

∼

−→ Y ⊗KG K[G/U]

(see [DiMi20, Prop. 5.1.8]). As both functors are thus given by tensor products with projective right modules, the following is immediate (see [Ge01, Lemma 3.4]): G Corollary 3.1.6 The functors RLP and ∗RG LP are exact, and they preserve direct sums and projectives.

Example 3.1.7 In the set-up of Example 3.1.2, the Borel subgroup B = BF is a parabolic subgroup of G with Levi complement T = TF . Let θ ∈ Irr(T). Then Proposition 2.2.7 shows that in this case RTGB (θ) is the same as the Deligne–Lusztig G (θ). We choose a slightly different notation here to conform with the character RT,U existing literature. Example 3.1.8 Assume that Y is a KG-module with character χ and P = U.L is a parabolic subgroup of G. Then the character of ∗RG LP (Y ) is given by 1 ∗ G R LP ( χ)(l) = χ(lu) for l ∈ L. |U| u ∈U This follows easily from the definition of ∗RG LP (Y ) as CY (U). In particular this shows (K that for KG the trivial KG-module, ∗RG G ) = K L is the trivial KL-module. LP Note that if J ⊆ I ⊆ S then UI  UJ  PJ  PI and LJ is a standard Levi subgroup of the parabolic subgroup VJ .LJ of LI corresponding to the subset J ⊆ I, where VJ = UJ /UI . Thus, if Q = V .M  P = U.L are parabolic subgroups such that M  L then M is a Levi subgroup of the parabolic subgroup L ∩ Q of L with Levi decomposition L ∩ Q = (V ∩ L).M. With this notation the Harish-Chandra functors satisfy natural transitivity properties:

216

Harish-Chandra Theories

Proposition 3.1.9 (Transitivity) Let Q  P be parabolic subgroups of G with Levi complements M, L respectively such that M  L. Then G L ◦ RM RLP L∩Q (X) ∗R L M L∩Q

◦

∗RG (Y ) LP



RG M Q (X)

for all X ∈ KM-mod,



∗RG M Q (Y )

for all Y ∈ KG-mod.

See Dipper–Fleischmann [DF92, Lemma 1.12], or [DiMi20, Prop. 5.1.11] for the case of K of characteristic 0. Furthermore, the two functors are adjoint to one another: Proposition 3.1.10 (Adjointness) Let P = U.L be a parabolic subgroup of G, X ∈ KL-mod and Y ∈ KG-mod. Then G HomKG (RLP (X), Y ) 

HomKL (X, ∗RG LP (Y )),

G HomKG (Y, RLP (X)) 

HomKL (∗RG LP (Y ), X),

as K-vector spaces. See [DF92, 1.8]. It is a crucial property of Harish-Chandra induction and restriction that like ordinary induction and restriction they are intertwined by a Mackey type formula: Theorem 3.1.11 (Mackey formula) Let P, Q be parabolic subgroups of G with Levi complements L, M respectively. Then for all X ∈ KM-mod we have  ∗ G L ∗ M R LP ◦ RG RL∩ w M L∩ w Q ◦ ad(w) ◦ R L w ∩M P w ∩M (X), M Q (X)  w ∈P\G/Q

where w runs over a system of P–Q double coset representatives in G. See [DF92, Thm. 1.4], or [DiMi20, Thm. 5.2.1] for characters. The set P\G/Q can be expressed in terms of the Weyl groups: Let I, J ⊆ S and PI , PJ the corresponding standard parabolic subgroups of G. Then there is a natural system DI J of double coset representatives for WI \W/WJ consisting of elements of minimal length in their respective double cosets. With this, { w | w ∈ DI J } is a system of PI –PJ double coset representatives in G, see [CuRe87, §64C], or [DiMi20, Prop. 3.2.3(ii)] in the case of finite reductive groups. Example 3.1.12 Observe that a subgroup L of G may be a Levi subgroup of two non-conjugate parabolic subgroups: Let q be a prime power and G = GL3 (q), with the BN-pair consisting of the upper triangular invertible matrices B and the monomial matrices N in G (see Example 3.1.4). Then W = s1, s2  S3 . Now it is easily seen that the two standard parabolic subgroups Pi corresponding to {si } ⊂ S, i = 1, 2, are not conjugate in G (in fact, one of them is a point stabiliser while the other is the stabiliser of a hyperplane; modulo G-conjugation they are interchanged

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217

by the transpose-inverse automorphism of G), but their standard Levi subgroups L1, L2 are G conjugate. Nevertheless it can be shown that Harish-Chandra induction and restriction are independent of the parabolic subgroup P containing a given Levi subgroup L (see [DD93, 5.2] or [HoLe94, Thm. 1.1], as well as [DiMi20, Thm. 5.3.1] for characters): Theorem 3.1.13 Let P, P  be parabolic subgroups of G containing the same Levi subgroup L. Then G RLP (X) ∗RG (Y ) LP



G RLP  (X)

for all X ∈ KL-mod,



∗RG LP (Y )

for all Y ∈ KG-mod.

Proof for characters (Deligne, see [LuSp79, Thm. 2.4]) We proceed by induction on the rank of G. Note that we may assume that L is strictly smaller than G. Now for χ a character of L the Mackey formula 3.1.11 shows that the scalar product   ∗ G   G G G χ, R LP (RLP RLP ( χ), RLP  ( χ) =  ( χ)) only involves Harish-Chandra induction and restriction in groups of smaller rank than G, so does not depend on the chosen parabolic subgroups. Thus, writing G ( χ) we obtain ψ P = RLP ψ P, ψ P = ψ P, ψ P = ψ P, ψ P , which implies that ψ P − ψ P has norm zero and so ψ P = ψ P . The claim for Harish-Chandra restriction follows by adjunction.  G ∗ G Thus, from now on we will write RLG and ∗RG L in place of RLP and R LP , as these constructions do not depend on the choice of parabolic subgroup P containing L as a Levi subgroup. We define an order relation 1 on the set of pairs

L (G) := {(L, X) | L  G Levi subgroup, X ∈ KL-mod simple} as follows: (L, X) 1 (M, Y ) if and only if L  M and X is isomorphic to a composition factor of the socle of ∗R LM (Y ). Observe that due to Proposition 3.1.9 this relation is transitive. Definition 3.1.14 A pair (L, X) ∈ L (G) is called a cuspidal pair in G if it is minimal with respect to the partial order 1 . Then X is called a cuspidal KLmodule. Thus, X ∈ KG-mod is cuspidal if and only if ∗RG L (X) = 0 for every proper Levi subgroup L < G. For a cuspidal pair (L, X) ∈ L (G) the corresponding Harish-Chandra series IrrK (G, (L, X)) is defined to be the set of all simple KG-modules Y (up to isomorphism) such that

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(1) L is minimal such that ∗RG L (Y )  0, and (2) X is a composition factor of ∗RG L (Y ) (see [Hi93, p. 224]). Observe that IrrK (G, (L, X)) only depends on the G-conjugacy class of (L, X). But note that at this point it is not clear that Harish-Chandra series are disjoint, nor even that they are non-empty. Example 3.1.15 Assume that G has a BN-pair with trivial Weyl group, that is, with N = B = G. Then the only parabolic subgroup of G, as well as the only Levi subgroup of G, is G itself. Thus, every simple KG-module is cuspidal. More generally, if G has an algebraic BN-pair in characteristic p (see 1.1.14) then B = O p (B).T, where T = B ∩ N, is the smallest parabolic subgroup, and by the previous observation, all of its simple modules are cuspidal. Thus, any (T, X) with X a simple KT-module, is a cuspidal pair. Moreover, according to Example 3.1.8 the trivial KG-module KG lies in the Harish-Chandra series of (T, KT ). The union of the Harish-Chandra series IrrK (G, (T, X)) with X ∈ KT-mod simple is called the principal series of G. Proposition 3.1.16 Let Y be a simple KG-module and (L, X) be a cuspidal pair for G. Then the following are equivalent: (i) Y ∈ IrrK (G, (L, X)); (ii) Y is contained in the socle of RLG (X); (iii) Y is contained in the head of RLG (X). See [Hi93, Thm. 5.8]. The following is an easy consequence: Corollary 3.1.17 The Harish-Chandra series partition IrrK (G). More precisely, IrrK (G, (L, X)) is non-empty for every cuspidal pair (L, X) in G, and IrrK (G, (L, X)) ∩ IrrK (G, (M, Y )) =  for all cuspidal pairs (M, Y ) of G not G-conjugate to (L, X). For the further study of Harish-Chandra series we need the following important object. Let (L, X) be a cuspidal pair in G and set Y := RLG (X). Then HG (L, X) := EndKG (Y )opp is called the Hecke algebra associated with (L, X). Note that by Theorem 3.1.13 we obtain an isomorphic algebra if we choose another parabolic subgroup containing L for the definition of RLG . The corresponding Hom-functor is now defined by FK : KG-mod −→ HG (L, X)-mod,

Z → FK (Z) := HomKG (Y, Z),

where HG (L, X) acts on FK (Z) via h. f := f ◦ h for h ∈ HG (L, X) and f ∈ FK (Z),

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219

and a KG-homomorphism ϕ : Z → Z  is sent to FK (ϕ) = ϕ∗ with ϕ∗ ( f ) := ϕ ◦ f for f ∈ FK (Z). The theory of endomorphism algebras now yields (see Harish-Chandra [HaCh] for K of characteristic 0, and [GHM96, Thm. 2.4] for the general case): Theorem 3.1.18 a bijection

For any cuspidal pair (L, X) in G the Hom-functor FK induces

IrrK (G, (L, X)) −→ IrrK (HG (L, X)),

Z → HomKG (Y, Z),

from the Harish-Chandra series above (L, X) to the set of simple HG (L, X)-modules up to isomorphism. This induces a bijection between the set of isomorphism classes of simple KGmodules and the set of equivalence classes of triples (L, X, Ψ) where L is a Levi subgroup of G, X is a simple cuspidal KL-module and Ψ a simple HG (L, X)-module (where (L, X) has to be taken modulo G-conjugation and Ψ up to isomorphism). Thus, Harish-Chandra theory provides the following inductive approach to a classification of the simple KG-modules: first determine the cuspidal pairs (L, X) in G, then for each such cuspidal pair (up to conjugation) parametrise the members of the corresponding Harish-Chandra series IrrK (G, (L, X)). While the first problem cannot be solved purely in the framework of Harish-Chandra theory, there are powerful results about the structure of Harish-Chandra series, which we will describe now. 3.1.19 In order to formulate the relevant result, we need to recall the construction and some basic facts about Iwahori–Hecke algebras for finite Coxeter groups. Let (W, S) be a Coxeter system with a finite Coxeter group W and a distinguished set S of involutive Coxeter generators of W (see e.g. [Bou68, IV,§1] or [GePf00, §1]). For s, t ∈ S we write mst for the order of st. We denote the corresponding length function on W by l, so l(s) = 1 for all s ∈ S, and more generally l(w) is the length of the shortest expression of w ∈ W as a product of elements from S. Let v = (vs | s ∈ S) be a set of indeterminates such that vs = vt whenever s, t ∈ S are conjugate in W. We will write xs := v2s and x = (xs | s ∈ S). The generic Iwahori–Hecke algebra H (W, x) of W over the ring A := Z[v±1 ] := Z[vs, v−1 s | s ∈ S] is the associative A-algebra with identity generated by elements Ts , s ∈ S, subject to the relations (Ts − xs )(Ts + 1) = 0

for s ∈ S,

Ts Tt Ts · · · = Tt Ts Tt · · · ( mst terms each)

for s, t ∈ S.

For w ∈ W with an expression w = s1 · · · sr (si ∈ S) of shortest possible length r = l(w) (a reduced expression for w) we set Tw := Ts1 · · · Tsr ∈ H (W, x). It is an important fact that this does not depend on the choice of reduced expression for w (Matsumoto’s Lemma, see [GePf00, Thm. 1.2.2]). Then H (W, x) is a free

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A-module of rank |W | with basis {Tw | w ∈ W }, and the multiplication in H (W, x) is described by the rules Tw Ts =

Tws

if l(ws) = l(w) + 1,

xs Tws + (xs − 1)Tw

if l(ws) = l(w) − 1,

for w ∈ W, s ∈ S (see [GePf00, Lemma 4.4.3]). It follows easily from this that the generic Iwahori–Hecke algebra for a product of Coxeter groups is just the product of the Iwahori–Hecke algebras of the factors (see [GePf00, Ex. 8.4]). 3.1.20 For any homomorphism ϕ : A → R from A into a commutative ring R we denote by HR (W, ϕ(x)) := H (W, x) ⊗ A R the specialisation of H (W, x) along ϕ, and we also write ϕ for the induced map on H (W, x). Obviously, such a specialisation is uniquely determined by specifying the images ϕ(vs ), s ∈ S. By a slight abuse of notation we will sometimes also denote the corresponding specialisation by H (W, ϕ(x)). Any such specialisation is called an Iwahori–Hecke algebra associated to the Coxeter group W. An important example is the so-called one-parameter Hecke algebra H (W, x) obtained by the specialisation √ ±1 √ vs → x for all s ∈ S ϕx : A → Z[ x ], √ (so ϕx (xs ) = x), for an indeterminate x. Another example is given by the specialisation ϕ1 : A → Q, vs → 1 for all s ∈ S. The images ϕ1 (Ts ), s ∈ S, then satisfy the relations ϕ1 (Ts )2 = 1

for s ∈ S,

ϕ1 (Ts )ϕ1 (Tt ) · · · = ϕ1 (Tt )ϕ1 (Ts ) · · · ( mst terms each)

for s, t ∈ S,

so H (W, ϕ1 (x)) is the rational group algebra Q[W] of W. Thus, H (W, x) has a semisimple specialisation and so is itself a semisimple algebra over Frac(A) = Q(v). Let QW  C denote the character field of the reflection representation of W. (So QW = Q if W is a Weyl group.) It can be shown moreover that H (W, x) is split over QW (v) ([GePf00, Thm. 9.3.5]). Thus, by a fundamental result of Tits (Tits’ deformation theorem, see [CuRe87, Thm. 68.17] or [GePf00, Thm. 7.4.6]), we have that HQW (v) (W, x) = H (W, x) ⊗ A QW (v)  QW (v)[W]. In particular, the specialisation ϕ1 induces a natural one-to-one correspondence Irr(H (W, x)) −→ Irr(W),

φ → φ1,

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221

between the irreducible characters of HQW (v) (W, x) and Irr(W). This can be described as follows: the values of the irreducible characters of HQW (v) (W, x) on the basis {Tw | w ∈ W } are all contained in QW [v] (see [GePf00, Prop. 7.3.8]) and so can be specialised according to ϕ1 . The image will then be the character of an irreducible representation of QW [W] which we denote by φ1 . Remark 3.1.21 Benson and Curtis [BeCu72] showed that all irreducible representations of the one-parameter Iwahori–Hecke algebra H (W, x) of an irreducible Weyl group W can in fact already be realised over Q(x), except for the two 512-dimensional irreducible representations for W of type E7 and the four 4096-dimensional irreducible representations for W of type E8 , for which a square root of the parameters is needed (see [Lu81a] and also [GePf00, Thm. 9.3.5, Ex. 9.3.4(a) and 9.2.3] and the further references given there). These so-called exceptional characters will also play a special role in the decomposition of unipotent characters, see for example Theorem 4.2.16. This rationality assertion remains true for the generic Iwahori-Hecke algebra H (W, x), except for the two 2-dimensional irreducible representations for W the Weyl group of type G2 that are non-rational. Thus, except for these eight irreducible representations, we can work with the set of parameters x and do not need to worry about their square roots v. Example 3.1.22 (Index and sign representation) The irreducible representation of H (W, x) specialising to the trivial character of W is the index representation ind : H (W, x) → A defined by ind(Ts ) = xs, r so ind(Tw ) = i=1 xsi if w = s1 · · · sr , with si ∈ S, is any reduced expression of w ∈ W. The sign representation of the reflection group W is the specialisation under ϕ1 of the sign representation ε : H (W, x) → A,

ε(Tw ) = (−1)l(w) .

More generally, let ϕ : A → R be any specialisation to a field R such that HR (W, ϕ(x)) is split semisimple. Then application of the procedure explained above yields natural bijections 1−1

1−1

Irr(HR (W, ϕ(x))) ←→ Irr(H (W, x)) ←→ Irr(W)

(∗)

by specialisation of character values. Example 3.1.23 The specialisations of H (W, x) of importance to us here are all √ of the following form: Let q be a prime power with positive square root q and √ define ϕ : A → R by ϕ(vs ) = q as for integers as (s ∈ S), so ϕ(xs ) = q as . Then

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Harish-Chandra Theories

from (∗) we obtain natural bijections φq ←− φ −→ φ1 . These will become relevant in the description of the decomposition of HarishChandra induction (see e.g. Example 3.2.6). 3.1.24 The setting of Iwahori–Hecke algebras is not quite general enough to describe the endomorphism algebras of induced cuspidal representations. Let W1 be a finite Coxeter group with distinguished set of Coxeter generators S1 and Ω a finite group with a homomorphism Ω → Aut(W1, S1 ), where Aut(W1, S1 ) denotes the subgroup of automorphisms of W1 that stabilise S1 . Denote by W = W1  Ω the corresponding semidirect product. This is called an extended Coxeter group. The length function l on W1 extends to W by decreeing that l(w) = 0 for all w ∈ Ω. Let v = (vs | s ∈ S1 ) be a set of indeterminates such that vs = vt whenever s, t ∈ S1 are conjugate in W. We set xs := v2s and x = (xs | s ∈ S1 ). Let μ : W × W → K× be a 2-cocycle that is trivial on W1 × W1 . Let O ⊆ K be the integral closure of the subring generated by the values of μ. The generic twisted extended Iwahori–Hecke algebra H μ (W1  Ω, x) of W with respect to μ over the ring Aμ := O[v±1 ] := O[vs, v−1 s | s ∈ S1 ] is the free A-module with basis {Tw | w ∈ W } and multiplication given by Tw Tω = μ(w, ω)Twω

and Tω Tw = μ(ω, w)Tωw

for all w ∈ W and ω ∈ Ω, and Tw Ts =

Tws

if l(ws) = l(w) + 1,

xs Tws + (xs − 1)Tw

if l(ws) = l(w) − 1,

for all w ∈ W1 , s ∈ S1 (see [Ca85, §10.8]). In particular, the subalgebra w ∈W1 OTw of H μ (W1  Ω, x) is isomorphic to the Iwahori–Hecke algebra HO (W1, x) for the Coxeter group W1 with parameters x. Under the specialisation ϕ1 : A → K, vs → 1 for all s ∈ S1 , H μ (W1  Ω, x) maps to the twisted group ring Kμ [W]. Thus, if Kμ [W] is semisimple, which is the case for example if K has characteristic 0, then so is the generic twisted extended Iwahori–Hecke algebra H μ (W1  Ω, x). Let K  be a finite extension of K(v) over which H μ (W1 Ω, x) splits. Then again by Tits’ deformation theorem (see [CuRe87, Thm. 68.17] or [GePf00, Thm. 7.4.6]) we have that μ

HK (W1  Ω, x) = H μ (W1  Ω, x) ⊗ A K   Kμ [W], and so the specialisation ϕ1 naturally induces a one-to-one correspondence Irr(H μ (W1  Ω, x)) −→ Irr(Kμ [W]),

φ → φ1 .

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223

3.1.25 Let (L, X) be a cuspidal pair in G. Then the endomorphism algebra EndKG (RLG (X))opp is also called the Hecke algebra of X in G. It is closely related to an Iwahori–Hecke algebra of a finite Coxeter group. To describe it more precisely, we need to introduce the relative Weyl group of a cuspidal pair. Assume that I ⊆ S. Then by [Ca85, Lemma 9.2.1] the normaliser of the parabolic subgroup WI in W has a semidirect product decomposition NW (WI ) = WI  CI

with CI := {w ∈ W | w(ΔI ) = ΔI },

with ΔI the simple roots corresponding to I, and the map NG (LI ) ∩ N → NW (WI ), n → nT, induces an isomorphism WG (LI ) := (NG (LI ) ∩ N)LI /LI  NW (WI )/WI  CI . The group WG (LI ) is called the relative Weyl group of LI in G. Example 3.1.26 The relative Weyl group of LI in G is in general different from NG (LI )/LI ; for example assume that G = GLn (2) with the BN-pair as in Example 3.1.4, then the Levi complement L of the smallest parabolic subgroup B = U.T of G satisfies L = T = 1, so NG (L )/L = G, while WG (L ) = (G ∩ N)T/T = N/T = W  Sn . If G = GF as in Example 3.1.2 then the relative Weyl group of L = LF is also given by WG (L) = NG (L)F /LF = NG (L)/L, see also Definition 3.5.8. The groups NW (WI ) are described in detail in [Ho80]. Definition 3.1.27 The relative Weyl group of a cuspidal pair (LI , X) in G is defined as WG (LI , X) :={g ∈ (NG (LI ) ∩ N)LI | ad(g)(X)  X as KLI -module}/LI

{w ∈ CI | ad(w)(X)  X as KLI -module}, a subgroup of WG (LI ). If L is an arbitrary Levi subgroup of G, then there is g ∈ G g and I ⊆ S such that L = LI , and we define the relative Weyl groups WG (L) and WG (L, X) by transport of structure. Note that this is well defined, since it can be g shown that if I, J ⊆ S are such that LI = LJ then there is w ∈ W with I w = J (see [Ca85, Prop. 9.2.2]). The endomorphism algebra HG (L, X) = EndKG (RLG (X))opp of a Harish-Chandra induced cuspidal KL-module X can now be described as follows: Theorem 3.1.28 Let (L, X) be a cuspidal pair in G. Then there is a natural semidirect product decomposition WG (L, X) = W1  Ω with (W1, S1 ) a Coxeter group, and a 2-cocycle μ : WG (L, X) × WG (L, X) → K× such that HG (L, X)

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Harish-Chandra Theories

is isomorphic to the twisted extended Iwahori–Hecke algebra H μ (W1  Ω, q) for suitable parameters q = (qs )s ∈S1 ⊆ K× , with qs  1 for all s ∈ S1 . In particular dimK HG (L, X) = |WG (L, X)| = |W1 | · |Ω|  KTw is isomorphic to the Iwahori–Hecke algebra HK (W1, q) and the subalgebra w ∈W1

for W1 with parameters q. Moreover HG (L, X) is a symmetric algebra. Special cases of this for K of characteristic 0 were first proved by Iwahori, HarishChandra, Lusztig and Kilmoyer. The general case of characteristic 0 was established by Howlett and Lehrer [HoLe80], and building on their work, by Geck, Hiss and Malle [GHM96, §3] in arbitrary characteristic prime to p; the above formulation is due to Ackermann [Ac02]. The fact that HG (L, X) is symmetric was shown in [GeHi97]. For a complete description of the Hecke algebra in Theorem 3.1.28 one needs to know its parameters. Remarkably, it turns out that they are determined locally, as we will now explain. Assume, as we may, that L = L I is a standard Levi subgroup for some subset I ⊆ S and that X is a cuspidal KL I -module such that WG (LI , X)  1. The construction of WG (LI , X) yields a distinguished Coxeter system (W1, S1 ) for the Coxeter group W1 such that for every standard generator s ∈ S1 there exists a minimal subset J ⊆ S containing I with W LJ (LI , X) = s , the Weyl group of type A1 . Then the parameter qs of the Hecke algebra HG (LI , X) at s is determined already by the situation when G = LJ (see [GHM96, (3.14)]). Here we have by [GHM96, Lemmas 3.15–3.17]: Proposition 3.1.29 (Parameters of the Hecke algebra) Let (L, X) be a cuspidal pair in G such that WG (L, X) is the Coxeter group of type A1 , with Hecke algebra HG (L, X) = H (W(A1 ), q). Then: (a) RLG (X) is indecomposable if and only if q = −1. In this case the characteristic of K divides dimK (RLG (X)). (b) Else, RLG (X) is the direct sum of two non-isomorphic simple KG-modules Y1, Y2 and (up to possibly interchanging Y1, Y2 ) we have dimK (Y2 ) = q dimK (Y1 ). In particular, if K has characteristic 0, or more generally, if the characteristic of K does not divide dimK (RLG (X)) = |G : L| p dim(X), then the parameter q of HG (L, X) is given by q = dimK (Y2 )/dimK (Y1 ). Example 3.1.30 Let G be a finite algebraic BN-pair of rank 1. The BN-pair axioms show that G then acts doubly transitively on the cosets of B, with N the

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225

setwise stabiliser of a pair of points, and T = B ∩ N the 2-point stabiliser. Let X = KT be the trivial KT-module. Then by definition Y := RTG (X) is the permutation module on the cosets of B. It is well known that for K of characteristic  dividing |G : B|, this permutation module Y has (at least) two trivial constituents, one in the head and one in the socle. So then Y is not semisimple and we are in case (a) of Proposition 3.1.29. On the other hand, if dimK (Y ) = |G : B| is prime to , then according to Proposition 3.1.29, the KG-module Y has two simple summands. (In characteristic 0 this is true for any 2-transitive group, but need not hold in positive characteristic.) In particular, the non-trivial constituent of the permutation character remains irreducible modulo . In this case by Proposition 3.1.29(b) and the subsequent remark, the parameter of the Hecke algebra HG (T, X) is given by (dimK (RTG (X)) − 1)/1 = |G : B| − 1 = |U|, where U = O p (B). Indeed, as G is doubly transitive with point stabiliser B and 2-point stabiliser T, we have |G : B| = |B : T | + 1 = |U| + 1. For example, if G = SL2 (q) then |G : B| = q + 1, whence the parameter equals q in this case. Remark 3.1.31 The extended Iwahori–Hecke algebras in 3.1.24 do not only occur as endomorphism algebras of induced cuspidal modules in finite reductive groups with non-connected centre as we will see in the next section, but also for disconnected groups of Lie type, see e.g. Digne–Michel [DiMi85] or Malle [Ma91, §1].

3.2 Harish-Chandra Theory for Groups of Lie Type 3.2.1 The most complete results on Harish-Chandra series are available in the case of representations in characteristic 0, and when G is a finite group of Lie type. We will therefore assume for the rest of this chapter that we are in the setting of Example 3.1.2, that is, K is a field of characteristic 0 and G is obtained as the group of fixed points G = GF of a connected reductive algebraic group G over F p under a Steinberg map F : G → G, with its natural algebraic BN-pair in characteristic p coming from the one of G. Recall that this means that the Weyl group of GF is W = WF , the group of F-fixed points of the Weyl group of a maximally split torus of G, with Coxeter generators S in bijection with the F-orbits on the set of Coxeter generators S˜ of W (see e.g. [MaTe11, §23]). We write q for the absolute value of the eigenvalues of F on the character group of an F-stable maximal torus of G. Moreover let us assume for simplicity that the field K of characteristic 0 is a splitting field for G and all of its subgroups. Then KG is split semisimple, and we can work with K-characters of G instead of KG-modules. In analogy to Deligne– Lusztig induction and in view of Example 3.1.7 we will employ the notation RLG

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G ∗ G F and ∗RG L in place of RL and R L if L = L is the group of F-fixed points of a split Levi subgroup L  G, and also use these to denote the induced linear maps on the corresponding character rings. We will also write E (G, (L, X)) in place of IrrK (G, (L, X)) for the Harish-Chandra series of a cuspidal pair (L, X) in G, as this is closer to Lusztig’s notation for characters of finite reductive groups.

A first connection to the Deligne–Lusztig theory presented in Section 2.2 is given by the fact that in the case of groups of Lie type, there is a criterion for cuspidality that can be phrased purely in terms of uniform functions, that is, of Deligne–Lusztig characters, see [Lu77b, Prop. 2.18]: Proposition 3.2.2 (Uniform criterion for cuspidality) Let G = GF be a finite group of Lie type and ρ ∈ Irr(G). Then ρ is cuspidal if and only if ∗RG T (ρ) = 0 for all F-stable maximal tori T contained in some proper (1-)split Levi subgroup of G. Proof Assume that ∗RG T (ρ)  0 for some F-stable maximal torus T of a proper 1∗ L ∗ G split Levi subgroup L < G. Then we have ∗RG T (ρ) = RT ( RL (ρ)) by the transitivity G ∗ statement in Proposition 2.2.7. This implies RL (ρ)  0, whence ρ is not cuspidal. Conversely, if ρ is not cuspidal, then there is some proper 1-split Levi subgroup L < ∗ G ∗ G G such that ∗RG L (ρ)  0, and thus as RL (ρ) is a character of L, RL (ρ)(1)  0. Since the characteristic function of the identity element is uniform by Proposition 2.3.23, there must be some F-stable maximal torus T  L and some θ ∈ Irr(TF ) with ∗ L ∗ G L ∗ G θ, ∗RG T (ρ) = θ, RT ( RL (ρ)) = RT (θ), RL (ρ)  0,

and so ∗RG T (ρ)  0.



According to Lemma 3.1.3 the 1-split Levi subgroups are just the centralisers in G of split tori. So another way to phrase the condition on T in the preceding statement is as follows: an F-stable maximal torus T does not lie in a proper 1-split Levi subgroup of G if and only if its maximal split subtorus is central in G. 3.2.3 The uniform criterion provides strong information on the degree polynomials of cuspidal characters. For this, we need to define the relative rank of a Levi subgroup g of G: If L = LI for some g ∈ G and some standard Levi subgroup of G associated to a subset I ⊆ S, then the relative rank of L is defined as r(L) := |I |. This is closely related to the notion of relative F-rank from Definition 2.2.11. Indeed, by definition the relative F-rank rF (G) of G is the dimension of the q-eigenspace of ϕR on XR = X ⊗Z R, where X is the character group of a maximally split torus of G, hence rF (G) is equal to dim(XRσ ) where σ := q1 ϕR is the automorphism of finite order associated to F. Now assume that G is semisimple. Then according to [MaTe11, Thm. C.5] we have that dim(XRσ ) = |S| equals the number of simple reflections for the Coxeter group W = WF = Wσ . Thus, if L = LI˜  G is a 1-split Levi subgroup,

3.2 Harish-Chandra Theory for Groups of Lie Type

227

˜ then LF is the corresponding to the F-stable set I˜ ⊆ S˜ of simple reflections of S, standard Levi subgroup LI of G corresponding to the subset I ⊆ S in bijection with ˜ and by the above we similarly have rF ([L, L]) = | I˜σ | = |I |. the set of σ-orbits I, F Thus r(L ) = rF ([L, L]) and so in particular (−1)r(L) = ε[L,L] . Corollary 3.2.4 the form

Let λ ∈ Irr(G) be cuspidal. Then the degree polynomial of λ has Dλ = (q − 1)r(G) f (q)

where f ∈ Q[q] is not divisible by q − 1. Proof

By Definition 2.3.25 the degree polynomial of λ is given by

|G| 1 θ . (−1)l(w) Rw , λ q−N Dλ = |W| |Tw | (w,θ)∈X(W,σ)

As λ is cuspidal, Proposition 3.2.2 shows that we only need to sum over w ∈ W such that Tw is not contained in any proper 1-split Levi subgroup of G. Since the centraliser of a split subtorus of Tw is a 1-split Levi subgroup by definition and contains Tw , the maximal split subtorus of Tw must be central. In particular, the power of q − 1 dividing |Tw | is the same as for the order polynomial of Z◦ (G). As G = [G, G]Z◦ (G) this shows that the polynomial |G|/|Tw | must be divisible by (q − 1)r(G) . So Dλ is divisible at least by this power of q − 1. On the other hand, the degree polynomial of λ divides the order polynomial of [G, G], and so it cannot be divisible by a higher power of q − 1.  It will become apparent in Corollary 3.2.21 that the converse of this statement also holds. In characteristic 0, a very explicit description of the endomorphism algebra of a Harish-Chandra induced cuspidal module is available, considerably sharpening Theorem 3.1.28: Theorem 3.2.5 Assume that G = GF is a finite group of Lie type and (L, X) is a cuspidal pair of G. Then HG (L, X) := EndKG (RLG (X))opp  K[WG (L, X)], the group ring of WG (L, X) over K. In particular, there is a bijection 1−1

G : IrrK (WG (L, X)) ←→ E (G, (L, X)). I L,X

If G has connected centre, or if X is unipotent, then WG (L, X) is a Coxeter group and HG (L, X) is the associated Iwahori–Hecke algebra for suitable parameters q all of which are positive integral powers of q.

228

Harish-Chandra Theories

Lusztig showed that HG (L, X) is an Iwahori–Hecke algebra if X is unipotent [Lu76c, Cor. 5.12] or if G has connected centre [Lu84a, Thm. 8.6], with parameters of the stated form. Geck [Ge93b, Cor. 2] settled the general case by showing that in the situation of the theorem, RLG (X) always has a constituent with multiplicity 1, so HG (L, X) has a 1-dimensional representation, and thus the 2-cocycle from Theorem 3.1.28 is trivial. The statement about the parameters can be found in [Lu84a, §8], see also [Ca85, §10.5]. Thus, from Theorem 3.1.18 we obtain a parametrisation of the irreducible Kcharacters of G by triples (L, λ, φ), with L a Levi subgroup of G, λ ∈ Irr(L) cuspidal, and φ ∈ Irr(WG (L, λ)), where (L, λ) is taken modulo G-conjugation. This parametrisation is effective as the cuspidal irreducible characters of all finite reductive groups are known: they can be obtained by applying the uniform criterion from Proposition 3.2.2 to Lusztig’s decomposition of RTG ([Lu84a, Thm. 4.23]; see Theorem 4.2.16). Example 3.2.6 (The unipotent principal series) Let G = GF with natural BNpair B = BF and N = NG (T)F , where T  B is an F-stable maximal torus in an F-stable Borel subgroup of G, and set T := TF = BF ∩ N. According to Example 3.1.15 all irreducible KT-modules are cuspidal, hence in particular so is the trivial module with character 1T . By Example 3.1.7, RTG (1T ) is a Deligne–Lusztig character, hence by Definition 2.3.8 all constituents of RTG (1T ) are unipotent. The elements of the Harish-Chandra series E (G, (T, 1)) are called the unipotent principal series characters of G. In this way Harish-Chandra theory provides an approach to at least a part of the unipotent characters of finite reductive groups. We now describe the parameters of the corresponding endomorphism algebra, which by Theorem 3.2.5 is an Iwahori–Hecke algebra. These are determined locally and can be read off from the result of Example 3.1.30: The relative Weyl group is WG (T, 1T ) = WG (T) = Wσ = W with Coxeter system (W, S), and the parameter of H (T, 1T ) at s ∈ S is given by |Us |, where Us = O p (Bs ) for a Borel subgroup Bs of the standard Levi subgroup L {s } of rank 1 corresponding to the subset {s} ⊆ S. These orders can be computed from the underlying root system of G as explained for example in [MaTe11, §23.2]. For example, for G a group of split type obtained from a standard Frobenius map with respect to an Fq -structure, all parameters are equal to the size q of the underlying field of G. More generally, the parameters are equal to ql(ws ) , where ws is the longest element of the parabolic subgroup WI of W, where I is the F-orbit on the set of Coxeter generators of W corresponding to s. The choice of a square root of q now establishes a canonical bijection between the unipotent principal series E (G, (T, 1)) and Irr(Wσ ) by combining the bijection from Theorem 3.2.5 induced by the Hom-functor (see Theorem 3.1.18) with the canonical bijections in 3.1.23.

3.2 Harish-Chandra Theory for Groups of Lie Type

229

The values for the various twisted types are collected in Table 3.1 (see [MaTe11, Table 23.2]). Observe that we may easily reduce to the case when G is simple: all characters in the unipotent principal series are trivial on the centre, so we may pass to a group of adjoint type, and this has a direct decomposition as explained in Corollary 1.5.16. On the other hand, the corresponding Iwahori–Hecke algebra is the product of the Iwahori–Hecke algebras corresponding to the simple factors (see 3.1.19). Table 3.1 Parameters for the principal series in twisted types G

2A 2n−1 (q) 2A (q) 2n 2D (q) n 3D (q) 4

WG (T) Bn Bn Bn−1 G2

parameters q; q2, . . . , q2 q3 ; q2, . . . , q2 q2 ; q, . . . , q q3 ; q, . . . , q

G

2E (q) 6 2B (q2 ) 2 2 G (q2 ) 2 2F (q2 ) 4

WG (T) F4 A1 A1 I2 (8)

parameters q2, q2 ; q, q q4 q6 q4 ; q2

The Ree groups 2F4 (q2 ) are the only instance in which a relative Weyl group for the principal series is a real reflection group which is not a Weyl group, namely the dihedral group I2 (8) of order 16. The parameters are given in the order induced by the labelling on the Dynkin diagrams in Table 1.1. Theorem 3.2.5 can be refined to relate the multiplicities in Harish-Chandra induced characters to the decomposition of induced characters in the corresponding relative Weyl groups: Theorem 3.2.7 (Howlett–Lehrer Comparison Theorem) pair in G. Then the collection of bijections

Let (L, λ) be a cuspidal

M I L,λ : IrrK (W M (L, λ)) −→ E (M, (L, λ))

from Theorem 3.2.5, where M runs over split Levi subgroups L  M  G, can be chosen such that the diagrams G I L, λ

ZIrrK (WG (L, λ)) . ⏐ Ind ⏐

−−−−→

ZIrrK (W M (L, λ))

−−−−→

M I L, λ

ZE (G, (L, λ)) . ⏐ RG ⏐ M ZE (M, (L, λ))

commute for all M, where Ind denotes ordinary induction. See [HoLe83, Thm. 5.9], and [GHM96, Prop. 2.7] for a version in arbitrary non-defining characteristic. Thus the decomposition of Harish-Chandra induction

230

Harish-Chandra Theories

(and restriction) can be computed purely inside relative Weyl groups. We will see a far-reaching generalisation, for unipotent characters, in Theorem 4.6.21. ˜ 3.2.8 We next relate Harish-Chandra theory to regular embeddings. Let G → G ˜ with connected centre and same be a regular embedding of G into a group G ˜ of the Steinberg map on derived subgroup as G and denote by F an extension to G F F ˜ ˜ ˜ G (see Definition 1.7.1). Then G = GZ(G) and G /G is abelian. Furthermore, ˜ is an F-stable maximal torus of G ˜ in the F-stable Borel subgroup T˜ = TZ(G) F ˜ and the natural map NG (T) → N ˜ (T) ˜ F induces an isomorphism B˜ = BZ(G), G ˜ F sending simple ˜ F between the Weyl groups of G = GF and G˜ = G WF → W reflections to simple reflections. Now if P is an F-stable parabolic subgroup of G, ˜ is an F-stable parabolic subgroup of G, ˜ with the same unipotent then P˜ := PZ(G) ˜ is an F-stable ˜ := LZ(G) radical as P, and for L an F-stable Levi subgroup of P, L ˜ This defines a natural bijection between the sets of split Levi Levi subgroup of P. ˜ subgroups of G and of G. ˜ be a regular embedding. Let L be a split Levi Proposition 3.2.9 Let G → G ˜ Then we have ˜ subgroup of G and L the corresponding split Levi subgroup of G. ˜

G IndG G ◦ RL ˜

Res LL ◦ ∗RG L˜ ˜

= =

˜

˜

Ind LL ◦ ∗RG L

˜

G ResG G ◦ RL˜

RLG˜ ◦ Ind LL,

∗RG L

◦ ResG G,

˜

˜

˜

= =

˜ ∗RG L˜ RLG

˜

◦ IndG G, ˜

◦ Res LL .

Proof The first formula is immediate from the definition of RLG as induction and inflation over the same kernel commute, and the second follows as taking fixed points with respect to the same normal subgroup commutes with induction. The third and fourth are obtained by adjunction.  The following statement is now immediate from the definitions and Proposition 3.2.9 (see [Bo06, Prop. 12.1]): Corollary 3.2.10 In the situation of Proposition 3.2.9, let ρ ∈ Irr(G) and suppose ˜ lies above ρ. Then ρ is cuspidal if and only if ρ˜ is. that ρ˜ ∈ Irr(G) 3.2.11 Let (L, X) be a cuspidal pair in G. Since the endomorphism algebra EndKG (RLG (X)) is semisimple, the degrees of the irreducible constituents of RLG (X) can be computed from the Schur elements of the natural symmetrising form on the symmetric algebra EndKG (RLG (X))opp = HG (L, X). Now assume that we are in the setting of 3.1.24; that is, W1 is a finite Coxeter group with distinguished set of Coxeter generators S1 , Ω a finite group with a homomorphism Ω → Aut(W1, S1 ), and W = W1 Ω is the corresponding semidirect product. Let H (W1  Ω, x) be the associated generic extended Iwahori–Hecke algebra over A := Z[v±1 ] with respect to the trivial 2-cocycle, as introduced in 3.1.24.

3.2 Harish-Chandra Theory for Groups of Lie Type

231

Then H (W1 Ω, x) is symmetric; more precisely, it carries a natural non-degenerate symmetrising form τ : H (W1  Ω, x) → A defined by τ(Tw ) =

1

if w = 1,

0

else,

(3.1)

on the A-basis {Tw | w ∈ W } of H (W1  Ω, x) from 3.1.24 (see [Ca85, §10.9], or also [GePf00, Prop. 8.1.1] for the case Ω = 1). Any such trace form can be written uniquely as a linear combination of irreducible characters of H (W1  Ω, x) with non-zero coefficients (see [GePf00, Thm. 7.2.6]), say

1 τ= φ, cφ φ ∈Irr(H (W1 Ω,x))

where Irr(H (W1  Ω, x) denotes the set of characters of irreducible representations of H (W1  Ω, x) over a splitting field K  as introduced in 3.1.24. The non-zero elements cφ ∈ AW , where AW is the integral closure of A in K , are called the Schur elements of H (W1  Ω, x) (with respect to τ). In the case Ω = 1, when W = W1 is a Coxeter group, we saw in 3.1.20 that we may choose K  = QW (v) as a splitting field, in which case we have AW = ZW [v±1 ], with ZW the ring of integers in QW . Example 3.2.12 Under the specialisation ϕ1 to the group algebra of W, the trace form on H (W, x) becomes the regular character of W divided by |W |, so the Schur elements of the specialisation Z[W] = H (W, ϕ1 (x)) are given by ϕ1 (cφ ) = |W |/φ(1). Definition 3.2.13 The 1-parameter specialisation ϕq (cind ) of the Schur element cind for the index representation from Example 3.1.22 is called the Poincaré polynomial of W. By the previous example it specialises to |W | under ϕ1 , so can be thought of as a ‘quantisation’ of the order of W. In view of Example 3.2.12 the quotient Dφ := cind /cφ is called the generic degree of φ ∈ Irr(W). We thus have ϕ1 (Dφ ) = φ(1), so Dφ is a ‘quantisation’ of the character degree φ(1). Note that by definition the trivial character of W always has generic degree equal to 1. Clifford theory provides a simple way to relate the Schur elements of an extended Iwahori–Hecke algebra H (W1  Ω, x) to those of the Iwahori–Hecke algebra H (W1, x) of the underlying Coxeter group W1 : Proposition 3.2.14 Let φ ∈ Irr(H (W1  Ω, x)). Then we have cφ φ(1) = |Ω|cφ1 φ1 (1) for all φ1 ∈ Irr(H (W1, x)) which occur in the restriction of φ to H (W1, x).

232

Harish-Chandra Theories

This is shown in [Ge00, Prop. 4.6] for the equal-parameter case, but the proof remains valid in our more general setting, see also [Ma95, Lemma 5.11]. It is therefore sufficient to know Schur elements for Iwahori–Hecke algebras. Example 3.2.15 Assume that W = W1 × W2 is a direct product of two Coxeter groups W1, W2 . Write τi for the natural symmetrising form on H (Wi, xi ), i = 1, 2. Then the natural symmetrising form on H (W, x) = H (W1, x1 ) ⊗ H (W2, x2 ), x = (x1, x2 ), is given by τ = τ1 ⊗ τ2 and the above definitions show that the corresponding Schur elements and generic degrees are multiplicative: cφ1 ⊗φ2 (x) = cφ1 (x1 ) cφ2 (x2 )

and

Dφ1 ⊗φ2 = Dφ1 Dφ2

for all φi ∈ Irr(Wi ), i = 1, 2. 3.2.16 The Schur elements of all finite irreducible Coxeter groups are known explicitly (see e.g. [Ca85, §13.5] for Weyl groups, where the generic degrees of H (W, x) are given). (The general case reduces to the one of irreducible groups by Example 3.2.15.) They were calculated by Hoefsmit [Ho74] for the classical types, and by Kilmoyer, Surowski [Su77, Su78], Lusztig [Lu79a], and Benson [Be79] for the exceptional types. Lusztig [Lu77a, 9.6] observed that the Schur elements for type Bn (and thus for type Dn ) could also be obtained by interpolation from the known unipotent character degrees of the general unitary groups, using Theorem 3.2.18 together with the fact that the Hecke algebra of type Bn occurs as endomorphism algebra of Harish-Chandra induced cuspidal modules for infinitely many different choices of parameters. See also Iancu [Ia01] for a different proof for type Bn using Markov traces, and Geck–Iancu–Malle [GIM00], Mathas [Mat04] and Rui [Ru01] for the generalisation to imprimitive complex reflection groups. Observe that the Schur elements are determined for example as the unique solution of the linear system of equations obtained by evaluating the defining equation 3.2.11(3.1) of τ on some basis of H (W, x), hence from a knowledge of the character table of H (W, x). It is a remarkable fact, first explicitly stated by Chlouveraki [Chl09], that all Schur elements of Iwahori–Hecke algebras have the following form: they are products of cyclotomic polynomials evaluated at monomials in the parameters of the Iwahori– Hecke algebra, times a monomial in the parameters, possibly times an integer. While this phenomenon can be explained rigorously for example for type An−1 by the fact that the specialisations at q of the Schur elements divide the order of the finite groups GLn (q), for an infinitude of prime powers q (see [CuRe87, Thm. 68.31]), no such explanation is available for example for the 2-parameter Iwahori–Hecke algebra of type F4 . A general proof of this observation using the theory of rational Cherednik algebras has been given by Rouquier [Rou08, Thm. 3.5].

3.2 Harish-Chandra Theory for Groups of Lie Type

233

Example 3.2.17 Let W = Sn , the symmetric group of degree n. The irreducible characters of W are labelled by the partitions α n. Here all reflections are conjugate so the Iwahori–Hecke algebra of W has just a single parameter x. Let α = (α1  α2 . . .  αr ) n be a partition. The Schur element cα of φα ∈ Irr(Sn ) is then given by n  xli − 1 , cα = x−a(α) x−1 i=1 where li denotes the hook length at the ith box of the Young diagram of α, and  a(α) = i< j (αi − α j ) is the a-invariant of α (see Definition 4.1.10, and also [Ca85, §13.5] and [Ma95, Bem. 3.12]). For example, for the trivial and the sign character of Sn , which are labelled by the partitions (n) and (1)n respectively, we obtain the Schur elements n n   n xi − 1 xi − 1 and c(1)n = x−( 2 ) c(n) = x−1 x−1 i=1 i=1 and thus the generic degree D(1)n = c(n) /c(1)n = x( 2 ) for the sign character. n

We can now give a degree formula for Harish-Chandra induction in terms of G : Irr(WG (L, λ)) → E (G, (L, λ)) from Theorem 3.2.5 (depending the bijection I L,λ on a choice of specialisation of the parameters x of the generic Hecke algebra of WG (L, λ) and thus on a choice of isomorphism HG (L, λ)  K[WG (L, λ)]). Theorem 3.2.18 (Degree formula) Let (L, λ) be a cuspidal pair in G. Then for G φ ∈ Irr(WG (L, λ)) the degree of ρ = I L,λ (φ) ∈ E (G, (L, λ)) is given by ρ(1) = RLG (λ)(1) cφ (q)−1 = λ(1)

|G| p cφ (q)−1 , |L| p

where q are the parameters of the relative Hecke algebra HG (L, λ). See [Ca85, Thm. 10.11.5], or [CuRe87, Prop. 68.30(iii)] for the characters in the principal series. Let us point out an immediate consequence: Corollary 3.2.19 In the situation of Theorem 3.2.18 if ρ lies in the HarishChandra series above (L, λ) then ρ(1) p divides RLG (λ)(1). In particular ρ(1) divides |G : Z(L)|. Proof The first claim follows from the formula for ρ(1) in Theorem 3.2.18 together with the fact that the Schur elements lie in AW (see 3.2.11), so their specialisations at powers of p are integral up to powers of p. As |Z(L)| is prime to p the second part then is an immediate consequence of the first and the well-known property of irreducible characters λ ∈ Irr(L) that λ(1) divides |L : Z(L)|. 

234

Harish-Chandra Theories

In order to be able to apply Theorem 3.2.18, the parameters of the Hecke algebra HG (L, λ) have to be known. They can be determined locally, as described in Proposition 3.1.29. Note that over a field of characteristic 0 we are always in the case (b). The parameters are known explicitly in all cases by the work of Lusztig [Lu84a, Thm. 8.6]. For unipotent characters they will be given in Table 4.8. We now relate the degree formula to regular embeddings. Choose a regular ˜ Then for L  G an F-stable Levi subgroup and L˜ = LZ(G) ˜ embedding G → G. F F F F ˜ ˜ ˜ the corresponding Levi subgroup of G, observe that | G : G | = | L : L |. Let λ ∈ Irr(LF ) be cuspidal and ρ ∈ E (GF , (L, λ)). By Theorem 1.7.15 induction of ˜ F and from LF to L ˜ F is multiplicity free, so we have characters from GF to G ˜

Ind LL (λ) =

a

λ˜i,

˜

IndG G (ρ) =

b

i=1

ρ˜ j ,

j=1

˜ F ). Setting λ˜ := λ˜1 ˜ F ) and ρ˜ j ∈ Irr(G for distinct irreducible characters λ˜i ∈ Irr(L ˜ we have λ is cuspidal by Corollary 3.2.10 and we may choose notation such that ˜ F , (L, ˜ λ)). ˜ Then aρ(1)/λ(1) = b ρ(1)/ ˜ ρ˜ := ρ1 ∈ E (G ˜ λ(1). So we obtain: Corollary 3.2.20 In the situation of Theorem 3.2.18 and with the above notation, G (φ) ∈ E (G, (L, λ)) is given by for φ ∈ Irr(WG (L, λ)) the degree of ρ = I L,λ ρ(1) = d RLG (λ)(1) cφ˜ (q)−1 = d λ(1) ˜

˜

˜

|G| p c ˜ (q)−1 , |L| p φ ˜

G L L where d = a/b = IndG G (λ), IndG (λ) /Ind L (ρ), Ind L (ρ) and cφ˜ denotes the Schur ˜ λ) ˜ in G, ˜ with parameters q. element of an extension φ˜ of φ to the Hecke algebra of ( L,

Observe that this formula is generic in the sense that it relates the degree polynomials Dλ of λ and Dρ of ρ by Dρ = d Dλ

|G|q c ˜ (q)−1 , |L|q φ

where cφ˜ (q) denotes the 1-parameter specialisation of cφ˜ through which the specialisation to q factors (see Example 3.1.23). We obtain the following generalisation and converse of Corollary 3.2.4: Corollary 3.2.21 Let ρ ∈ Irr(G) lie in the Harish-Chandra series of the cuspidal pair (L, λ). Then the degree polynomial of ρ has the form Dρ = (q − 1)r(L

F)

f (q),

where f ∈ Q[q] is not divisible by q − 1. F In particular ρ is cuspidal if and only if (q − 1)r(G ) is the precise power of q − 1 dividing its degree polynomial.

3.2 Harish-Chandra Theory for Groups of Lie Type

235

Proof First assume that G has connected centre. Then by Theorem 3.2.5 the parameters q of HG (L, λ) are integral powers of q, say, q = (qs | s ∈ S) with qs = q as . Thus the corresponding specialisation ϕq : Z[v±1 ] → Z, xs → qs = q as , factors through a specialisation ϕq : Z[v±1 ] → Z[q±1 ],

xs → qas .

Let cφ,q = ϕq (cφ ), φ ∈ Irr(WG (L, λ)), denote the specialised Schur elements. These lie in Z[q±1 ], so have no pole at q = 1. But the specialisation ϕ1 to the group algebra of the relative Weyl group WG (L, λ) also factorises through ϕq , and by Example 3.2.12 it sends the Schur elements to the non-zero elements |WG (L, λ)|/φ(1), so the cφ,q do not have a zero at q = 1 either. Since L is a 1-split Levi subgroup of G, it contains a maximal split subtorus of G by Lemma 3.1.3 and so the order polynomials of L and G are divisible by the same power of q − 1. Since the Schur elements cφ,q have neither a zero nor a pole at q = 1, the form of Dρ now follows by Corollary 3.2.4 combined with Theorem 3.2.18. The last claim follows with Corollary 3.2.4 from the observation that r(LF ) = |I | < |S| = r(GF ) for any proper split Levi subgroup L of G (if LF is conjugate to the standard Levi subgroup LI for I  S). ˜ be a regular embedding and be L ˜  G the In the general case, let G → G F ˜ ˜ ˜ F-stable Levi subgroup with L = L ∩ G, λ ∈ Irr(L ) a cuspidal character lying ˜ F ) in E (G ˜ F , λ) ˜ above ρ. Then the degree polynomials of ρ above λ and ρ˜ ∈ Irr(G and of ρ˜ differ by an integer factor by Remark 2.3.27(c) and hence the claim for ρ follows from the one for ρ. ˜  A formula for values of RLG and ∗RG L on arbitrary elements will be given in Theorem 3.3.12 in the more general setting of Lusztig induction. We now clarify the relation between cuspidality and Jordan decomposition. For this let (G∗, F) be dual to (G, F) (see Definition 1.5.17). (For simplicity we denote the Steinberg map on the dual group G∗ again by F.) The following result (which first appeared in special cases in [Lu77a, 7.8]) shows that in order to understand the cuspidal characters of all connected reductive groups it is enough to classify unipotent cuspidal characters: Theorem 3.2.22 Let s ∈ G∗F be semisimple. Let ρ ∈ E (GF , s) and let ψ be in a CG∗ (s)F -orbit of unipotent characters of CG◦ ∗ (s)F corresponding to ρ under Jordan decomposition (see Theorem 2.6.22). Then ρ is cuspidal if and only if (1) ψ is cuspidal, and (2) Z◦ (G∗ ) and Z◦ (CG◦ ∗ (s)) have the same Fq -rank, that is, the maximal split subtorus of Z(CG◦ ∗ (s)) is contained in Z◦ (G∗ ). Proof

˜ be a regular embedding. Then by (See [CE99, Prop. 1.10].) Let i : G → G

236

Harish-Chandra Theories

Corollary 3.2.10 cuspidality of ρ and ψ is equivalent to cuspidality of corresponding ˜ F and C ˜ ∗ (s˜)F respectively, where s˜ ∈ G ˜ ∗F is a preimage of s under characters of G G ˜ ∗  G∗ dual to i. Clearly the condition (2) in the statement the epimorphism i ∗ : G is also preserved by regular embeddings, so we may and will assume for the proof that G has connected centre and thus CG∗ (s) is connected. As the regular character regG is uniform by Proposition 2.3.23, there exists an Fstable maximal torus T  G with RTG (θ), ρ  0 for some θ ∈ Irr(TF ). Moreover, as ρ ∈ E (G, s), the pair (T, θ) must lie in the geometric conjugacy class of (T∗, s), where T∗ is an F-stable maximal torus of G∗ dual to T (see Corollary 2.5.14 and Definition 2.5.17). If ρ is cuspidal, then by the uniform criterion in Proposition 3.2.2 the maximal split subtorus of T must be contained in Z(G), and thus the maximal split subtorus of T∗ lies in Z(G∗ ). Since s ∈ T∗ we also have Z◦ (CG∗ (s))  T∗ , so we obtain condition (2). Conversely, if (2) is satisfied, then the maximal split subtorus of T lies in Z(G) if and only if the maximal split subtorus of T∗ lies in Z(CG∗ (s)). But then ρ satisfies the uniform criterion for cuspidality if and only if ψ does, since   C ∗ (s)   G RT (θ), ρ = ± RT∗G (1), ψ by the fundamental property of Jordan decomposition for groups with connected centre (see Theorem 2.6.4).  A special case of this was already shown in [Lu77a, Prop. 7.9].

3.3 Lusztig Induction and Restriction We introduce a far-reaching common generalisation of both Harish-Chandra induction and restriction and of Deligne–Lusztig characters introduced by Lusztig [Lu76b]. It provides a generalised induction from a family of subgroups encompassing the split Levi subgroups used in Harish-Chandra induction as well as the maximal tori featuring in Deligne–Lusztig induction: the F-fixed points of arbitrary F-stable Levi subgroups. Throughout this section, let G denote a connected reductive algebraic group over an algebraic closure of a finite field of characteristic p, with a Steinberg endomorphism F : G → G. We let q denote the absolute value of the eigenvalues of F on the character group of an F-stable maximal torus of G. As before K denotes a sufficiently large field of characteristic 0. 3.3.1 We start by classifying F-stable Levi subgroups of G in terms of combinatorial data by generalising the corresponding classification of F-stable maximal tori from 1.6.4. Fix a maximally split torus T0 of G contained in an F-stable Borel ˜ Let L  G be subgroup of G, with corresponding set of simple reflections S.

3.3 Lusztig Induction and Restriction

237

an F-stable Levi subgroup of a (not necessarily F-stable) parabolic subgroup P of G. Choose an F-stable maximal torus T  L. Then there is x ∈ G such that L = xLI x −1 for the standard Levi subgroup LI corresponding to a subset I ⊆ S˜ and moreover T = xT0 x −1 . Then we have F(x)T0 F(x)−1 = F(T) = T = xT0 x −1 F(x)F(LI )F(x)

−1

and

−1

= F(L) = L = xLI x .

The first equation shows that x −1 F(x) ∈ NG (T0 ) and so x −1 F(x) = w for some

F(I) w −1 = LI and w ∈ W. In the second equation, we have F(LI ) = LF(I) , so wL hence wF(I) = I. Thus, L determines a pair (I, w) with I ⊆ S and w ∈ W with wF(I) = I. We then say that L is a Levi subgroup of type (I, w). Conversely, any such pair (I, w) defines an F-stable Levi subgroup xLI x −1 of G, where x ∈ G

The complete root datum for a Levi subgroup of type is such that x −1 F(x) = w.

(I, w) is of the form (X, RI , Y, RI∨ ), wWI , where RI is the parabolic subsystem of R determined by I with Weyl group WI = I . It is easily seen that this construction sets up a bijection between GF -conjugacy classes of F-stable Levi subgroups of G and equivalence classes of pairs (I, w) as before, with (I, w), (I, w ) equivalent if and only if there is v ∈ W with vI = I  and vw = w  F(v); see [Lu77a, 7.2]. Assume that L is 1-split. Then it contains a maximally split torus, hence up to conjugation we have T0  L and thus we can choose w = 1 in our construction. Thus, the 1-split Levi subgroups are parametrised by the equivalence classes of pairs of the form (I, 1), as is also clear from the BN-pair structure of G. In the Chevie-system [MiChv] the various rational forms of a standard Levi subgroup can be obtained with the command Twistings. Recall the construction of Deligne–Lusztig characters in Definition 2.2.6 starting from an F-stable maximal torus T of G contained in a not necessarily F-stable Borel subgroup B. Then RTG was defined from the -adic cohomology of the preimage of the unipotent radical U of B under the Lang–Steinberg map L . Definition 3.3.2 (Lusztig [Lu76b]) Let L be an F-stable Levi subgroup of a (not necessarily F-stable) parabolic subgroup P of G. Then the unipotent radical Y := Ru (P) of P satisfies condition (∗) in 2.2.4 with respect to the finite subgroup H := LF . Thus, for any λ ∈ Irr(LF ) we obtain a virtual character of GF , called the Lusztig induced character of λ, defined by

1 G RLP (λ)(g) : = F L (g, l), L −1 (Y) λ(l) |L | l ∈L F

= (−1)i Trace g ∗, Hic (L −1 (Y), Q )λ i0

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Harish-Chandra Theories

for g ∈ GF , where Hic (L −1 (Y), Q )λ denotes the λ-isotypic part of the Q LF module Hic (L −1 (Y), Q ) and g ∗ is the automorphism induced by g on it (see 2.2.1). We denote by ∗RG LP the adjoint map, Lusztig restriction, which sends characters of F G to virtual characters of LF , given by



∗ RLP (ρ)(l) = (−1)i Trace l ∗, ρ Hic (L −1 (Y), Q ) i0

for ρ ∈ Irr(GF ), l ∈ LF , where ρ Hic (L −1 (Y), Q ) denotes the ρ-isotypic part of the KGF -module Hic (L −1 (Y), Q ). Thus we have     G for all λ ∈ Irr(LF ), ρ ∈ Irr(GF ). RLP (λ), ρ = λ, ∗RG LP (ρ) G twisted induction, a term which is sometimes still Lusztig originally called RLP used. G (λ) and ∗RLP (ρ) only Note that the values of the generalised characters RLP involve roots of unity in Q , so (after a suitable identification) can be considered as elements of our chosen large enough field K of characteristic 0. G It is clear from the definition that RLP is a generalisation of the Deligne–Lusztig characters, which arise in the special situation that the parabolic subgroup P is minimal possible, that is, P is a Borel subgroup and L an F-stable maximal torus, ∗ G and ∗RG LP is a generalisation of RT . But Lusztig induction also generalises HarishChandra induction:

Proposition 3.3.3 Assume that L is an F-stable Levi subgroup of an F-stable G parabolic subgroup P of G, that is, L is a split Levi subgroup of G. Then RLP G F F ∗ is just Harish-Chandra induction from L to G and RLP is Harish-Chandra restriction. See the remarks after [DiMi20, Prop. 9.1.4]. This result justifies our use of the same symbol for both. Example 3.3.4 In generalisation of Example 3.1.8 we have ∗RG L (1G ) = 1L for any F-stable Levi subgroup L  G, see [DiMi20, proof of Cor. 10.1.7]. 3.3.5 We briefly discuss another realisation of Lusztig induction from [Lu84a, p. 215] (see also [BoRo93, §5]), generalising the alternative model for Deligne– Lusztig characters introduced in 2.3.18. Fix a maximally split torus T0 of G and an F-stable Borel subgroup B containing T0 . Let L  G be an F-stable Levi subgroup, of type (I, w) for some I ⊆ S˜ and w ∈ W with wF(I) = I (see 3.3.1). As discussed in 2.3.18, F  : G → G,

g → wF(g) w −1,

is again a Steinberg map on G and conjugation with x defines an isomorphism

3.3 Lusztig Induction and Restriction

239

F

Let LI be the standard Levi GF  G , where x ∈ G is such that x −1 F(x) = w. subgroup of G corresponding to I. Then LI is F -stable, and we set 

LI [w] := LFI = {l ∈ LI | F(l) = w −1 l w}. Note that LI [w] is a finite subgroup of L I that depends on w, but not on the repres (Another common notation for this subgroup is LwF entative w. I .) Then conjugation −1 by x defines an isomorphism LF  LI [w]. Let PI be the standard parabolic subgroup of G corresponding to I and hence containing L I , and YI := Ru (PI ) its unipotent radical. Then YI w satisfies the condition (∗) in 2.2.4 with respect to the finite subgroup LI [w]. So as before we obtain virtual characters of GF by setting, for any λ ∈ Irr(LI [w]),

λ (g) := (−1)i Trace g ∗, Hic (L −1 (YI ), Q )λ for g ∈ GF . RI,w i0 θ (the case With an identical proof as in Lemma 2.3.19 for the virtual characters Rw λ when I = , so LI = L = T0 ) we then have that RI,w does not depend on the choice

and furthermore, with the parabolic subgroup P = xPI x −1 of a representative w, containing L, λ G = RLP (xλ) RI,w

where

λ(l) := λ(x −1 l x) for l ∈ LF .

x

As for Harish-Chandra induction (see Proposition 3.1.9), Lusztig induction and restriction are transitive: Theorem 3.3.6 (Transitivity) Let Q  P be parabolic subgroups of G with Fstable Levi subgroups M, L respectively, such that M  L. Then G L ◦ RML∩Q (ψ) RLP ∗RL ML∩Q

=

G RMQ (ψ)

for all ψ ∈ Irr(MF ),

◦ ∗RG LP (ρ) =

∗RG MQ (ρ)

for all ρ ∈ Irr(GF ).

See [DiMi20, Prop. 9.1.8]. It is generally expected that Lusztig induction and restriction do also satisfy a Mackey formula, but this has not yet been proved in full generality. It is true in the case that both parabolic subgroups are F-stable by Theorem 3.1.11, it follows from the work of Deligne and Lusztig if one of the two Levi subgroups is a maximal torus (see [DeLu83, Thm. 7] or [DiMi20, Thm. 9.2.6]), and it has also been shown if q is large enough [Bo98]. Results of Shoji can then be used to deduce that the Mackey formula always holds for unipotent characters, see [BMM93, Thm. 1.35]. (Observe that Shoji’s proofs [Sho87] need the assumption that q  −1 (mod 3) for G of type E8 . But in fact by [Sho87, Rem. 3.3] this assumption is not necessary for the case of unipotent characters.) The most far-reaching result to date was obtained

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Harish-Chandra Theories

by Bonnafé–Michel [BoMi11, Thm.] using sophisticated computational techniques, and building upon that by Taylor [Ta18a]: Theorem 3.3.7 (Mackey formula) Let P, Q be parabolic subgroups of G with F-stable Levi complements L, M respectively. Let ψ ∈ Irr(MF ) and assume that one of the following conditions is satisfied: either L or M is a maximal torus; ψ is unipotent; F is a Frobenius map and q > 2; F is a Frobenius map and GF does not possess a component of type 2E6 , E7 or E8 ; or (5) F is a Frobenius map, Z(G) is connected and G does not have a factor of type E8 .

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

Then ∗ G RLP

G ◦ RMQ (ψ) =

L ∗ M RL∩ wML∩ w Q ◦ ad(w) ◦ RL w ∩MP w ∩M (ψ),

w ∈L F \S (L,M) F /M F

where w runs over a system of LF –MF double coset representatives in S (L, M)F := {g ∈ GF | L ∩ g M contains a maximal torus of G}. G and As in the Harish-Chandra case, this result immediately implies that RLP are independent of the particular parabolic subgroup P containing the Fstable Levi subgroup L, unless possibly when we are in one of the excluded cases in Theorem 3.3.7:

∗RG LP

Theorem 3.3.8 In the notation of Theorem 3.3.7, assume that any one of the assumptions (1)–(5) of that theorem are satisfied, or that the centre of G is connected. G Then RLP and ∗RG LP are independent of the parabolic subgroup P containing L. The statement for connected centre had been shown by Shoji [Sho96, Thm. 4.2] in the case that the underlying characteristic is almost good for G. This latter assumption has now been removed by Lusztig by showing the ‘cleanness’ of character sheaves, see e.g., [Ge18, §7]. G ∗ G We will from now on write RLG and ∗RG L in place of RLP and RLP respectively whenever the result will not depend on the choice of P, for example whenever the Mackey formula is known to hold. Remark 3.3.9 Since the Mackey formula holds if one Levi subgroup is a maximal torus, it follows by transitivity of Lusztig induction that the Mackey formula, and thus also the independence of Lusztig induction from the chosen parabolic subgroup, holds on the subspace of uniform functions for arbitrary L and M.

3.3 Lusztig Induction and Restriction

241

By transitivity Lusztig induction sends uniform functions to uniform functions. The Mackey formula implies the following stronger property: Proposition 3.3.10 Let G be connected reductive. For all F-stable Levi subgroups L  G we have G L πun ◦ RLG = RLG ◦ πun

and

L ∗ G G πun ◦ ∗RG L = RL ◦ πun,

that is, Lusztig induction and restriction commute with uniform projection. L ( f ) be the part orthogonal Proof For a class function f on LF let f ⊥ := f − πun to the space of uniform functions. By transitivity of Lusztig induction (see TheL ( f )) is uniform. We claim that RG ( f ⊥ ) is orthogonal orem 3.3.6) we have that RLG (πun L to all uniform functions. Indeed, for any F-stable maximal torus T  G and any θ ∈ Irr(TF ) we have     G G ⊥ . RT (θ), RLG ( f ⊥ ) = ∗RG L (RT (θ)), f G By the Mackey formula, which holds as T is a maximal torus, ∗RG L (RT (θ)) is a linear L  F combination of Deligne–Lusztig characters RT (θ ) of L and hence orthogonal L ( f ) + f ⊥ ) = RG (π L ( f )) + RG ( f ⊥ ) is the decomposition to f ⊥ . So RLG ( f ) = RLG (πun L un L into the uniform part and the part orthogonal to it. The second formula follows by adjunction. 

In contrast to the situation for Deligne–Lusztig characters and for Harish-Chandra induction, the decomposition of Lusztig induction RLG is not yet known in general. Nevertheless, quite substantial partial results are known in important special cases, for example in the case of unipotent characters, or when G has connected centre, see Section 4.6. Lusztig induction satisfies a character formula generalising Theorem 2.2.16 for RTG . For this we need a generalisation of the Green functions introduced there: Definition 3.3.11 Let L be an F-stable Levi subgroup of a parabolic subgroup P of G with Levi decomposition P = UL. Then

1 F F (−1)i Trace (u, v) | Hic (L −1 (U)) , QG LP : Guni × Luni → Q, (u, v) → |LF | i0 is called the associated 2-parameter Green function. Here Guni denotes the set of unipotent elements of G. F = {1} and If L = T is a maximal torus, so P = B is a Borel subgroup, then Luni G G F the defining formula shows that QTB (u, 1) = RT (1)(u) (u ∈ Guni ), which is the (1-parameter) Green function introduced in Definition 2.2.15. See Lusztig [Lu90, Thm. 1.14] for a more general result, and also Corollary 3.3.19.

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Harish-Chandra Theories

G As for Lusztig induction, we will write QG L in place of Q LP whenever the result does not depend on P.

Theorem 3.3.12 (Character formula) Then RLG (ψ)(g) = |LF |

1 |CG◦ (s)F |

Let L be an F-stable Levi subgroup of G.

|Ch◦L (s)F |

F v ∈Ch◦ (s)uni

h ∈G F |s h ∈L

C ◦ (s) (u, v −1 ) hψ(sv), (s) h

QCG◦

L

L

for ψ ∈ Irr(LF ) and g ∈ GF with Jordan decomposition g = su, and |C ◦ (s)F | C ◦ (s) ∗ G RL (ρ)(l) = L◦ F QCG◦ (s) (u, v −1 ) ρ(su), L |CG (s) | ◦ F u ∈CG (s)uni

for ρ ∈ Irr(GF ) and l ∈ LF with Jordan decomposition l = sv. For proofs, see [DiMi83, Thm. 2.2], [Lu86a, Prop. 6.2], [Lu90, 1.7(b)] or [DiMi20, Prop. 10.1.2]. The right-hand side of the character formula for ∗RG L does not change when replacing G by CG◦ (s) and L by CL◦ (s), so this immediately implies: Corollary 3.3.13 Let L be an F-stable Levi subgroup of G and ρ ∈ Irr(GF ). Let s ∈ GF be semisimple. Then C ◦ (s)

L ∗ G ∗ G G (ResC ◦ (s) F ◦ RL )(ρ)(g) = ( RC ◦ (s) ◦ ResC ◦ (s) F )(ρ)(g) F L

for any g ∈

LF

L

F

G

with semisimple part s. In particular, if CG◦ (s)  L then ∗ G RL (ρ)(g)

= ρ(g).

Observe that by [DiMi20, Prop. 3.5.3], CL (s)/CL◦ (s) consists of semisimple elements, so any g ∈ LF with semisimple part s lies in the connected component CL◦ (s) of the centraliser of s, hence the first formula makes sense. The second part follows from the first as then CG◦ (s) = CL◦ (s). Let L  G be an F-stable Levi subgroup. Then for ρ ∈ Irr(GF ) we may write



   ∗ G ∗ G ρ, RLG (λ) λ. RL (ρ) = RL (ρ), λ λ = λ∈Irr(L F )

λ∈Irr(L F )

Thus, from Corollary 3.3.13 we obtain Schewe’s formula [Sche85, Satz 1.3]

  ρ(g) = ρ, RLG (λ) λ(g) λ∈Irr(L F )

for any g ∈ GF with Jordan decomposition g = su with CG◦ (s)  L. The character formula, together with the fact that the Green functions take values in Q, also implies compatibility with Galois automorphisms:

3.3 Lusztig Induction and Restriction

243

Corollary 3.3.14 Let L  G be an F-stable Levi subgroup and σ be a field automorphism of K. Then we have σ(RLG (λ)) = RLG (σ(λ))

and

∗ G σ(∗RG L (ρ)) = RL (σ(ρ))

for all λ ∈ Irr(LF ), ρ ∈ Irr(GF ). This allows us to deduce the following action of Galois automorphisms on Lusztig series: Proposition 3.3.15 Let s ∈ G∗F be semisimple and σ be a field automorphism of K. Then σ(E (GF , s)) = E (GF , sr ) where r ∈ N is such that σ(ζ) = ζ r for all |GF |th roots of unity ζ. Proof By definition ρ ∈ E (GF , s) if there is a pair (T, θ) ∈ X(G, F), with T  G an F-stable maximal torus and θ ∈ Irr(TF ), corresponding to (T∗, s) ∈ Y(G∗, F) via Corollary 2.5.14 such that ρ, RTG (θ)  0. Now by assumption we have σ(θ) = θ r , and according to Remark 2.5.15 the pair (T, θ r ) corresponds to (T∗, sr ), so we obtain       σ(ρ), RTG∗ (sr ) = σ(ρ), RTG (θ r ) = σ(ρ), RTG (σ(θ))     = σ(ρ), σ(RTG (θ)) = ρ, RTG (θ)  0, where the third equality holds by Corollary 3.3.14. Hence σ(ρ) ∈ E (GF , sr ).



For a finite group H denote by CF(H) p the set of p-constant class functions on H, that is, class functions f such that f (h) = f (h p ) for any h ∈ H. The following result, which is strongly reminiscent of a similar assertion for ordinary induction and restriction, is easily deduced from the character formula (see [DiMi20, Prop. 10.1.6]): Proposition 3.3.16 Let f ∈ CF(GF ) p and L be an F-stable Levi subgroup of G. Then

F ( f ) = RLG (π) ⊗ f for all π ∈ CF(LF ), RLG π ⊗ ResG L FF ∗RG (η) ⊗ ResG ( f ) = ∗RG for all η ∈ CF(GF ). L L (η ⊗ f ) LF In particular ∗ G RL ( f )

F

= ResG ( f ). LF

Proof The first claim is immediate from the character formula in Theorem 3.3.12; from this the second is obtained by adjunction. The last assertion follows by choosing η = 1G in the preceding formula and using  that ∗RG L (1G ) = 1L by Example 3.3.4.

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Harish-Chandra Theories

In particular the previous result shows that for f ∈ CF(GF ) p , G RLG (π ⊗ ∗RG L ( f )) = RL (π) ⊗ f

and

∗ G RL (η)

∗ G ⊗ ∗RG L ( f ) = RL (η ⊗ f )

for all π ∈ CF(LF ) and η ∈ CF(GF ). For applications to the block theory of finite reductive groups, the following consequence of the character formula is fundamental; here for an -element g ∈ G and a prime , g

d : CF(GF ) → CF(CG (g)F ) denotes the generalised decomposition map defined by f (gh) if h ∈ CG (g) F,

g

d ( f )(h) =

0

else

(see e.g. [CE04, Def. 5.7]). Proposition 3.3.17 Let  be a prime different from p and L  G be an F-stable Levi subgroup. Then C ◦ (s)

∗ G s d s ◦ ∗RG L = RC ◦ (s) ◦ d L

for all (semisimple) -elements s ∈

LF .

Observe that the image of d s consists of class functions on CG (s)F vanishing outside CG (s) F , and the latter is contained in CG◦ (s)F (see e.g. [DiMi20, Prop. 3.5.3]), so the composition on the right-hand side of the formula does make sense. The claim now readily follows from the character formula for ∗RG L as given in Corollary 3.3.13. The degree of Lusztig induced characters is given by the same formula as in the Harish-Chandra case, but this is somewhat harder to show (see [Lu76b, Prop. 12]): Proposition 3.3.18 Let L be an F-stable Levi subgroup of G. Then RLG (λ)(1) = εG εL |GF : LF | p λ(1)

for all λ ∈ Irr(LF ).

Thus in particular ∗ G RL (regG )

= εG εL |GF : LF | p reg L .

The proof only uses the degree formula in terms of Deligne–Lusztig characters in Proposition 2.3.23 and the Mackey formula in the case (1) of Theorem 3.3.7, see [DiMi20, Prop. 10.2.9]. The second statement follows by applying the first to obtain G F F λ, ∗RG L (regG ) = RL (λ), regG = εG εL |G : L | p λ, reg L

for all λ ∈ Irr(LF ). In contrast there is no simple formula for the degree of a Lusztig restricted character!

3.3 Lusztig Induction and Restriction

245

The previous formula allows us to determine certain values of the 2-parameter Green functions introduced in 3.3.11: F be Corollary 3.3.19 Let L be an F-stable Levi subgroup of G and v ∈ Luni unipotent. The 2-parameter Green function satisfies

QG L (1, v) =

εG εL |GF : LF | p

if v = 1,

0

otherwise.

Proof Apply the character formula in Theorem 3.3.12 to the formula for ∗RG L (regG ) in Proposition 3.3.18 at the element v.  We now relate Lusztig induction and restriction to the rational Lusztig series introduced in Definition 2.6.1. For this, let (G∗, F) be in duality with (G, F) as in Definition 1.5.17. Let L  G be an F-stable Levi subgroup of G of type (I, w) (see 3.3.1), and let L be an F-stable Levi subgroup of G∗ of type (I ∗, (w ∗ )−1 ), where I ∗ = {s∗ | s ∈ I}. (Observe that the map (I, w) → (I ∗, (w ∗ )−1 ) induces a bijection between GF -conjugacy classes of F-stable Levi subgroups of G and G∗F -conjugacy classes of F-stable Levi subgroups of G∗ .) Then the complete root



data of L and L  are given by (X, RI , Y, RI∨ ), wWI and (Y, RI∨, X, RI ), (w ∗ )−1 W∗I respectively, with RI the parabolic subsystem of R determined by I. Thus, L has the same complete root datum as a group (L∗, F) dual to (L, F) and hence L   L∗ . In this way we can and will identify the dual L∗ of L with an F-stable Levi subgroup of G∗ (up to conjugation). Proposition 3.3.20 semisimple. Then

Let L be an F-stable Levi subgroup of G and s ∈ L∗F be

RLG (λ) ∈ ZE (GF , s)

for all λ ∈ E (LF , s),

that is, Lusztig induction preserves rational Lusztig series. Furthermore, for all F ρ ∈ E (GF , s), any constituent of ∗RG L (ρ) lies in a Lusztig series E (L , t) for some ∗F ∗F t ∈ L in the G -conjugacy class of s. This is shown in [Lu76b, Cor. 6], see also [Bo06, Thm. 11.10, Cor. 11.11], for F a Frobenius endomorphism; see [BoRo93, Thm. 10.3] for the case of Steinberg maps. It requires consideration of the individual cohomology groups, not just the alternating trace. Note that the second statement follows from the first by adjunction. Corollary 3.3.21 Lusztig series are unions of Harish-Chandra series. Proof Let (LF , λ) be a cuspidal pair, with λ ∈ E (LF , s) for some semisimple element s ∈ L∗F . Now if ρ ∈ Irr(GF ) lies in the Harish-Chandra series of (LF , λ), then by definition RLG (λ), ρ  0 and hence ρ ∈ E (GF , s) by Proposition 3.3.20.  Thus indeed E (GF , (LF , λ)) ⊆ E (GF , s).

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Harish-Chandra Theories

We now return to Lusztig’s important Jordan decomposition of characters stated in Section 2.6, in particular concerning its connection and interrelation with Lusztig induction. The next result gives a conceptual construction of the Jordan decomposition in most cases, that is, for Lusztig series parametrised by semisimple elements of the dual group whose centraliser lies inside a proper F-stable Levi subgroup. In fact, it can be shown that for a semisimple algebraic group G there is only a finite number of (F-stable) semisimple conjugacy classes of G∗ whose centraliser is not contained in a proper (F-stable) Levi subgroup of G∗ ; they are called quasi-isolated. See Bonnafé [Bo05] for a classification of such elements. Theorem 3.3.22 (Lusztig [Lu76b, Prop. 10], [Lu77a, (7.9)]) Let L be an F-stable Levi subgroup of G and s ∈ G∗F be a semisimple element such that CG∗ (s)F  L∗F . Then RLG induces a bijection of rational Lusztig series E (LF , s) −→ E (GF , s),

λ → εL εG RLG (λ).

If moreover we have that CG∗ (s)F = L∗F then the composition JsG := ( ⊗ sˆ−1 ) ◦ εL εG (RLG )−1 : E (GF , s) → E (LF , s) → Uch(LF ) (where ( ⊗ sˆ−1 ) denotes tensoring with the linear character sˆ−1 ) is a Jordan decomposition as in Theorem 2.6.22. Note that we don’t assume CG∗ (s) = L∗ in the second part; so CG∗ (s) might in fact be disconnected, as long as CG∗ (s)F = CG◦ ∗ (s)F . In fact, as first conjectured by Broué, even more is true in this situation: for any prime   p such that s is of  -order, RLG induces a Morita equivalence between the union of -blocks of LF containing characters from E (LF , s) and the union of -blocks of GF containing characters from E (GF , s) (see Bonnafé–Rouquier [BoRo93, Thm. 10.7], from which the above result follows for arbitrary Steinberg endomorphisms, as well as [CE04, §12], and Bonnafé–Dat–Rouquier [BDR15] for an even more far-reaching result). We obtain the following partial compatibility between Jordan decomposition and Harish-Chandra induction; see also Corollary 4.7.6 for a stronger statement for groups with connected centre. Corollary 3.3.23 Let (L, λ) be a cuspidal pair in G with λ ∈ E (LF , s) for some s ∈ L∗F and let ρ ∈ Irr(GF ) lie in the Harish-Chandra series above (L, λ). Assume that CG∗ (s)  G∗s for some F-stable Levi subgroup Gs of G and set L∗s := CL∗ (s) = L∗ ∩ G∗s , a Levi subgroup of G∗s . Let ρs ∈ E (GsF , s) and λs ∈ E (LsF , s) with G (ρs ) ρ = εGs εG RG s

and

λ = εLs εL RLLs (λs )

as in Theorem 3.3.22. Then (Ls, λs ) is a cuspidal pair, ρs lies in the Harish-Chandra

3.3 Lusztig Induction and Restriction series above (Ls, λs ) and



247

   RLG (λ), ρ = RLGss (λs ), ρs .

That is, in the situation of Theorem 3.3.22, Jordan decomposition preserves HarishChandra series. Proof First of all, ρ ∈ E (GF , s), ρs ∈ E (GsF , s) and λs ∈ E (LsF , s) by Proposition 3.3.20. Since L∗ is split in G∗ we have L∗ = CG∗ (Z ◦ (L∗ )1 ) and thus L∗s = G∗s ∩ L∗ = CG∗s (Z ◦ (L∗ )1 ) is also a split Levi subgroup of G∗s by Lemma 3.1.3. Moreover, as λ is cuspidal, so is λs by Theorem 3.2.22. Now we calculate      G Gs  εGs εG RLGss (λs ), ρs = RLGss (λs ), ∗RG G s (ρ) = RG s (RL s (λs )), ρ     = RLG (RLLs (λs )), ρ = εLs εL RLG (λ), ρ  0. In particular, this shows that ρs lies in the Harish-Chandra series above (Ls, λs ). Moreover, as both multiplicities are non-negative, the signs must cancel.  As a consequence of Proposition 3.3.20 and Theorem 3.3.22 the problem of decomposing Lusztig induction reduces to cases of characters lying in quasi-isolated Lusztig series, of various F-stable Levi subgroups of G. For unipotent characters this decomposition has been determined completely up to one small indeterminacy, see Section 4.6. Further partial results were obtained by Kessar–Malle [KeMa13] and Hollenbach [Ho19] for certain d-split Levi subgroups (to be introduced in Section 3.5) and quasi-isolated series in exceptional groups of Lie type. The problem of decomposing RLG for arbitrary characters would reduce to the unipotent case if it were known in general that Lusztig induction and restriction commute with Jordan decomposition of characters, or more precisely, that a Jordan decomposition can always be chosen having this property. At least for groups with connected centre this is now known except for very few situations in exceptional groups. This will be explained in Section 4.7 once we have collected detailed information on unipotent characters. We next discuss the behaviour of Lusztig induction with respect to regular embeddings. For this, as in Remark 2.3.16, let us first consider a slightly more general situation. Let G also be connected reductive and F  : G → G be a Steinberg map. Let f : G → G be an isotypy (see 1.3.21), that is, f is a homomorphism of   f (G), and assume further that algebraic groups such that ker( f )  Z(G) and Gder   f ◦ F = F ◦ f . Then the map L → L := f (L)Z(G) sets up a bijection between the F-stable Levi subgroups of G and the F -stable Levi subgroups of G, with inverse given by L := f −1 (L)  G for L an F -stable Levi subgroup of G. Proposition 3.3.24

In the above setting, let L be an F-stable Levi subgroup of

248

Harish-Chandra Theories

G and L := f (L)Z(G)  G, an F -stable Levi subgroup of G. If ker( f ) is connected then we have 

fG∗ ◦ RLG  fL∗ ◦ ∗RG L

= =

RLG ◦ fL∗, ∗RG ◦ f ∗, L G

where fG∗ (ψ) = ψ ◦ f for class functions ψ on G  F , and fL∗ (η) = η ◦ f for class functions η on L  F . See [Bo00, Cor. 2.1.3] and [DiMi90, Cor. 9.2] for two different proofs. ˜ is a regular embedding (see Definition 1.7.1). This Now assume that i : G → G ∗ = ResG˜ and i ∗ = Res L˜ . Thus the above is, in particular, an isotypy, and we have iG L G L yields the following generalisation of Proposition 3.2.9: Corollary 3.3.25 Let L be an F-stable Levi subgroup of G with corresponding ˜  G. ˜ Then we have ˜ = LZ(G) Levi subgroup L ˜

G IndG G ◦ RL ˜

Res LL ◦ ∗RG L˜ ˜

= =

˜

˜

Ind LL ◦ ∗RG L

˜

G ResG G ◦ RL˜

RLG˜ ◦ Ind LL,

∗RG L

◦ ResG G,

˜

˜

˜

= =

˜ ∗RG L˜ RLG

˜

◦ IndG G, ˜

◦ Res LL .

Here, the first two formulas follow from the second two by adjunction, see [Bo00, (2.1.4)–(2.1.7)]. Another basic and useful property of Lusztig induction (and restriction) is its commutation with restriction of scalars, which is sometimes useful in reduction statements. For this let G = G1 × · · · × Gr be a direct product of isomorphic simple groups Gi and F : G → G a Steinberg endomorphism such that F(Gi ) = Gi+1 (with the indices taken modulo r). Thus G is F-simple in the sense of 1.5.14. Then F r stabilises each G1 and induces a Steinberg endomorphism on it. Let L1  G1 be an F r -stable Levi subgroup and Li := F i−1 (L1 )  Gi . Denote by π1∗ the inflation r of class functions from G1F to GF . Proposition 3.3.26 Lusztig induction commutes with restriction of scalars, that is, in the above situation we have π1∗ ◦ RLG11 = RLG ◦ π1∗ . This is shown in [Ta19, Prop. 8.3], see also [DiMi14, Prop. 7.17].

3.4 Duality and the Steinberg Character The Harish-Chandra induction and restriction functors can be used to define an important duality operation on the space of K-valued class functions on a finite reductive group, see [Al79, Cu80, DeLu82, Kaw82] and also [Lu17]. We continue the set-up of the previous section, that is, G is a connected reductive algebraic group over an algebraically closed field of characteristic p with a Steinberg endomorphism

3.4 Duality and the Steinberg Character

249

F : G → G, and we set G := GF . Furthermore, S denotes the set of Coxeter generators of the Weyl group W = WF of GF with respect to some maximally split torus of G. Definition 3.4.1 The Alvis–Curtis–Kawanaka–Lusztig duality operator on the space of class functions CF(G) on G is defined as

DG := (−1) |I | RLGI ◦ ∗RG LI . I ⊆S

As the functors RLG and ∗RG L are adjoint to one another by Proposition 3.1.10 it easily follows that DG is self-adjoint: Proposition 3.4.2 We have ρ, DG (ψ) = DG (ρ), ψ for all class functions ρ, ψ ∈ CF(G), that is, DG is self-adjoint. The third and fourth formulae in Proposition 3.2.9 show that duality commutes with regular embeddings: ˜ be a regular embedding. Then Proposition 3.4.3 Let G → G ˜

˜

G ResG ˜ = DG ◦ ResG . G ◦ DG

The Mackey formula can be used to show the following commutation with Lusztig induction: Theorem 3.4.4 Let L be an F-stable Levi subgroup of G such that the Mackey formula 3.3.7 holds for RLG . Then εG DG ◦RLG = εL RLG ◦ DL and ∗ G εG ∗RG L ◦ DG = εL DL ◦ RL .

In particular for all F-stable maximal tori T of G we have εG DG ◦RTG = εT RTG . See [DiMi20, Thms. 7.2.4 and 10.2.1]. The proof of the first assertion uses only transitivity of RLG and the Mackey formula, the second claim follows from the first by adjunction, and the third by using that the Mackey formula holds if one of the Levi subgroups is a torus and that by definition duality is the identity on the characters of a torus. We conclude from this: Corollary 3.4.5 The operator DG ◦ DG is the identity on CF(G).

250 Proof

Harish-Chandra Theories (See [DiMi20, Cor. 7.2.8].) From the definition and Theorem 3.4.4 we have

(−1) |I | RLGI ◦ ∗RG DG ◦ DG = L I ◦ DG I ⊆S

=

I ⊆S

=

(−1) |I | RLGI ◦ DL I ◦∗RG LI (−1) |I |

J ⊆I

I ⊆S

=

J ⊆S

(−1)

|J |

(−1) |J | RLGI ◦ RLLJI ◦ ∗RLLJI ◦ ∗RG LI

I ⊇J

(−1) |I | RLGJ ◦ ∗RG LJ ,

using transitivity and the Mackey formula for Harish-Chandra induction. The claim  now follows as clearly I ⊇J (−1) |I | = 0 for all proper subsets J of S.  Since Lusztig series are defined in terms of multiplicities in Deligne–Lusztig characters (see Definition 2.6.1) the previous theorem also immediately implies the preservation of rational Lusztig series: Corollary 3.4.6 If ρ ∈ E (G, s) then ± DG (ρ) ∈ E (G, s). A fundamental property of the duality operation is that it sends irreducible characters to irreducible characters up to sign; recall from 3.2.3 that the relative rank r(L) of a Levi subgroup L = LF of G satisfies (−1)r(L) = ε[L,L] . Proposition 3.4.7 The duality DG permutes the irreducible characters of G up to sign. More precisely, if ρ ∈ E (G, (L, λ)) for a cuspidal pair (L, λ) in G then (−1)r(L) DG (ρ) is again irreducible and also lies in E (G, (L, λ)). The irreducibility up to sign was first shown in [Al79, Thm. 4.2], see also [DiMi20, Cor. 7.2.9]. It is immediate from the definition that DL (λ) = (−1)r(L) λ for a cuspidal character λ of L. It then follows by adjointness (Proposition 3.1.10) and commutation with DG (Theorem 3.4.4) that       ∗ G DG (ρ), RLG (λ) = ∗RG L (DG (ρ)), λ = DL ( RL (ρ)), λ   ∗ G  = ∗RG L (ρ), DL (λ) = ± RL (ρ), λ  0, so DG (ρ) lies in the Harish-Chandra series of (L, λ) as claimed. This can be made even more precise. For this, let ε denote the sign character G of the Coxeter group WG (L, λ) and recall the bijection I L,λ : IrrK (WG (L, λ)) → E (G, (L, λ)) from Theorem 3.2.5. Theorem 3.4.8 Let (L, λ) be a cuspidal pair in G. We have G G (φ)) = (−1)r(L) I L,λ (ε ⊗ φ) DG (I L,λ

for all φ ∈ Irr(WG (L, λ)).

3.4 Duality and the Steinberg Character

251

See [HoLe83, Thm. 7.5], or [CuRe87, Thm. 71.14] for the case of the principal series. We will have more to say about the degree of DG (ρ) in Propositions 3.4.20 and 3.4.21 below. Duality gives a particularly elegant approach, due to Curtis [Cu66], to the Steinberg character, originally introduced by Steinberg [St57] in the case of groups of Lie type, of a finite group with an algebraic BN-pair. By Example 3.1.8 we have that ∗RG L (1G ) = 1L for any Levi subgroup L  G. Applying this to L = T we see that 1G ∈ E (G, (T, 1T )). Since r(T) = 0 it follows from Proposition 3.4.7 that DG (1G ) is an irreducible character. Definition 3.4.9 The irreducible character StG := DG (1G ), where 1G is the trivial character of G, is called the Steinberg character of G. According to Theorem 3.4.8 the Steinberg character StG lies in the principal series E (G, (T, 1T )) of G and is labelled by the sign character of the Weyl group WG (T, 1G ) = W. Moreover, as 1G is a uniform function, the same holds for StG by construction; more explicitly we have 1 StG = (−1)l(w) Rw |W| w ∈W (see Example 2.3.23). Since duality commutes with Lusztig induction by Theorem 3.4.4 we conclude via Remark 3.3.9 that ∗ G RL (StG )

∗ G = ∗RG L (DG (1G )) = εL εG DL ( RL (1G ))

= εL εG DL (1L ) = εL εG StL for any F-stable Levi subgroup L  G. The values of the Steinberg character are easy to describe: Proposition 3.4.10 Let G = GF be a finite group of Lie type and g ∈ G. Then StG (g) =

εG εCG◦ (g) |CG◦ (g)F | p

if g is semisimple,

0

else.

See [DiMi20, Cor. 7.4.4]. In particular the Steinberg character StG has degree StG (1) = |G| p , so it is an irreducible character of G of p-defect zero. Now assume that G is perfect (which is often the case for G of simply connected type, see e.g. [MaTe11, Thm. 24.17]). Then it can be shown using the representation theory of algebraic groups that all other irreducible characters of G = GF have positive pdefect. (This also follows from Lusztig’s classification of irreducible characters.) Indeed, Weyl’s character formula shows that all other p-modular irreducible representations of G have degree less than |G| p . Since any complex irreducible character

252

Harish-Chandra Theories

of G of degree divisible by |G| p remains irreducible under p-modular reduction, no other complex irreducible character of G can have p-defect zero. The Steinberg character relates Lusztig induction to ordinary induction: Corollary 3.4.11 Let L be an F-stable Levi subgroup of G. Then F

for all λ ∈ Irr(LF ),

F

for all ρ ∈ Irr(GF ).

εG RLG (λ) ⊗ StG =

εL IndG (λ ⊗ StL ) LF

εG ∗RG L (ρ ⊗ StG ) =

εL ResG (ρ) ⊗ StL LF

Proof Application of the character formula 3.3.12 and Proposition 3.4.10 show that in order to verify the second formula we need to prove that εCG◦ (s)

|CL◦ (s)F |

◦ (s) CG Q (1, v −1 ) ◦ ◦ C |CG (s)F | p L (s)

ρ(s) =

εCL◦ (s) |CL◦ (s)F | p ρ(s)

if v = 1,

0

else,

F centralising s. But this for any semisimple element s ∈ LF and unipotent v ∈ Luni is just Corollary 3.3.19 applied to the F-stable Levi subgroup CL◦ (s) of CG◦ (s). The first formula follows from the second by adjunction. 

3.4.12 Recall the semisimple characters ρ ∈ S0 (GF ) of GF introduced in Definition 2.6.9, characterised by the property that ρ, ΔG  0, where ΔG is the class function taking values ΔG (g) =

|Z◦ (G)F |ql

if g ∈ O0F ,

0

otherwise,

with l the semisimple rank of G and O0 ⊆ G the class of regular unipotent elements. Definition 3.4.13 Let ρ ∈ Irr(GF ). Then ρ is a regular character of GF if ± DG (ρ) is a semisimple character. We write S 0 (GF ) for the set of all regular characters of GF . This notion should not be confused with the (in general reducible) regular character of an abstract finite group. Example 3.4.14 According to 2.6.9 the trivial character is semisimple and thus the Steinberg character StG is an example of a regular character. 3.4.15 Assume that Z(G) is connected. Let (G∗, F) be dual to (G, F) (see Definition 1.5.17). Then the (rational) Lusztig series of any semisimple element s ∈ G∗F contains a unique semisimple character ρs (see 2.6.10). By Theorems 2.6.11 and 3.4.4 we obtain  G      RT∗ (s), DG (ρs ) = DG (RTG∗ (s)), ρs = εG εT∗ RTG∗ (s), ρs = εH εT∗ ,

3.4 Duality and the Steinberg Character

253

where H = CG∗ (s). The degree formula in Theorem 2.6.2 then gives   |GF | |GF | DG (ρs )(1) = F εG εT∗ RTG∗ (s), DG (ρs ) = F εG εH, |H | T∗ |H | T∗ H where the sum runs over all F-stable maximal tori of H; thus the regular character corresponding to ρs is ρs := εG εH DG (ρs ) ∈ Irr(GF ). It lies in the same (rational) Lusztig series E (GF , s) as ρs . We have the following analogue of Theorem 2.6.11: Theorem 3.4.16 (See [DeLu76, Thm. 7]) Assume that Z(G) is connected and let s ∈ G∗ F be semisimple. Let ρs ∈ S 0 (GF ) be the unique regular character that belongs to E (GF , s). Then the following hold. (a) For any F-stable maximal torus T∗  G∗ with s ∈ T∗ , RTG∗ (s), ρs = εG εT∗ . (b) Let H := CG∗ (s). Then ρs (1) = |G∗F : HF | p · |HF | p . (c) Via the Jordan decomposition of characters in Theorem 2.6.4, the character ρs ∈ E (GF , s) corresponds to the Steinberg character of HF . Proof (a) was already shown above, and (b) follows from the above formula for DG (ρs )(1) with Remark 2.5.16. (c) Let ψ ∈ Uch(HF ) correspond to ρs via a bijection as in Theorem 2.6.4. Let ∗ T ⊆ H be an F-stable maximal torus. Using (a) we obtain RTH∗ (1T∗ ), ψ = εG εH RTG∗ (s), ρs = εH εT∗ . Hence, Example 2.2.27 shows that ψ must be the Steinberg character of HF .



Regular characters also satisfy the analogue of Corollary 2.6.18. For this let ˜ ∗ → G∗ . ˜ be a regular embedding with dual epimorphism i ∗ : G i : G → G Corollary 3.4.17 Let s ∈ G∗ be semisimple and let s˜ ∈ G˜ ∗ be such that i ∗ (s˜) = s. ˜ be the unique regular character in E (G, ˜ s˜). Let ρ˜ ∈ Irr(G) (a) The regular characters in E (G, s) are precisely the irreducible constituents of ρ| ˜ G. (b) Let ρ ∈ E (G, s) be a regular character. Then RTG∗ (s), ρ = ±1 for any F-stable maximal torus T∗ ⊆ G∗ such that s ∈ T∗ . Proof Let ρ ∈ E (G, s) be regular. Then there exists a semisimple character ρ ∈ E (G, s) with ρ = ± DG (ρ). By Corollary 2.6.18 this is a constituent of ρ˜  |G for the ˜ has connected centre, E (G, ˜ s˜). As G ˜ s˜) (unique) semisimple character ρ˜  ∈ E (G,  also contains a unique regular character, so that ρ˜ = ± DG˜ ( ρ˜ ). Thus we have G   ResG ˜ ρ = ±ResG ˜ ( ρ˜ )), ρ = ±DG (ResG ( ρ˜ )), ρ G ( ρ), G (DG ˜

˜

˜

G    = ±ResG G ( ρ˜ ), DG (ρ) = ±ResG ( ρ˜ ), ρ  0, ˜

˜

254

Harish-Chandra Theories

where the second equality holds by Proposition 3.2.9; so ρ is a constituent of ρ| ˜ G. ˜ The same sequence of equations also shows that any constituent of ResG ( ρ) ˜ is in G fact regular. In (b), as ± DG (ρ) is semisimple by definition, we have       G RT∗ (s), ρ = DG (RTG∗ (s)), DG (ρ) = ± RTG∗ (s), DG (ρ) = ±1 

by Corollary 2.6.18(b). Remark 3.4.18 the dual

Assume G has connected centre. By Theorems 2.6.11 and 3.4.16 ΓG := DG (ΔG ) =

ρs

s ∈G∗F /∼

of ΔG is an actual character (where the sum runs over a system of representatives of the semisimple conjugacy classes in G∗F ). It is the so-called Gelfand–Graev character of GF . It can be constructed as an induced character from a suitable unipotent subgroup of GF . Applying DG to the formula for ΔG in Remark 2.6.12 we obtain the expression

1 ΓG = F εG εT∗ |T∗ F |RTG∗ (s) |G | (T∗,s)∈Y(G∗,F) for the Gelfand–Graev character of GF . The Gelfand–Graev character, together with its generalisations, the so-called generalised Gelfand–Graev characters, plays a crucial role in the study of decomposition matrices of finite groups of Lie type for non-defining primes, see e.g. [GeHi97], [DuMa18]. See also [Ca85, §8.1]; [DiMi20, §12.3] discuss Gelfand– Graev characters in the non-connected centre case, leading to a different notion of generalised Gelfand–Graev characters. Let Guni ⊂ G denote the set of unipotent elements of G (i.e., the set of pelements), and δu the characteristic function of Guni , taking value 1 on elements of Guni and zero elsewhere. Then the dual of the regular character regG of G is given as follows (see [DiMi20, Cor. 7.4.5]): Corollary 3.4.19 We have DG (regG ) = |G| p δu . Proof The function δu has the property that δu (g) = δu (g p ) for all g ∈ G, where g p denotes the p-part of g. It then follows from Proposition 3.3.16 (with f = δu G G ∗ G and π = 1L ) that RLG (ResG L (δu )) = RL (1L ) ⊗ δu and RL (δu ) = Res L (δu ) for any F-stable Levi subgroup L  G. But then DG (δu ) = DG (1G ) ⊗ δu and so DG (|G| p δu ) = |G| p DG (1G ) ⊗ δu = |G| p StG ⊗ δu = regG

3.4 Duality and the Steinberg Character

255 

by Proposition 3.4.10. From this, we find with Proposition 3.4.2 and Corollary 3.4.5 that   |G| = regG, regG = DG (regG ), DG (regG ) = |G| 2p δu, δu = |G| 2p |Guni |/|G|.

This shows that G has exactly |Guni | = |G| 2p unipotent elements, a result originally due to Steinberg [St68, Thm. 15.1]. As a further consequence we obtain (see [CuRe87, Thm. 71.11], and also [Lu84a, (8.5.12)]): Proposition 3.4.20 Let ρ ∈ Irr(G). Then DG (ρ)(1) p = ±ρ(1) p, that is, the degrees of ρ and ± DG (ρ) differ only by a power of p. Proof

From Corollary 3.4.19 and Proposition 3.4.2 we get

  ρ(u) = |G|ρ, δu =|G| DG (ρ), DG (δu ) u ∈Guni

  =|G| p DG (ρ), regG = |G| p DG (ρ)(1).

 Now ρ(1)−1 u ∈Guni ρ(u) is a sum of values of central characters on the p-singular classes of G, hence an algebraic integer. Thus |G| p DG (ρ)(1)/ρ(1) ∈ Z, and, replacing ρ by ± DG (ρ) in this argument and using that D2G = id by Corollary 3.4.5, we  also have |G| p ρ(1)/DG (ρ)(1) ∈ Z. The claim follows from this. This has the following nice explanation due to Alvis [Al82, Cor. 3.6]: Proposition 3.4.21 are related by

Let ρ ∈ Irr(G). Then the degree polynomials of ρ and DG (ρ) DDG (ρ) (q) = q N Dρ (q−1 ).

Proof Recall that the order polynomial |G| p is a product of cyclotomic polynomials of degree N + r, where N = |R+ | and r = rnk(G). Similarly, the order |T| for any maximal torus T of G is a product of cyclotomic polynomials of degree r dividing |G| p , and moreover the number of factors q − 1 in this polynomial equals r(T) by Corollary 3.2.21. Now if Φ is a cyclotomic polynomial, then Φ(q−1 ) = ±q−d Φ(q), where d = deg(Φ) and the sign is positive unless Φ = q − 1. Thus if we let fT := |G| p /|T| ∈ Q[q] then q N fT (q−1 ) = εT εG fT (q). Now by Remark 2.2.28 the uniform projection of ρ can be written as a sum over conjugacy classes of pairs (T, θ) ∈ X(G, F)

εT εG a(T, θ) RTG (θ), (T,θ)

256

Harish-Chandra Theories

with rational coefficients a(T, θ) = ρ, RTG (θ) /RTG (θ), RTG (θ) . Similarly, the uniform projection of DG (ρ) is given by

εT εG b(T, θ) RTG (θ) (T,θ)

  with RTG (θ), RTG (θ) b(T, θ) = DG (ρ), RTG (θ) . By Proposition 3.4.2 and the commutation property of DG with RTG from Theorem 3.4.4 this latter quantity equals       ρ, DG (RTG (θ)) = εT εG ρ, RTG (θ) = εT εG RTG (θ), RTG (θ) a(T, θ), 



whence b(T, θ) = εT εG a(T, θ). Evaluation at the identity element thus gives

a(T, θ) RTG (θ)(1) DG (ρ)(1) =εT εG (T,θ)

=εT εG

a(T, θ) fT (q) = q N

(T,θ)

a(T, θ) fT (q−1 ),

(T,θ)

the degree polynomial of ρ evaluated at

q−1 .



Let qaρ denote the precise power of q dividing the degree polynomial of ρ, and Aρ its degree. Then the preceding result shows that aDG (ρ) = N − Aρ . We will have more to say about the invariants aρ and Aρ in Section 4.1. Proposition 3.4.22 Assume that G has only components of classical type, F is a Frobenius map and that Z(G) is connected. Then the Jordan decomposition for G commutes with DG . Proof Let s ∈ G∗ be semisimple and H := CG∗ (s). By our assumption on Z(G), H is connected. We write Js : E (GF , s) → Uch(HF ) for a Jordan decomposition as in Theorem 2.6.4. Then for all ρ ∈ E (GF , s), all F-stable maximal tori T  H∗ and all θ ∈ Irr(TF ) we have       DH (Js (ρ)), RTH (θ) = Js (ρ), DH (RTH (θ)) = Js (ρ), RTH (DT (θ))     = εG εH ρ, RTG (DT (θ)) = εG εH ρ, DG (RTG (θ))     = εG εH DG (ρ), RTG (θ) = Js (DG (ρ)), RTH (θ) . So DH (Js (ρ)) and Js (DG (ρ)) have the same multiplicities in all Deligne–Lusztig characters of H. Now note that H has only components of classical type. But the unipotent characters of groups of classical type are uniquely determined by their multiplicities in Deligne–Lusztig characters, see Theorem 4.4.23. Thus DH (Js (ρ)) = Js (DG (ρ)) as claimed. 

3.5 d-Harish-Chandra Theories

257

In fact the proof shows that the assertion holds for any Lusztig series E (GF , s) for G of classical type without assumption on Z(G) as long as CG∗ (s) is connected. Note that Proposition 3.4.22 also follows immediately from the (deeper) result of Enguehard (see Theorem 4.7.2) on the commutation of Jordan decomposition with Lusztig induction. It can be shown that the statement continues to hold for groups with connected centre and with no components of type E7 or E8 by using that DG preserves the so-called eigenvalues of Frobenius (see 4.2.21) and appealing to Corollary 4.5.4.

3.5 d-Harish-Chandra Theories As a consequence of the investigation of Brauer -blocks of finite groups of Lie type it became apparent that Harish-Chandra theory has an important generalisation in terms of Lusztig induction, which is obtained by replacing the set of 1-split Levi subgroups by d-split Levi subgroups and Harish-Chandra induction by Lusztig induction. This has proved to be of fundamental importance for example in the understanding and the description of the block theory of the finite groups of Lie type. While it is expected that these d-Harish-Chandra theories share many of the properties of ordinary Harish-Chandra theory, this has at present only been proved in particular circumstances. We start by defining the relevant subgroups, which were first introduced in [BrMa92]. 3.5.1 As before let G be connected reductive with a Steinberg map F. First assume that F is not very twisted. An F-stable torus T of G is called a d-torus (of (G, F)) if its complete root datum ((X, , Y, ), ϕ) (see Definition 1.6.10) is such that the characteristic polynomial of ϕ on Y is a power of the dth cyclotomic polynomial Φd . So, T is a d-torus if and only if its order polynomial is a power of the dth cyclotomic k polynomial Φd : there is a  0 such that |TF | = Φd (q k )a for all k ≡ 1 (mod d), with a = dim T/φ(d) where here φ denotes the Euler totient function. Alternatively, the torus T is a d-torus if and only if it splits completely over Fq d and no non-trivial subtorus of it splits over any proper subfield of Fq d . If T is an F-stable torus in G we denote by Td its maximal d-subtorus. This is well defined as the following description shows: In terms of complete root data, if T has complete root datum

((X, , Y, ), ϕ), then the root datum of Td is given by (X , , Y , ), ϕ  , where X  is the largest quotient of X on which ϕ has characteristic polynomial a power of Φd , Y  is the kernel of Φd (ϕ) on Y , and ϕ  the map on X  induced by ϕ. It is well known (see e.g. [MaTe11, Prop. 12.10]) that the centraliser of a torus in a connected reductive group is a Levi subgroup. The centralisers in G of d-tori

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are called d-split Levi subgroups of G. So in particular G itself is d-split as the centraliser of its trivial subtorus. Observe that d-split Levi subgroups are F-stable, being the centralisers of F-stable d-tori. It is clear from the definition that d-tori of (G, F) become 1-tori in (G, F d ). Thus, d-split Levi subgroups of (G, F) are 1-split if considered with respect to F d . Example 3.5.2 In the case d = 1 Lemma 3.1.3 shows that 1-split Levi subgroups are exactly what we had called (1-)split Levi subgroups in Example 3.1.2. 3.5.3 The above definitions have to be modified slightly if the Steinberg map is very twisted (see 1.6.2). For simplicity assume here that G is an F-stable Levi subgroup of some simple algebraic group with a very twisted Steinberg map. Let p denote the underlying characteristic of G. We consider the set Ψp consisting of q2 − 1 together √ with the irreducible factors over Z[ p] of dth cyclotomic polynomials Φd over Q with d  3. Then for Φ ∈ Ψp , a Φ-torus of G is an F-stable torus with complete root datum ((X, , Y, ), ϕ) such that the characteristic polynomial of ϕ on Y ⊗Z R is a power of Φ, and Φ-split Levi subgroups are the centralisers in G of Φ-tori (see [BrMa92, 3.F] and [EM17]). By a slight abuse of notation we will also call d-tori and d-split Levi subgroups these Φ-tori and Φ-split Levi subgroups respectively, where d is the order of the zeros of Φ, respectively d = 1 when Φ = q2 − 1. Example 3.5.4 In the very twisted case, when p = 2, the set Ψ2 contains the polynomials of degree 2 √ √ q2 − 1, q2 + 1, Φ8 := q2 + 2q + 1, Φ8 := q2 − 2q + 1, with d = 1, 4, 8, 8 respectively, as well as the polynomials of degree 4 √ √ √ √   := q4 + 2q3 + q2 + 2q + 1, Φ24 := q4 − 2q3 + q2 − 2q + 1, Φ24 with d = 24. These occur as factors of the order polynomial of groups of type 2F4 . When p = 3, Ψ3 contains for example the polynomials √ √   := q2 + 3q + 1, Φ12 := q2 − 3q + 1, Φ12 with d = 12. These occur as factors of the order polynomial of groups of type 2 G2 . Observe that evaluating any of these polynomials at the (positive) square root of an odd power p2 f +1 of p yields positive integral values. In order to make the exposition simpler we will from now on assume that F is not very twisted. Nevertheless, all statements, suitably modified, continue to hold in the very twisted case. Any Levi subgroup L of G is the centraliser of its connected centre: L = CG (Z◦ (L)) (see [DiMi20, Prop. 3.4.6]). The d-split Levi subgroups are characterised as follows:

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259

Proposition 3.5.5 An F-stable Levi subgroup L of G is d-split if and only if it satisfies L = CG (Z◦ (L)d ), where Z◦ (L)d denotes the maximal d-torus of Z◦ (L). Proof Note that Z◦ (L) is an F-stable torus of G, so its d-part is defined and its centraliser is a Levi subgroup. Clearly any Levi subgroup with the stated property is d-split by definition. Conversely, if L is the centraliser of a d-torus S, then S  Z◦ (L), so L = CG (S)  CG (Z◦ (L)d )  CG (Z◦ (L)) = L, proving the claim.



3.5.6 We now describe how to construct d-tori of G in terms of complete root data.

For this let (X, R, Y, R ), ϕW be the complete root datum of (G, F). If w ∈ W then

Y  = kerY (Φd (ϕw)) is a pure sublattice of Y and (X , , Y , ), ϕw , with X  the largest quotient of X on which ϕ has minimal polynomial Φd , is the complete root datum of a d-torus of G since, by construction, ϕw|Y  has characteristic polynomial a power of Φd . The d-tori constructed from elements w ∈ W with kerY (Φd (ϕw)) of maximal possible rank are called Sylow d-tori of G. It is clear from the order formula in 1.6.8(b) that the order polynomial of a Sylow d-torus of G is precisely the Φd -part of the order polynomial of G. The justification for this name comes from the fact that, as first observed by Broué–Malle [BrMa92, Thm. 3.4], the d-tori of G satisfy a Sylow theory. This is a consequence of Springer’s theory of eigenspaces of elements in finite reflection √ groups [Spr74]. For this, let ζd = exp(2π −1/d) be a primitive dth root of unity. For w ∈ W denote by V(ϕw, ζd ) the ζd -eigenspace of ϕw on V := Y ⊗Z C. Then the eigenspaces V(ϕw, ζd ) of maximal possible dimension are all conjugate under W, and any such eigenspace is contained in one of maximal dimension. Example 3.5.7 (Regular numbers) A minimal d-split Levi subgroup of G is, by our definitions, just the centraliser of a Sylow d-torus. Now observe that this is itself a (maximal) torus if and only if the maximal ζd -eigenspaces V(ϕw, ζd ) of elements ϕw (with w ∈ W) in the natural reflection representation have trivial centraliser in W. This happens if and only if V(ϕw, ζd ) contains a regular vector, that is, a vector not fixed by any element of W. By a result of Steinberg [St64, Thm. 1.5], these are exactly the vectors not lying in any reflecting hyperplane of W. Elements ϕw ∈ ϕW with a ζd -eigenspace containing regular vectors are called d-regular and d is then called a regular number for ϕW (see [Spr74]). As with ordinary Harish-Chandra theory an important ingredient for d-HarishChandra theory is given by relative Weyl groups: Definition 3.5.8 Let L be an F-stable Levi subgroup of G. The relative Weyl group of L in G is WG (L) := NG (L)F /LF .

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The relative Weyl group of an F-stable Levi subgroup can be described purely in terms of the complete root datum. Assume that L has complete root datum

(X, RL, Y, RL∨ ), ϕWL . Then WG (L) = NW (ϕWL )/WL . Thus for 1-split Levi subgroups L = LF of G = GF this agrees with the relative Weyl group WG (L) as defined in 3.1.25, see Example 3.1.26.

Now let d  1 and assume that L is d-split. Let (X , , Y , ), ϕw be the complete root datum of the Sylow d-subtorus Z◦ (L)d of Z◦ (L), so in particular Y  = kerY (Φd (ϕw)) (see 3.5.1). Then WG (L) acts on V = Y  ⊗Z C, and thus on the ζd -eigenspace V(ϕw, ζd ) of ϕw in V. A key feature of the theory is the fact that the relative Weyl group WG (L) of a minimal d-split Levi subgroup L is again a reflection group on V(ϕw, ζd ), but not necessarily any longer a real one: Proposition 3.5.9 Let Sd be a Sylow d-torus of G with complete root datum

(X, , Y, ), ϕw and L = CG (Sd ) be the corresponding minimal d-split Levi subgroup. Then WG (L) acts faithfully as a complex reflection group on V(ϕw, ζd ). Moreover, if the root system of G is indecomposable, then WG (L) is irreducible on V(ϕw, ζd ). This was shown first by Springer [Spr74, Thms. 4.2(iii) and 6.4(iii)] for the case that the centraliser of a Sylow d-torus is a maximal torus and then by Lehrer– Springer [LeSp99, Thms. 3.4 and 5.1] for the general case using the invariant theory of reflection groups. Their results are actually more precise; for example they show how the degrees of the complex reflection group WG (L) can be read off from those of W. The irreducibility statement is due to the same authors [LeSp99b, Thm. A]. See also [LeTa09, Thm. 11.15 and 11.38] for more elementary proofs. Remark 3.5.10 Some of these proofs rely on case-by-case considerations using the Shephard–Todd classification of the finite irreducible complex reflection groups [ST54]. These groups comprise several infinite series and a further 34 exceptional types usually denoted G4, . . . , G37 . Here, G4, . . . , G22 are in dimension 2, G23, . . . , G27 in dimension 3, G28, . . . , G32 in dimension 4, and G33, . . . , G37 are in dimensions 5,6,6,7,8 respectively. The finite Coxeter groups of exceptional types are contained in this list as W(H3 ) = G23, W(F4 ) = G28, W(H4 ) = G30, W(E6 ) = G35, W(E7 ) = G36, W(E8 ) = G37 . The three infinite series are the cyclic groups (in dimension 1), the symmetric groups Sn , n  3, in dimension n − 1, and the imprimitive groups G(de, e, n) with n, de  2 and (de, e, n)  (2, 2, 2). Here G(de, e, n) is a normal subgroup of index e of the wreath product G(d, 1, n)  Cd ' Sn . The dihedral Weyl groups W(I2m ) of order 2m

3.5 d-Harish-Chandra Theories

261

occur in this series as the groups G(m, m, 2). In particular the group G(e, e, 2) is isomorphic to the Weyl group of type G2 . The relative Weyl groups of the minimal d-split Levi subgroups of quasi-simple finite reductive groups are described in [BrMa93, Tab. 3.6] and in [BMM93, Tab. 1 and 3]. Example 3.5.11 We illustrate Proposition 3.5.9 by giving the non-cyclic relative Weyl groups (from [BMM93, Tab. 1 and 3]) of minimal d-split Levi subgroups in the simple exceptional groups in Table 3.2 in terms of their Shephard–Todd labels; see Remark 3.5.10. Table 3.2 Non-cyclic relative Weyl groups of minimal d-split Levi subgroups in exceptional types

G2 3D 4 2F 4 F4 E6 2E 6 E7 E8

d=1 G(6, 6, 2) G(6, 6, 2) G(8, 8, 2) G28 G35 G28 G36 G37

2 G(6, 6, 2) G(6, 6, 2) G28 G28 G35 G36 G37

3

4

5

G4 G5 G25 G5 G26 G32

6

8

10

12

G16

G10

G4 G12 G8 G8 G8 G8 G31

G8

G16

G5 G5 G25 G26 G32

G9

The relative Weyl groups in simple groups of classical type will be described in Examples 3.5.14 and 3.5.15. As noted above, minimal d-split Levi subgroups are centralisers of Sylow dtori. The previous result enables us to describe the complete lattice of d-split Levi subgroups, in analogy to the case d = 1 considered in Lemma 3.1.3: Proposition 3.5.12 Let Ld be a minimal d-split Levi subgroup of G. The d-split Levi subgroups of G containing Ld are in bijection with the parabolic subgroups of the reflection group WG (Ld ). Here, by definition parabolic subgroups of a complex reflection group are the centralisers of subspaces in its reflection representation. By a theorem of Steinberg [St64, Thm. 1.5], any such centraliser is generated by the reflections it contains, hence in particular it is itself a reflection group. Proof Let Sd be a Sylow d-torus of G such that Ld = CG (Sd ), and let L  Ld be

d-split. Then S = Z◦ (L)d is a d-torus contained in Sd . Thus, if (Xd, , Yd, ), ϕw is

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the complete root datum of Sd , then S has complete root datum (X , , Y , ), ϕw for some pure sublattice Y   Yd . Hence, V (ϕw, ζd ) := kerY  ⊗C (ϕw − ζd )  V(ϕw, ζd ) := kerYd ⊗C (ϕw − ζd ). As L = CG (S) by Proposition 3.5.5, the complete root datum of L has the form

(X, RL, Y, RL∨ ), ϕwWL , where RL consists of the roots of G that are trivial on Y . Thus, in its action on V(ϕw, ζd ), WL is the centraliser in WG (Ld ) of the subspace V (ϕw, ζd ), hence a parabolic subgroup of WG (Ld ). Conversely, a parabolic subgroup P of WG (Ld ) is the centraliser of a subspace V  of V(ϕw, ζd ). This defines a d-torus S with cocharacters Y  such that V  = kerY  ⊗C (ϕw − ζd ), and thus its centraliser L = CG (S), a d-split Levi subgroup of G.  It is no longer true for general complex reflection groups that the parabolic subgroups can be described in terms of standard parabolic subgroups corresponding to subsets of a suitable fixed minimal generating system. Still, it turns out that this description continues to hold for the complex reflection groups occurring in our setting as WG (Ld ). The parabolic subgroups of all finite complex reflection groups have been determined; for the exceptional types see for example the lists given in [OrTe92, App. C]. Example 3.5.13 Let G be simple of type E7 with standard Frobenius map, and consider d = 4. Here, the centraliser L4 = CG (S4 ) of a Sylow 4-torus S4 of G has rational form Φ24 .A1 (q)3 and its relative Weyl group is the complex reflection group G8 in the Shephard–Todd notation, of order 96 (see Table 3.2). This reflection group has just one conjugacy class of proper non-trivial parabolic subgroups, all isomorphic to the cyclic group C4 , and the 4-split Levi subgroups containing L4 corresponding to these by Proposition 3.5.12 are of rational type Φ4 .2D4 (q).A1 (q). Example 3.5.14 We determine the d-split Levi subgroups in G = GLn .

(a) First let (X , , Y , ), w be the complete root datum of an n-torus (in the sense of 3.5.1) of GLn with split Frobenius map. Then at least one eigenvalue of w on V = Y  ⊗Z C has to be a primitive nth root of unity. The only elements with this property of W = Sn in its natural permutation representation are the n-cycles (which comprise in particular the Coxeter elements of W). Thus any non-trivial n-torus S of GLn must have complete root datum such that w is an n-cycle and Y  is the maximal sublattice of Y = Zn such that all eigenvalues of w on Y  ⊗Z C are primitive nth roots of unity (so Y  has rank ϕ(n)). The centraliser CG (S)F is then a Coxeter torus of GLn (q) of order q n − 1, isomorphic to GL1 (q n ). (b) Still taking for F the standard Frobenius map, more generally it is easily seen by considering cycle shapes of elements in the Weyl group Sn that for arbitrary

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263

d, the elements with a maximal ζd -eigenspace are precisely those with cycle type consisting of a disjoint d-cycles and a disjoint partition of r, where n = ad + r with 0  r < d. Then a Sylow d-torus of GLn has centraliser Ld of rational form GL1 (q d )a .GLr (q) and its relative Weyl group is WG (Ld )  Cd ' Sa , the imprimitive complex reflection group usually denoted G(d, 1, a). The minimal parabolic subgroups of WG (Ld ) up to conjugacy are Cd and C2 , generated by reflections of order d and 2 respectively, with corresponding d-split Levi subgroups GLd (q) × GL1 (q d )a−1 × GLr (q),

GL1 (q d )a−2 × GL2 (q d ) × GLr (q),

respectively, where the latter only occurs when a  2. More generally, according to the description given in Proposition 3.5.12 the d-split Levi subgroups of GLn have rational form GLn1 (q d ) × · · · × GLns (q d ) × GLt (q) for some n1, . . . , ns, t  0 with d(n1 + · · · + ns ) + t = n. The d-tori and d-split Levi subgroups of SLn are now obtained by intersecting those of GLn with SLn , and the ones of PGLn are the images under the canonical map of those in GLn . (c) From the above it is easy to read off the corresponding results for GUn (q) since its complete root datum is obtained from the one of GLn (q) by just replacing the automorphism ϕ = id by −id (see Example 1.6.18(b)). Now from the transformation property of cyclotomic polynomials Φd (q) upon replacing q by −q, this immediately implies that the d-tori of GUn (q) have rational forms obtained from those of the d -tori of GLn (q) by replacing q by −q (Ennola duality), where ⎧ 2d if d is odd, ⎪ ⎪ ⎨ ⎪ d = d/2 if d ≡ 2 (mod 4), ⎪ ⎪ ⎪d if d ≡ 0 (mod 4), ⎩ 

and accordingly for the d-split Levi subgroups. In particular, the (1-)split Levi subgroups of GUn (q) are obtained from the 2-split Levi subgroups of GLn (q) by replacing q by −q, and the Weyl group of GUn (q), that is, the relative Weyl group of a Sylow 1-torus of GUn (q) is isomorphic to the relative Weyl group of a Sylow 2-torus of GLn (q), of type Bm with m = $n/2%. Example 3.5.15 We next describe the d-split Levi subgroups for classical groups. This is most easily done for the matrix group versions with 1-dimensional centre and simple derived subgroup. So let G be of classical type and F : G → G be a Steinberg map such that [G, G]F ∈ {SO2n+1 (q), Sp2n (q), SO±2n (q)} is of classical type. First let d  1 be odd. Then the d-split Levi subgroups in G have rational

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forms GLn1 (q d ) × · · · × GLns (q d ) × HF for some n1, . . . , ns  0, where H is of the same classical type as G and of rank  t such that t + d ni = n. This follows by a straightforward calculation in the respective Weyl groups, similar to the case of GLn discussed in Example 3.5.14, see also [BMM93, p. 49]. In particular the minimal d-split Levi subgroups have rational form GL1 (q d )a .HF where n = ad + r and 0  r < d, with relative Weyl group the complex reflection group G(2d, 1, a)  C2d ' Sa . If d is even, then set e := d/2. In this case the d-split Levi subgroups in G have rational forms GUn1 (q e ) × · · · × GUns (q e ) × HF for some n1, . . . , ns  0, where H is of the same classical type as G and of rank  t such that t + e ni = n, except that here the twisting induced by F on H is  opposite to the one on G if G is of type Dn and ni is odd (see [BMM93, p. 52]). The minimal d-split Levi subgroups of G now have rational form GU1 (q e )a .HF where n = ae + r and 0  r < e, with relative Weyl group G(2e, 1, a)  C2e ' Sa , unless [G, G]F = SO+2n (q) and r = 0, in which case it has type G(2e, 2, a), a normal subgroup of index 2 in G(2e, 1, a). Example 3.5.16 Finally we give the rational structure of d-split Levi subgroups in simple groups of exceptional type. The 1-split Levi subgroups are just the Levi subgroups of F-stable parabolic subgroups and hence easily described via the Dynkin diagram of G. The 2-split Levi subgroups are obtained from these by Ennola duality. If the relative Weyl group of a Sylow d-torus Td is cyclic then according to Proposition 3.5.12 the only d-split Levi subgroups, up to conjugacy, are CG (Td ) and G itself. All the remaining cases are listed in Table 3.3. Here a cyclotomic polynomial Φd stands for the group of F-fixed points of a d-torus√of generic√order Φd . The notation for factors of cyclotomic polynomials over Q( 2) and Q( 3) is as introduced in Example 3.5.4. (In Chevie [MiChv] they can be obtained by the command SplitLevis.) We now turn to representation theoretic properties in the d-split setting. First, generalising the 1-split case from Theorem 3.1.11 and the subsequent remarks, the Mackey formula takes a particularly simple form in this setting, see [BMM93, Thm. 1.35]: Theorem 3.5.17 Let M1, M2 be d-split Levi subgroups of G containing the minimal d-split Levi subgroup L. Let ψ ∈ Irr(M1F ) and assume that one of the assump-

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Table 3.3 Rational type of proper d-split Levi subgroups, d  3, in simple exceptional groups GF 4 (q)

d 3 6 3 F4 (q) 4 6 2F (q2 ) 4 4 8 8  E6 (q) 3 4 6 2E (q) 3 6 4 6 3 E7 (q) 3D

4 6 E8 (q)

3 4 5 6 8 10 12

LF Φ23, Φ3 .A2 (q) Φ26, Φ6 .2A2 (q) Φ23, Φ3 .A2 (q) Φ24, Φ4 .B2 (q) Φ26, Φ6 .2A2 (q) Φ24, Φ4 .A1 (q2 ) Φ8 2, Φ8 .2B2 (q2 ) Φ8 2, Φ8 .2B2 (q2 ) Φ33, Φ23 .A2 (q), Φ3 .A2 (q)2, Φ3 .3D4 (q) Φ24 Φ21, Φ4 Φ1 .2A3 (q) Φ26 Φ3, Φ6 Φ3 .2A2 (q), Φ6 .A2 (q2 ) Φ23 Φ6, Φ3 Φ6 .A2 (q), Φ3 .A2 (q2 ) Φ24 Φ22, Φ4 Φ2 .A3 (q) Φ36, Φ26 .2A2 (q), Φ6 .2A2 (q)2, Φ6 .3D4 (q) Φ33 Φ1, Φ23 Φ1 .A2 (q), Φ23 .A1 (q3 ), Φ3 Φ1 .3D4 (q), Φ3 .A5 (q), Φ3 .A2 (q)A1 (q3 ) 2 Φ4 .A1 (q)3, Φ4 .2D4 (q)A1 (q) Φ36 Φ2, Φ26 Φ2 .2A2 (q), Φ26 .A1 (q3 ), Φ6 Φ2 .3D4 (q), Φ6 .2A5 (q), Φ6 .2A2 (q)A1 (q3 ) Φ43, Φ33 .A2 (q), Φ23 .A2 (q)2, Φ23 .3D4 (q), Φ3 .3D4 (q)A2 (q), Φ3 .E6 (q) Φ44, Φ34 .A1 (q2 ), Φ24 .2A2 (q2 ), Φ24 .A1 (q2 )2, Φ24 .D4 (q), Φ4 .2A3 (q2 ), Φ4 .2D6 (q), Φ4 .A1 (q2 ).2A2 (q2 ) Φ25, Φ5 .A4 (q) Φ46, Φ63 .2A2 (q), Φ26 .2A2 (q)2, Φ26 .3D4 (q), Φ6 .3D4 (q)2A2 (q), Φ6 .2E6 (q) 2 Φ8, Φ8 .2D4 (q) Φ210, Φ10 .2A4 (q) Φ212, Φ12 .3D4 (q)

tions of Theorem 3.3.7 is satisfied. Then

M2 ∗ G G 1 RM2 ◦ RM (ψ) = RM ◦ ad(w) ◦ ∗RM (ψ), w M w ∩M1 1 2 ∩ M1 w

2

where the sum runs over a system of WM2 (L)F –WM1 (L)F double coset representatives in WG (L)F . The proof of that result hinges on the following observation (see [BMM93, Prop. 1.24]), which we will use again later:

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Lemma 3.5.18 Let M1, M2 be d-split Levi subgroups of G. If M := M1 ∩ M2 contains an F-stable maximal torus T of G then M is also d-split and in particular contains a minimal d-split Levi subgroup. Proof Let Si be the Sylow d-torus of Z ◦ (Mi ), for i = 1, 2. This is contained in every maximal torus of Mi , hence in T. So S := S1 S2 is a d-torus in T. It is now  easy to see that M = CG (S) and hence is a d-split Levi subgroup. The following natural generalisation of Definition 3.1.14 of cuspidal characters is fundamental in the setting of modular representation theory of finite reductive groups: Definition 3.5.19 Let d  1. An irreducible character ρ ∈ Irr(GF ) is d-cuspidal if ∗RG L (ρ) = 0 for all proper d-split Levi subgroups L of G. A pair (L, λ), where L is a d-split Levi subgroup of G and λ ∈ Irr(LF ) is d-cuspidal is called a d-cuspidal pair of G. Example 3.5.20 (a) With this notation, 1-cuspidal characters of G = GF are just characters of cuspidal KG-modules as introduced in Definition 3.1.14. d (b) Assume that F defines an Fq -structure on G. Then the group GF inherits an Fq -structure, but it can also be considered as a group over Fq d . By 3.5.1 the d-split Levi subgroups of G with respect to the Frobenius map F are the 1-split Levi subgroups of G with respect to F d . It follows that the d-cuspidal characters of d d GF considered as a group defined over Fq are the (1-)cuspidal characters of GF considered over Fq d . In analogy to Proposition 3.2.2 d-cuspidal characters enjoy the following uniform property: Proposition 3.5.21 Let ρ ∈ Irr(GF ) be d-cuspidal. Then ∗RG T (ρ) = 0 for all F-stable maximal tori T of G contained in some proper d-split Levi subgroup of G. The proof of this statement is identical to that of Proposition 3.2.2, simply replacing (1-)split and (1-)cuspidal by d-split and d-cuspidal throughout. But note that the proof of the reverse direction in Proposition 3.2.2 does not go through here ∗ G since it might be that ∗RG L (ρ)(1) = 0 even though RL (ρ)  0. We will say something about the converse in 3.5.24 below. Again, by Proposition 3.5.5 the condition on T in Proposition 3.5.21 can be reformulated as follows: An F-stable maximal torus T of G does not lie in a proper d-split Levi subgroup of G if and only if its Sylow d-torus Td is central in G. There is also a weak analogue of the characterisation in Theorem 3.2.22 of cuspidal characters in terms of Jordan decomposition; for this let us say that a character ρ ∈ Irr(GF ) has property (Ud ) if it satisfies the conclusion of the uniform

3.5 d-Harish-Chandra Theories

267

criterion for d-cuspidality in Proposition 3.5.21. Thus, in particular, any d-cuspidal character has property (Ud ). ˜ be a regular embedding. Let L˜ be an F-stable Levi Lemma 3.5.22 Let G → G F ˜ and λ˜ ∈ Irr(L ˜ ˜ ). Let L = L∩G and let λ be an irreducible constituent subgroup of G L˜ F ˜ ˜ λ) ˜ has property of ResL F (λ). Then (L, λ) has property (Ud ) for G if and only if (L, ˜ (Ud ) for G. This is shown in [CE99, Prop. 1.10] using a certain amount of case-by-case analysis. As a direct consequence they obtain: Proposition 3.5.23 Let s ∈ G∗ be semisimple. Let ρ ∈ E (G, s) and let ψ be in a CG∗ (s)F -orbit of unipotent characters of CG◦ ∗ (s)F corresponding to ρ under Jordan decomposition (see Theorem 2.6.22). Then ρ has property (Ud ) if and only if (1) ψ is d-cuspidal; and (2) Z◦ (G∗ )d = Z◦ (CG◦ ∗ (s))d , that is, the Sylow d-torus of Z◦ (CG◦ ∗ (s)) lies in Z(G∗ ). If the centre of G is connected, this follows exactly as for the case d = 1 shown in the proof of Theorem 3.2.22. The reduction to this situation for property (Ud ) is given by Lemma 3.5.22 and for properties (1) and (2) it follows by elementary arguments from the properties of Jordan decomposition, see [CE99, Prop. 1.10]. Characters with the property in Proposition 3.5.23 have been baptised d-Jordan cuspidal in [KeMa15, Def. 2.1]. 3.5.24 We define two relations on the set of d-cuspidal pairs of G in terms of Lusztig induction, generalising the relation 1 from Definition 3.1.14: We say that (M, μ) d (L, λ) for d-cuspidal pairs (M, μ), (L, λ) of G if M  L and there is a L (μ)  0 parabolic subgroup P of L with Levi complement M such that λ, RMP (see [BMM93, Def. 3.1]). The transitive closure of d is denoted *d . So, by definition ρ ∈ Irr(GF ) is d-cuspidal if (G, ρ) is minimal with respect to *d . As with ordinary Harish-Chandra theory, we can now define the d-Harish-Chandra series E (GF , (L, λ)) above a d-cuspidal pair as consisting of those ρ ∈ Irr(GF ) such that (L, λ) d (G, ρ). Note that it is not clear from our definition whether the d-Harish-Chandra series partition Irr(GF ) (but see Theorem 4.6.21 for the case of unipotent characters). Still, let us point out the following immediate consequence of Proposition 3.3.20: Corollary 3.5.25 Let (L, λ) be a d-cuspidal pair of G for some d  1, and assume that λ ∈ E (GF , s) for some s ∈ L∗F . Then the d-Harish-Chandra series E (GF , (L, λ)) is contained in E (GF , s). It is conjectured in [CE99, 1.11] that the relations d and *d coincide, that the minimal pairs with respect to *d below a fixed ρ ∈ Irr(GF ) are all GF -conjugate,

268

Harish-Chandra Theories

and that the criterion for d-cuspidality in Proposition 3.5.21 is also sufficient, that is, d-cuspidality is determined by a uniform condition. In contrast to the situation for Harish-Chandra induction and ordinary cuspidality, none of these statements is known to hold in general, though. Apart from the case d = 1 (see Corollary 3.1.17), and from the case of unipotent characters (see Theorem 4.6.20) the validity of the stated conjectures has been shown by Cabanes–Enguehard [CE99, Thm. 4.2] for characters lying in Lusztig series E (G, s) with s ∈ G∗F a semisimple  -element, for primes   5 different from p such that q has order d modulo , by using sophisticated methods from block theory. Some quasi-isolated cases in exceptional type groups have been settled by Kessar–Malle [KeMa13]. See Corollary 4.7.8 for the case of simple groups with connected centre. Let us mention another compatibility of d-cuspidality with Jordan decomposition that was shown in [KeMa15, Prop. 4.1]. Let G be connected reductive and assume that F is a Frobenius map. Let s ∈ G∗ F , and G1  G an F-stable Levi subgroup with dual G∗1 containing CG∗ (s). For (L1, λ1 ) a d-cuspidal pair of G1 below E (G1F , s) set G ΨG (L1, λ1 ) := (L, λ) with L := CG (Z◦ (L1 )d ) and λ := εL εL1 RLL1 (λ1 ). 1

Observe that λ ∈ Irr(LF ) by Theorem 3.3.22. G Proposition 3.5.26 Assume that the Mackey formula holds for GF . Then ΨG 1 F defines a bijection between the set of d-cuspidal pairs of G1 below E (G1 , s) and the set of d-cuspidal pairs of G below E (GF , s). G is given as follows: let (L, λ) be d-cuspidal in G below The inverse of ΨG 1 F ∗ ∗ E (G , s). Set L1 := L ∩ G∗1 and observe that L∗1  L∗ ∩ CG∗ (s) = CL∗ (s), so ∗RL (λ) has a unique constituent λ by Theorem 3.3.22. Then (L , λ ) is d-cuspidal. 1 1 1 L1

As in the case d = 1 we still have the following consequence of Proposition 3.5.21: Corollary 3.5.27 has the form

Let λ ∈ Irr(G) be d-cuspidal. Then the degree polynomial of λ Dλ = Φd (q)a(d) f (q)

where a(d) is the precise power of Φd dividing the order polynomial of [G, G] and f ∈ Q[q] is not divisible by Φd (q). Proof The proof is identical to the one for Corollary 3.2.4, only replacing (1-)split and (1-)cuspidal by d-split and d-cuspidal throughout, q−1 by Φd , and the reference to Proposition 3.2.2 to one to Proposition 3.5.21. 

3.5 d-Harish-Chandra Theories

269

Definition 3.5.28 Let (L, λ) be a d-cuspidal pair in G. The relative Weyl group of (L, λ) is WG (L, λ) := NG F (L, λ)/LF (a subgroup of the relative Weyl group WG (L) introduced in 3.5.8). We saw in Theorem 3.2.5 that relative Weyl groups of unipotent (1-)cuspidal pairs are always Coxeter groups, and in Proposition 3.5.9 that relative Weyl groups of d-split Levi subgroups are complex reflection groups. It is a remarkable fact, for which no conceptual proof is known, that the relative Weyl groups of unipotent d-cuspidal pairs (L, λ) are also complex reflection groups. They are described in all cases in [BrMa93, §3] and [BMM93, Tab. 3]. Example 3.5.29 We describe the relative Weyl groups of unipotent d-cuspidal pairs in classical types. (a) First let G = GLn with GF = GLn (q). The d-split Levi subgroups L of G were described in Example 3.5.14(b): they have rational form LF  GLn1 (q d ) × · · · × GLns (q d ) × GLt (q)

 with t +d ni = n. In order for L to have a d-cuspidal character, all factors must have such a character. According to the observation in 3.5.1 and Example 3.5.20(b) the d-cuspidal characters of GLni (q d ) are just its cuspidal characters if we consider it as a group over Fqd . Now it follows from Example 2.4.20 that the group GLn (q) only has cuspidal unipotent characters when n = 0, so LF possesses d-cuspidal unipotent characters only when it has rational form GL1 (q d )a × GLt (q) with n = ad + t for some a, t  0. We will actually see in Corollary 4.6.5 that the d-cuspidal unipotent characters of GLt (q) are exactly the characters parametrised by a partition of t that is a d-core. Since any automorphism of GLt (q) fixes all unipotent characters, see Theorem 4.5.11, the relative Weyl group of any such d-cuspidal character λ ∈ Uch(LF ) satisfies WG (L, λ) = WG (L), hence equals the complex reflection group G(d, 1, a). The same line of argument can be used when GF = GUn (q) to conclude that again we have WG (L, λ) = WG (L) for all d-cuspidal unipotent characters of d-split Levi subgroups L of G. (b) Now assume that G is of classical type as in Example 3.5.15. For d  1 we set e := d when d is odd, and e := d/2 when d is even. Using Example 3.5.15 we conclude as above that the only d-split Levi subgroups possessing a d-cuspidal unipotent character are of type (qe +(−1)d )a ×HF with H of rank t where n = ae+t. The d-cuspidal characters of HF will be described explicitly in Corollary 4.6.16. By Theorem 4.5.11, unless HF is of untwisted type Dn , all unipotent characters are invariant under outer automorphisms, so the relative Weyl groups again satisfy WG (L, λ) = WG (L) for all d-cuspidal unipotent characters of d-split Levi subgroups

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Harish-Chandra Theories

L  G. On the other hand, when HF is of untwisted type Dn with n even, there exist d-cuspidal unipotent characters λ ∈ Uch(LF ) labelled by so-called degenerate symbols for which WG (L, λ) is a subgroup of index 2 in WG (L), see [BMM93, p. 51] and also Section 4.4. Example 3.5.30 We continue Example 3.5.13 with the principal 4-series in G of type E7 . We saw that the relative Weyl group WG (L4 ) of a Sylow 4-centraliser L4 in G is isomorphic to the complex reflection group G8 . This group has two orbits of length 3 and two fixed points on the set Uch(L4F ) of unipotent characters of L4F = Φ24 .A1 (q)3 . Thus, for any (necessarily 4-cuspidal) unipotent character λ of L4 in an orbit of length 3, the relative Weyl group WG (L4, λ) is a proper subgroup of WG (L4 ). It turns out to be the imprimitive complex reflection group C4 ' S2 = G(4, 1, 2) (in the Shephard–Todd notation) in each case, see [BMM93, Tab. 1]. We will have more to say about d-Harish-Chandra theory for unipotent characters in Section 4.6.

4 Unipotent Characters

Lusztig’s Jordan decomposition of characters discussed in Section 2.6 underlines the importance of understanding the unipotent characters of finite reductive groups. It shows that many questions on arbitrary irreducible characters can be reduced to problems on unipotent characters. Thus, unipotent characters carry basic information on the ordinary representation theory of finite reductive groups. In fact, much is known about unipotent characters and they are easier to understand in purely combinatorial terms than other characters. In this chapter we collect various important properties of unipotent characters of (nearly) simple groups of Lie type. As unipotent characters are well behaved with respect to almost direct products of finite reductive groups, this is sufficient for many purposes. The parametrisation of unipotent characters and the description of their properties are closely related to the irreducible characters of Weyl groups. We hence first recall in Section 4.1 some of the properties of and concepts around Weyl group characters. In Section 4.2 we introduce the notion of families of unipotent characters and associated Fourier matrices which are then used to state Lusztig’s fundamental decomposition theorem (Theorem 4.2.16). We continue by defining the Frobenius eigenvalues attached to unipotent characters and explaining their relation with the Fourier matrices. In Sections 4.3–4.5 we give more detailed information for the various simple types. For groups of type A, which are considered in Section 4.3, the associated combinatorics is entirely in terms of partitions and highly reminiscent of the representation theory of symmetric groups. As one might expect, the properties of unipotent characters of groups of classical type also have a quite combinatorial flavour. This can be encoded in suitable combinatorial parameters, the so-called Lusztig symbols, in terms of which the properties of unipotent characters of classical groups are then expounded in Section 4.4. On the other hand, the unipotent characters of groups of exceptional type have mainly to be studied case-by-case, 271

272

Unipotent Characters

and this is done in Section 4.5. We conclude that section by collecting some general properties that can be read off from the explicit classification of unipotent characters, for example concerning rationality and on the action of automorphisms. In Section 4.6 we discuss the decomposition of Lusztig induction of unipotent characters, giving in particular a new proof of Asai’s formula, and from this derive the d-Harish-Chandra theories for unipotent characters. With this knowledge we then return to Lusztig’s Jordan decomposition of characters in Section 4.7 and present the known results on its commutation with Lusztig induction. In the final Section 4.8 we give a brief introduction to the as yet quite incomplete representation theory of finite reductive disconnected groups. Throughout this chapter, p denotes the underlying characteristic of all considered algebraic groups.

4.1 Characters of Weyl Groups In this section we collect some basic properties of irreducible characters of finite Weyl groups that will be relevant for the description of unipotent characters. We do not aim to develop the theory from scratch, an introduction with proofs can be found for example in [GePf00]. Let us point out that several crucial facts in the representation theory of finite Weyl groups rely on an explicit case-by-case analysis of the various irreducible types as no general proofs have been found to date. Throughout this section let W be a finite Weyl group, with generating set S of simple reflections and corresponding length function l : W → Z0 . Let T = {wsw −1 | w ∈ W, s ∈ S} denote the set of reflections in W, and let Irr(W) be the set of (complex) irreducible characters of W. Following Lusztig [Lu79b], we introduce two functions φ → aφ and φ → bφ on Irr(W). 4.1.1 The b-invariant bφ of φ ∈ Irr(W) is defined as the smallest integer i  0 such that φ occurs in the character of the ith symmetric power of the natural reflection representation of W. For example, if 1W denotes the trivial character and ε denotes the sign character of W, then b1W = 0

and

bε = |T | = l(w0 )

where w0 ∈ W is the longest element (see [GePf00, 5.3.1(a)]). Let W   W be a subgroup generated by reflections and let T  = W  ∩ T. Let ε  be the sign character of W . By a result due to Macdonald (see [GePf00, 5.2.11]), there is a unique

4.1 Characters of Weyl Groups

273

φ ∈ Irr(W) such that bφ = |T  | and  IndW W  (ε ) = φ + (sum of various ψ ∈ Irr(W) such that bψ > bφ ).

We shall denote this character by  φ := jW W  (ε ).

This j-induction can be used to systematically construct all the irreducible characters of Weyl groups W of type An−1 , Bn and Dn ; it also provides a convenient way of labelling the characters of W of exceptional type. Example 4.1.2 Let n  1 and W = Sn be the symmetric group, where the generators are the basic transpositions si = (i, i + 1) for 1  i  n − 1. (We also set S0 = {1}.) It is well known that the irreducible characters of Sn are parametrised by the partitions of n; we write this as Irr(Sn ) = {φα | α

n}.

This labelling is determined as follows; see, for example, [GePf00, 5.4.7]. Given a partition α n, let α∗ denote the transpose partition. Let Sα∗  Sn be the corresponding Young subgroup; we have Sα∗  Sα1∗ × · · · × Sαk∗ , where α1∗, . . . , αk∗ are the parts of α∗ . Let εα∗ be the sign character of Sα∗ . Then n ∗ φα = jS S ∗ (εα ) α

and

bφ α = n(α) :=

l

(i − 1)αi

i=1

(as in Example 2.8.7) where α = (α1  · · ·  αl ). For example, φ(n) is the trivial n ∗ character and φ(1 ) is the sign character ε. We have εφα = φα for all α n (see [GePf00, 5.4.9]). Example 4.1.3 Let n  1 and Wn be a Coxeter group of type Bn , with generators {t, s1, s2, . . . , sn−1 } and Coxeter diagram given as follows: t i

s1 i

s2 i

· · ·

sn−1 i

(We also set W0 = {1}.) The irreducible characters of Wn are parametrised by pairs of partitions (α, β) such that |α| + | β| = n. We write this as Irr(Wn ) = {φ(α,β) | (α, β) For (α, β)

n}.

n, there is a reflection subgroup Wα,β  Wn of type Dα1 × Dα2 × · · · × Dαl × Bβ1 × Bβ2 × · · · × Bβk

where α = (α1  · · ·  αl ) and β = (β1  · · ·  βk ), and D1 is understood as the

274

Unipotent Characters

empty Dynkin diagram. Let εα,β be the sign character of Wα,β . Then, by [GePf00, 5.5.1, 5.5.3], we have n φ(α,β) = jW Wα, β (εα,β )

and

bφ(α, β) = 2n(α) + 2n(β) + | β|

with n(α) as defined in Example 4.1.2. Note also that Wn  (Z/2Z)n  Sn , which leads to an alternative description of Irr(Wn ) in terms of Clifford theory. Further∗ ∗ more, we have εφ(α,β) = φ(β ,α ) for all (α, β) n (see [GePf00, 5.5.6]). To fix the notation, the character table for the case n = 2 is printed in Table 4.1; here C denotes the Cartan matrix of W(B2 ).

Table 4.1 The character table of W(B2 ) B2 φ(2,−)

S = {t, s1 }  C=

2 −1

−2 2

1W =  = φ(−,11) φ(11,−) φ(−,2) φ(1,1)



bφ 0 4 2 2 1

1 1 1 1 1 2

t 1 −1 −1 1 0

s1 1 −1 1 −1 0

ts1 1 1 −1 −1 0

(ts1 )2 1 1 1 1 −2

Example 4.1.4 Let n  2 and Wn be a Coxeter group of type Dn , with generators u, s1, . . . , sn−1 and Coxeter diagram given as follows: s1

i PP

u

 i

s P P 2i  

s3 i

· · ·

sn−1 i

(By convention, we also set W0 = W1 = {1}.) We can identify Wn with a reflection subgroup of index 2 in a Weyl group of type Bn . Indeed, let Wn be as in Example 4.1.3, with generators t, s1, . . . , sn−1 . Then the assignment u → ts1 t,

si → si

for 1  i  n − 1,

defines an embedding Wn → Wn . Observe that t acts on Wn as the non-trivial graph automorphism σ of order 2 interchanging u and s1 , thus identifying Wn with Wn σ . This provides a convenient setting for classifying the irreducible characters of Wn . Given (α, β) n, we denote by φ[α,β] the restriction of φ(α,β) ∈ Irr(Wn ) to Wn . Then we have (see [GePf00, 5.6.1, 5.6.2]): (a) If α  β, then φ[α,β] = φ[β,α] ∈ Irr(Wn ), and bφ[α, β] = 2n(α) + 2n(β) + min{|α|, | β|} = min{bφ(α, β) , bφ(β, α) }.

4.1 Characters of Weyl Groups

275

(b) If α = β, then φ[α,β] = φ[α,+] + φ[α,−] where φ[α,+] , φ[α,−] are distinct irreducible characters of Wn . Here, bφ[α,+] = bφ[α,−] = 4n(α) + n/2 = bφ(α, α) . Obviously, by Clifford theory all irreducible characters of Wn arise in this way. Of course, case (b) can only occur if n is even. In this case, the two characters φ[α,±] can explicitly be specified as follows; see [Lu84a, 4.6.2]. Let Hn+ = s1, s2, . . . , sn−1

and

Hn− = u, s2, . . . , sn−1

be two maximal parabolic subgroups of Wn isomorphic to Sn . Let α n/2 and S2α∗ be the corresponding Young subgroup in Sn where 2α∗ denotes the partition of n obtained by multiplying all parts of α∗ by 2. We have corresponding parabolic +  H + and H −  H − . Then subgroups H2α ∗ n n 2α∗ W

+ φ[α,+] = j H n+ (ε2α ∗) 2α ∗

and

W

− φ[α,−] = j H n− (ε2α ∗) 2α ∗

± denotes the sign character of H ± , respectively. (This is also discussed where ε2α ∗ 2α∗ in [GePf00, §5.6] but [GePf00, 5.6.3] has to be reformulated as above.) The effect of tensoring with the sign character ε of Wn immediately follows from the formulae in Example 4.1.3 for type Bn , except for the cases in (b). So let n be even and α n/2. Then we have  [α∗,+] if n/2 is even, φ [α,+] = εφ ∗ if n/2 is odd. φ[α ,−]

(See [Ge13, 3.5]; this was stated incorrectly in [GePf00, 5.6.5].) Example 4.1.5 Let W be of exceptional type G2 , F4 , E6 , E7 or E8 . Then, from the explicit knowledge of the irreducible characters of W one sees that: • If W is of type E6 , E7 or E8 , then each φ ∈ Irr(W) is uniquely determined by the pair (φ(1), bφ ). • If W is of type G2 , then each φ ∈ Irr(W) is uniquely determined by the pair (φ(1), bφ ) together with the tuple of values {φ(s) | s ∈ S}, see Table 4.2. • For W of type F4 , there are eight pairs of characters with equal invariants (φ(1), bφ ); seven of these can be distinguished by their values on reflections, see Table 4.3 where we give one character of each pair. These are precisely those pairs of characters swapped by the non-trivial graph automorphism of the Coxeter diagram of F4 (see also Theorem 4.5.11). Here, notation is chosen such that the first character in each pair takes smaller value on s1 than the second (and  is defined to be the exterior square of the thus larger value on s3 ). Finally, φ6,6  is its tensor product with character φ4,1 of the reflection representation, and φ6,6   ), see [Lu84a, 4.10]. (or with φ1,12 φ1,12

276

Unipotent Characters

Explicit tables defining the notation can also be found in [GePf00, App. C].

Table 4.2 The character table of W(G2 ) S = {s1, s2 }  C=

2 −3

−1 2

G2 1W ε  φ1,3  φ1,3 φ2,1 φ2,2



bφ 0 6 3 3 1 2

1 1 1 1 1 2 2

s1 1 −1 −1 1 0 0

s2 1 −1 1 −1 0 0

(s1 s2 )2 1 1 1 1 −1 −1

s1 s2 1 1 −1 −1 1 −1

(s1 s2 )3 1 1 −1 −1 −2 2

Table 4.3 Part of the character table of W(F4 ) S = {s1, s2, s3, s4 } 2  −1  C= 0  0

−1 2 −2 0

0 −1 2 −1

0 0 −1 2

   

F4  φ1,12  φ2,4  φ2,16  φ4,7  φ8,3  φ8,9  φ9,6

bφ 12 4 16 7 3 9 6

1 1 2 2 4 8 8 9

s1 −1 . −2 −2 . −4 −3

s3 1 2 . 2 4 . 3

4.1.6 The b-invariant is also encoded in the so-called fake degrees. To define these let V be the natural reflection representation of W over R. This induces a representation of W on the symmetric algebra S(V) of V. Observe that S(V) carries a natural grading by placing V in degree 1. Denote by S(V)W + the ideal generated by the W-invariants in S(V) of positive degree and by RW := S(V)/S(V)W + the and thus coinvariant algebra. The grading on S(V) induces a grading on S(V)W +i for the ith graded component of R , so R i . It also on RW . Write RW = R W W i W is known that RW affords a graded version of the regular representation of W (see [Bou68, V.5.2, Thm. 2(ii)]). The fake degree of φ ∈ Irr(W) is the graded multiplicity

i Pφ := φ, RW qi ∈ Z[q]. i0

It is clear from our definitions that the precise power of q dividing Pφ is qbφ . We define Bφ to be the degree in q of Pφ , so that Pφ (q) = cφ qbφ + linear combination of higher powers of q = cφ qBφ + linear combination of lower powers of q

4.1 Characters of Weyl Groups

277

for some cφ, cφ ∈ Z>0 . The relevance of fake degrees of Weyl groups lies in the fact that, as we will see in Proposition 4.2.5, they are the degree polynomials of the unipotent uniform almost characters of a corresponding (untwisted) finite reductive group. Example 4.1.7 (Fake degrees for type An−1 ) Let us describe the fake degrees of the irreducible characters of the Weyl group Sn of type An−1 . The fake degree of φα ∈ Irr(Sn ) labelled by the partition α = (α1  · · ·  αm ) is given by n 

Pφ α =

(qi − 1)

 (qu j − qui ) i< j

i=1

r −1 r −2 q( 2 )+( 2 )+...

ui  (qk − 1) i

k=1

where (u1, . . . , ur ), with ui := αi + i − 1, is the β-set associated to α (see e.g. [Ma95, Bem. 2.10]). Example 4.1.8 (Fake degrees for classical types) Let u = (u1 < · · · < um+d ), v = (v1 < · · · < vm ) be two strictly increasing sequences of non-negative integers, with d ∈ {0, 1}. Define   qv1 +...+vm i< j (q2u j − q2ui ) i< j (q2v j − q2vi ) . Pd (u, v) := ui vi   2(m−1)+d 2(m−2)+d + +··· 2k 2k ) ( ) ( 2 2 q (q − 1) (q − 1) i

k=1

i

k=1

Now let Wn be a Weyl group of type Bn . As explained in Example 4.1.3 the irreducible characters of Wn are labelled by pairs of partitions of n. For (α, β) n such a pair, add zeros to the parts of α or β such that α has one more part than β, and denote by u = (u1 < · · · < um+1 ), v = (v1 < · · · < vm ) the corresponding β-sets. Then the fake degree of φ(α,β) is given by Pφ(α, β) =

n 

(q2i − 1) P1 (u, v)

i=1

(see e.g. [Lu77a, Lemma 2.4] or [Ma95, Bem. 2.10]). Now let Wn be of type Dn , with n  2. The irreducible characters of Wn were described in Example 4.1.4 as φ[α,β] for (α, β) n, respectively φ[α,±] for α n/2. Let (α, β) be a pair of partitions of n with corresponding β-sets u = (u1 < · · · < um ), v = (v1 < · · · < vm ). Then the fake degree of φ[α,β] is given by Pφ[α, β] = (qn − 1)

n−1  (q2i − 1) (P0 (u, v) + P0 (v, u)) i=1

278

Unipotent Characters

if α  β, and by Pφ[α,±] = (qn − 1)

n−1 

(q2i − 1) P0 (u, v)

i=1

when α = β (see e.g. [Lu77a, Lemma 2.7(iii)] or [Ma95, Bem. 5.6]). The following semi-palindromicity property of fake degrees of Weyl groups with respect to tensoring with the sign character can easily be proved from their definition, see [Ca85, Prop. 11.1.2] for example; here N = |T | denotes the number of reflections in W: Lemma 4.1.9 For φ ∈ Irr(W) we have Pεφ (q) = q N Pφ (q−1 ). In fact, most fake degrees turn out to be palindromic, see Remark 4.1.16 below. We now define the second function on Irr(W). For this, observe that the set of reflections T is a union of rational conjugacy classes of W, so by an elementary result of Burnside from character theory we have

φ(t) ωφ := ∈Z for all φ ∈ Irr(W). φ(1) t ∈T For J ⊆ S, let WJ := J be the corresponding parabolic subgroup. If I ⊆ J ⊆ S then we denote by IndJI the induction of characters from WI to WJ . Definition 4.1.10 Define the a-function Irr(W) → Z0 , φ → aφ , inductively as follows. If W = {1}, then Irr(W) = {1W } and we set a1W := 0. Now assume that W  {1} and that the function ψ → aψ has already been defined for the irreducible characters of all proper parabolic subgroups of W. Then, for any φ ∈ Irr(W), define aφ := max{aψ | ψ ∈ Irr(WJ ) where J  S and IndSJ (ψ), φ  0}. Finally, set aφ :=

aφ

 −ω aεφ φ

 − a  ω , if aεφ φ φ

otherwise,

where, as before, ε denotes the sign character of W. Remark 4.1.11 Note that ωεφ = −ωφ for all φ ∈ Irr(W). With this one immediately checks that the a-function has the following properties: aφ  aφ  0

and

aεφ − aφ = ωφ

for all φ ∈ Irr(W).

This also shows that aφ  aψ if ψ ∈ Irr(WJ ) with J ⊆ S and IndSJ (ψ), φ  0. It is also immediate from the definition that if W = W1 × W2 is a direct product

4.1 Characters of Weyl Groups

279

of Weyl groups then aφ1 φ2 = aφ1 + aφ2 for all φi ∈ Irr(Wi ), i = 1, 2 (see [GePf00, 6.5.4]). Example 4.1.12 The trivial character has a-invariant a1W = 0 and a straightforward computation shows that for the sign character ε it equals aε = l(w0 ) = |T | where w0 ∈ W is the longest element; we have 0  aφ  l(w0 ) for all φ ∈ Irr(W). Example 4.1.13 Let W = Sn . Then, with the notation in Example 4.1.2, we have aφ α = bφ α = n(α) for all α n, see [GePf00, 6.5.8]. The a-invariants in the other Weyl groups of classical type will be given in 4.4.10. Lusztig [Lu79b, Lu03b] originally defined ‘a-invariants’ aφ using the generic degrees of the 1-parameter Iwahori–Hecke algebra associated with W (see 4.1.14). It was shown in [Ge11, Rem. 4.3] that this is equivalent to Definition 4.1.10. 4.1.14 For φ ∈ Irr(W), we define a further invariant by Aφ := l(w0 ) − ωφ − aφ = l(w0 ) − a φ, with w0 the longest element in W. The a- and A-invariants are related to the generic degrees Dφ ∈ Q[q] (where q is an indeterminate) introduced in Definition 3.2.13 in terms of the associated 1-parameter generic Iwahori–Hecke algebra H (W, q) (that is, the specialisation of the generic Iwahori–Hecke algebra with parameters xs = q at each reflection s ∈ S, see 3.1.20) in exactly the same way as the b-invariant is related to the fake degrees. Namely, we have Dφ (q) = fφ−1 qaφ + linear combination of higher powers of q = fφ−1 q Aφ + linear combination of lower powers of q, where fφ is a positive integer; see Lusztig [Lu84a, 4.1]. In view of Example 3.2.15 this shows that if W = W1 × W2 is a direct product of Weyl groups, then Aφ1 φ2 = Aφ1 + Aφ2 for all φi ∈ Irr(Wi ), i = 1, 2. The generic degrees satisfy the following semi-palindromicity property, which is completely analogous to that for fake degrees stated in Lemma 4.1.9. It can easily be proved from their definition, see [Ca85, Prop. 11.3.2] for example; here N = |T | again denotes the number of reflections in W: Lemma 4.1.15 For φ ∈ Irr(W) we have Dεφ (q) = q N Dφ (q−1 ). Remark 4.1.16 Recall our observation in 3.2.16 that the generic degrees are products of cyclotomic polynomials Φd . As they map to the character degrees of W under the specialisation q → 1, they are not divisible by Φ1 = q − 1. Since all

280

Unipotent Characters

cyclotomic polynomials Φd with d  2 are palindromic this shows that we actually also have the palindromicity Dφ (q) = qaφ +Aφ Dφ (q−1 )

for all φ ∈ Irr(W).

As first observed by Beynon and Lusztig [BeLu78, Prop. A] the corresponding result does not quite hold for the fake degrees. Almost all fake degrees turn out to actually be palindromic, that is, they satisfy Pφ (q) = q N −ωφ Pφ (q−1 ). The only characters of irreducible Weyl groups for which this fails to hold are precisely those for which the corresponding irreducible character of the 1-parameter Iwahori–Hecke √ algebra H (W, q) does not take values in Q(q), but in Q( q), see Remark 3.1.21. A conceptual explanation for this palindromicity property was subsequently found by Opdam [Op95]. Recall that the non-rational characters in question are the two irreducible characters of degree 512 for W of type E7 and the four irreducible characters of degree 4096 for W of type E8 , see [BeCu72]. Moreover, unless we are in these exceptional cases, we then have bφ + Bφ = aφ + Aφ

(φ ∈ Irr(W) non-exceptional)

by [Lu79b, (2.2)]. This formula fails to hold in the excluded cases. The a- and b-invariants can also be defined in an analogous way for finite complex reflection groups. A generalisation of the above observations to these groups is discussed in [Ma99, Thm. 6.5]. Definition 4.1.17 Following [Lu84a, 4.2], [Ge12b, 2.10], define a relation + on Irr(W) inductively as follows. For W = {1}, the trivial character is related to itself. Now assume that W  {1} and that + has already been defined for all proper parabolic subgroups of W. Let φ, φ  ∈ Irr(W). Then we write φ + φ  if there is a sequence φ = φ0, φ1, . . . , φm = φ  in Irr(W) such that, for each i ∈ {1, . . . , m}, the following condition is satisfied. There exists a subset Ii  S and ψi, ψi ∈ Irr(WIi ), where ψi + ψi within Irr(WIi ), such that either IndSIi (ψi), φi  0

and

aψi = aφi ,

IndSIi (ψi), εφi−1  0

and

aψi = aεφi−1 .

IndSIi (ψi ), φi−1  0, or IndSIi (ψi ), εφi  0,

We say that φ, φ  belong to the same family in Irr(W), if φ + φ  and φ  + φ; this defines an equivalence relation on Irr(W) and thus a partition of Irr(W) into the various families. (By [Ge12b, 4.4], this definition of families via the relation + does indeed coincide with Lusztig’s original definition in [Lu84a, 4.2].) Remark 4.1.18

The definition immediately implies that, if φ, φ  ∈ Irr(W) are

4.1 Characters of Weyl Groups

281

such that φ + φ , then εφ  + εφ. In particular, if F ⊆ Irr(W) is a family, then εF := {εφ | φ ∈ F } also is a family. It also follows from the definition that if W = W1 × W2 is a product of two Weyl groups, then φ1  φ2, φ1  φ2 ∈ Irr(W) lie in the same family of Irr(W) if and only if φi, φi ∈ Irr(Wi ) are in the same family in Irr(Wi ) for i = 1, 2 (see [GePf00, 6.5.4]). Families are closely related to the a-function: Proposition 4.1.19 (See [Lu84a, 4.14.1], [Ge12b, 4.4]) If φ, φ  ∈ Irr(W) are such that φ + φ , then aφ  aφ . In particular, the function φ → aφ is constant on the families of Irr(W). The original proof of this result relies on a case-by-case verification. One can give a general proof, based on deep results about Kazhdan–Lusztig cells; see [Lu87b]. A more general statement, involving weight functions on W, is established in [GeIa12, §9]. By 4.1.14 we have Aφ = l(w0 ) − aεφ . So Remark 4.1.18 and Proposition 4.1.19 imply that we also have Aφ  Aφ whenever φ + φ , with equality if φ, φ  belong to the same family. Note however that we may well have aφ = aφ and Aφ = Aφ for two characters φ, φ  ∈ Irr(W) not lying in the same family. The smallest example already occurs in type A1 × A1 . The subdivision of Irr(W) into families has been determined explicitly by Lusztig in each case, see the results given later in this chapter. From this, it is possible to verify the following fact, for which no a priori proof seems to be known: Proposition 4.1.20 We have aφ  bφ  Bφ  Aφ for all φ ∈ Irr(W). Moreover, in any family F ⊆ Irr(W) there is a unique character φ ∈ F such that aφ = bφ . This character is called the special character in that family. See [Lu79b, (2.1)] for the inequalities. In fact, it suffices to check the first inequality; the second is clear by definition, and then the last one follows from the first as Aφ = N − aεφ  N − bεφ = Bφ by Lemmas 4.1.15 and 4.1.9. The notion of special characters was first introduced by Lusztig [Lu79b, p. 324], and in [Lu82a, Sec. 12] he observes that each family contains a unique special character. A new characterisation was given in [Lu18b]. Note that if φ is special and non-exceptional (in the sense of Remark 3.1.21) then we also have Aφ = Bφ by Remark 4.1.16. The family in type E7 and the two families in type E8 containing exceptional characters are called exceptional; for those the previous conclusion fails to hold. Remark 4.1.21

By 4.1.14 and Proposition 4.1.20, 0  aφ  Aφ  l(w0 ) for all

282

Unipotent Characters

φ ∈ Irr(W). The explicit determination of aφ shows that the first inequality is strict unless φ = 1W is the trivial character, and thus the last inequality is strict unless φ = ε is the sign character, while the middle inequality turns out to be strict unless φ is a product of trivial or sign characters of the various irreducible factors of W. Thus, only in the latter cases the generic degree Dφ is a monomial. Example 4.1.22 (Families and special characters in type An−1 ) Let W = Sn be the Weyl group of type An−1 . Then all families in Irr(W) are singletons, that is, all φ ∈ Irr(W) lie in a family of their own and (hence) all of them are special [Lu84a, (4.4)]. In particular aφ α = bψ α = n(α), see Example 4.1.2. Example 4.1.23 (Families and special characters in type Bn ) Let Wn be a Coxeter group of type Bn . As explained in Example 4.1.3 the irreducible characters of Wn are labelled by pairs of partitions (α, β) of n. Given such (α, β), add zero parts to α and β such that α = (α1  · · ·  αn+1 ) has n + 1 parts and β = (β1  · · ·  βn ) has n parts, and consider the corresponding β-sets x =(x1, . . . , xn+1 )

with xi = αi + i − 1,

y =(y1, . . . , yn )

with yi = βi + i − 1.

Then φ(α,β) is special if and only if x1  y1  x2  · · ·  yn  xn+1 . Furthermore,   two characters φ(α,β), φ(α ,β ) ∈ Irr(Wn ) lie in the same family if and only if the corresponding β-sets (x, y) and (x , y ) have the same multiset of entries [Lu84a, (4.5.3) and (4.5.6)]. Example 4.1.24 (Families and special characters in type Dn ) Let n  2 and Wn a Coxeter group of type Dn . The irreducible characters of Wn have been described in Example 4.1.4. For (α, β) a pair of partitions of n add parts of size 0 so that both α and β have exactly n parts, and denote by (x, y) the corresponding pair of β-sets. If α = β then both φ[α,+] and φ[α,−] are special, and each lies in a singleton family. If α  β then φ[α,β] is special if and only if x1  y1  · · ·  xn  yn (after possibly   interchanging x and y). Furthermore, φ[α,β], φ[α ,β ] ∈ Irr(Wn ) lie in the same family if and only if the corresponding β-sets (x, y) and (x , y ) have the same multiset of entries [Lu84a, (4.6.4) and (4.6.10)]. See [Lu84a, (4.8)–(4.13)] for families and special characters for the irreducible Weyl groups of exceptional type. Remark 4.1.25 It turns out that the relation aφ  bφ is no longer satisfied for all characters of complex reflection groups; in fact, validity of this relation is one of several equivalent ways to characterise the so-called spetsial reflection groups, see [Ma00, Prop. 8.1]. The notion of families in Irr(W) and of special characters has been extended to these spetsial reflection groups, see [BrKi02, MaRo03, Chl09, Chl10].

4.1 Characters of Weyl Groups

283

4.1.26 We next extend some of the previous notions to Weyl groups with an automorphism. Recall that V denotes the natural real reflection representation of W so that we can view W  GL(V), and let σ ∈ NGL(V ) (W) be such that it induces a Coxeter group automorphism of W, also denoted by σ, that is, σ satisfies σ(S) = S. Then in particular σ respects the length function on W and it induces a graph automorphism of the Coxeter diagram of W. Conversely it can easily be seen that every graph automorphism of the Coxeter diagram of W is induced by a suitable σ ∈ NGL(V ) (W). Following Lusztig [Lu84a, 3.1] we say that σ is ordinary if, whenever s  s  in S lie in the same σ-orbit, then ss  has order 2 or 3. Equivalently, this happens if and only if σ is not only an automorphism of the Coxeter diagram, but even of any Dynkin diagram associated to (W, S). In particular for irreducible Weyl groups, the non-ordinary graph automorphisms are precisely those for types B2, G2 and F4 . As σ stabilises the reflection representation V of W, in its induced action on Irr(W) it preserves the b-invariants, and it also acts in a natural way on the symmetric algebra S(V) and on the coinvariant algebra RW . This allows us to define σ-twisted fake degrees. For this let W˜ be the semidirect product of W with σ . Thus, W˜ = W, σ ˜ we have σ(w) = σwσ −1 for all w ∈ W. Let φ ∈ Irr(W) be σ-stable, and and, in W, denote by φ˜ a σ-extension of φ as in 2.1.7. The fake degree of φ˜ (or σ-twisted fake degree of φ) is then the graded multiplicity

i ˜ RW Pφ˜ := φ, σ qi ∈ Z[q], i

where  , σ is the scalar product on the space of σ-class functions (see 2.1.7). For σ = idV this specialises to our original definition of the fake degree Pφ in 4.1.6. ˜ then its values on the Note that if we choose a different extension φ˜1 of φ to W, ˜ coset W σ differ from those of φ by a fixed root of unity, and thus Pφ˜ and Pφ˜1 differ by that same root of unity. In particular the σ-twisted fake degree is determined by φ and σ up to a root of unity of order dividing the order of σ. See Proposition 4.2.5 for the interpretation of twisted fake degrees in the representation theory of finite reductive groups. Example 4.1.27 Consider the situation when σ = −w0 ∈ NGL(V ) (W). Let φ˜ be the ˜ ˜ σ-extension of φ ∈ Irr(W) such that φ(ww = φ(w) for all w ∈ W. As 0 ) = φ(−wσ) i i of the coinvariant w0 σ = −idV , it acts by (−1) on the homogeneous component RW

284

Unipotent Characters

algebra. Then we have

1 ˜ i i ˜ RW φ(ww0 σ)Trace(ww0 σ, RW Pφ˜ (q) = ) qi φ, σ qi = |W | i i w ∈W 1 i = φ(w)Trace(w, RW ) (−q)i |W | i w ∈W

i = φ, RW (−q)i = Pφ (−q), i

so the twisted fake degree Pφ˜ (q) is obtained from Pφ (q) by replacing q by −q. This case occurs for the non-trivial graph automorphism of W of type An with n  2, of type D2n+1 with n  1 and of type E6 . Example 4.1.28 Let Wn denote the Weyl group of type Dn , n  2, and let σ : Wn → Wn be the automorphism of Wn induced by the symmetry of order 2 of the Dynkin diagram, interchanging the Coxeter generators u, s1 in the labelling introduced in Example 4.1.4. According to our description in that example, the σinvariant characters of Wn are the ones labelled φ[α,β] with α  β. Let φ˜[α,β] (w) := φ(α,β) (wσ) for w ∈ Wn , where φ(α,β) is one of the two extensions of φ[α,β] to Wn .σ = Wn , the Weyl group of type Bn . Let u = (u1 < · · · < um ), v = (v1 < · · · < vm ) be corresponding β-sets of equal length. Then the twisted fake degree of φ˜(α,β) is given by Pφ˜ [α, β] = (qn + 1)

n−1 

(q2i − 1) P0 (u, v) − P0 (v, u) i=1

with P0 (u, v) as defined in Example 4.1.8 (see [Lu77a, Lemma 2.7(ii)] or [Ma95, Bem. 5.6]). Observe that this expression does indeed depend on the choice of σextension of φ[α,β] to Wn : if we define φ˜[α,β] using φ(β,α) instead, the twisted fake degree is multiplied by −1. Also observe that if n is odd, Pφ˜ [α, β] (q) = ±Pφ[α, β] (−q) as already seen in Example 4.1.27. As the Coxeter group automorphism σ also sends parabolic subgroups of W to parabolic subgroups, it moreover preserves a-invariants and the relation + on Irr(W). Hence σ sends families in Irr(W) to families. Here we have the following observation (see [Lu84a, 4.17]) for which again no a priori proof seems to be known: Proposition 4.1.29 Assume that the automorphism σ of W as in 4.1.26 is ordinary. If F ⊆ Irr(W) is a σ-stable family then all elements of F are σ-stable. This no longer holds for the non-ordinary graph automorphisms of the Weyl groups W of types B2, G2 and F4 : Let F ⊆ Irr(W) be a σ-stable family with |F | > 1. If |F | = 3, which occurs in types B2 and F4 , then |F σ | = 1; if |F | = 4,

4.2 Families of Unipotent Characters and Fourier Matrices

285

which only occurs in type G2 , then |F σ | = 2, and if |F | = 11, which happens in F4 , then |F σ | = 5. But in all of these cases, it is still true that the special character in the family is σ-stable. Remark 4.1.30 The constructions in 4.1.26 depend on the choice of σ-extensions of the characters φ ∈ Irr(W)σ . It was already shown in Proposition 2.1.14 that any φ ∈ Irr(W)σ has a σ-extension φ˜ with values lying in R. Note that φ˜ may not be uniquely determined but, clearly, there are at most two possibilities for φ˜ with this property. It is sometimes convenient to fix a particular choice of σ-extension φ˜ of φ. ˜ In Lusztig [LuCS, IV, 17.2] has introduced the notion of a preferred σ-extension φ. the cases where W is an irreducible Weyl group and σ is ordinary, these are defined as follows. • If σ = idV , then φ˜ = φ. • If σ acts by conjugation with w0 and W is of type An (n  2) or E6 , then φ˜ is the ˜ 0 ) = (−1)aφ φ(1) where aφ is the a-invariant of unique σ-extension such that φ(w φ from Definition 4.1.10. • If σ has order 3 and W is of type D4 , then φ˜ is the unique σ-extension with values in Z. • If σ has order 2 and W is of type Dn (n  4) then W˜ can be identified with a Weyl group of type Bn (see Example 4.1.4) and the irreducible characters of W˜ that remain irreducible upon restriction to W are parametrised by pairs of partitions (α, β) of n with α  β. The preferred σ-extension of the character φ = φ[α,β] is then defined as follows. Interchange α, β if necessary such that the following holds: we add parts of size 0 to α or β so that both have the same number of parts. Order the parts of each increasingly and let k denote the first index at which ˜ αk  βk . Then βk < αk . Then φ˜ is defined by φ(w) := φ(α,β) (wσ) for w ∈ W. For W of type B2 , G2 or F4 and σ non-ordinary, we take as preferred extensions the ones described in Examples 2.1.12, 2.1.13, 2.1.15 and Remark 2.8.19

4.2 Families of Unipotent Characters and Fourier Matrices We now explain how the combinatorial notions introduced before connect with the representation theory of finite reductive groups. Let G be connected reductive and F : G → G a Steinberg endomorphism. Recall from Definition 2.3.8 that an irreducible character ρ ∈ Irr(GF ) is called unipotent if RTG (1), ρ  0 for some F-stable maximal torus T of G. The set of unipotent characters of GF is denoted Uch(GF ). It was explained in Chapter 2 that these play a distinguished role in the

286

Unipotent Characters

whole theory via Lusztig’s Jordan decomposition. In this section we gather several important constructions related to unipotent characters. Remark 4.2.1 Let us begin this section by explaining how the classification of Uch(GF ) is reduced to the case where G is a simple algebraic group of adjoint type. (See [Lu76c, 1.18] and [Lu77b, 3.15]). First, since G/Z(G) is semisimple, there exists a surjective homomorphism of algebraic groups π : G → Gad with central kernel and where Gad is a semisimple group of adjoint type (see Proposition 1.5.8 and [St67, p. 45/64]). Furthermore, there exists a Steinberg map F : Gad → Gad such that F ◦ π = π ◦ F. (See Proposition 1.5.9(b) and [St68, 9.16].) Hence, we F (but note that this is not necessarily obtain a group homomorphism π : GF → Gad surjective any more). By [DeLu76, 7.10], this induces a bijection ∼

F Uch(Gad ) −→ Uch(GF ),

ρ → ρ ◦ π.

Now by Proposition 1.5.10 we can write Gad = G1 × · · · × Gr where each Gi is semisimple of adjoint type, F-stable and F-simple, that is, Gi is a direct product of simple algebraic groups that are cyclically permuted by F. Let hi  1 be the number of simple factors in Gi , and let Hi be one of these. Then F hi (Hi ) = Hi and ιi : Hi → Gi,

g → gF(g) · · · F hi −1 (g),

is an injective homomorphism of algebraic groups that restricts to an isomorphism ∼ ιi : HiFi −→ GiF where we denote Fi := F hi |Hi : Hi → Hi (see Lemma 1.5.15) so F  HF1 × · · · × HFr . Let f : G × · · · × G → G be the product map. that Gad 1 r ad r 1 Then, finally, it is shown in [Lu76c, 1.18] that f and the homomorphisms ι1, . . . , ιr induce bijections ∼

∼

F ) −→ Uch(G1F ) × · · · × Uch(GrF ) −→ Uch(H1F1 ) × · · · × Uch(HrFr ), Uch(Gad

such that if χ maps to χ1  · · ·  χr then χ(1) = χ1 (1) · · · χr (1). By definition the Hi are simple algebraic groups and Fi : Hi → Hi are Steinberg maps. Thus, the classification of Uch(GF ) is reduced to the case where G is simple. A similar reduction applies to various invariants attached to unipotent characters. For this reason we will, when convenient, just explain the situation for the case when G is simple. The preceding argument also shows that the classification of unipotent characters is independent of the isogeny type. 4.2.2 In order to describe the multiplicities of unipotent characters in Deligne– Lusztig characters we now introduce an important partition, due to Lusztig, of the set of unipotent characters of a finite reductive group into so-called families. For this, recall that F induces a Coxeter group automorphism σ of the Weyl group W of G (see 1.6.1). Now define a graph on the set of vertices Uch(GF ) as follows:

4.2 Families of Unipotent Characters and Fourier Matrices

287

two unipotent characters ρ1, ρ2 ∈ Uch(GF ) are joined if and only if there is a σstable irreducible character φ ∈ Irr(W)σ such that Rφ˜ , ρi  0 for i = 1, 2 for the almost character Rφ˜ associated to some σ-extension φ˜ of φ (see Definition 2.4.17). (Observe that this does not depend on the chosen σ-extensions.) Then the sets of vertices of the various connected components of this graph are called the families in Uch(GF ). Clearly the families play an important role in understanding the relation between almost characters and unipotent characters, so in the decomposition of the Deligne– Lusztig characters RTG (1). The above procedure induces a corresponding subdivision of Irr(W)σ by declaring that two characters φ, φ  ∈ Irr(W)σ are equivalent if and only if Rφ˜ , Rφ˜  have unipotent constituents lying in the same family of Uch(GF ), for some (all) σ˜ φ˜ of φ, φ  respectively. The choice of notation is justified by the extensions φ, following: Proposition 4.2.3 Assume that σ is ordinary. The equivalence classes in Irr(W)σ as defined above are precisely the σ-stable families in Irr(W) defined in 4.1.17. This is proved in [Lu84a]. Indeed, in the case where F is a Frobenius map [Lu84a, 6.17] shows that the equivalence relation on Irr(W)σ defined above is at least as coarse as the subdivision into families. On the other hand, by Corollary 4.2.19 below all unipotent characters in a given family occur as constituents of the almost character of the special character (see 4.1.20) in a σ-stable family of Irr(W). Thus, both relation must agree. The fact that a similar statement is also true for the Suzuki and Ree groups is implicitly contained in the remarks in [Lu84a, §14.2]. Here, the equivalence classes in Irr(W)σ consist exactly of the σ-fixed points in σ-stable families in Irr(W). This shows that the families in Uch(GF ) are closely related to the purely combinatorial notion of families in the Weyl group introduced in Definition 4.1.17. There is a way to read off the families of unipotent characters from the character table of GF , in terms of the unipotent support introduced in Theorem 2.7.15: Theorem 4.2.4 (Geck–Malle) Two unipotent characters of GF lie in the same family if and only if they have the same unipotent support. This was shown in [GeMa00, Prop. 4.2 and Cor. 5.2] after having been conjectured by Lusztig [Lu80b, Problem II]. The following is an elementary computation in the invariant theory of Weyl groups (see [Lu77b, 3.16 and 3.19] or [BrMa92, Prop. 1.6’]): Proposition 4.2.5

Let G be connected reductive with a Steinberg map F. Let W

288

Unipotent Characters

be the Weyl group of G and σ the automorphism of W induced by F. Then for φ ∈ Irr(W)σ with σ-extension φ˜ we have Pφ˜ = Dφ˜ , that is, the fake degree coincides with the degree polynomial of the corresponding unipotent almost character (see Remark 2.4.31). 4.2.6 Recall from Definition 2.3.25 that the degrees of unipotent characters can be written as polynomials in q. As in Remark 2.3.26 for ρ ∈ Uch(GF ) we denote by aρ the precise power of q dividing this polynomial, and by Aρ its degree in q, so that Aρ

Dρ = ci qi for suitable ci ∈ Q. i=a ρ

These invariants coincide with the a- and A-invariants of the associated family in Irr(W): Proposition 4.2.7 The a- and A-values on a family of Uch(GF ) agree with the aand A-values of the corresponding family of Irr(W). In particular both are constant on families. Proof Let U ⊆ Uch(GF ) be a family of unipotent characters and F ⊆ Irr(W)σ the corresponding family in Irr(W). By Definition 2.3.25 the degree polynomial of ρ is given by

1 |G| θ (−1)l(w) Rw , ρ q−N Dρ = . |W| |Tw | (w,θ)∈X(W,σ)

Now ρ is unipotent so by Definition 2.3.8 only terms with θ = 1T will contribute to the sum. Hence using the definition of the almost characters Rφ˜ in Remark 2.4.17 we obtain |G| 1 (−1)l(w) Rw, ρ q−N Dρ = |W| w ∈W |Tw |

|G| 1 ˜ (−1)l(w) = φ(w) Rφ˜ , ρ q−N |W| w ∈W |T w| σ =

φ ∈Irr(W) σ

=

φ ∈Irr(W) σ

φ ∈Irr(W)

Rφ˜ , ρ

|G| 1 ˜ q−N (−1)l(w) φ(w) |W| w ∈W |Tw |

Rφ˜ , ρ DRφ˜ .

By Proposition 4.2.5 the degree of DRφ˜ is given by the fake degree Pφ˜ , and by

4.2 Families of Unipotent Characters and Fourier Matrices

289

Proposition 4.1.20 the special character φ ∈ F is the unique one with aφ = bφ , while all other b-values in F are strictly bigger, and by Corollary 4.2.19 below we have Rφ˜ , ρ  0 for all ρ ∈ U . Thus aρ = aφ is constant on U . The argument for A is entirely similar, using again Proposition 4.1.20 and the subsequent remark, or alternatively Proposition 4.2.8 together with Proposition 3.4.21.  Proposition 4.2.8 The Alvis–Curtis–Kawanaka–Lusztig duality DG (see Definition 3.4.1) sends families in Uch(GF ) to families. Moreover, special unipotent characters are sent to special unipotent characters, except for the exceptional families in types E7 and E8 . Here we call special unipotent character in a family U ⊆ Uch(GF ) the unipotent principal series character in U labelled by the special character in the corresponding family of Irr(WF ). Proof The first assertion is immediate from the definition of families and the commutation of DG with RTG from Theorem 3.4.4. For the second claim, if U ⊆ Uch(GF ) is a family and ρ ∈ U is labelled by the special character φ in the corresponding family F ⊆ Irr(WF ), then DG (ρ) is labelled by  ⊗ φ ∈ Irr(WF ) by Theorem 3.4.8, where  ∈ Irr(WF ) is the sign character. Now  ⊗ φ is again special by the remarks after Proposition 4.1.20 unless F is an exceptional family in types E7 or E8 .  We now turn to the problem of describing the multiplicities of the unipotent characters in the unipotent almost characters. Observe that by the very definition of families, a unipotent character can occur as a constituent of some almost character Rφ˜ only if the restriction φ of φ˜ lies in the corresponding family of Irr(W). So the base change matrix from almost characters to unipotent characters is blockdiagonal according to families, and it suffices to consider the decomposition problem family by family. Here Lusztig has discovered an intriguing combinatorics which we describe next; see also [Lu87b, 2.5] for a different interpretation in terms of equivariant vector bundles. 4.2.9 (Lusztig’s non-abelian Fourier transform) Let G be a finite group. Write M (G ) for the set of pairs (x, σ), where x ∈ G and σ ∈ Irr(CG (x)), modulo the equivalence relation (x, σ) ∼ (g x, σ g ) for g ∈ G , where σ g (gy) := σ(y) for y ∈ CG (x). Observe that if G = G1 × G2 is a direct product, then M (G ) can naturally be identified with M (G1 ) × M (G2 ). Following Lusztig [Lu79a, §4] we define a pairing { , } : M (G ) × M (G ) → C by

1 {(x, σ), (y, τ)} := σ(gyg −1 )τ(g −1 x −1 g). |CG (x)| |CG (y)| g ∈G xgyg−1 =gyg−1 x

290

Unipotent Characters

Let MG denote the operator on the space of (complex-valued) functions on M (G ) defined by

(MG f )(x, σ) := {(x, σ), (y, τ)} f (y, τ) (y,τ)∈M (G )

for (x, σ) ∈ M (G ). This is called the non-abelian Fourier transform associated to G . Alternatively we can also consider the endomorphism of the vector space CM (G ) defined on the basis by

(y, τ) → {(x, σ), (y, τ)} (x, σ), (x,σ)∈M (G )

but the first point of view will be more convenient later. Lemma 4.2.10 The operator MG is hermitian, that is, {(x, σ), (y, τ)} = {(y, τ), (x, σ)}

for all

(x, σ), (y, τ) ∈ M (G ),

and M2G = 1. Proof The first claim follows directly from the definition of M(G ), the second is an easy consequence of the orthogonality relations for irreducible characters.  4.2.11 Another interpretation of M (G ) can be given in terms of the category C (G ) of finite-dimensional G -graded CG -modules, that is, the objects of C (G ) are CG -modules V with a decomposition  V= Vx x ∈G

such that g.Vx = Vg x for all x, g ∈ G , and the morphisms in C (G ) are CG -linear maps f : V → V  such that f (Vx )  Vx for all x ∈ G . Equivalently, this can be described as the category of finite-dimensional modules for the Drinfeld double of G , see [Dr87]. Observe that any graded component Vx , x ∈ G , of a G -graded CG module V carries a representation of the centraliser CG (x). It is easily seen that C (G ) is semisimple, and the simple objects are obtained as follows (see e.g., [Br17, §8.1]): For x ∈ G let U be an irreducible CCG (x)-module. Then V(x, U) := t ∈G V(x, U)t , with V(x, U)t :=

g⊗U

if t = gxg −1,

0

else,

is an irreducible G -graded CG -module. The V(x, U) provide a complete set of nonisomorphic simple objects in C (G ) when x runs over a system of representatives for the conjugacy classes of G , and U over a complete system of representatives for the isomorphism classes of irreducible CCG (x)-modules. Thus, M (G ) as introduced above is in bijection with the set of isomorphism

4.2 Families of Unipotent Characters and Fourier Matrices

291

classes of simple objects in C (G ), by sending (x, σ) to V(x, U) where U affords the character σ. Remark 4.2.12 (Mellin transform) Digne and Michel [DiMi85, VII.3] introduced another basis for the space CM (G ). Let G be a finite group and denote by N (G ) the set of pairs (x, z) ∈ G × G with xz = zx, taken modulo simultaneous G -conjugation. Clearly, |N (G )| = |M (G )|. To (x, z) ∈ N (G ) we associate

σ(z)(z, σ) ∈ CM (G ). e(x,z) := σ ∈Irr(CG (x))

From the orthogonality relations for Irr(CG (x)) one gets that {e(x,z) | (x, z) ∈ N (G )} is also a basis of CM (G ). (The base change is called Mellin transform by Digne–Michel.) A straightforward computation shows that the matrix of the nonabelian Fourier transform on CM (G ) induced by MG takes a particularly simple form when expressed with respect to this basis (see [DiMi85, Prop. 3.1]): Proposition 4.2.13 We have MG e(x,z) = e(z −1, x −1 ) for all (x, z) ∈ N (G ). 4.2.14 For twisted groups we need to introduce an extension of the set-up described in 4.2.9, see [Lu84a, 4.16]. Assume that G˜ is a finite group with a normal subgroup G  G˜ of index c such that G˜ is a split cyclic extension of G . For a fixed generator of G˜/G let G  denote its full preimage in G˜, a coset of G . To this we associate two sets M = M (G  G˜) and M = M (G  G˜) as follows. The set M consists of all pairs (x, σ) where x ∈ G is such that CG˜ (x) ∩ G    and σ ∈ Irr(CG˜ (x)) has irreducible restriction to CG (x), taken again modulo the equivalence relation (x, σ) ∼ (g x, σ g ) for g ∈ G˜. The cyclic group Irr(G˜/G ) of linear characters of G˜/G acts on M by ε : (y, τ) → (y, τ ⊗ ε) for ε ∈ Irr(G˜/G ). ¯ with x ∈ G  and σ ¯ ∈ Irr(CG (x)) an The set M consists of all pairs (x, σ) irreducible character, taken modulo the same equivalence relation as for M . Observe that CG˜ (x) is a central cyclic extension of CG (x) by x, so all irreducible characters of CG (x) extend to CG˜ (x). In the case that G = G˜ we have M = M = M (G˜) and we recover the situation in 4.2.9. The pairing { , } on M (G˜) introduced in 4.2.9 now induces a pairing { , } : M × M → C,

{(x, σ), ¯ (y, τ)} := c{(x, σ), (y, τ)},

for any fixed extension σ of σ ¯ to CG˜ (x). Consider the operator MG G˜ on the space of functions M (G˜) → C with support contained in M , defined by

¯ := {(x, σ), ¯ (y, τ)} f (y, τ), (MG G˜ f )(x, σ) (y,τ)/∼

292

Unipotent Characters

where the sum runs over a system of representatives of the Irr(G˜/G )-orbits on M . It defines an isomorphism onto the space of functions on M , with inverse given by

(M−1 f )(y, τ) = {(x, σ), ¯ (y, τ)} f (x, σ). ¯ G  G˜ (x, σ)∈ ¯ M

4.2.15 Now return to our setting with G connected reductive with a Steinberg map F : G → G and assume that the automorphism σ induced by F on the Weyl group W is ordinary. In [Lu84a, §4] Lusztig defines the following data: to each family U ⊆ Uch(GF ) of unipotent characters with corresponding σ-stable family F ⊆ Irr(W)σ he associates finite groups GU  G˜U , where | G˜U : GU | = c is the order of the automorphism σ, and defines a bijection U → M (GU  G˜U ), ρ → x¯ρ , and an injection F → M (GU  G˜U ), φ → xφ . An interpretation of the group GU is given without proof in [Lu84a, 13.1.3] and then substantiated in [Lu13a]. Moreover, if ρ is a unipotent character lying in the (ordinary) Harish-Chandra series of a cuspidal unipotent character of a split Levi subgroup L  G, then let Δ( x¯ρ ) := [L,L] = (−1)r(L

F)

(see [Lu84a, 6.7 and Prop. 6.20]). By Proposition 3.4.7 this is exactly the sign by which DL (ρ) differs from being a true character. It turns out that for simple groups G with a split Frobenius F, the sign Δ( x¯ρ ) is always equal to 1, except for the nonprincipal series characters in the exceptional families in types E7 and E8 already encountered in Remark 3.1.21 and also discussed in 4.1.16: these are exactly the families in which the principal series characters correspond to representations of the equal-parameter Iwahori–Hecke algebra H (W, q) that cannot be realised over Q(q). With these data the following fundamental decomposition theorem holds: Theorem 4.2.16 (Lusztig [Lu84a, Thm. 4.23]) Assume that σ is ordinary. Let U ⊆ Uch(GF ) be a family of unipotent characters with corresponding σ-stable family F ⊆ Irr(W)σ . Then we have ρ, Rφ˜ = Δ( x¯ρ ){ x¯ρ, xφ } ˜ for all ρ ∈ U and φ ∈ Irr(W)σ with preferred σ-extensions φ. Observe that the set of unipotent characters Uch(GF ) as well as the multiplicities Rφ˜ , ρ are thus generic, that is, they depend only on the complete root datum underlying (G, F), so the same is true for the subdivision of Uch(GF ) into families. By the definition of almost characters the multiplicities Δ( x¯ρ ){ x¯ρ, xφ } are related to the coefficients m(w, x¯ρ ) mentioned in Theorem 2.4.1 through the σ-character table of W.

4.2 Families of Unipotent Characters and Fourier Matrices

293

Remark 4.2.17 The matrix M(U ) := ({ x¯ρ, x}), with ρ ∈ U , x ∈ M (GU ), is called the non-abelian Fourier transform matrix associated to the family U . Theorem 4.2.16 immediately implies the following formula for almost characters (see [Lu84a, Cor. 4.24]):

Rφ˜ = Δ( x¯ρ ){ x¯ρ, xφ } ρ, ρ∈U

involving the rectangular submatrix ({ x¯ρ, xφ }) of M(U ) with ρ ∈ U and φ ∈ F . The orthonormal set of class functions

Rx = Δ( x¯ρ ){ x¯ρ, x} ρ (x ∈ M (GU )) ρ∈U

is called the unipotent almost characters of GF . As already mentioned in Remark 2.7.26, Lusztig conjectured that each Rx should have a geometric interpretation, as the characteristic function of an F-stable character sheaf on G (suitably normalised); that is, the Fourier matrix M(U ) should, up to scalars, describe the base change from characteristic functions to unipotent characters. This is known to be true by Shoji’s work [Sho95]. (The restrictions on the characteristic in [Sho95] can now be removed thanks to [Lu12a].) See [Ge18, §7] for a more in-depth discussion. The proof of Theorem 4.2.16 (for arbitrary characters in place of unipotent ones) occupies most of the book [Lu84a]. Lusztig had previously already shown this for unipotent characters in the case when the underlying field Fq is sufficiently large in [Lu80a, Lu81c, Lu82b]. Let us comment somewhat more on this result. The finite group GU turns out in all cases to be a direct product of symmetric groups Sn , with 1  n  5. Moreover, S5 only occurs if G has a factor of type E8 , and then only for one family, and S4 only occurs if G has a factor of type F4 , again for a single family. If G is of classical type, all groups GU are elementary abelian 2-groups. The sizes of the occurring Fourier matrices are given by |M (C2e )| = 22e, |M (S3 )| = 8, |M (S4 )| = 21, |M (S5 )| = 39. Let us point out that although all symmetric groups have rational character tables, the corresponding Fourier transform matrices M(U ) may contain non-rational entries (and in fact do so for G = S5 which occurs in type E8 ). Remark 4.2.18 A trivial consequence of this explicit decomposition is the existence of a Jordan decomposition E (GF , 1) → E (G∗F , 1) as claimed in Theorem 2.6.4, at least for unipotent characters. Indeed, as pointed out in Remark 2.6.5 there is an immediate reduction to the case of simple groups using the arguments in Remark 4.2.1. Then one uses that the adjoint quotients of G and G∗ are isomorphic except when G is of type B or C. In the latter case, though, the combinatorial

294

Unipotent Characters

data describing the multiplicities of unipotent characters in almost characters in Theorem 4.2.16 are the same in both types, thus there exists a bijection as claimed. As a consequence of the explicit Decomposition Theorem 4.2.16 Lusztig [Lu84a, l. 1 on p. 133] observed: Corollary 4.2.19 Let U ⊆ Uch(GF ) be a family of unipotent characters. Then there is a σ-extension of the special character in the corresponding σ-stable family in Irr(W) such that all χ ∈ U occur with positive multiplicity in the corresponding almost character. Example 4.2.20 (a) It easily follows from the definitions given in 4.2.9 and in 4.2.14 that in the case when GU = C2e all entries of the corresponding 22e × 22e Fourier transform matrix are of the form ±2−e . (b) We give the Fourier transform matrices for the two smallest non-trivial cases, the symmetric groups S2  C2 and S3 in Tables 4.4 and 4.5. As in previous tables, an entry “.” stands for “0”. Here we write g2 for the transposition (1, 2) and g3 for the 3-cycle (1, 2, 3). Also, ε denotes the sign character of S2 or S3 , r is the reflection character of S3 and θ, θ 2 denote the two non-trivial characters of the cyclic group generated by g3 . We see that by Corollary 4.2.19 for families with group S2 or S3 the special character must always correspond to the pair (1, 1). Table 4.4 Fourier transform matrix for S2

1 2

 (1, 1)   (1, ε)  (g2, 1) (g2, ε)

(1, 1) 1 1 1 1

(1, ε) 1 1 −1 −1

(g2, 1) 1 −1 1 −1

(g2, ε) 1  −1  −1  1 

Table 4.5 Fourier transform matrix for S3      1  6      

(1, 1) (g2, 1) (1, r) (g3, 1) (1, ε) (g2, ε) (g3, θ) (g3, θ 2 )

(1, 1) 1 3 2 2 1 3 2 2

(g2, 1) 3 3 . . −3 −3 . .

(1, r) 2 . 4 −2 2 . −2 −2

(g3, 1) 2 . −2 4 2 . −2 −2

(1, ε) 1 −3 2 2 1 −3 2 2

(g2, ε) 3 −3 . . −3 3 . .

(g3, θ) 2 . −2 −2 2 . 4 −2

(g3, θ 2 ) 2 . −2 −2 2 . −2 4

             

4.2 Families of Unipotent Characters and Fourier Matrices

295

In the exceptional 4-element families of E7 and E8 , the sign Δ( x¯ρ ) equals −1 on the two elements (g2, 1), (g2, ε) (see [Lu84a, 4.14]). See [Lu84a, 4.15] or [Ca85, 13.6] for those parts of the matrices for S4 and S5 that are relevant in Theorem 4.2.16. The complete matrices can also easily be obtained in Chevie [MiChv]. 4.2.21 A further important invariant of unipotent characters are their so-called Frobenius eigenvalues, which we define now. Let δ > 0 be the smallest integer such that F δ acts trivially on the Weyl group W of G. Let ρ ∈ Uch(GF ). First, since ρ occurs with non-zero multiplicity in some Deligne–Lusztig generalised character, × it is clear that there exist some w ∈ W, i  0 and μ ∈ Q such that ρ occurs in the character of the generalised μ-eigenspace of F δ on the -adic cohomology group Hci (Xw, Q ). Now Lusztig [Lu77b, 3.9] showed that μ is uniquely determined by ρ, independently of w and i, up to a factor which is an integral power of q δ . Furthermore, by [DiMi85, III.2.3], there is a well-defined root of unity ωρ ∈ Q and a well-defined element λρ ∈ {1, q δ/2 } such that μ = ωρ λρ q sδ

for some integer s  0.

Moreover, λρ is an integral power of q unless ρ lies in an exceptional family in type E7 or E8 (see [DiMi85, III.2.3 in conjunction with II.3.4] and also [GeMa03, Lemma 4.4 and Ex. 4.6–4.8]). The root of unity ωρ is called the Frobenius eigenvalue of ρ. This is closely related to the parametrisation of unipotent characters occurring in Lusztig’s Decomposition Theorem 4.2.16. For (x, σ) ∈ M (G ) let ω(x,σ) := σ(x)/σ(1) be the scalar by which the central element x acts in any representation with character σ. Proposition 4.2.22 Assume that F is split. Let U ⊆ Uch(GF ) be a non-exceptional family of unipotent characters with corresponding group GU . Then the bijection ∼ U → M (GU ) in Theorem 4.2.16 can be chosen such that if ρ ↔ (x, σ) ∈ M (GU ) then ωρ = ω(x,σ) . This shows that Frobenius eigenvalues, at least in the split case, are also generic data, only depending on the underlying complete root datum. The explicit results presented later show that this statement remains true for general Steinberg maps. See [Lu84a, Thm. 11.2] (and also [DiMi85, Prop. 1.5]) for the proof of the above result. It relies, among other things, on the fact that Frobenius eigenvalues are preserved by Harish-Chandra induction: Proposition 4.2.23 Let ρ ∈ Uch(GF ) lie in the Harish-Chandra series of the cuspidal pair (L, λ). Then the Frobenius eigenvalues ωλ and ωρ coincide. See [Lu77b, 3.33] and [Lu84a, 11.3]. If instead we consider the d-Harish-Chandra

296

Unipotent Characters

series of a d-cuspidal character of some d-split Levi subgroup L  G (see 3.5.24), then there still is a relation between Frobenius eigenvalues, but in general they are no longer preserved, see [Ma95, Satz 4.21] and also [BMM14, Axiom 4.31(d)]. 4.2.24 We now present an axiomatic framework for the Fourier matrices and Frobenius eigenvalues attached to unipotent characters of a series of finite reductive groups which was first spelled out in this form by Geck–Malle [GeMa03], and which was mainly inspired by the empirical data obtained for spetsial imprimitive complex reflection groups in [Ma95] and those for primitive groups which were constructed in 1994 but published only much later in [BMM14]. Let G be a connected reductive group with a Steinberg endomorphism F and denote by σ the automorphism of the Weyl group W of G induced by G. Let F ⊆ Irr(W) be a σ-invariant family and Uch(F ) ⊂ Uch(GF ) the corresponding family of unipotent characters. If σ is ordinary in the sense of 4.1.26, then Lusztig’s Decomposition Theorem 4.2.16 defines a Fourier matrix M = M(Uch(F )) for this family, depending only on the family F and σ. In the remaining cases (i.e., types B2 , G2 and F4 with a non-ordinary σ) such a matrix was obtained in [GeMa03], see Theorem 4.5.2. In all cases the columns of M are indexed by Uch(F ), the rows are indexed by the almost characters Alm(F ) corresponding to this family (that is, characteristic functions of F-stable unipotent character sheaves on G, see Remarks 2.7.26 and 4.2.17). Let us write M = M(F, σ) = (aλμ ), where (λ, μ) ∈ Alm(F ) × Uch(F ), for this Fourier matrix. Following Asai [As84c], define the twisting operator t1∗ on class functions f on F G by t1∗ ( f )(x) := f (yxy −1 )

for x ∈ GF ,

where y ∈ G is such that x = y −1 F(y). Any uniform almost character is an eigenvector for t1∗ with eigenvalue 1. More generally, for any F-stable character sheaf K on G (see 2.7.24) the corresponding characteristic function χK,φ is also an eigenvector of t1∗ , where the corresponding eigenvalue does not depend on σ (see Eftekhari [Ef94] and also Shoji [Sho95, I.§3]). Let Fr1 = Fr1 (F, σ) be the diagonal matrix of eigenvalues of t1∗ on the almost characters in Alm(F ), and Fr2 := Fr2 (F, σ) the diagonal matrix of Frobenius eigenvalues of the unipotent characters in Uch(F ). By the results of Lusztig [Lu84a, 11.2] and Shoji [Sho95, I.§3] we have Fr1 = Fr2 when σ = 1, in which case we just write Fr for this matrix. These eigenvalues are known explicitly in all cases, see [GeMa03, Rem. 4.9 and Thm. 4.11] and the references given there. Finally, let Δ be the permutation matrix describing complex conjugation on Alm(F ) (hence also on the eigenvalues of t1∗ on Alm(F )). Then we may observe the following general properties satisfied by all these matrices (see [GeMa03, Thm. 6.9]):

4.3 Unipotent Characters in Type A

297

Theorem 4.2.25 In the notation introduced above we have: (F1) M transforms the vector of unipotent degrees in Uch(F ) to the vector of fake degrees (extended by zeros) associated to F . (F2) All entries aλμ of M are real. (F3) We have M · Mtr = Mtr · M = 1. (F4) Let λ0 be the row index of the uniform almost character corresponding to the special character in F . Then all entries in that row are non-zero. (F5) The structure constants

aλκ aμκ aνκ aνλμ := aλ0 κ κ ∈Uch(F )

are rational integers for all λ, μ, ν ∈ Alm(F ). (F6) If σ = 1 then M = Mtr , Δ · M = M · Δ and (Fr · Δ · M)3 = 1.

2 (F6’) If σ  1, σ 2 = 1 then Fr2 · Mtr · Fr−1 1 · M = 1.

3 (F6”) If σ  1, σ 3 = 1 then Fr2 · Mtr · Fr−1 1 · M = 1. This assertion may be derived from the explicitly known data. In fact, the result stated in [GeMa03] is more general as it also applies to the unipotent degrees, Fourier matrices and Frobenius eigenvalues associated to the non-rational finite Coxeter groups of types I2 (m), H3 and H4 . See also [Ma95] and [BMM14] for analogous results pertaining to the unipotent degrees associated to spetsial complex reflection groups.

4.3 Unipotent Characters in Type A Here we describe the properties of the unipotent characters of groups of type A. That is, we let G = SLn and F : G → G a Frobenius endomorphism, so that the finite group GF of fixed points is either a finite special linear group SLn (q) or a finite special unitary group SUn (q), with q a power of the prime p. According to Remark 4.2.1 all the results stated in this section also hold for all groups isogenous to G, so for example for PGLn (q), respectively PGUn (q). We start off with the case that F is an untwisted Frobenius map, so GF = SLn (q). The unipotent characters of GLn (q) were first computed by Steinberg [St51a] (notwithstanding the fact that the notion of unipotent character was only introduced later). According to Remark 4.2.1 this also yields the parametrisation and degrees of unipotent characters of SLn (q). By Example 2.4.20 all unipotent characters lie in the principal Harish-Chandra series, so by Theorem 3.2.5 they are in bijection with the irreducible characters of the Weyl group W = Sn of G, the symmetric group of degree n. We will denote the unipotent character labelled by

298

Unipotent Characters

the partition α n by ρα . By the degree formula in Theorem 3.2.18 its degree can be expressed in terms of the corresponding Schur element of the 1-parameter Hecke algebra H (W, q), which was given in Example 3.2.17; the a-value was given in Example 4.1.2. We obtain: Proposition 4.3.1 (Hook formula for type A) The degree of the unipotent character ρα of SLn (q) labelled by α = (α1  · · ·  αm ) n is given by ρα (1) = q a(α)

n  qi − 1 , q li − 1 i=1

where li is the length of the hook at the ith box of the Young diagram of α, and m (m − i)αi . a(α) = n(α) = i=1 It is obvious from this that under the substitution q → 1 the degree of the unipotent character ρα of SLn (q) labelled by α specialises to the degree of the irreducible character of the symmetric group Sn labelled by the same partition α. There is another formula in terms of β-sets (see Example 4.1.7: the β-set of the partition α = (α1  α2  · · ·  αm ) is given by (α1, α2 + 1, . . . , αm + m − 1)). Proposition 4.3.2 The degree of the unipotent character ρα of SLn (q) labelled by α n with β-set (x1, . . . , xm ) is given by n   (qi − 1) (q x j − q xi )

ρα (1) =

i=1

i< j

m−1 m−2 q( 2 )+( 2 )+...

xi  (q k − 1) i

.

k=1

This follows by elementary calculations from Proposition 4.3.1, see [Ca85, 13.8]. Example 4.3.3 The degrees of the unipotent characters of SLn (q), 2  n  5, are as given in Table 4.6, where Φd denotes the dth cyclotomic polynomial evaluated at q. One standard application of the hook formula is the determination of unipotent characters of -defect 0 for a prime  different from the defining characteristic p. Let e be the order of q modulo . Then  divides (q f − 1)/(q e − 1) if and only if f is divisible by e. Now by definition an irreducible character of a finite group is of -defect zero if its degree is divisible by the full -part of the group order. Proposition 4.3.1 then leads to the following combinatorial criterion: Corollary 4.3.4 Let ρ be the unipotent character of SLn (q) labelled by the partition α n. Let  be a prime different from p, and e the order of q modulo . Then ρ is of

4.3 Unipotent Characters in Type A

299

Table 4.6 Unipotent characters of SL2 (q), SL3 (q), SL4 (q) and SL5 (q) α 2 12

ρα (1) 1 q

α 3 21 13

ρα (1) 1 qΦ2 q3

α 4 31 22 212 14

ρα (1) 1 qΦ3 q2 Φ4 q3 Φ3 q6

α 5 41 32 312 22 1 213 15

ρα (1) 1 qΦ2 Φ4 q2 Φ5 3 q Φ3 Φ4 q4 Φ5 6 q Φ2 Φ4 q10

-defect zero if and only if α is an e-core, that is, it has no hook of length (divisible by) e. This is in complete analogy to the result that an irreducible character of the symmetric group Sn is of -defect zero if and only if its parametrising partition does not have a hook of length (divisible by) . The situation for the prime  = 2 is slightly different, because both q − 1 and q + 1 are even when q is odd. Yet it is immediate from the hook formula that the degree polynomial of any unipotent character of SLn (q) is not divisible by q − 1, so that indeed for  = 2 the group SLn (q), n  2 and q odd, has no unipotent characters of 2-defect zero. As already stated above, all unipotent characters of SLn (q) lie in the principal series, so the only case with a cuspidal unipotent character occurs for n = 0, that is, when G is the trivial group. It then follows from Proposition 4.2.23 that all unipotent characters have their associated Frobenius eigenvalue equal to 1. We saw in Corollary 2.4.19 that the unipotent characters for groups of type A all lie in singleton families. Thus, all Fourier matrices are the 1-by-1 identity matrix. This also explains why the unipotent degrees (as given in Proposition 4.3.2) agree with the fake degrees given in Example 4.1.7. All unipotent characters of groups of type A are uniform; so by transitivity the decomposition of the Lusztig functor RLG on the space spanned by unipotent characters can in principle be computed from the known decomposition of Deligne– Lusztig characters. We will give a description of this decomposition in the situation of d-Harish-Chandra theory, that is, for d-split Levi subgroups L, in Section 4.6. Now consider the case when F is twisted, so that GF = SUn (q). Here it was shown by Lusztig–Srinivasan [LuSr77, Thm. 2.2] that the unipotent characters are again naturally labelled by partitions α of n; from their result it follows (see for example [Lu77a, Rem. 9.5]) that the degree of the unipotent character ρα of SUn (q) labelled by α is obtained from the degree of the corresponding unipotent character of SLn (q) by formally replacing q by −q, and adjusting the sign if necessary. This

300

Unipotent Characters

result is called Ennola duality since it is a particular instance of a conjecture made by Ennola [Enn63], see Theorem 2.8.17. So in particular we have: Proposition 4.3.5 (Hook formula for type 2A) The degree of the unipotent character ρα of SUn (q) labelled by α n is given by ρα (1) = q a(α)

n  qi − (−1)i , qli − (−1)li i=1

where li is the length of the hook at the ith box of the Young diagram of α, and m (m − i)αi as for SLn (q). a(α) = n(α) = i=1 Clearly, the analogue of the formula in Proposition 4.3.2 obtained by replacing q by −q then also holds for SUn (q). On the other hand, the subdivision of unipotent characters into (ordinary) HarishChandra series is more interesting here. The group SUn (q) has a cuspidal unipotent

character if and only if n is a triangular number n = 2t for some t  1, and then it

is labelled by the triangular partition δt = (1, 2, . . . , t − 1) of n. Now if n = 2m + 2t for some m  0 and t  1 there exists an F-stable 1-split Levi subgroup Ln−2m of SLn of type An−2m−1 corresponding to the F-stable subset of simple reflections sm+1, . . . , sn−m−1 in the set-up of Example 3.1.4. We write ρt for the cuspidal F labelled by δt . Then we have: unipotent character of Ln−2m

Proposition 4.3.6 Let n = 2m + 2t for some m  0, t  1. The Harish-Chandra series E (SUn (q), (Ln−2m, ρt )) consists of the unipotent characters ρα such that α n has 2-core δt . The relative Weyl group of (Ln−2m, ρt ) has type Bm , and the parameters of the associated Iwahori–Hecke algebra are given by q2, q2t−1 . See [FoSr90, App.]. This can also be seen as a consequence of Ennola duality applied to the 2-Harish-Chandra theory for SLn (q), see Section 4.6. Again by Corollary 2.4.19 the unipotent characters for groups SUn (q) all lie in singleton families. The Frobenius eigenvalues (see 4.2.21) of unipotent characters of unitary groups are given as follows: Proposition 4.3.7 Let ρ be a cuspidal unipotent character of G = SUn (q) labelled

by the 2-core λ n = 2t . Then the Frobenius eigenvalue ωρ attached to ρ satisfies DG (ρ) = ωρ ρ, that is ωρ = (−1)r(G) =

1

if t ≡ −1, 0, 1, 2

−1 if t ≡ 3, 4, 5, 6

(mod 8), (mod 8).

See Rem. (a) after Thm. 3.34 in [Lu77b] for q sufficiently large. The claim for arbitrary q also follows with the proof of [GeMa03, Thm. 4.11] applied to the case of 2An .

4.4 Unipotent Characters in Classical Types

301

4.4 Unipotent Characters in Classical Types 4.4.1 The unipotent characters of the other classical groups are best parametrised by combinatorial objects introduced by Lusztig, the so-called symbols. These symbols are natural generalisations of partitions, or rather their β-sets, and provide a very intuitive way to label the unipotent characters of groups of classical types, to understand their distribution into families, to encode arithmetical properties of their degrees, but also to describe the decomposition of Lusztig induction, the distribution into -blocks and further properties. Moreover, symbols also serve as a tool to describe unipotent conjugacy classes of classical groups and to describe the generalised Springer correspondence, in bad as well as in good characteristic, and even for disconnected groups. Thus, symbols provide the right generalisation of the concept of partitions, which govern the representation theory of the symmetric groups as well as of the general linear and unitary groups. Symbols were first introduced by Lusztig in [Lu77a] to parametrise unipotent characters of groups of classical type. Later, Lusztig [Lu84c, §11] showed that they are also very convenient to describe the generalised Springer correspondence, see also [LuSp85]. The formalism of hooks and cohooks of symbols was introduced by Olsson [Ol84]. Symbols should be thought of as encoding pairs of partitions which are shifted with respect to each other in a well-defined way. Recall that the β-set associated to a partition (α1  · · ·  αr ) is the strictly increasing sequence (x1 < · · · < xr ) with entries xi := αi +i −1. Conversely, a finite sequence (x1 < · · · < xr ) of non-negative integers encodes the partition (α1  · · ·  αr ) with αi := xi − i + 1 (where we first remove leading zeros). For integers a, b  0 and for d ∈ Z, we consider the following set X˜da,b of symbols: The elements of X˜da,b are pairs S = (X, Y ) of finite sequences X = (x1, . . . , xr ) and Y = (y1, . . . , ys ) of non-negative integers, subject to the conditions x j  x j−1 + a + b

for 2  j  r,

y j  y j−1 + a + b

for 2  j  s,

y1  b, r − s = d. Note that for b = 0 the third condition is empty. Symbols are often displayed in the form     X x 1 x2 . . . S= = , y 1 y2 . . . Y and we then say that X, Y are the rows of S.

302

Unipotent Characters

For S = (X, Y ) a symbol, we call d = d(S) = |X | − |Y | its defect. The content of S is defined by |S| := |X | + |Y | and the rank by rnk(S) :=

x ∈S

0 x − (a + b)

1 0 1 |S| (|S| − 1)2 −b , 4 2

where $z% := max{n ∈ Z | n  z} denotes the Gauß brackets. It is easy to check that rnk(S) ∈ N0 for any S ∈ X˜da,b . On X˜da,b we have the shift operation which to S = (X, Y ) associates the symbol

S  = {0} ∪ (X + a + b), {b} ∪ (Y + a + b) ∈ X˜da,b . It is clear that the defect of symbols is invariant under shift. We write Xda,b for the classes in X˜da,b under the equivalence relation defined by the symmetric and transitive hull of the shift operation. Similarly, an easy calculation shows that the a,b for the set of classes of symbols rank is invariant under shift, and we write Xd,n in Xda,b of rank n. Note that also the contents is invariant modulo 2 under shift. It a,b is easy to see that Xd,n is a finite set for any choice of parameters a, b  0, d ∈ Z and n  0. 0,0 we have Example 4.4.2 Consider the case that a = b = 0. Then for S ∈ Xd,n  0,0 rnk(S) = x ∈S x. So for any d ∈ Z and n  0, the set Xd,n is in natural bijection with the set of bipartitions of n as follows: Let (α, β) n. By adding zero parts to α or 0,0 β we may assume that α has exactly d more entries than β. Then (α, β) ∈ Xd,n . It is easy to see that this assignment defines a bijection. Thus, according to Example 4.1.3 0,0 this provides a natural parametrisation Xd,n → Irr(Wn ) of the irreducible characters of the Weyl group of type Bn in terms of symbols of rank n and fixed defect d.

4.4.3 An element S = (X, Y ) ∈ Xda,b is called distinguished if d ∈ {0, 1} and x1  y1  x2  · · ·  ys (and ys  xs+1 if d = 1). Note that this definition is independent of the chosen representative in its class. Two (classes of) symbols are called similar if for suitable representatives Si = (Xi, Yi ), i = 1, 2, the multisets of a,b , entries X1 ∪ Y1 and X2 ∪ Y2 of entries coincide. Thus for any symbol S ∈ Xd,n a,b d ∈ {0, 1}, there exists a unique distinguished symbol in Xd,n up to shift which is similar to S. A further important invariant of symbol classes is the genus of a symbol S ∈ X˜da,b

4.4 Unipotent Characters in Classical Types

303

defined by



t+1 4t−1  if |S| = 2t + 1, a 2t 4t+1 3 +b 2 3 g(S) := min(x, y) − 



⎪ ⎪ {x,y } if |S| = 2t + b 2t 4t+1 ⎪ a 2t 4t−5 3 3 ⎩ (see [GeMa00, (2.22)]). Here the sum extends over all 2-element subsets of the multiset X ∪ Y of entries of S = (X, Y ). Thus we have g(S) ∈ N0 for any S ∈ X˜da,b . Furthermore, the following is easily checked:

⎧ ⎪ ⎪ ⎨ ⎪

Lemma 4.4.4 The genus is invariant under shift, hence well defined on Xda,b . 4.4.5 (Cuspidal symbols, addition) The core or 1-core of a symbol S = (X, Y ) in X˜da,b is the symbol core(S) = (X , Y ) ∈ X˜da,b with X  = (0, a + b, . . . , (r − 1)(a + b)),

Y  = (b, a + 2b, . . . , (s − 1)a + sb),

where r = |X |, s = |Y |. Thus the defect d(core(S)) = d(S) remains unchanged while rnk(core(S))  rnk(S). If equality holds, then core(S) = S. Such symbols S will be called cuspidal. Note that the equivalence class of any cuspidal symbol contains a representative with either X =  or Y = . For any fixed defect d there exists an addition 





a,b a ,b a+a ,b+b + : Xd,n × Xd,n −→ Xd,n+n  



defined by component-wise addition of representatives of the same length. For fixed a,b a, b, d, n and a cuspidal symbol S ∈ Xd,k of rank rnk(S) = k let X(S) be the subset a,b with core S. The following observation is then of (classes of) symbols in Xd,n immediate: a,b Lemma 4.4.6 Let S = (X, Y ) ∈ Xd,k be cuspidal. Then addition of S defines a natural bijection 1−1

0,0 a,b −→ X(S) ⊆ Xd,n , Xd,n−k

S  → S  + S.

0,0 Example 4.4.7 By Example 4.4.2 the set Xd,n−k can be naturally identified with the set of pairs of partitions of n − k, which in turn by Example 4.1.3 is in natural bijection with the set Irr(Wn−k ) of irreducible characters of the Weyl group Wn−k of type Bn−k . Thus Lemma 4.4.6 gives a natural parametrisation of X(S) by irreducible characters of Wn−k .

4.4.8 If the parameter b equals 0 then the cyclic group of order 2 acts on Xda,0 ∪ a,0 X−d by interchanging the two rows, that is, S = (X, Y ) → S tr := (Y, X). The symbol S is called degenerate if it is a fixed point under this action, that is, if X = Y . Clearly, degenerate symbols necessarily have defect d = 0. Let Yda denote the set

304

Unipotent Characters

a,0 modulo this action. Again, the rank, the of equivalence classes in Xda,0 ∪ X−d genus and the contents modulo 2 are well defined on elements of Yda , the cores of equivalent symbols are equivalent, and if a symbol is cuspidal, then so are all equivalent symbols. The defect is well defined up to sign. We say that S ∈ Yda a the is distinguished if one of S, S tr is distinguished. As before, we denote by Yd,n a subset of symbol classes in Yd of rank n. a be cuspidal of rank k and denote by Y (S) Example 4.4.9 Let S = (X, Y ) ∈ Yd,k a with core S. Note that the only degenerate the subset of (classes of) symbols in Yd,n core in this case is equivalent to (−; −) and this occurs if and only if d = 0. First a is naturally in bijection assume that S is not degenerate, that is, d  0. Then Yd,n a,0 and addition again defines a bijection with Xd,n 1−1

0,0 a Xd,n−k −→ Y (S) ⊆ Yd,n ,

S  → S  + S,

which by Example 4.4.7 induces a bijection between Irr(Wn−k ) and Y (S). On the other hand, when S is degenerate, hence equivalent to the symbol (−; −) of defect d = 0 and rank 0, then we obtain a bijection 1−1

0 a Y0,n −→ Y (S) ⊆ Y0,n ,

S  → S  + S.

According to Example 4.1.4 this sets up a map Irr(Wn ) → Y (S) from the irreducible a with core characters of the Weyl group Wn of type Dn to the symbol classes in Y0,n S which is one-to-one, except that both characters in a pair {φ[α,±] } are sent to the same (degenerate) symbol class. 4.4.10 As for partitions, there is a notion of hooks of symbols, but now they come in two different flavours. Let d  1. A d-hook h of a symbol S = (X, Y ) is an entry x  d of S with either x ∈ X, x − d  X, or x ∈ Y , x − d  Y . Thus, a hook of S is nothing else but a hook of the partition with associated β-set either X or Y . The length of the hook h is l(h) := d. Removing the d-hook h at x leads to the symbol S \ h := (X , Y ) with X  = X \ {x} ∪ {x − d} (respectively S \ h := (X, Y ) with Y  = Y \ {x} ∪ {x − d}). We then also say that S is obtained from S \ h by adding the d-hook h. Attached to the hook h is the sign h := (−1)m

where m :=

|{y ∈ X | x − d < y < x}|

if x ∈ X,

|{y ∈ Y | x − d < y < x}|

if x ∈ Y .

A symbol having no d-hooks is called a d-core. A moment’s thought shows that a d-core can not possess any hooks of length divisible by d. A d-cohook c of S is an entry x  d of S with either x ∈ X, x − d  Y , or x ∈ Y , x − d  X. Again its length is defined to be l(c) = d. Removing the d-cohook c at x from S leads to the symbol S \ c := (X , Y ) with X  = X \ {x}, Y  = Y ∪ {x − d}

4.4 Unipotent Characters in Classical Types

305

(respectively Y  = Y \ {x}, X  = X ∪ {x − d}). Again, S is called the symbol obtained by adding the d-cohook c to S \ c. Attached to the cohook c is the sign c := (−1)m , where m :=

|{y ∈ X | y < x}| + |{y ∈ Y | y < x − d}|

if x ∈ X,

|{y ∈ Y | y < x}| + |{y ∈ X | y < x − d}|

if x ∈ Y .

A symbol without d-cohooks is a d-cocore. A d-cocore does not possess any cohooks of length an odd multiple of d, nor any hook of length a multiple of 2d. Note that the multiset of hook lengths and of cohook lengths of a symbol is invariant under shift and hence constant on equivalence classes. Clearly, removing a d-hook or a d-cohook from a symbol of rank n yields a symbol of rank n − d. The following is obvious from our definitions: Proposition 4.4.11 Let S be a symbol and d  1. Then successively removing d-hooks (respectively d-cohooks) from S as often as possible leads to a uniquely determined symbol, called the d-core (respectively d-cocore) of S. These notions will play a fundamental role in the combinatorial description of unipotent character degrees but also in the decomposition of Lusztig induction. Remark 4.4.12 The combinatorics of symbols introduced above has a natural generalisation to so-called e-symbols, that is, symbols with e  2 rows. These play a central role in the parametrisation of unipotent characters attached to the imprimitive complex reflection groups G(d, 1, n) and G(d, d, n), extending the combinatorics of unipotent characters of classical groups, see [Ma95] for details. The unipotent characters of the classical groups of types B, C and D, that is, the symplectic and various orthogonal groups, were determined by Lusztig [Lu77a, Thm. 8.2]. He showed that they can naturally be parametrised by suitable equivalence classes of symbols: Theorem 4.4.13 (Lusztig) The unipotent characters of a finite classical group of 1 with rank n are labelled by equivalence classes of symbols of rank n in Yd,n (1) odd defect d for types Bn and Cn , (2) defect d ≡ 0 (mod 4) for type Dn , (3) defect d ≡ 2 (mod 4) for type 2Dn , where in type Dn any class of degenerate symbols labels two unipotent characters. Example 4.4.14 (a) The symbols in Yd1 of rank 2 and odd defect have representatives up to equivalence given by             2 02 12 01 012 , , , , , . 1 0 2 012 12

306

Unipotent Characters

In particular, Sp4 (q) and SO5 (q) both have six unipotent characters. (b) The symbols in Yd1 of rank 4 and defect d ≡ 2 (mod 4) are given up to equivalence by             04 31 014 013 012 023 , , , , , , 1 2 3 1         123 0124 0123 01234 , , , . 0 12 13 123 In particular, SO−8 (q) has ten unipotent characters. (c) If n = 2m is even, then there exist degenerate symbols in Y01 of rank n. They are in natural bijection with partitions of m, by sending a partition λ m to the symbol with two equal rows each consisting of the β-set of λ. For example, for SO+8 (q) there are two pairs of unipotent characters, labelled by each of the degenerate symbols     2 13 and . 2 13 The degrees of the unipotent characters are given by a formula analogous to the one for type A in Proposition 4.3.2. Again, we state the result for one particular isogeny type, but according to Remark 4.2.1 it is the same for any other group isogenous to this. Proposition 4.4.15 The degree of the unipotent character ρS of a classical group G = Sp2n (q), SO2n+1 (q), or SO±2n (q) labelled by a symbol S = (X, Y ) = (x1 < · · · < xr ; y1 < · · · < ys ) is given by    (q x j − q xi ) (q y j − q yi ) (q xi + q y j ) |G|q ρS (1) =

i< j

i< j

i, j

yi xi   b(S) (r +s−2 +(r +s−4 +··· 2k ) ) 2 2 2 q (q − 1) (q2k − 1) i

where

2 b(S) =

r

i

k=1

(r + s − 1)/2

3

k=1

if X  Y, else.

It is an easy exercise to check that this expression is indeed constant on equivalence classes. It is also clear from this that the unipotent characters in types Bn and Cn are not only labelled by the same set, but also have the same degrees. Example 4.4.16 The degrees of the unipotent characters of Sp4 (q) and of SO−8 (q) for which the symbols were shown in Example 4.4.14 are as given in Table 4.7.

4.4 Unipotent Characters in Classical Types Table 4.7 Unipotent characters of SO−8 (q). . . S 0 4

3 1

0 1 4

0 11 3

0 21 2

3

ρS (1) 1 qΦ8 q2 Φ3 Φ6 1 q3 Φ Φ 3 8 2 1 q3 Φ Φ 6 8 2

S 0 2 3

1 12 3

0 102 4

0 11 22 3

0 11 23 3 4

123

307

. . . and of Sp4 (q)

ρS (1) 1 q3 Φ Φ 3 8 2 1 q3 Φ Φ 6 8 2 q6 Φ3 Φ6 q 7 Φ8 q12

S 2

0 2

112

001

0 12 2

0 1 2

12

ρS (1) 1 1 qΦ2 2 2 1 qΦ 4 2 1 qΦ 4 2 1 qΦ2 2 1 q4

For many applications concerning character degrees an analogue of the hook formula for SLn (q) in Proposition 4.3.1 is more useful than the degree formula given in Proposition 4.4.15, since it reflects the arithmetic properties of the degrees much more clearly. Here, for a symbol S = (X, Y ) we set

 |X | + |Y | − 2i 

, min{x, y} − a(S) := 2 i1 {x,y } ⊆S

where the first sum runs over all 2-element subsets of the multiset X ∪ Y of entries of S (this is just the genus of S as introduced in 4.4.3, so in particular is independent of the chosen representative in an equivalence class by Lemma 4.4.4). Proposition 4.4.17 (Hook formula for classical types) The degree of the unipotent character ρS of a classical group G = Sp2n (q), SO2n+1 (q), or SO±2n (q) labelled by a symbol S = (X, Y ) is given by ρS (1) = q a(S)

2

b (S)

|G|q   , l(h) (q − 1) (ql(c) + 1) h

c

where the products in the denominator run over all hooks h, and cohooks c of S, and b(S) =

$(|X | + |Y | − 1)/2% − |X ∩ Y |

if X  Y,

0

else.

This formula was first given by Olsson [Ol84, Prop. 5]; see also Malle [Ma95, Bem. 3.12 and 6.8] for a proof in the more general setting of imprimitive complex reflection groups. As the multisets of hook lengths and cohook lengths are invariant under equivalence, this expression is indeed constant on equivalence classes. As in type A, the hook formula yields an easy criterion for a unipotent character of a classical group to be of -defect 0. Observe for this that for a prime  > 2 different from the defining characteristic p, if e denotes the order of q modulo ,

308

Unipotent Characters

then q f + 1 is divisible by  if and only if e is even and 2 f is an odd multiple of e. The following combinatorial criterion is then immediate from Proposition 4.4.17: Corollary 4.4.18 Let ρ be a unipotent character of a classical group Sp2n (q), SO2n+1 (q), or SO±2n (q), labelled by a symbol S. Let 2 <  be a prime different from p, and e the order of q modulo . Then ρ is of -defect zero if and only if (1) S is an e-core, that is, it has no hook of length (divisible by) e, if e is odd; (2) S is an e/2-cocore, that is, it has no cohook of length e/2, if e is even. Again, the situation is slightly more complicated for the prime  = 2. Here again it turns out that no unipotent character is of 2-defect zero when q is odd. Example 4.4.19 Let G = Sp2n (q) or G = SO2n+1 (q) and consider primes  such that q has order 2n modulo . Then from Corollary 4.4.18(1) an easy argument shows that the unipotent characters of G not of -defect zero are the ones labelled by the symbols   1 ... a S= (0  a  n), 0 1 ... a−1 n and



0 S= 1

1 ...

... a−1

a

n

 (1  a  n − 1).

Thus, there are 2n unipotent characters of G not of -defect 0 (this was first used in [MSW94, Thm. 2.3]; see also [Ma88] for an application of such considerations in constructive Galois theory, and the survey [Ma14] for more recent applications). Now assume that G = SO−2n (q), and let  be as above. Then the unipotent characters not of -defect zero are the n different unipotent characters labelled by   0 1 ... a n S= (0  a  n − 1) 1 ... a (see [MSW94, Thm. 2.5]). The combinatorics of symbols makes it easy to describe the partition of the unipotent characters into families (introduced in 4.2.2); for this let us say that a symbol S is reduced if at most one of its rows contains an entry 0. By applying suitable shifts it is easily seen that any equivalence class of symbols contains (at least) one reduced one. Proposition 4.4.20 (Families in classical groups) Sp2n (q), SO2n+1 (q), or SO±2n (q).

Let G be a classical group

4.4 Unipotent Characters in Classical Types

309

(a) Two unipotent characters ρ, ρ ∈ Uch(G) lie in the same family if and only if their reduced labelling symbols are similar, that is, if their multisets of entries agree. The two unipotent characters labelled by a degenerate symbol each lie in a singleton family. (b) For U ⊆ Uch(G) a non-degenerate family, denote by k the number of entries of any symbol S belonging to U that occur in exactly one row of S. Then |U | = 22m and GU  C2m , with m = $ k−1 2 %. See [Lu84a, (4.5.6) and (4.6.10)] for (a) and [Lu84a, p. 88 and 94] for (b). 4.4.21 (Fourier matrices in classical types) We next describe the Fourier matrices for families of unipotent characters in classical groups. Let G be isogenous to a simple group with F-fixed points one of Sp2n (q), SO2n+1 (q), SO+2n (q) or SO−2n (q). Fix a family U ⊆ Uch(G) of unipotent characters of G. If U contains a unipotent character labelled by a degenerate symbol, then by the preceding proposition we have |U | = 1, and the corresponding Fourier matrix is the 1-by-1 identity matrix. So assume that this is not the case. Choose, as we may by Proposition 4.4.20, a set S = S (U ) of representative 1 for the characters ρ ∈ U all having the same multiset of symbols S = (X, Y ) ∈ Yd,n entries, and such that |X | ≡ |Y | + d (mod 4), where for types Sp2n (q) and SO2n+1 (q) we have that d ≡ 1 (mod 4), while for type SO−2n (q) we have d ≡ 2 (mod 8). We then have |S | = 22m where 2m + 1 and 2m + 2, respectively, denote the number of entries that appear only once in any symbol S ∈ S (see Proposition 4.4.20(b)). We set k := $(|X | + |Y |)/2%. Let F be the corresponding family of Irr(W) (which is σ-stable in the case of SO−2n (q)). For Sp2n (q) and SO2n+1 (q) according to Lemma 4.4.6 addition of the 1 induces an embedding of F into S . For SO+ (q) note cuspidal symbol (0; −) ∈ Y1,0 2n that no member of F will be degenerate. Thus again addition of the cuspidal symbol 1 defines an embedding of F into S . In both cases we set S  := S . (−; −) ∈ Y0,0 1 For G = SO−2n (q) we let S  be a set of representative symbols S = (X, Y ) ∈ Y0,n having the same multiset of entries as the symbols in S . Recall from 4.4.8 that the σ-stable irreducible characters of W are naturally identified (via addition of the 1 ) with elements of Y 1 . This defines an injection of cuspidal symbol (−; −) ∈ Y0,0 0,n  F into S in this case. For S = (X, Y ) ∈ S ∪ S  define   |Y | c(S) := |{(x, y) ∈ X × Y | x < y}| + . 2 The Fourier transform matrix of U is then given by (S, S  )S ∈S ,S ∈S  (see [Lu77a, §1] and [Ma95, Satz 4.17 and 6.26]), where for (S = (X, Y ), S  = (X , Y )) ∈ S ×S 

310

Unipotent Characters

we set   1 (−1)c(S)+c(S )+k+|Y∩Y | . 2m 4.4.22 Let us reformulate the above expression for the Fourier matrices. First of all, assume that the common multiset of entries of the symbols in S ∪ S  contains an entry z appearing twice. Then denoting by S1 = (X1, Y1 ) the symbol obtained by removing that entry z from both rows of S ∈ S ∪ S , we see that     |Y1 | |Y1 | + 1 − c(S)−c(S1 ) =|{x ∈ X1 | x < z}| + |{y ∈ Y1 | z < y}| + 2 2

S, S  :=

=|{x ∈ X1 | x < z}| − |{y ∈ Y1 | y < z}| + 2|Y1 | ≡|{x ∈ X1  Y1 | x < z}|

(mod 2)

is independent of S. Since k + |Y ∩Y  | also remains unchanged modulo 2, the Fourier matrix does not change if we remove all double entries in all symbols in S ∪ S . Moreover, the precise value of the entries (which are now all different) does not matter. Hence we may assume that X ∪Y = {0, . . . , |S| − 1} for all (X, Y ) ∈ S ∪ S . Then we get   |Y | c(S) =|{(x, y) ∈ X × Y | x < y}| + 2

y =|{(x, y) ∈ (X ∪ Y ) × Y | x < y}| = y ∈Y

which is congruent to the number of odd entries in Y modulo 2. Write o(Z) for the number of odd entries in a finite set Z ⊂ Z, and e(Z) for the number of even entries. Then for (X, Y ) ∈ S , (X , Y ) ∈ S  we have o(Y ) + o(Y ) = o(Y ∪ Y ) + o(Y ∩ Y ), |Y ∩ Y  | = o(Y ∩ Y ) + e(Y ∩ Y ) and o(X ∪ Y ) = k, so we have the following congruences c(S) + c(S ) + k + |Y ∩ Y  | ≡ o(Y ) + o(Y ) + k + o(Y ∩ Y ) + e(Y ∩ Y ) ≡ o(Y ∪ Y ) + o(X ∪ Y ) + e(Y ∩ Y ) ≡ o(X ∩ X ) + e(Y ∩ Y ) = |Y # ∩ Y # | modulo 2, where we write Y # := Y , {1, 3, . . . , 2k − 1} for the symmetric difference, so Y # contains all even entries in Y and all odd integers in X. This proves Lusztig’s original formula [Lu84a, 4.15 and 4.18] # # 1 (−1) |Y ∩Y | m 2 for the coefficients of the Fourier matrix.

S, S  =

An immediate consequence of this explicit description is the following important

4.4 Unipotent Characters in Classical Types

311

observation, which was first shown in [DiMi90, Prop. 6.3], and stated without proof in [Lu88, Proof of Prop. 8.1] and in [Lu02, Lemma 1.7]: Theorem 4.4.23 Let G be connected reductive such that all simple components are of classical type and F : G → G a Frobenius endomorphism. Then the unipotent characters of GF are uniquely determined by their uniform projections, that is, by their multiplicities in the uniform almost characters. Proof First assume G is simple. There is nothing to prove in type A, since by Corollary 2.4.19 there the unipotent characters coincide with uniform almost characters, and hence in particular are uniform. For the types Bn , Cn , Dn and 2Dn we argue as follows. Since the value on the identity element is a uniform function it is sufficient to see that unipotent characters are uniquely determined in their family by their degrees. Let ρ be labelled by a symbol S = (X, Y ). The degree formula in Proposition 4.4.15 shows that it would be enough to see that    (q j − qi ) (q j − qi ) (qi + q j ) {i< j } ⊆X

{i< j } ⊆Y

(i, j)∈X×Y

distinguishes characters in a family U . Now by 4.4.22 the Fourier matrix of U as well as the set of columns indexed by almost characters in U do not depend on the exact values of the multi-set X ∪ Y of entries of the symbols in U , only on their number. Thus, it suffices to show the above assertion in just one family with a given Fourier, for which we may choose these entries arbitrarily. Since the powers of q are Zariski dense in the integers, it thus suffices to show that the polynomial    (Z j − Zi ) (Z j − Zi ) (Zi + Z j ) (∗) {i< j } ⊆X

{i< j } ⊆Y

(i, j)∈X×Y

with indeterminates Zi , determines {X, Y }. But this is clear since any polynomial ring is a UFD. Finally, for GF of type 3D4 all unipotent characters have distinct degree (see [Ca85, §13.9]), hence necessarily distinct uniform projection. It now follows by Corollary 4.2.19 and Example 4.2.20 that for G simple any two unipotent characters with proportional uniform projection must already be equal. Then using the considerations in Remark 4.2.1 there is a straightforward reduction of the general case to the one treated above.  Corollary 4.4.24 Let G be connected reductive such that all simple components are of classical type and F : G → G a Frobenius endomorphism. Then all unipotent characters of G are rational valued. Proof This is immediate from Theorem 4.4.23 and the fact that the unipotent almost characters, being rational linear combinations of the RTG (1T ), are rational. 

312

Unipotent Characters

The conclusion of the previous theorem as well as of its corollary both fail to hold for any group GF of exceptional type apart from 3D4 (q); see Theorem 4.5.3. Remark 4.4.25 (a) Digne–Michel [DiMi90, Prop. 6.3] propose a more explicit proof of Theorem 4.4.23: let ρ be a unipotent character, labelled by a symbol S say. Let Rφ˜ be an almost character in the same family corresponding to the σ-extension of a character φ ∈ Irr(W)σ labelled by the symbol S . Then in the notation introduced # # above, S, S  = 2−m (−1) |Y ∩Y | where 22m is the size of the corresponding family, and Y, Y  are the second rows of the symbols S, S  respectively. First assume that G is of type Bn or Cn . For x ∈ Z := X ∪ Y choose a partition Y1 ∪ Y2 = Z \ {x} into two subsets of equal cardinality m and let S1, S2 be the two symbols in the family with second row Y1, Y2 respectively. Then by construction these parametrise uniform almost characters. It is then easy to see that |Y # ∩ Y1# | ≡ |Y # ∩ Y2# |

mod 2

if and only if

x  Y #.

Hence Y and so S is uniquely determined by the multiplicities in almost characters. For types Dn and 2Dn , for any 2-element subset {x, y} ⊆ Z = X ∪ Y choose subsets Y1, Y2 ⊂ Z of cardinality m + 1 with Y1 , Y2 = {x, y}. Then |Y # ∩ Y1# | + |Y # ∩ Y2# | ≡ |Y # ∩ {x, y}|

mod 2.

Since the two symbols (X, Y ), (Y, X) parametrise the same character, we may always arrange things so that x  Y , and then again Y is determined by the multiplicities in the almost characters. (b) For families U with |U | = 22e  28 = 64 elements another easy argument is as follows: one can check that in these cases more than half of the symbols in the family lie in the principal series, that is, the space of uniform class functions in the family has dimension more than half of the whole space of class functions. If two unipotent characters have the same uniform projection, then their difference f will involve less than 22e−1 almost characters in that family, all with multiplicity absolutely bounded by 21−e , so f has norm less than 2, a contradiction. We have an even stronger statement that we will need later: Proposition 4.4.26 Let G be connected reductive such that all simple components are of classical type and F : G → G be a Frobenius endomorphism. Let U ⊂ Uch(GF ) be a family with |U | > 4. Then the various sums of two distinct unipotent characters from U are uniquely determined by their uniform projections. Proof As in the proof of Theorem 4.4.23 we may reduce to the case that G is simple. For types A and 3D4 all families have at most 4 elements. For the remaining types of classical groups we will again argue via the degrees of characters. For this

4.4 Unipotent Characters in Classical Types

313

set I = X ∪ Y (where again we may assume that X, Y are disjoint) and rewrite the polynomial in (*) above as  f (X, Y ) := (Z j − i j Zi ) {i< j } ⊆I

with i j := −1 if (i, j) ∈ X × Y ∪ Y × X and i j := 1 otherwise. For S  = (X , Y ) the symbol of a second unipotent character in the family we have  (Z j − ij Zi ) f (X , Y ) = {i< j } ⊆I

with ij defined as the i j but now in terms of X , Y  in place of X, Y . Then we claim that the polynomial f (X, Y ) + f (X , Y ) uniquely determines the set  then the specialisation of both f (X, Y ) {{X, Y }, {X , Y  }}. If k, l ∈ I with kl = kl   and f (X , Y ) and hence of their sum at Zk = kl Zl vanishes. On the other hand, if  then the specialisation of f (X, Y ) + f (X , Y  ) at Z = ± Z gives kl  kl k kl l f (X , Y )| Zk =k l Zl  0,

respectively

f (X, Y )| Zk =−k l Zl  0,

and from these we can determine the sets X, Y and X , Y  up to the positions of k, l. Observe that |I |  5 since |U | > 4; but then if S  S  there are at least three distinct  , so the above allows us to retrieve {{X, Y }, {X , Y  }}. k < l with kl  kl  It can easily be seen that the conclusion fails for 4-element families. Example 4.4.27 (a) Application of Proposition 4.4.20 to the symbols in Example 4.4.14(a) shows that the unipotent characters of Sp4 (q) fall into three families, two of them singletons containing the trivial and the Steinberg character respectively, and one family U containing the four characters labelled by         02 12 01 , , , . 1 0 2 012 The corresponding family F ⊂ Irr(W) contains the characters φ(α,β) with (α, β) = (1, 1), (12, −), (−, 2) respectively, which by the procedure described in 4.4.21 correspond to the first three symbols listed above in this order. The distinguished symbol, labelling the special character in that family, is the first one. Here, the parameter m attached to this family U as in Proposition 4.4.20 equals 1 and so GU  C2 and indeed |U | = 4. By 4.4.22 the Fourier matrix is given by 1 1 1 1 1 1 1 −1 −1 . 2 1 −1 1 −1 1 −1 −1 1 

314

Unipotent Characters

(b) The fourteen unipotent characters of SO+8 (q) fall into ten singleton families, and one 4-element family U labelled by the symbols         13 12 23 , , , , 02 03 01 0123 and GU  C2 . The corresponding family F ⊂ Irr(W) contains the characters φ[α,β] with (α, β) = (21, 1), (12, 2), (22, −), which are sent to the first three symbols in this order. The special character is the first one, and the Fourier matrix is as given in (a). (c) The ten unipotent characters of SO−8 (q) as described in Example 4.4.14(b) fall into six singleton families, and one 4-element family U labelled by the symbols         013 012 023 123 , , , , 2 3 1 0 where again we have GU  C2 . The corresponding σ-stable family F ⊂ Irr(W) contains the characters φ[α,β] with (α, β) = (21, 1), (12, 2), (22, −), which by the procedure described in 4.4.21 are identified to the symbols       13 12 23 , , 02 03 01 in this order. The special character is the first one. The Fourier matrix with respect to these orderings and these choices of representatives is again as given in (a). The description of Harish-Chandra series is also very intuitive in terms of cores of symbols as introduced in 4.4.5. For this, we first describe the cuspidal unipotent characters; they turn out to be very rare (see Lusztig [Lu77a, Thm. 8.2]): Theorem 4.4.28 A classical group of rank n has a cuspidal unipotent character if and only if (1) n = s(s + 1) for some s  1 for types Bn and Cn (n  2), (2) n = (2s)2 for some s  1 for type Dn (n  4), (3) n = (2s + 1)2 for some s  1 for type 2Dn (n  4). In each of these cases, there is a unique such character, and it is labelled by a cuspidal symbol (in the sense of 4.4.5). It is easy to check from the definition in 4.4.5 that there is a unique cuspidal symbol (up to equivalence) under the conditions stated in the theorem, and none otherwise. Now recall from Example 3.5.15 that 1-split Levi subgroups of simple groups G of classical type have at most one factor not of untwisted type A, which is then of the same type as G. As a group of untwisted type An only possesses cuspidal unipotent characters when n = 0 (see Section 4.3), that is, when it is a torus, a 1-split Levi subgroup of a simple classical group can have a cuspidal unipotent character

4.4 Unipotent Characters in Classical Types

315

only if its derived subgroup is again a simple classical group, whose rank m has the form m(s) = s(s + 1), respectively (2s)2, (2s + 1)2 , according to the type of group, as given in Theorem 4.4.28. So the Harish-Chandra series of unipotent characters of a simple classical group of rank n are in bijection with the set of integers s such that m(s)  n. The following is in [Lu77a, 8.10]: Proposition 4.4.29 (Harish-Chandra series in classical groups) Let G be a classical group of rank n, and S a cuspidal symbol of a Levi subgroup of rank n − m of G. Then the Harish-Chandra series above the corresponding cuspidal character ρS consists of the unipotent characters of G labelled by symbols with core S. If S is non-degenerate, then 0 1 −→ Yd,n , Yd,m

S  → S + S ,

defines a natural bijection of Irr(W(Bm )) with the Harish-Chandra series above ρS . If S = (−; −) is degenerate, and so G is of type Dn and m = n, then 0 1 −→ Y0,n , Y0,n

S  → S + S ,

defines a natural bijection of Irr(W(Dn )) with the Harish-Chandra series above ρS . Example 4.4.30 We continue Examples 4.4.14 and 4.4.27. (a) An application of Proposition 4.4.29 shows that five out of the six unipotent characters of Sp4 (q) listed in Example 4.4.14(a) lie in the principal series, while the one labelled by   012 lying in the four-element family is cuspidal. By Proposition 4.4.29 the Harish-Chandra series in Sp8 (q) above this cuspidal unipotent character of Sp4 (q) contains the unipotent characters with labels           014 023 0124 0123 01234 , , , , . 1 2 12 (b) On the other hand, all ten unipotent characters of SO−8 (q) lie in the HarishChandra series of the cuspidal character labelled by   01 of a Levi subgroup of type SO−2 (q). But the latter is a torus, so all ten unipotent characters lie in the principal series. This shows that even non-singleton families may consist solely of principal series characters. It is easily seen that this latter phenomenon can happen only in groups of twisted type. In untwisted groups, each family does contain at least one.

316

Unipotent Characters

The parameters of the associated Hecke algebras (for all types) have been determined by Lusztig, see [Lu77b, Tab. II]: Theorem 4.4.31 (Lusztig) Let G be simple with Steinberg endomorphism F. Then the parameters of the Iwahori–Hecke algebras associated to cuspidal unipotent characters of 1-split Levi subgroups of G are as given in Table 4.8.

Table 4.8 Parameters for Hecke algebras for cuspidal unipotent characters GF untwisted 2A (q) 2n+( s+1 2 )−1

(n  1, s  0) Bn+s2 +s (q) (n  1, s  1) Cn+s2 +s (q) (n  1, s  1) Dn+s2 (q) (n  1, s  2 even) 2D n+s 2 (q) (n  1, s  3 odd) 2D (q) n (n  4) 3D (q) 4 F4 (q) E6 (q) 2E (q) 6 E7 (q) E8 (q) 2B (q2 ) 2 2 G (q2 ) 2 2F (q2 ) 4

[L, L]F 1 2A ( s+1)−1 (q)

WG F (L, λ) W Bn

Bs2 +s (q)

Bn

q2s+1 ; q, . . . , q

Cs2 +s (q)

Bn

q2s+1 ; q, . . . , q

Ds2 (q)

Bn

q2s ; q, . . . , q

s 2 (q)

Bn

q2s ; q, . . . , q

1

Bn−1

q2 ; q, . . . , q

1 B2 (q) D4 (q) 1 2A (q) 5 D4 (q) E6 (q) D4 (q) E6 (q) E7 (q) 1 1 1 2B (q2 ) 2

G2 B2 A2 F4 A1 B3 A1 F4 G2 A1 A1 A1 I2 (8) A1

q3 ; q q3 ; q3 q4, q4 q2, q2 ; q, q q9 q; q4, q4 q9 q, q; q4, q4 q9 ; q q15 q4 q6 q4 ; q2 q12

2

2D

parameters all q q2s+1 ; q2, . . . , q2

This extends Table 3.1 which gave the parameters for the principal series. Recall from Proposition 4.2.23 that Frobenius eigenvalues are determined by the Harish-Chandra source of a character. Thus, to describe the Frobenius eigenvalues of unipotent characters it is sufficient to do this for the cuspidal characters.

4.5 Unipotent Characters in Exceptional Types

317

Proposition 4.4.32 Let ρ be a cuspidal unipotent character of a classical group labelled by the cuspidal symbol S. Then the Frobenius eigenvalue of ρ equals

⎧ (−1) $(s+1)/2% if S = 0 ... 2s , ⎪ ⎪ ⎨ ⎪

ωρ = (−1)s/2 if S = 0 ... 2s−1 , s even, ⎪

⎪ ⎪1 if S = 0 ... 2s−1 , s odd. ⎩ This was proved for sufficiently large q in [Lu81c, Prop. 6.6] for symplectic and odd-dimensional orthogonal groups, in [Lu82b, 3.18] for even-dimensional orthogonal groups of split type, and then without restriction on q in [Lu84a, 11.3]. For even-dimensional orthogonal groups of twisted type it was shown in [GeMa03, Thm. 4.11]. In Table 4.9 we give the Frobenius eigenvalues of cuspidal characters for all simple groups of classical type and rank at most 15. Table 4.9 Frobenius eigenvalues of some cuspidal unipotent characters

S ωρ

B2 0 1 2

D4 0 1 2 3

B6, C6 0 1 2 3 4

2D 9 0 1 2 3 4 5

B12, C12 0 1 2 3 4 5 6

−1

−1

−1

1

1

4.5 Unipotent Characters in Exceptional Types In view of the fact that the data for unipotent characters of exceptional groups of Lie type are not only given in printed form in [Lu84a, App.] and [Ca85, §13.8], but are also readily available electronically through the CHEVIE system [MiChv] we refrain from giving complete tables of the degrees of the unipotent characters in this case. On the other hand, neither [Lu84a] nor [Ca85] give explicitly the Frobenius eigenvalues of the unipotent characters, and these are somewhat difficult to find in the literature, so we include them here. According to Proposition 4.2.23 they are constant on Harish-Chandra series, so it is sufficient for this to list the various Harish-Chandra series together with their Frobenius eigenvalues. The Frobenius eigenvalues of unipotent characters were determined by Lusztig [Lu76c] for the constituents of Deligne–Lusztig characters for Coxeter tori. In [Lu77b, Thm. 3.34] he restricts the possible values in the general case and in [Lu84a, Chap. 11] he settles completely the case of split groups. The remaining eigenvalues were determined in [GeMa03].

318

Unipotent Characters

Theorem 4.5.1 The Frobenius eigenvalues of the Harish-Chandra series of unipotent characters in simple groups of exceptional type are as given in Tables 4.10–4.14. √ Here, for d  1, we denote by ζd := exp(2π −1/d) a primitive dth root of unity. The Frobenius eigenvalues of the cuspidal characters in groups of classical type occurring as Levi subgroups can be found in Table 4.9. Table 4.10 Unipotent Harish-Chandra series in G2 (q) and. . . WG (L, λ) W(G2 )

([L, L], λ) (1, 1)

. . . in 3D4 (q) |E | 6

ωρ 1

+4 cuspidals with ωρ = 1, −1, ζ3, ζ32

WG (L, λ) W(G2 )

([L, L], λ) (1, 1)

|E | 6

ωρ 1

+2 cuspidals with ωρ = 1, −1

Table 4.11 Unipotent Harish-Chandra series in F4 (q) ([L, L], λ) (1, 1) (B2, B2 [−1])

WG (L, λ) W(F4 ) W(B2 )

|E (G, (L, λ))| 25 5

ωρ 1 −1

+7 cuspidals with ωρ = 1, 1, −1, ζ3, ζ32, ζ4, −ζ4 .

Table 4.12 Unipotent Harish-Chandra series in E6 (q) and. . . ([L, L], λ) (1, 1) (D4, D4 [−1])

WG (L, λ) W(E6 ) W(A2 )

. . . in 2E6 (q) |E | 25 3

+2 cuspidals with ωρ = ζ3, ζ32

ωρ 1 −1

([L, L], λ) (1, 1) (2A5, 2A5 [−1])

WG (L, λ) W(F4 ) W(A1 )

|E | 25 2

ωρ 1 −1

+3 cuspidals with ωρ = 1, ζ3, ζ32

We include the complete data for unipotent characters of groups of very twisted type, as the table for 2F4 (q2 ) in [Ca85, §13.8] contains misprints (and neither the Frobenius eigenvalues nor the Fourier matrices or labels for cuspidal characters are given there). The Harish-Chandra labels are given via the characters of the relative Hecke √ algebra√as in [MiChv]. Recall our notation for cyclotomic polynomials over Q( 2) and Q( 3) from Example 3.5.4 (which differs from the one used in [Ca85, §13]).

4.5 Unipotent Characters in Exceptional Types

319

Table 4.13 Unipotent Harish-Chandra series in E7 (q) ([L, L], λ) (1, 1) (D4, D4 [−1]) (E6, E6 [ζ3 ]) (E6, E6 [ζ32 ])

WG (L, λ) W(E7 ) W(C3 ) W(A1 ) W(A1 )

|E (G, (L, λ))| 60 10 2 2

ωρ 1 −1 ζ3 ζ32

+2 cuspidals with ωρ = ζ4, −ζ4 .

Table 4.14 Unipotent Harish-Chandra series in E8 (q) ([L, L], λ) (1, 1) (D4, D4 [−1]) (E6, E6 [ζ3 ]) (E6, E6 [ζ32 ]) (E7, E7 [ζ4 ]) (E7, E7 [−ζ4 ])

WG (L, λ) W(E8 ) W(F4 ) W(G2 ) W(G2 ) W(A1 ) W(A1 )

|E (G, (L, λ))| 112 25 6 6 2 2

ωρ 1 −1 ζ3 ζ32 ζ4 −ζ4

+13 cuspidals with ωρ = 1, 1, −1, ζ3, ζ32, ζ4, −ζ4, ζ5, ζ52, ζ53, ζ54, −ζ3, −ζ32 .

Theorem 4.5.2 Let (G, F) be such that GF is a Suzuki group 2B2 (q2 ), a Ree group 2 G (q 2 ) or a Ree group 2F (q 2 ). Let F ⊆ Irr(W) be an F-stable family. Then the 2 4 number of unipotent characters in E (GF | F ) is 1, 2, 6 or 13. (a) If Uch(GF | F ) contains 2 characters, the associated Fourier matrix is given by   1 1 1 . M(F ) = √ 2 1 −1 (b) In type 2 G2 (q2 ) there is a six-element family of unipotent characters. The associated Fourier matrix is given by √ √ 3 . 2 1 3 1 √ √    1 − 3 . 2 1 − 3   √ √  1  1 3 2 . −1 − 3  √ √ . M(F ) = √  2 . −1 3  2 3 1 − 3  2 . . −2 2 .   . −2 . −2 .   2 Here, the rows are labelled by ρ2, . . . , ρ7 from Table 4.15, and the columns are

320

Unipotent Characters Table 4.15 Unipotent characters of 2 G2 (q2 ). . . HC-label φ1,0 2 G I [ζ 5 ] 2 12 2 G I [ζ 7 ] 2 12 2 G I I [ζ 5 ] 2 12 2 G I I [ζ 7 ] 2 12 2 G [ζ ] 2 4 2 G [−ζ ] 2 4

φ1,6

ρ(1) 1 1 qΦ Φ Φ  √ 1 2 12

2 3 1 qΦ Φ Φ  √ 1 2 12 2 3 1 qΦ Φ Φ  √ 1 2 12 2 3 1 qΦ Φ Φ  √ 1 2 12 2 3 √1 qΦ1 Φ2 Φ4 3 √1 qΦ1 Φ2 Φ4 3 q6

ωρ 1 5 ζ12

. . . and of 2B2 (q2 ) HC-label φ1,0 2B [ζ 3 ] 2 8

7 ζ12

2B [ζ 5 ] 2 8

5 ζ12

φ1,4

ρ(1) 1 √1 qΦ1 Φ2 2 √1 qΦ1 Φ2 2 q4

ωρ 1 ζ83 ζ85 1

7 ζ12

ζ4 −ζ4 1

Table 4.16 Unipotent characters of 2F4 (q2 ) HC-label φ1,0 2B [ζ 3 ], 1 2 8 2B [ζ 5 ], 1 2 8  φ1,4

φ2,1 φ2,3 φ2,2 2F I [−1] 4 2F I I [−1] 4

2F I I I [−1] 4

ρ(1) 1 √1 qΦ1 Φ2 Φ2 Φ12 4

2 √1 qΦ1 Φ2 Φ2 Φ12 4 2 q2 Φ12 Φ24 1 q4 Φ2 Φ Φ  2 Φ  4 4 12 8 24 1 q4 Φ2 Φ Φ  2 Φ  4 4 12 8 24 1 q4 Φ2 Φ 2 8 24 1 q4 Φ2 Φ2 Φ2 Φ 6 1 2 4 24 1 4 2 2 2  12 q Φ1 Φ2 Φ12 Φ8 Φ24 1 q4 Φ2 Φ2 Φ Φ  2 Φ  12 1 2 12 8 24

ωρ 1 ζ83

HC-label 2F I [ζ ] 4 4 2F I [−ζ ] 4 4

ζ85

2F I I [ζ ] 4 4 2F I I [−ζ ] 4 4 2F [−ζ ] 4 3 2F [−ζ 2 ] 4 3 2F IV [−1] 4  φ1,4 2B [ζ 3 ],  2 8 2B [ζ 5 ],  2 8

1 1 1 1 −1 −1 −1

φ1,8

ρ(1)

1 q4 Φ2 Φ2 Φ2 Φ Φ  4 1 2 4 12 24 1 q4 Φ2 Φ2 Φ2 Φ Φ  4 1 2 4 12 24 1 q4 Φ2 Φ2 Φ2 Φ Φ  4 1 2 4 12 24 1 q4 Φ2 Φ2 Φ2 Φ Φ  4 1 2 4 12 24 1 q4 Φ2 Φ2 Φ2 Φ2 3 1 2 4 8 1 q4 Φ2 Φ2 Φ2 Φ2 3 1 2 4 8 1 q4 Φ2 Φ2 Φ Φ 3 1 2 12 24 q10 Φ12 Φ24 √1 q13 Φ1 Φ2 Φ2 Φ12 4 2 √1 q13 Φ1 Φ2 Φ2 Φ12 4 2 q24

ωρ ζ4 −ζ4 ζ4 −ζ4 −ζ3 −ζ32 −1 1 ζ83 ζ85 1

labelled by suitable almost characters; with the first two lying in the principal series. (c) In type 2F4 (q2 ) there is a 13-element family of unipotent characters. The associated Fourier matrix M(F ) is given in Table 4.17. This was proved in [GeMa03, Thm. 5.4] by explicitly decomposing the unipotent almost characters into unipotent characters, using the known character tables. By inspection of the Fourier matrices one obtains the following partial analogue of Theorem 4.4.23 (the statement in [DiMi90, Prop. 6.3] is not quite complete): Theorem 4.5.3

Let G be simple such that GF is of exceptional type. Then the

4.5 Unipotent Characters in Exceptional Types

321

Table 4.17 The Fourier transform matrix in type 2F4 3 3  6  2  1   1 1  3 12  3  3  3  4  4  4

3 . 3 . 6 . −6 −4 −3 4 −3 4 −3 . −3 . −3 . −3 . . 4 . 4 . −8

−6 −6 . −4 −2 −2 . . . . 4 4 4

√ 3√2 −3 2 . . √ −3√2 3√2 3√2 3√2 −3√2 −3 2 . . .

√ 3√2 −3 2 . . √ 3√2 −3√2 −3√2 3√2 3√2 −3 2 . . .

√ 3√2 −3 2 . . √ 3√2 −3√2 3√2 −3√2 −3√2 3 2 . . .

√ 3√2 −3 2 . . √ −3√2 3√2 −3√2 −3√2 3√2 3 2 . . .

3 3 −6 −6 −3 −3 3 3 3 3 . . .

−3 . . . −3 . . .  6 . . .  −2 −4 −4 .  −1 4 4 .   −1 4 4 .  3 . . 6  3 . . −6  3 . . 6  3 . . −6  −4 4 −8 .  −4 −8 4 .  −4 4 4 .

The rows are labelled by the unipotent characters ρ5, . . . , ρ17 from Table 4.16. ˜ 1, The columns are labelled by the almost characters in the principal series 12 ˜ 1 , and eight further almost characters not in the principal series. 4˜ 1, 6˜ 1, 6˜ 2, 16

unipotent characters ρ of GF are uniquely determined by their uniform projections, that is, by their multiplicities in the uniform almost characters except in the following cases: (1) ρ is labelled by a character (x, σ) of the Drinfeld double corresponding to the family of ρ (see 4.2.9) such that σ is non-rational linear of odd order; (2) ρ belongs to an exceptional family in types E7 or E8 ; or (3) GF is very twisted. For a family corresponding to the symmetric group S3 there is one such pair of non-rational characters of the Drinfeld double; for a family corresponding to the symmetric group S4 there are two pairs; for a family corresponding to the symmetric group S5 there are five pairs and one quadruple. The four unipotent characters in an exceptional family fall into two pairs, in each of which both characters have the same restriction to the space of uniform functions. Corollary 4.5.4 (Digne–Michel [DiMi90, Prop. 6.4]) Let G be simple with a Steinberg map F such that GF is of exceptional type. Then the unipotent characters of GF are uniquely determined by their uniform projections, together with their eigenvalue of Frobenius and, for characters in the principal series belonging to an exceptional family, by the associated character of the Iwahori–Hecke algebra.

322

Unipotent Characters

Proof We only need to discuss the exceptions (1)–(3) in Theorem 4.5.3. For these the statement can be checked from the explicit knowledge of the Frobenius eigenvalues; for example, for the Suzuki and Ree groups the lists given in Tables 4.15 and 4.16 show that their unipotent characters are distinguished by their degrees (hence by their uniform projection) together with the associated Frobenius eigenvalues.  This result plays a crucial role in Digne and Michel’s construction of a unique Jordan decomposition, see Theorem 4.7.1. Next we consider the rationality properties of unipotent characters. For χ an irreducible character of a finite group we denote by Q( χ) its character field, that is, the field generated by the values of χ. The following result first stated by Geck [Ge03b, Thm. 1.4] gives a nice connection to Frobenius eigenvalues: Proposition 4.5.5 Let G be simple with a Steinberg map F and ρ ∈ Uch(GF ). Then Q(ρ) = Q(ωρ ), unless ρ lies in the principal series in an exceptional family √ for E7 or E8 . In the latter case we have Q(ρ) = Q( q). Proof Let ρ ∈ Uch(GF ) be a unipotent character with Frobenius eigenvalue ωρ . By definition there exists an F-stable maximal torus T  G such that the Deligne– Lusztig character RTG (1T ) has ρ as a constituent. Then there is an -adic cohomology group M of the associated Deligne–Lusztig variety whose ρ-isotypic component Mρ is non-zero, and F δ acts on it by ωρ times a half-integral power of q. In fact this is an integral power of q unless ρ lies in an exceptional family of a group of type E7 or E8 (see 4.2.21). The actions of GF and F δ on M commute, and any Galois automorphism σ of Q /Q sends the ωρ -eigenspace of F δ to the σ(ωρ ) eigenspace, and the ρ-isotypic component to the σ(ρ)-isotypic component. Thus σ(ρ)  ρ if σ(ωρ )  ωρ . Since for any m > 2 there is some prime   p such that Q does not contain a primitive mth root of unity, this argument shows that Q(ωρ )  Q(ρ) unless ρ lies in an exceptional family. For the converse, note that the Deligne–Lusztig characters RTG (1T ) are rational valued (see 2.2.1). So the same holds for all unipotent characters that are determined by their uniform projections. This was already used in Corollary 4.4.24 to see that all unipotent characters for G of classical type and F a Frobenius map are rational. By the same argument this holds for those unipotent characters of groups of exceptional type which are not in one of the exceptions (1)–(3) of Theorem 4.5.3. In case (1) of that result, the number of unipotent characters with non-rational Frobenius eigenvalue is exactly equal to the length of the Galois orbit of that root of unity, so our claim follows. This also applies to case (3) when F is not a Frobenius map. Finally, the characters lying in an exceptional family but not in the principal series also satisfy this property. The last assertion requires some further considerations and is shown in [Ge03b, Prop. 5.6]. 

4.5 Unipotent Characters in Exceptional Types

323

With this we obtain the following complement to Corollary 4.4.24: Corollary 4.5.6 Let G be simple and F : G → G a Steinberg map. Then a unipotent character ρ of G is rational valued if and only if the following hold: (1) its associated Frobenius eigenvalue is ±1; and (2) if ρ lies in the principal series in an exceptional family in type E7 or E8 then the associated character of the Iwahori–Hecke algebra is rational. Proof

This is an immediate consequence of Proposition 4.5.5.



For G an arbitrary connected reductive group, Remark 4.2.1 gives an immediate reduction of this question to the case when G is simple. Remark 4.5.7 The much more difficult problem of determining the Schur index has also been solved completely for unipotent characters, see [Ohm96, Lu02, Ge03b, Ge03c, Ge05a]. The restrictions in the last cited paper can now be removed using [Ta16]. It turns out that the Schur index is always at most 2, and there do exist unipotent characters for which this value is attained. Rationality properties of general irreducible characters are studied in the papers [Ge03b] and [TiZa04], using rather different methods; see also Theorem 4.7.9. We now describe some general consequences that can be derived from the explicit knowledge of the properties of unipotent characters. The results of Lusztig described above show in particular that unipotent characters are generic, that is, their classification, their degrees and many of their further properties only depend on the underlying complete root datum. This can be formalised as follows: Theorem 4.5.8 function

Let G be a complete root datum. There exists a set Uch(G) and a D : Uch(G) −→ Q[q],

γ → Dγ,

such that for any admissible choice of prime power q (and hence of connected reductive group G and Steinberg map F corresponding to G) there is a bijection ψqG : Uch(G) → Uch(GF ) with Dγ = DψqG (γ) , so ψqG (γ)(1) = Dγ (q). Here, Uch(G) is called the set of generic unipotent characters of G. As we saw above, the partition of unipotent characters into Harish-Chandra series and into families is also described in combinatorial terms, only depending on the complete root datum, hence these are generic as well; that is, they already exist at the level of Uch(G). This observation has led Broué, Malle and Michel to introduce similar combinatorial sets of unipotent characters attached to certain types of complex reflection groups (baptised ‘spetsial’ in [Ma00]), see e.g. [Ma95, BMM99, BMM14], which should play the role of unipotent characters of yet to be discovered new objects called ‘spetses’ [Ma98]. Independently, Lusztig [Lu93] had defined unipotent

324

Unipotent Characters

degrees for those finite irreducible Coxeter groups that are not Weyl groups, of types H3, H4 and I2 (m) with m = 5 or m  7. We take this opportunity to recall some properties of the degree polynomials introduced in 2.3.25: Proposition 4.5.9 The degree polynomial of ρ ∈ Uch(GF ) is of the form Dρ =

1 Aρ (q ± · · · ± qaρ ) nρ

for a real number nρ > 0, and nρ Dρ is a product of cyclotomic polynomials. Here, aρ = 0 if and only if ρ = 1G , Aρ = l(w0 ) is maximal if and only if ρ = StG , and aρ < Aρ unless ρ is a tensor product of trivial and Steinberg characters of the various simple factors of GF . Furthermore, if F is a Frobenius map then nρ is an integer dividing the order |GU | of the group GU attached to the family U ⊆ Uch(GF ) of ρ. Proof The form of Dρ follows from 4.1.14 in conjunction with the fact that a- and A-values are constant on families of unipotent characters by Proposition 4.2.7. By Proposition 4.1.20 only the (trailing term of the) fake degree of the special character in U contributes to the trailing term of Dρ . But the coefficient of the trailing term of the fake degree is 1 for special characters and so the coefficient of qaρ equals the entry in the Fourier matrix at the special character. The real number nρ was introduced in Remark 2.3.26. For F a Frobenius map its properties then follow from the explicit description of Fourier matrices by inspection. The second assertion is a consequence of Remark 4.1.21 together with the fact that the trivial and the sign character always lie in singleton families.  It is relatively straightforward from this to see that for G simple the Steinberg character is the only unipotent character of p-power degree larger than 1. In fact, all irreducible characters of prime power degree of quasi-simple groups of Lie type have been determined, see [MZ01]. It can be shown that also Lusztig induction and restriction behave generically on unipotent characters: Theorem 4.5.10 Let G be a complete root datum. For each Levi sub-datum L of G there exist linear maps RLG : ZUch(L) −→ ZUch(G), ∗ G RL

: ZUch(G) −→ ZUch(L),

such that for any admissible choice of connected reductive groups L  G and Steinberg map F (and hence prime power q) corresponding to L and G we have RLG ◦ ψqL = ψqG ◦ RLG

and

∗ G RL

◦ ψqG = ψqL ◦ ∗RG L.

4.5 Unipotent Characters in Exceptional Types

325

This follows from results of Shoji [Sho87], see [BMM93, Thm. 1.33]. For classical groups it can also be derived from the explicit decomposition formulas of Asai, see Section 4.6. A further consequence of the explicit descriptions concerns the action of automorphisms on unipotent characters. It turns out to be generic as well. Theorem 4.5.11 Let G be simple with Steinberg endomorphism F. Let ρ be a unipotent character of GF . Then ρ is Aut(GF )-invariant, except in the following cases: (a) In GF = Dn (q) with even n  4, the graph automorphism of order 2 interchanges the two unipotent characters in all pairs labelled by the same degenerate symbol of defect 0 and rank n. (b) In GF = D4 (q) the graph automorphism of order 3 has two non-trivial orbits consisting of the unipotent characters labelled by the symbols             2 2 14 12 12 124 , , , , , . 2 2 01 12 12 014 (c) In GF = Sp4 (22 f +1 ), f  0, the graph automorphism of order 2 interchanges the two unipotent principal series characters labelled by the symbols     12 01 , . 0 2 (d) In GF = G2 (32 f +1 ), f  0, the graph automorphism of order 2 interchanges the  , φ  } two unipotent principal series characters labelled by the characters {φ1,3 1,3 of the Weyl group W(G2 ). (e) In GF = F4 (22 f +1 ), f  0, the graph automorphism of order 2 has eight orbits of length 2, consisting of the unipotent characters labelled by         {φ8,3 , φ8,3 }, {φ8,9 , φ8,9 }, {φ2,4 , φ2,4 }, {φ2,16 , φ2,16 },       {φ9,6 , φ9,6 }, {φ1,12 , φ1,12 }, {φ4,7 , φ4,7 }, {(B2,  ), (B2,  )}.

Here the notation for the unipotent characters of groups of type F4 is taken from [Ca85, 13.8]. This result follows rather easily from Lusztig’s description of the unipotent characters. For example for G of classical type, the unipotent characters are uniquely determined by their multiplicities in the Deligne–Lusztig characters RTG (1) (see Theorem 4.4.23), and the action of automorphisms on the latter is readily described. For exceptional types, this statement is not always true, but then ad hoc arguments can be used, for example based on Corollary 4.5.4. See e.g. [Ma07, Prop. 3.7 and 3.9]. Based on the previous theorem the following extension property was shown in [Ma08, Thm. 2.4]:

326

Unipotent Characters

Theorem 4.5.12 (Malle) Let G be simple with Steinberg endomorphism F. Then any unipotent character of GF extends to its inertia group in Aut(GF ). For the proof note that by our standard reductions we may choose G to be of adjoint type. Now the outer automorphism groups of simple groups of Lie type are known (see e.g. [MaTe11, Thm. 24.24]); in particular one finds that the group Out(GF ) is cyclic (in which case the claim follows by elementary character theory) unless possibly when F is untwisted and the Dynkin diagram of G has a non-trivial graph automorphism, that is, G is of type An , Dn or E6 . The latter groups can be dealt with via a case-by-case analysis. In the general case the action of automorphisms on the irreducible characters of a finite reductive group is not yet known. Problematic here are the characters of groups with disconnected centre that fuse under a diagonal automorphism. Partial results for cuspidal characters can be found for example in [Ma17b].

4.6 Decomposition of RLG and d-Harish-Chandra Series While the decomposition of Deligne–Lusztig characters was determined completely by Lusztig in [Lu84a] — see his Decomposition Theorem 4.2.16 in conjunction with Jordan decomposition in Theorem 2.6.4 — the problem of decomposing Lusztig induction (and restriction) is not yet solved in full generality. In this section we present some partial results, which include in particular the case of unipotent characters. From this, we will deduce the decomposition for characters in arbitrary Lusztig series, at least for classical groups with connected centre, in the next section. 4.6.1 Let us start off by making some easy reductions. Let G be connected reductive and F : G → G be a Steinberg map. First, by transitivity of Lusztig induction (see Theorem 3.3.6) it is clearly sufficient to know the decomposition of RLG for the maximal proper F-stable Levi subgroups L < G. We claim that in this case L is necessarily d-split for some d  1. Indeed, since L = CG (Z◦ (L)) (see [DiMi20, Prop. 3.4.6]) and L < G is proper, there must be some d such that the Sylow d-torus Sd  Z◦ (L) of Z◦ (L) (see 3.5.6) is not contained in Z(G). Then by maximality of L we have L = CG (Sd ), whence L is d-split. Thus, for the question of decomposing RLG we may assume that L < G is a maximal d-split Levi subgroup. Here we first consider the case of unipotent characters. Then we may further reduce to the situation when G is simple: Write [G, G] = G1 · · · Gr with each Gi an F-simple group. Then L = Z◦ (G)(L ∩ G1 ) · · · (L ∩ Gr ) where Li := L ∩ Gi is an F-stable Levi subgroup of Gi . Since by assumption L is maximal in G there is a unique i such that Li  Gi , say i = 1. By Remark 4.2.1 any λ ∈ Uch(LF ) is of the form λ = λ1  · · ·  λr with λ1 ∈ Uch(L1F ) and λi ∈ Uch(GiF ) for i > 1. Then we

4.6 Decomposition of RLG and d-Harish-Chandra Series

327

have RLG (λ) = RLG11 (λ1 )  λ2  · · ·  λr and thus it is sufficient to understand RLG11 (λ1 ).

Note that here ||RLG (λ)|| = ||RLG11 (λ1 )|| and

G1 G2 Gr G πun (RLG (λ)) = πun (RLG11 (λ1 ))  πun (λ2 )  · · ·  πun (λr ). m Now if G1 = j=1 G1j is a product of m simple groups G1j permuted transitively  by F, then we may write L1 = m j=1 L1j with L1j := L1 ∩ G1j maximal in G1j , and by Proposition 3.3.26 Lusztig induction commutes with restriction of scalars from L1 to L11 , and from G1 to G11 , as does uniform projection. Thus, we may assume that G is simple.

As already remarked in Section 3.5 the d-Harish-Chandra theories for unipotent characters are rather well understood. In fact, the decomposition of Lusztig induction of unipotent characters from certain maximal d-split Levi subgroups has been determined explicitly. For groups of classical type this is a result of Asai (for which we present a new proof, see Theorem 4.6.9), for groups of exceptional type the decomposition was computed by Broué–Malle–Michel [BMM93], up to some small indeterminacies. 4.6.2 We first discuss the decomposition of Lusztig induction for unipotent characters in groups of type A. Here the d-split Levi subgroups were described in Example 3.5.14. Now by Proposition 3.5.12 the maximal d-split Levi subgroups of G = GLn above a fixed Sylow d-torus Sd are in bijection with the maximal parabolic subgroups of the relative Weyl group of Sd . Thus for the split Frobenius map they are of rational type GLm (q d ).GLn−dm (q) for some 1  m  n/d. We give the explicit decomposition for the case when m = 1, the others can then in principle be derived from that one inductively. Let us recall a Murnaghan–Nakayama type rule for restriction of irreducible characters in certain wreath products. The complex reflection group denoted G(b, 1, m) is isomorphic to the wreath product of the cyclic group of order b with the symmetric group Sm . The irreducible characters of G(b, 1, m) are parametrised by b-multi-partitions α of m (see e.g. [Os54, §2] or [Ma95, 2A]). Observe that for any 1  d  m, G(b, 1, m) has a parabolic subgroup of type Sd × G(b, 1, m − d). By a hook of a multi-partition α = (α1, . . . , αb ) we mean a hook h of one of the αi . The leg length f (h) of h is defined to be the difference between the row indices of the Young diagram of αi where the hook ends and where it starts. The following generalisation of the classical Murnaghan–Nakayama rule (see e.g. [JaKe81, 2.4.7]) relating characters of symmetric groups to those of suitable Young subgroups was shown by Osima [Os54, Thm. 7]. Proposition 4.6.3

Let b, d  1, φα ∈ Irr(G(b, 1, m)) be parametrised by the

328

Unipotent Characters

b-multipartition α

m. Then for any d-cycle x ∈ Sd and any y ∈ G(b, 1, m − d)

(−1) f (h) φα\h (y) φα (xy) = h

where the sum runs over all d-hooks h of α and α \ h denotes the multi-partition obtained by removing h from α. From the case b = 1 we obtain the following result of Fong and Srinivasan [FoSr86, Thm. (2A)]: Proposition 4.6.4 Let (G, F) be simple of rational type An−1 (q) and let 1  d  n. Let L = CG (Td ) where Td is an F-stable torus of G with TFd  GL1 (q d ), so |TFd | = q d − 1. Let ρα ∈ Uch(GF ) be the unipotent character labelled by the partition α n. Then

∗ G RL (ρα ) = (−1) f (h) ρα\h h d-hook

where the sum runs over all d-hooks h of α. Proof The derived subgroup [L, L] is simple of rational type An−d−1 (q). Let ρβ be the unipotent character of LF parametrised by β n − d. By Example 2.4.20 the unipotent characters of GF are almost characters, so 1 α 1 β G φ (w)Rw and ρβ = φ (v)RvL, ρα = |W| w ∈W |WL | v ∈W L

where W = Sn is the Weyl group of G, WL = Sn−d is the Weyl group of L and where RvL denotes the Deligne–Lusztig character of LF for a maximal torus of the form TFd TvF with Tv  [L, L] a maximal torus of type v. Thus    G G L  1 α |WL | ρα, RLG (ρβ ) = φ (w)φβ (v) Rw , RL (Rv ) . (∗) |W| v ∈W w ∈W L

G where uv ∈ S is the product of v ∈ S By transitivity, RLG (RvL ) = Ruv n n−d with a disjoint d-cycle u ∈ Sd . By Example 2.3.22 this has non-zero scalar product with G if and only if w and uv are conjugate in W, so have the same cycle type. Thus Rw (∗) becomes

 G G 1 |W : CW (uv)|φα (uv)φβ (v) Rw , Ruv = φα (uv)φβ (v). |W| v ∈W v ∈W L

L

By Proposition 4.6.3 applied with b = 1, this yields

   ∗ G RL (ρα ), ρβ = ρα, RLG (ρβ ) = (−1) f (h) φα\h, φβ h d-hook

and the only non-zero term in the sum occurs for α \ h = β.



4.6 Decomposition of RLG and d-Harish-Chandra Series

329

The following description of d-cuspidal unipotent characters and of d-HarishChandra series is now immediate: Corollary 4.6.5 Let (G, F) be simple of rational type An−1 (q) and let d  1. (a) A unipotent character of GF is d-cuspidal if and only if it is labelled by a d-core. (b) Let L  G be d-split with [L, L] of rational type An−wd−1 (q) for some w  0, and let λ be a d-cuspidal unipotent character of LF labelled by the d-core β n − wd. Then the d-Harish-Chandra series E (GF , (L, λ)) consists of the unipotent characters of GF labelled by partitions of n with d-core β. For the unitary groups the same formula as in Proposition 4.6.4 holds but with signs adjusted suitably: using that all unipotent characters are uniform and that the complete root datum for 2An−1 (q) is obtained by replacing F by −F in the one for An−1 (q), we obtain the formula from that for An−1 (q) by formally replacing q by −q. Here, for a unipotent character ρ, Aρ denotes the A-value introduced in 4.2.6. Proposition 4.6.6 Let (G, F) be of rational type 2An−1 (q) and let d  1. Let L = CG (Td ) where Td is an F-stable torus of G with |TFd | = q d − (−1)d . Let ρα ∈ Uch(GF ) be labelled by the partition α n. Then

f (h)+A ρ α\h (−1) ρα\h (−1) Aρ α ∗RG L (ρα ) = h d-hook

where the sum runs over the d-hooks h of α. Note that here L is d -split in G, where ⎧ 2d if d is odd, ⎪ ⎪ ⎨ ⎪  d = d/2 if d ≡ 2 (mod 4), ⎪ ⎪ ⎪d if d ≡ 0 (mod 4) ⎩ (compare also Example 3.5.14(c)). Thus to obtain the analogue of Corollary 4.6.5 for the unitary groups one just needs to replace d-hooks by d -hooks: Corollary 4.6.7 Let (G, F) be of rational type 2An−1 (q) and let d  1. (a) A unipotent character of GF is d-cuspidal if and only if it is labelled by a d -core. (b) Let L  G be d-split with [L, L] of rational type 2An−wd −1 (q) for some w  0, and let λ be a d-cuspidal unipotent character of LF labelled by the d -core β n − wd . Then the d-Harish-Chandra series E (GF , (L, λ)) consists of the unipotent characters of GF labelled by partitions of n with d -core β.

330

Unipotent Characters

4.6.8 We now turn to the symplectic and orthogonal groups. Let G be simple of classical type B, C or D and F : G → G a Frobenius endomorphism such that GF is not of rational type 3D4 . Let d be a positive integer. We worked out the structure of d-split Levi subgroups in Example 3.5.15. Again, the maximal ones above a fixed Sylow d-torus Sd are in bijection with maximal parabolic subgroups in the relative Weyl group of Sd . First, if d is odd then it is easily seen that any maximal d-split Levi subgroup has the rational form GLm (q d ).HF

for some m  1,

where HF is a classical group of the same type as GF with rnk(H) + dm = rnk(G). If d = 2e is even, the maximal d-split Levi subgroups have rational form GUm (q e ).HF

for some m  1,

where again H is a classical group of the same type as G with rnk(H) + em = rnk(G). Here, however, the rational types of HF and GF differ if G is of type Dn : HF is twisted if and only if GF is untwisted. Now observe with Example 3.5.20 that the d-cuspidal unipotent characters of GLm (q d ) are in bijection with the 1cuspidal unipotent characters of GLm (q), of which there are none except for m = 1 by Example 2.4.20, so the only maximal d-split Levi subgroups of G possibly possessing a d-cuspidal unipotent character are of the rational form GL1 (q d ).HF ,

with rnk(H) = rnk(G) − d

GU1 (q e ).HF ,

with rnk(H) = rnk(G) − e

if d is odd, and

if d = 2e is even, respectively. We will describe Lusztig induction from these d-split Levi subgroups. We present the fundamental result of Asai on the decomposition of Lusztig induction as formulated in [FoSr86, (3.1), (3.2)]. This is a summary of results from [As84a, 2.8], [As84b, 1.5] and [As85, 2.2.3], see also [LM16, Thm. 3.2]. As pointed out there, a sign is missing in the formula [FoSr86, (3.2)] in the case of Dn , as can be seen for example by evaluating it at the trivial character. Theorem 4.6.9 (Asai) Let G be simple of type Bn, Cn or Dn with a Frobenius map F such that GF is not of type 3D4 . For S a symbol we write ρS for the corresponding unipotent character if S is not degenerate, and for the sum of the two corresponding characters if S is degenerate. (a) Let d be odd and L = CG (Td ) where Td  G is an F-stable torus with

4.6 Decomposition of RLG and d-Harish-Chandra Series

331

|TFd | = q d − 1. Then for ρS ∈ Uch(LF ) we have

RLG (ρS ) = h ρS , (S,h)

where the sum runs over all symbols S  with a d-hook h such that S = S  \ h. (b) Let d  1 and L = CG (Td ) where Td  G is an F-stable torus with |TFd | = q d + 1. Then for ρS ∈ Uch(LF ) we have

 c ρS  , RLG (ρS ) = (−1)δ (S,c)

where the sum runs over all symbols S  with a d-cohook c such that S = S  \ c, and δ = 0 for types Bn, Cn , δ = 1 for type Dn . Note that in the theorem [L, L]F is a classical group of rank n − d of the same rational type as GF , except that removing cohooks changes the defect of symbols modulo 4, so in type Dn interchanges the twisted and untwisted types. We give a purely combinatorial proof of Asai’s formula, only using the Mackey formula for unipotent characters for part (b). First, let us observe that part (a) follows just from ordinary Harish-Chandra theory. For this let G be as in Theorem 4.6.9 and d  1 odd. Consider the d-split Levi subgroup L = CG (T) = TG of G where T is an F-stable torus of G with TF  GL1 (q d ) and G is of the same type as G but of rank n − d. Let L1 = GLd G be the intermediate 1-split Levi subgroup of G with T  GLd . By transitivity of ∗ L1 ∗ G F Lusztig restriction, ∗RG L (ρ) = RL RL1 (ρ) for any class function ρ on G . Proof of Theorem 4.6.9(a) (See [As84a, Lemma 2.8.4].) Let ρ ∈ Uch(GF ) be labelled by the symbol S. Let (L0, λ0 ) be a Harish-Chandra source of ρ, that is, L0 is a 1-split Levi subgroup of G and λ0 ∈ Uch(L0F ) is cuspidal with ρ, RLG0 (λ0 )  0. By Theorem 4.4.28 then [L0, L0 ] is simple of the same classical type as G. If rnk([L0, L0 ]) > n − d then fewer than d 1-hooks can be removed successively from S, and hence S cannot have a d-hook. Moreover, L0 cannot be contained in L, so ∗RG (ρ) = 0, in accordance with the claim. L Else, we may assume after conjugation that L0 is contained in the maximal 1-split Levi subgroup L1 . Let S0 denote the symbol labelling λ0 and first assume that S0 is non-degenerate. Let W0 := WG (L0, λ0 )  W(Bn−k ) be the relative Weyl group of (L0, λ0 ) in G, and W1 := WL1 (L0, λ0 )  Sd × W(Bn−k−d ) the one in L1 (see Example 3.5.29). The irreducible characters of W0 are parametrised by bipartitions α n − k. By Proposition 4.4.29 addition of S0 defines the natural labelling Irr(W0 ) → E (GF , (L0, λ0 )),

φα → ρα,

of the Harish-Chandra series above λ0 , with ρα denoting the unipotent character

332

Unipotent Characters

labelled by the symbol S0 + α. Any ψ ∈ Irr(W1 ) is of the form ψ1  ψ2 with ψ1 ∈ Irr(Sd ) and ψ2 ∈ Irr(G(2, 1, n − k − d)), and correspondingly we obtain a parametrisation Irr(W1 ) → E (L1F , (L0, λ0 )), ψ1  ψ2 → ρψ1  ρα , if ψ2 is labelled by α n − k − d. Observe that addition of S0 commutes with adding or removing d-hooks. Let ρ correspond to the character φ ∈ Irr(W0 ). By the Howlett–Lehrer Comparison Theorem 3.2.7 the decomposition of ∗RG L1 (ρ) is given by

∗ G RL1 (ρ) = aψ ρψ where φ|W1 = aψ ψ. ψ

ψ ∈Irr(W1 )

∗ GL d d Now ∗RLL1 (ρψ1  ρψ2 ) = ∗RGL T (ρψ1 )  ρψ2 , and RT (ρψ1 ) is non-zero if and only if ψ1 does not vanish on the d-cycles, that is, if and only if it is parametrised by a hook partition. Thus we may set aψ = 0 unless ψ1 (x)  0 for x ∈ Sd a d-cycle. But then Proposition 4.6.3 applied to W0  G(2, 1, n − k) shows that aψ = (−1) f (h) if ψ2 is obtained from φ by removing the d-hook h, and aψ = 0 else. Since removing d-hooks commutes with the Harish-Chandra parametrisation, our claim follows. If S0 is degenerate, then L0 is of type Dk and the relative Weyl groups W0  W(Dn−k ), W1  Sd × W(Dn−k−d ) are normal subgroups of index 2 in the relative Weyl groups in the non-degenerate case. As described in Example 4.1.4 their irreducible characters are now labelled by unordered bipartitions, and an easy Clifford  theory argument shows that again the constituents of ∗RG L (ρ) are as claimed.

For the second part of Theorem 4.6.9 we establish two combinatorial statements. The first requires the Mackey formula. For simplicity we will assume that we are in the situation of part (b) of the theorem, but the arguments similarly apply to part (a). Let d  1. We write ad (S) for the number of different d-cohooks that can be added to a symbol S, and rd (S) for the number of d-cohooks that can be removed from S. We can reinterpret these numbers in terms of abacus diagrams. As in [Ma95, Bem. 3.4] the two rows of S = (X0, X1 ) can naturally be encoded in a 2d-runner abacus diagram A as follows: the ith runner of A has a bead at position j if Xi+j (mod 2) has an entry $i/2% + dj. Then adding a d-cohook to S corresponds to moving one bead in A one position down. Thus the number ad (S) of addable d-cohooks of S equals the number of beads in A such that the next position downwards is empty. Clearly, the lowest bead on any runner can always be moved down, which already gives 2d possibilities. All other addable d-cohooks correspond to an empty position on some runner such that some later position is occupied. Thus, they are in natural bijection with the removable d-cohooks of S. This shows that ad (S) = 2d + rd (S).

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333

The above considerations hold as long as the symbol S is not degenerate. If it is, then we only obtain half that many distinct symbols by adding a hook (or cohook), and also only half that many by removing one. (Note that neither operation can again result in a degenerate symbol.) Proposition 4.6.10 Let L  G be a 2d-split Levi subgroup as in Theorem 4.6.9(b) and ρ ∈ Uch(LF ) parametrised by the symbol S. Then ||RLG (ρ)|| 2 = ad (S) is as claimed by Asai’s formula. Proof First assume G is of type Bn or Cn . We apply the Mackey formula for dsplit Levi subgroups in Theorem 3.5.17. Let M be a minimal d-split Levi subgroup contained in L. Then we have  G     L  w G RL (ρ), RLG (ρ) = ρ, ∗RG RL∩wL (∗RL∩LwL (wρ)) L (RL (ρ)) = ρ, w

where the sum is over WL (M)–WL (M) double coset representatives w in WG (M). Here, WG (M) is the complex reflection group G(2d, 1, m), where dm = rnk(G) − rnk(M), and WL (M) is its maximal parabolic subgroup G(2d, 1, m − 1) (see Example 3.5.15). Since L is proper in G we have m  1. If m = 1 then G(2d, 1, 1) is cyclic of order 2d and there are 2d double coset  representatives  for the trivial group in G(2d, 1, 1), each with wL = L, leading to RLG (ρ), RLG (ρ) = 2d. Now assume that m  2. It is easy to see that there are exactly 2d + 1 double coset representatives w0, . . . , w2d , with WL (M) ∩ wiWL (M) = WL (M) for i = 1, . . . , 2d and WL (M) ∩ w0WL (M) = G(2d, 1, m − 2), so L0 := L ∩ w0L is a maximal 2d-split Levi subgroup of L. Thus we find  G    RL (ρ), RLG (ρ) = 2dρ, ρ + ρ, RLL0 (∗RLL0 (ρ)) . By induction, || ∗RLL0 (ρ)|| 2 = rd (S) and so ||RLG (ρ)|| 2 = 2d + rd (S), which is our claim by the considerations prior to this proposition. The above arguments remain valid for G of type Dn , except when ρ lies in the 2d-Harish-Chandra series above a 2d-cuspidal unipotent character parametrised by a degenerate symbol. (For this observe that if S is degenerate, then so is its d-cocore.) In this case WG (M) is the complex reflection group G(2d, 2, m), and WL (M) is its maximal parabolic subgroup G(2d, 2, m − 1). If m = 1, then G(2d, 2, 1) is cyclic of order d and there   are d double coset representatives for the trivial group, leading to RLG (ρ), RLG (ρ) = d. If m  2 then it is easy to see that again there are exactly 2d + 1 double coset representatives w0, . . . , w2d , all but w0 normalising L, while WL (M) ∩ w0WL (M) = G(2d, 2, m − 2). Arguing as before we again find  ||RLG (ρ)|| 2 = 2d + rd (S). We consider the 2d-split Levi subgroup L = TG  = CG (T) of G where T is an F-stable torus with TF  GU1 (q d ) and G is of the same type as G but of rank

334

Unipotent Characters

n − d. Let L1 = GLd G be the intermediate F-stable Levi subgroup of G with T  GLd and GLFd = GUd (q). We first determine Lusztig restriction on uniform almost characters; for this let W denote the Weyl group of G. Lemma 4.6.11 In the above setting, let φ ∈ Irr(W)F be parametrised by α and Rφ be the associated unipotent almost character of GF . Then

∗ G RL (Rφ ) = (−1) f (h) Rφ α\h , h

where the sum runs over all d-hooks h of α. Proof The Levi subgroup L1 has Weyl group W1  Sd × G(2, 1, n − d) and L has Weyl group WL  G(2, 1, n − d). Then ∗RG L1 (Rφ ) = Rφ1 , with φ1 = φ|W1 , and by transitivity of Lusztig induction ∗ G RL (Rφ )

∗ L1 = ∗RLL1 ∗RG L1 (Rφ ) = RL (Rφ1 ) = Rφ ,

with φ  the restriction of φ1 to the product C × W1 , with C the class of d-cycles in Sd . Applying Proposition 4.6.3 we thus find

(−1) f (h) φα\h, φ = h

where the sum runs over all d-hooks h of α, from which the claim follows.



Proposition 4.6.12 The uniform projection of Theorem 4.6.9(b) holds. Proof Let ρ ∈ Uch(GF ) be parametrised by the symbol S. We write T ∼ S if T is the symbol of some φ ∈ Irr(W) lying in the family of ρ. By Proposition 3.3.10 uniform projection commutes with Lusztig restriction, so

∗ G πun (∗RG S, T ∗RG L (ρ)) = RL (πun (ρ)) = L (RT ), T ∼S

where RT is the uniform almost character labelled by T and S, T is the Fourier coefficient. By Lemma 4.6.11 this equals

S, T (−1) f (h) RT \h (1) T ∼S

h

with the inner sum ranging over all d-hooks h of T. This should equal (−1)δ times



πun c ρS\c = c πun (ρS\c ) = c S \ c, T RT , (2) c

c

c

T ∼S\c

where the outer sum runs over d-cohooks c of S. In order to prove this, we will compare coefficients according to the d-cohooks of S at an entry x of S.

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335

Let us first assume that x − d is not an entry of S. Then any symbol T ∼ S also has a d-hook h at x, and we need to show that (−1) f (h) S, T = (−1)δ c S \ c, T \ h for all T ∼ S. This is a combinatorial exercise using 4.4.22. Now assume that x − d is an entry of S in the opposite row to x. Then there is no d-cohook in S at x and (2) gives no contribution. We get a contribution to (1) for all T ∼ S for which x, x − d lie in different rows. But these come in pairs, according to whether x is in the first row or the second row, and the corresponding terms in the sum, having opposite sign, cancel. Finally, assume that x − d lies in the same row of S as x. Then the contribution to (2) is the sum over all T ∼ S \ c. Note that here the size of the family of S \ c is smaller than that of S, and the coefficients in the Fourier matrix have twice the absolute value. We get a contribution to (1) for all T ∼ S that have x, x − d in distinct rows. Again, these come in pairs, and this time the two symbols give contributions with the same sign, thus matching with the doubled Fourier coefficients in (2). The proof is complete.  Based on Propositions 4.6.10 and 4.6.12 we now prove the second part of Asai’s formula. This is inspired by Enguehard [En13, Lemma 5.3.9], but note that the proof given there contains a serious gap, which is here amended by Proposition 4.4.26. Proof of Theorem 4.6.9(b) It suffices to compare the projection of RLG (ρ) and of Asai’s formula to any family U ⊂ Uch(GF ). By Proposition 4.6.12 the uniform projection of RLG (ρ) and of Asai’s formula agree. In Asai’s formula, the symbols parametrising the constituents of RLG (ρ) are obtained from the symbol of ρ by adding a d-cohook, that is, by increasing one of the entries by d and moving it to the other row. It is obvious that at most two of the resulting symbols can share the same multi-set of entries, and in the latter case, this multi-set has more distinct elements than the one for ρ, so its associated family in Uch(GF ) is bigger. On the other hand, since all unipotent characters have the same positive multiplicity in the special uniform almost character of a family, we can read off from the uniform projection whether the number of constituents of RLG (ρ) is one character or the sum or the difference of two of them, and the result will be the same as for Asai’s formula. Now by Proposition 4.6.10 the norm of RLG (ρ) agrees with the one in the formula of Asai, so we conclude that the projection to any family of either have the same number of constituents, either zero, one or two. Thus, the projection f of RLG (ρ) onto U must in fact be a linear combination of at most two unipotent characters. We are thus left to show that f is already determined G ( f ) and its norm being  2. If f has norm one then by its uniform projection πun by Theorem 4.4.23 there is exactly one unipotent character having this uniform

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Unipotent Characters

projection, and we are done. If f has norm 2 and U has more than four elements, our claim follows by Proposition 4.4.26. Finally assume that |U | = 4. Then by what we said above, ρ lies in a 1-element family of Uch(LF ) and hence is uniform, whence so is RLG (ρ). In particular it is orthogonal to the space of non-uniform functions and we conclude again.  Let us call (d-)Asai Levi subgroup any F-stable Levi subgroup of a simple group of classical type with rational form as occurring in the statement of Theorem 4.6.9. As a consequence of the above proof we see (this can be shown by the same methods to also hold for case (a) of Theorem 4.6.9): Corollary 4.6.13 Let L be an Asai Levi subgroup of a simple group G of classical type with a Frobenius map F and ρ ∈ Uch(LF ). Assume that (1) the uniform projection of RLG (ρ), and (2) the norm of RLG (ρ) are as in Asai’s formula. Then RLG (ρ) is as given by Theorem 4.6.9. Remark 4.6.14 The result of Corollary 4.6.13 can be reformulated to say: if L < G is an Asai Levi subgroup and ρ ∈ Uch(LF ), then RLG (ρ) is uniquely determined as the virtual character of minimal norm with the same uniform projection as RLG (ρ). This conclusion continues to hold for any Levi subgroup of G of type A, as there all unipotent characters are uniform, but also for GF = 3D4 (q). Indeed, in this case there are five maximal Levi subgroups up to conjugation, of rational types A1 (q3 )(q ± 1), A2 (q)(q2 + q + 1), 2A2 (q)(q2 − q + 1), and the Coxeter torus. All of them are of type A, so all of their unipotent characters are uniform and thus their Lusztig induction can be computed explicitly as linear combinations of Deligne–Lusztig characters. It turns out that in all cases RLG (ρ) is multiplicity-free, has at most two constituents in the 4-element family of GF , and the 1-dimensional space of functions in that family with zero uniform projection is orthogonal to all of them. Thus indeed RLG (ρ) has minimal norm among all virtual characters with the same uniform projection. Theorem 4.6.9 provides a recursive algorithm to compute the decomposition of for any F-stable Levi subgroup L of a group G of classical type:

RLG

Proposition 4.6.15 Let G be simple of classical type with a Frobenius map F and L  G be an F-stable Levi subgroup. Then for any ρ ∈ Uch(LF ) the decomposition of RLG (ρ) can be computed from Theorem 4.6.9. Proof For the case when GF = 3D4 (q) see Remark 4.6.14. Else, by the reduction laid out in 4.6.1 it is sufficient to consider maximal d-split Levi subgroups L of G for the various relevant d. By 4.6.8 these have rational type LF = GLm (±q d ).HF

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337

for some 1  m  n/d, where n denotes the rank of G, H is simple of the same classical type as G and where we write GLm (−q d ) := GUm (q d ). Let ρ ∈ Uch(LF ). Then ρ = ρ1  ρ2 with ρ1 ∈ Uch(GLm (±q d )) and ρ2 ∈ Uch(HF ). Now any unipotent character of GLm (±q d ) is uniform, so can be written as an (explicitly known) linear combination of Deligne–Lusztig characters for the various maximal tori of GLm (±q d ). By transitivity of Lusztig induction, to determine RLG (ρ) it is hence sufficient to decompose Lusztig induction from F-stable Levi subgroups of G of the form T.H, with TF a maximal torus of GLm (±q d ). By transitivity this can be obtained as a sequence of Lusztig inductions from dAsai Levi subgroups for various d. The decomposition of the latter is known by Theorem 4.6.9.  As remarked in 4.6.8, d-Asai Levi subgroups are the only maximal d-split Levi subgroups of simple groups of classical type possibly possessing d-cuspidal unipotent characters, so we obtain from Theorem 4.6.9: Corollary 4.6.16 Let G be simple of classical type with a Frobenius map F such that (G, F) is not of type 3D4 , and let d  1. (a) A unipotent character of GF parametrised by a symbol S is d-cuspidal if and only if S is a d-core if d is odd, or a d/2-cocore if d is even, respectively. (b) Let L  G be a d-split Levi subgroup, and let λ ∈ Uch(LF ) be a d-cuspidal unipotent character labelled by the d-core (respectively d/2-cocore) S. Then the d-Harish-Chandra series E (GF , (L, λ)) consists of the unipotent characters of GF labelled by symbols with d-core (respectively d/2-cocore) S. In particular, in contrast to the situation when d = 1 (considered in Theorem 4.4.28) a classical group will in general have many d-cuspidal unipotent characters. Note the similarity of this result with Corollary 4.4.18: thus, d-cuspidal unipotent characters are exactly those unipotent characters which are of -defect zero for any prime   3 such that q has order d modulo . This fact is a crucial ingredient in the classification of the unipotent -blocks of the finite classical groups, see Fong–Srinivasan [FoSr86, FoSr89], Broué–Malle–Michel [BMM93], and Cabanes–Enguehard [CE94]. 4.6.17 Considerations as in the proof of Theorem 4.6.9 have also been used in [BMM93] to determine the decomposition of RLG for d-split Levi subgroups L of simple groups G of exceptional type, in conjunction with the Mackey formula Theorem 3.3.7 for unipotent characters. We will not give this decomposition explicitly, it can be found in [BMM93, Tab. 2] (up to some small indeterminacies which we will comment on in Remark 4.6.19). A conceptual description will be given in Theorem 4.6.21. The proof is based upon the following analogue of Corollary 4.6.13:

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Unipotent Characters

Proposition 4.6.18 Let G be simple of exceptional type with a Steinberg morphism F, let L be a maximal F-stable Levi subgroup of G and λ ∈ Uch(LF ). Then RLG (λ) is uniquely determined by its uniform projection and its norm, up to ambiguities in G of type E7 or E8 arising from pairs of unipotent characters of L having the same uniform projection. The latter only occur when L < G has a component of type E6 or E7 and λ involves a unipotent character with non-rational Frobenius eigenvalue, or a principal series character in the exceptional family of E7 . Sketch of proof We start off by making some reductions. As pointed out in 4.6.1 a maximal F-stable Levi subgroup is necessarily d-split for some d  1. If λ is uniform, then so is RLG (λ), so certainly it is determined by its uniform projection and norm. In particular, this is the case if all simple factors of L are of type A. This already deals with the groups GF of types 2B2 , 2 G2 , G2 and 3D4 . Assume that G is of type F4 and F is a Frobenius map. By Table 3.3 the only maximal d-split Levi subgroups with not all factors of type A are of type B3 and C3 (with d = 1, 2) and of type B2 (with d = 4). Using that the families define an orthogonal decomposition of the space spanned by unipotent characters, one concludes that for any of these the uniform projection of RLG (λ), with λ ∈ Uch(LF ) not uniform, to a 4-element family in Uch(GF ) agrees with that of a single unipotent character and thus is determined by its norm being 1. The uniform projection restricted to the 21-element family U ⊆ Uch(GF ) associated to GU  S4 agrees with a linear combination of at most 5 unipotent characters. From the explicit Fourier matrix it can then be checked that this, together with the norm, determines RLG (λ): The pair of cuspidal unipotent characters of F4 (q) that have the primitive fourth roots of unity as associated Frobenius eigenvalues (so are algebraically conjugate by Proposition 4.5.5) possess the same uniform projection. From the norm and the uniform projection of RLG (λ) it follows that RLG (λ) must contain both characters with the same multiplicity. In type E6 the only relevant d-split Levi subgroups are of type dD4 with d = 1, 2, 3 and dD5 with d = 1, 2. Again, it turns out that the projection of RLG (λ) to 4-element families with λ not uniform agrees with that of a single unipotent character, while on the 8-element family U ⊆ Uch(GF ) associated to GU  S3 there are at most 4 constituents. Again this determines RLG (λ) uniquely, up to the individual multiplicities of the two cuspidal unipotent characters of E6 (q) which have the primitive third roots of unity as associated Frobenius eigenvalues. The norm condition shows that they must have the same multiplicity in all RLG (λ). The arguments for the other types are analogous. The only cases that can not be settled completely are those configurations mentioned in the statement.  Remark 4.6.19

(a) There is another way to resolve certain ambiguities about

4.6 Decomposition of RLG and d-Harish-Chandra Series

339

multiplicities of characters with the same uniform projection. If these characters have non-rational Frobenius eigenvalues, hence are Galois conjugate by Proposition 4.5.5, one can argue as follows: assume that λ ∈ Uch(LF ) is rational valued; then so is RLG (λ) by Corollary 3.3.14. Hence the multiplicities in RLG (λ) of all unipotent characters of GF from a fixed Galois orbit must agree. This applies for example in groups G of type F4 or E6 , showing that the multiplicities in all RLG (λ) of the cuspidal unipotent characters with non-rational Frobenius eigenvalues are the same in each pair. (b) This approach fails, though, when we consider Lusztig induction of the j two cuspidal unipotent characters λ j = 2E6 [ζ3 ], j = 1, 2, of 2E6 (q), where ζ3 = √ exp(2π −1/3). These are non-rational, with character field Q(λ j ) = Q(ζ3 ) (see Proposition 4.5.5). Since λ1, λ2 are not conjugate under any group automorphism of 2E6 (q) by Theorem 4.5.11, RLG (λ1 ) and RLG (λ1 )σ = RLG (λ1σ ) = RLG (λ2 ) differ for any group G containing a Levi subgroup L of rational type 2E6 , where σ is the non-trivial Galois automorphism of Q(ζ3 )/Q, that is, they are also non-rational. For GF = E7 (q) and L of rational type 2E6 (q).Φ2 the uniform projections show that RLG (λ1 ) + RLG (λ2 ) contains both pairs of non-rational unipotent characters of E7 (q), but this does not allow us to deduce their subdivision among RLG (λ1 ) and RLG (λ2 ). (c) In the latter case, the decomposition of RLG (λi ) can still be determined by a block-theoretic argument, namely we have j

j

j

RLG (2E6 [ζ3 ]) = E6 [ζ3 ], 1 − E6 [ζ3 ], 

for j = 1, 2

(see [KeMa19]). Similarly one can argue that for L of rational type E7 (q).Φ2 in G = E8 we have RLG (φ512,11 ) = φ4096,11 − φ4096,26

and

RLG (φ512,12 ) = φ4096,12 − φ4096,27

for the two principal series unipotent characters φ512,11, φ512,12 lying in the exceptional family of E7 (q). Among d-cuspidal unipotent pairs (L, λ) of simple groups this leaves only one j open case, viz. the decomposition of RLG (2E6 [ζ3 ]), j = 1, 2, for L of rational type 2E (q).Φ2 in G = E , see case “40+41" in [BMM93, Tab. 2]. If we move away from 6 8 2 d-cuspidal pairs, there are further open cases in E8 , such as the decomposition of j Lusztig induction to E8 (q) of λ j = E6 [ζ3 ], j = 1, 2, from a 3-split Levi subgroup of j rational type E6 (q).Φ3 , and of λ j = 2E6 [ζ3 ], j = 1, 2, from a 6-split Levi subgroup of rational type 2E6 (q).Φ6 . The explicit results on the decomposition of RLG for d-split Levi subgroups of simple groups allow one to verify the following important structural observation on unipotent d-Harish-Chandra series and the relations d and *d (introduced in 3.5.24), notwithstanding the yet unknown decompositions:

340

Unipotent Characters

Theorem 4.6.20 (Broué–Malle–Michel) Steinberg map F and d  1.

Let G be connected reductive with a

(a) The unipotent d-Harish-Chandra series partition the set of unipotent characters of GF , that is, Uch(GF ) = E (GF , (L, λ)), (L,λ)

where the union runs over a system of representatives of d-cuspidal unipotent pairs (L, λ) in G modulo GF -conjugation. (b) The relation d is transitive for unipotent characters, so d and *d agree. See [BMM93, Thms. 3.2 and 3.11]. That is, d-Harish-Chandra series of unipotent characters enjoy the same properties as usual Harish-Chandra series do by Corollary 3.1.17. This turns attention to the structure of the individual d-Harish-Chandra series. Here, we have the following natural generalisation of the Howlett–Lehrer Comparison Theorem 3.2.7. But as is the case for Theorem 4.6.20, at present there is no conceptual proof known for this fact; rather it is obtained by comparing the explicit decomposition of RLG with the induction in the corresponding relative Weyl groups WG (L, λ) = NG F (L, λ)/LF (see 3.5.8) in each individual case, see [BMM93, Thm. 3.2]: Theorem 4.6.21 (Comparison Theorem) Let G be connected reductive with Steinberg map F and d  1. For any d-cuspidal unipotent pair (L, λ) in G there exists a collection of isometries M IL,λ : ZIrr(WM (L, λ)) → ZE (MF , (L, λ)),

where M runs over d-split Levi subgroups L  M  G, such that the diagram G IL, λ

ZIrr(WG (L, λ)) . ⏐ Ind ⏐

−−−−→

ZIrr(WM (L, λ))

−−−−→ ZE (MF , (L, λ))

M IL, λ

ZE (GF , (L, λ)) . ⏐ RG ⏐ M

commutes for all M, where Ind denotes ordinary induction. In fact, the latter result can be thought of as a specialisation of a generic statement. The explicit results on the decomposition of RLG show that the classification of dcuspidal unipotent pairs (L, λ), their relative Weyl groups WG (L, λ) and the partition of Uch(GF ) into its d-Harish-Chandra series E (GF , (L, λ)) are all generic, as are the M . So for any complete root datum G and any d-cuspidal unipotent pair (L, λ) maps IL,λ

4.6 Decomposition of RLG and d-Harish-Chandra Series

341

M : ZIrr(W (L, λ)) → ZE (M, (L, λ)), of G there exists a collection of isometries IL,λ M where L  M  G runs over d-split Levi sub-data, such that the diagram G IL, λ

ZIrr(WG (L, λ)) . ⏐ Ind ⏐

−−−−→

ZIrr(WM (L, λ))

−−−−→

M IL, λ

ZE (G, (L, λ)) . ⏐ RG ⏐ M ZE (M, (L, λ))

commutes and specialises to the diagram in Theorem 4.6.21 for any choice of connected reductive group G and Steinberg map F corresponding to G. Example 4.6.22 Let Ld denote the centraliser of a Sylow d-torus of G. Since this is a minimal d-split Levi subgroup, all of its characters are d-cuspidal by definition. The d-Harish-Chandra series E (GF , (Ld, 1L d )) above the trivial character of LFd is called the d-principal series of GF . According to Theorem 4.6.21 it is in bijection with Irr(WG (Ld )). A formal consequence of Theorem 4.6.21 in conjunction with the Mackey formula is as follows (see [BMM93, Prop. 3.15]): Corollary 4.6.23 Let G be connected reductive, (L, λ) be a d-cuspidal unipotent pair in G and ρ ∈ E (GF , (L, λ)). Then

  w ∗ G RL (ρ) = ρ, RLG (λ) λ. w ∈WG (L)/WG (L,λ)

In particular ∗ G RL (ρ)(1)

  = ρ, RLG (λ) |WG (L) : WG (L, λ)| λ(1)  0.

The analogy of d-Harish-Chandra theory for unipotent characters with ordinary Harish-Chandra theory goes even further, as we have the complete analogue of Theorem 3.2.18 describing the degrees of the constituents of RLG in terms of suitable Schur elements. We first present the generic formulation in the case of Frobenius maps: Theorem 4.6.24 Let G be a complete root datum and (L, λ) a d-cuspidal pair of G. Then for any φ ∈ Irr(WG (L, λ)) there exists a Laurent polynomial cφ ∈ Q[q±1 ] G (φ) we have with zeros only at roots of unity or zero, such that for ρ = IL,λ Dρ = ±Dλ

|G|q −1 c |L|q φ

√ and moreover cφ (ζd ) = φ(1)/|WG (L, λ)|, where ζd := exp(2π −1/d).

342

Unipotent Characters

Thus, applying the maps ψqG from Theorem 4.5.8 we obtain: if G is a connected reductive group with Frobenius map F corresponding to G, and (L, λ) is a d-cuspidal unipotent pair in G, then for any φ ∈ Irr(WG (L, λ)) the degree of the associated G unipotent character ρ = IL,λ (φ) is given by ρ(1) = ±RLG (λ)(1) cφ (q)−1 = ±λ(1)

|GF | p cφ (q)−1 . |LF | p

In the case d = 1 (which is the content of Theorem 3.2.18) this was a consequence of the Howlett–Lehrer–Lusztig theory of Hecke algebras of induced cuspidal representations, and the Laurent polynomial cφ is then the Schur element of φ of the associated Iwahori–Hecke algebra. In the general case, Broué and Malle [BrMa93] introduced a so-called cyclotomic Hecke algebra attached to the complex reflection group WG (L, λ). The Laurent polynomials cφ should then be suitable specialisations of the Schur elements of this cyclotomic Hecke algebra with respect to a certain canonical trace form (specified in [BMM99, Thm.-Ass. 2]). The latter statement is conjectured to be true in general (see [BrMa93, (d-HV6)]). It has been proved for all but finitely many types, but a general proof is not known at present. Nevertheless, the existence of the polynomials cφ with the stated properties has been verified in [Ma95, Folg. 3.16 and 6.11] for the groups of classical type, and in [Ma97, Prop. 5.2], [Ma00, Prop. 7.1] for those of exceptional type. We also have the analogue of Corollary 3.2.21: Corollary 4.6.25 Let (L, λ) be a d-cuspidal unipotent pair in G and let ρ ∈ E (GF , (L, λ)). Then the degree polynomial of ρ satisfies Dρ = Φda d ([L,L]) f , where f ∈ Q[q] is not divisible by Φd , and ad ([L, L]) denotes the precise power of Φd dividing the order polynomial of [L, L]. Proof This is entirely analogous to the proof of Corollary 3.2.21. According to Theorem 4.6.24 the Laurent polynomials cφ have no zero (or pole) at ζd . Since L is a d-split Levi subgroup of G, it contains a Sylow d-torus of G by Proposition 3.5.5 and so its order polynomial is divisible by the same power of Φd as the one of G. The claim then follows by Corollary 3.5.27 in conjunction with Theorem 4.6.24.  The assertions of Theorem 4.6.24 and Corollary 4.6.25 continue to hold for very twisted Steinberg maps F (see [BrMa93, Folg. 5.11]) except that then the √ Schur elements cφ lie in Q( p)[q±1 ] and the Φd have to be replaced by cyclotomic √ polynomials over Q( p) as in 3.5.3.

4.7 On Lusztig’s Jordan Decomposition

343

4.7 On Lusztig’s Jordan Decomposition The explicit results on unipotent characters can be used to make Lusztig’s Jordan decomposition in Theorem 2.6.22 unique at least in certain cases. For example, in classical groups with connected centre, Theorem 4.4.23 shows that any irreducible character is uniquely determined by its uniform projection, so in this case there can exist at most one Jordan decomposition. More generally we have the following uniqueness result from [DiMi90, Thm. 7.1]: Theorem 4.7.1 (Digne–Michel) There exists a unique collection of bijections JsG : E (GF , s) −→ Uch(CG∗ (s)F ), where G runs over connected reductive groups with connected centre and Frobenius map F, and s ∈ G∗F is semisimple, satisfying the following, where we write H := CG∗ (s): (1) For any F-stable maximal torus T∗  H,  G    RT∗ (s), ρ = εG εH RTH∗ (1T∗ ), JsG (ρ)

for all ρ ∈ E (GF , s).

(2) If s = 1 and ρ ∈ E (GF , 1) is unipotent then (a) the Frobenius eigenvalues ωρ and ωJ G (ρ) are equal, and 1

(b) if ρ lies in the principal series then ρ and J1G (ρ) correspond to the same character of the Iwahori–Hecke algebra. G (ρ ⊗ zˆ) = J G (ρ) for ρ ∈ E (GF , s), where zˆ is the linear (3) If z ∈ Z(G∗F ) then Jsz s character of GF corresponding to z (see Proposition 2.5.20). (4) For any F-stable Levi subgroup L∗ of G∗ such that H  L∗ , with dual L  G, the following diagram commutes: JsG

E (GF , s) ↑ RLG

−→

E (LF , s)

−→

JsL

Uch(HF ) ↑ id Uch(HF )

(5) If G is of type E8 and H is of type E7 A1 (resp. E6 A2 ) and L  G is a Levi subgroup of type E7 (resp. E6 ) with dual L∗  H then the following diagram commutes: JsG

ZE (GF , s) ↑ RLG

−→

ZE (LF , s)c

−→

JsL

ZUch(HF ) ↑ RLH∗ ZUch(L∗F )c

where the index c denotes the subspace spanned by the cuspidal part of the corresponding Lusztig series.

344

Unipotent Characters

(6) For any F-stable central torus T1  Z(G) with corresponding natural epimorphism ϕ : G → G1 := G/T1 and for s1 ∈ G∗1 with s = ϕ∗ (s1 ) the following diagram commutes: E (GF , s) ↑

JsG

−→

Uch(HF ) ↓

G

Js11

E (G1F , s1 ) −→

Uch(H1F )

with H1 = CG∗1 (s) and where the vertical maps are just the inflation map along GF → G1F and the restriction along the embedding H1F → HF respectively.   G = JsGi i . (7) If G is a direct product i Gi of F-stable subgroups Gi then J si Sketch of proof Let us explain the main points in the proof by Digne and Michel. Their statement and proof are given in terms of the alternative description of Lusztig induction discussed in 3.3.5. In particular, they do not (need to) assume the Mackey formula since in that setting there’s always a well-defined standard parabolic subgroup containing a given standard Levi subgroup. In our case, the independence of Lusztig induction and restriction from the parabolic subgroup is guaranteed by Theorem 3.3.8. First, observe that CL∗ (s) is connected for all Levi subgroups L∗ of G∗ , since G has connected centre, see e.g. [Bo06, §8.B]. Now one needs to show that conditions (1)– (7) are compatible with each other and do specify Js uniquely. One starts with the case when Gder := [G, G] is F-simple, that is, Gder is a central product of simple algebraic groups permuted transitively by F. First, if s = 1 then conditions (1) and (2) do determine a unique bijection J1 by Theorem 4.4.23 when Gder has classical type, and by Corollary 4.5.4 when Gder is exceptional. If s  1 is central, then Js is determined by (3); note that (2) is not relevant here and that (1) continues to hold as RTG (sˆ ⊗ zˆ) = RTG (sˆ) ⊗ zˆ (see Proposition 2.5.21). Next suppose that H is contained in a proper F-stable Levi subgroup L∗ < G∗ with dual L < G. In this case RLG is an isometry from E (LF , s) to E (GF , s) by Theorem 3.3.22 which respects (1). Thus, as we may assume by induction on the rank of G that JsL exists and is unique, (4) specifies JsG uniquely. Again (3) holds as RLG (sˆ ⊗ zˆ) = RLG (sˆ) ⊗ zˆ by Proposition 3.3.16. In the case when Gder is F-simple it only remains to consider non-central elements s that are quasi-isolated in G∗ and in fact isolated, as we assume that Z(G) and hence CG∗ (s) is connected. In this case Digne–Michel show by explicit caseby-case computations [DiMi90, Lemma 7.2] that there is a unique bijection JsG satisfying (1)–(5). Note that (5) is only relevant for G of type E8 . Observe that the bijections Js for G obtained above do satisfy (6). This is clear

4.7 On Lusztig’s Jordan Decomposition

345

when Js has been determined by (1); it holds for J1 found with (2) since in this case ϕ induces an isomorphism between the relevant Deligne–Lusztig varieties, as well as between the Iwahori–Hecke algebras; in case we constructed Js through (3) it is again clear as zˆ factors through G1F ; and finally when we use (4) or (5) it follows as ϕ commutes with RLG by Proposition 3.3.24. Now consider the case of a general G. Then there are connected reductive groups Gi with Frobenius maps again denoted F such that (Gi )der is F-simple, and an  F-equivariant epimorphism i Gi → G with kernel a central torus. In this case, Js is uniquely determined by (6) and (7). Here one needs to show that the map obtained  in this way does not depend on the choice of central isotypy i Gi → G, which uses the fact that (6) holds for the Gi . Finally, one proves that this Js also satisfies the requirements (1)–(5), which is easy.  It is not known in general whether Lusztig induction and restriction commute with Jordan decomposition of characters, or more precisely, whether a Jordan decomposition can always be chosen such that it has this property. For classical groups with connected centre, this commutation was first proved by Fong and Srinivasan [FoSr89, App. A] based on results of Shoji. Later, Enguehard [En13, Prop. 5.3] sketched a proof which only uses the Mackey formula and Asai’s decomposition formula for Lusztig induction of unipotent characters. We present a slightly stronger version (whose proof is essentially the same): Theorem 4.7.2 Let G be connected reductive with a Frobenius endomorphism F : G → G and assume that the Mackey formula holds for GF . Let s ∈ G∗F be a semisimple element such that CG∗ (s) is connected and only has components of classical type A, B, C and D. Then for all F-stable Levi subgroups L∗  M∗  G∗ satisfying s ∈ L∗ , with duals L  M  G, the diagram ZE (MF , s) . ⏐ RLM ⏐ ZE (LF , s)

JsM

−−−→ JsL

−−−→

ZUch(CM∗ (s)F ) . C ∗ (s) ⏐ RCLM∗ (s) ⏐ ZUch(CL∗ (s)F )

commutes, where Js• denotes Jordan decompositions as in Theorem 2.6.22. Proof Since CG∗ (s) is connected by assumption, the centraliser of s in any Levi subgroup of G∗ is also connected (see [Bo06, §8.B]). Thus we have corresponding Jordan decompositions Js• as in Theorem 2.6.22. We let Ψ• denote its inverse. If CM∗ (s) = CL∗ (s) then we have Z◦ (L∗ )  Z◦ (CM∗ (s)) since L∗ = CG∗ (Z◦ (L∗ )) and so CL∗ (s) = CCG∗ (s) (Z◦ (L∗ )). Then RLM restricts to a bijection E (LF , s) → E (MF , s) (up to a global sign) which commutes with Jordan decomposition by Theorem 3.3.22, that is, we have RLM ◦ Ψ L = ΨM .

346

Unipotent Characters

So we may assume L < M is proper. We next claim that we may assume L is maximal in M. Indeed, let K be an F-stable Levi subgroup of G with L < K < M, C ∗ (s) M ◦ΨK and ΨK ◦RCK∗ (s) = RK ◦ΨL , with dual L∗ < K∗ < M∗ . Then ΨM ◦RC M∗ (s) = RK L CL∗ (s) K which implies the claim. By definition Jordan decomposition commutes with uniform projection, so we have M

C ∗ (s) C ∗ (s) C ∗ (s)

M πun ◦ ΨM ◦ RC M∗ (s) = ΨM ◦ πunM ◦ RC M∗ (s) = πun ◦ RLM ◦ ΨL . L

CM∗ (s)

That is, ΨM (RC

L∗ (s)

L

(λ)) has the same uniform projection as RLM (ΨL (λ)) for any C

∗ (s)

λ ∈ Uch(CL∗ (s)F ), and by Proposition 4.7.3, ψ := ΨM (RC M∗ (s) (λ)) has the same C

L

∗ (s)

norm as RC M∗ (s) (λ). Now by Corollary 4.6.13 and Remark 4.6.14, ψ is uniquely L determined as the element of minimal norm in ZE (MF , s) with the same uniform projection as ψ, so our claim follows.  The following was used in the previous proof: Proposition 4.7.3 Let G be connected reductive with a Steinberg map F and assume that the Mackey formula holds for GF . Let L  G be a d-split Levi subgroup for some d  1 and s ∈ L∗F be semisimple with CG∗ (s) connected. Assume that Lusztig induction commutes with Jordan decomposition for any proper d-split Levi subgroup of L. Then for any Jordan decomposition Js : E (LF , s) → Uch(CL∗ (s)F ) as in Theorem 2.6.22 we have C

∗ (s)

||RLG ( χ)|| = ||RC G∗ (s) (Js ( χ))|| L

Proof

for all χ ∈ E (LF , s).

Let χ ∈ E (LF , s). By the Mackey formula 3.3.7   G (R ( χ)), χ = n(g) ||RLG ( χ)|| 2 = ∗RG L L

(∗)

g∈S

where g runs over a system S of LF –LF double coset representatives in GF such g contains a maximal torus of G, and we have put n(g) := that  L L∗g gL:= g L ∩ L  RLg ( RLg ( χ)), χ . Let T0 be a maximally split torus of a minimal d-split Levi subgroup of L (and hence of G). By Lemma 3.5.18 we may assume that the elements of S are chosen in NG (T0 )F . Set W := NG (T0 )/T0 , the Weyl group of G. Let g ∈ S. If   g   g 0  n(g) = RLLg (∗RLLg (gχ)), χ = ∗RLLg (gχ), ∗RLLg ( χ) then there exists some common constituent ψ ∈ Irr(LgF ), that is, ∗ L    g   RLg ( χ), ψ  0 and ∗RLLg (gχ), ψ = ∗RLLg ( χ), ψ g  0. g

4.7 On Lusztig’s Jordan Decomposition

347

  L   Choose (T, θ) ∈ X(Lg, F) with RT g (θ), ψ  0, then also R (θ g ), ψ g  0, and ∗F hence (T, θ) and (Tg, θ g ) are geometrically conjugate in X(L, F). Let g ∗ T∗F 0 ∈ W ∗ F be the image of gT0 under the isomorphism W → W (see Remark 1.5.19). Then ∗ g∗ by Proposition 2.5.5 the corresponding pairs (T∗, s1 ) and ((T∗ )g , s1 ) in Y(L∗, F) are geometrically conjugate. That is, by Theorem 2.6.2 there are y ∗, z ∗ ∈ NL∗ (T∗0 )F such that y ∗ g ∗ z ∗ ∈ CG∗ (s)F . Replacing the representative g ∈ S by ygz we have thus shown: only those elements of S contribute to (∗) for which g ∗ ∈ CG∗ (s)F . Now note that CL∗ (s) = CCG∗ (s) (Z(L∗ )◦d ) is d-split in CG∗ (s), so arguing as before we see that we can choose a system Ss of CL∗ (s)F –CL∗ (s)F double coset representatives in CG∗ (s)F lying in the normaliser of a maximally split torus T∗s of a minimal d-split Levi subgroup of CL∗ (s). The preceding construction then shows that we may replace our reference torus T0 for S above such that T∗0 = T∗s . The above construction thus defines a surjective map g → g ∗ from {g ∈ S | n(g)  0} to Ss . It is now easy to check that it is also injective. Thus we obtain our claim once we have shown that n(g) = n(g ∗ ) for any pair (g, g ∗ ) as above. Let (g, g ∗ ) be such a pair. As g induces an F-equivariant isomorphism L → gL, g ∗ ∗ induces an F-equivariant isomorphism L∗ → g L∗ fixing s so that we may assume ∗ Js (gχ) = g λ, where λ = Js ( χ). Now if g ∈ NG (L)F then Lg = L and L∗g = L∗ , and ∗ n(g)  0 if and only if gχ = χ, if and only if g λ = λ, and in that case g Lg Tg

∗

n(g) = gχ, χ = 1 = g λ, λ = n(g ∗ ). Otherwise Lg = L ∩ gL < L is proper and d-split, so the inductive hypothesis applies to give Cg ∗L∗ (s) g∗ ( λ) L∗g (s)

g

Js g (∗RLLg (gχ)) = ∗RC L

whence n(g) =

and

L

Lg

∗ Cg∗L∗ (s) g∗ ∗ CL∗ (s)  RC ∗ (s) ( λ), RC ∗ (s) (λ) = n(g ∗ ). Lg

C ∗ (s)

Js g (∗RLLg ( χ)) = ∗RCL∗ (s) (λ),

Lg



Remark 4.7.4 Theorem 4.7.2 shows, using Proposition 4.7.3, that the equality of norms stated in the latter actually holds for all Levi subgroups, not just for d-split ones. Notice that in the proof of Theorem 4.7.2 the assumption on the components of CG∗ (s) being of classical type is only used in the very last paragraph when invoking Corollary 4.6.13. All of the preceding steps work in complete generality. In view of this, the above proof can be adapted to the case of exceptional groups; since here there may exist several Jordan decompositions, we need to fix one. Theorem 4.7.5 Let G be simple with connected centre, F : G → G a Steinberg map, and assume that the Mackey formula holds for GF . Let s ∈ G∗F be a semisimple

348

Unipotent Characters

element. Then for all F-stable Levi subgroups L∗  M∗  G∗ satisfying s ∈ L∗ , with duals L  M  G, the diagram ZE (MF , s) . ⏐ RLM ⏐ ZE (LF , s)

JsM

−−−→ ZUch(CM∗ (s)F ) . C ∗ (s) ⏐ RCLM∗ (s) ⏐ JsL −−−→ ZUch(CL∗ (s)F )

commutes, where Js• denotes the Jordan decomposition specified in Theorem 4.7.1, except possibly when G = M is of type E8 , CG∗ (s) is of type E6 A2 or E7 A1 and CL∗ (s) has a factor of type E6 or E7 . Proof First, if F is not a Frobenius map, and hence G is of type B2 , G2 or F4 in characteristic 2, 3 or 2 respectively, then the claim follows with Theorem 3.3.22 unless 1  s ∈ G∗F is isolated. But the only such semisimple elements are involutions in type G2 , with centraliser A21 , and elements of order 3 in F4 with centraliser of type A22 . In either case, CG∗ (s) only has factors of type A, and thus all of its unipotent characters are uniform. The commutation follows from Theorem 4.7.1(1). So now assume that F is a Frobenius endomorphism. Since G has connected centre, CG∗ (s) is connected and thus Theorem 4.7.2 applies. Hence we may assume that CM∗ (s) has some factor of exceptional type and thus in particular G itself is of exceptional type. If s = 1 then we have CM∗ (s) = M∗ and the commutation follows by (2) of Theorem 4.7.1 and the fact that Lusztig induction of unipotent characters in MF and M∗F are given by the same formulas (by Proposition 4.6.18 and Remark 4.6.19) unless we are in the excluded case that M = G = E8 and L has rational type 2E6 (q).Φ22 . But in the latter case, since G∗ = G, whatever the precise decomposition of RLG is, it will be the same on both sides. If 1  s ∈ Z(M∗ ) (and thus also s ∈ Z(L∗ )) we have CM∗ (s) = M∗ and the commutation follows by property (3) in Theorem 4.7.1 together with the previous case. Thus s is non-central in M∗ and since CM∗ (s) has an exceptional factor, M∗ can not be of type G2, F4 or E6 . If CM∗ (s) lies in a proper Levi subgroup of M∗ we are again done by property (4) in Theorem 4.7.1. Thus we may assume that s is isolated but not central in M∗ . Such elements are classified in [Bo05]. If M is of type E7 then there are no such centralisers with an exceptional factor. The only cases in M of type E8 (which then also forces G = E8 ) are those for elements of order 2 with centraliser structure E7 A1 , respectively of order 3 with centraliser structure E6 A2 . Set C := CM∗ (s), C1 := CL∗ (s). Proposition 4.6.18 shows that the decomposition of RCC1 (λ) for any λ ∈ Uch(C1F ) is uniquely determined by its norm and its uniform projection, except for indeterminacies coming from pairs of unipotent characters with the same uniform projection, and by Proposition 4.7.3 the same holds for RLG (ΨsL (λ)). If C1 is of classical type, then the indeterminacies can be resolved

4.7 On Lusztig’s Jordan Decomposition

349

again by using the arguments given in the proof of Proposition 4.6.18. Thus we may further assume that CL∗ (s) also has a factor of exceptional type and we reach the excluded configurations.  Something more can be said even in the excluded cases of Theorem 4.7.5. In fact, the proof of Proposition 4.6.18 shows that the decomposition of RLG (λ) is the same on the two sides of our diagram unless λ belongs to a pair of characters with the same uniform projection: those lying above a cuspidal unipotent character of E6 (q), 2E (q) or E (q) or involving the two principal series characters φ 6 7 511,11, φ512,12 of E7 (q). Corollary 4.7.6 Let G be simple with connected centre and F : G → G be a Steinberg map. Then Harish-Chandra induction commutes with some Jordan decomposition Js• as in Theorem 2.6.22. Proof We need to show that for all semisimple elements s ∈ G∗F the diagram in Theorem 4.7.5 commutes for all F-stable Levi subgroups L∗  M∗  G∗ , with s ∈ L∗ and L∗ 1-split in M∗ , for some choice of JsL . Now the Mackey formula holds for 1-split Levi subgroups by Theorem 3.1.11. So by Theorem 4.7.5 this claim holds unless M = G is of type E8 , s has centraliser CG∗ (s) of type E6 A2 or E7 A1 and CL∗ (s) has a factor of type E6 or E7 . The only Harish-Chandra series not covered by our arguments are the principal series for CG∗ (s) = E7 (q).A1 (q) and those series listed in Table 4.18. Let’s discuss these in turn. Table 4.18 Critical 1-Harish-Chandra series CG∗ (s)

CL∗ (s)

λ

E6 (q).A2 (q) 2E (q).2A (q) 2 6

E6 (q)Φ21

j E6 [ζ3 ], j = 1, 2 2E [ζ j ], j = 1, 2 6 3 j E6 [ζ3 ], j = 1, 2 E7 [±ζ4 ]

E7 (q).A1 (q) E7 (q).A1 (q)

2E (q)Φ Φ 1 2 6 E6 (q)Φ21 E7 (q)Φ1

WCG∗ (s) (CL∗ (s), λ)F W(A2 ) W(A1 ) W(A1 ) × W(A1 ) W(A1 )

For λ as in the table let χ ∈ E (LF , s) be such that JsL ( χ) = λ (for the Jordan decomposition for L from Theorem 4.7.1). Then χ is also cuspidal by Theorem 3.2.22. Furthermore the relative Weyl groups of χ and of its Jordan correspondent λ agree. By the Comparison Theorem 3.2.7, the multiplicities in the decompositions of C ∗ (s) RLG ( χ) and RC G∗ (s) (λ) are the same for characters on both sides whose uniform L projections correspond under Jordan decomposition. It follows that there is some choice of Jordan decomposition between E (GF , s) and Uch(CG∗ (s)F ) that makes the diagram commute. Finally, for the principal series in E7 (q).A1 (q) we need to choose a bijection for the

350

Unipotent Characters

two pairs of characters in Uch(CG∗ (s)F ) lying in an exceptional family and having the same uniform projection. Again the relative Weyl groups on both sides agree and thus also the multiplicities. So there exists a compatible choice of bijection.  Let us define an equivalence relation on the sets of unipotent characters of reductive subgroups H of a simple group of type E8 , as follows: if H is of classical type, all unipotent characters form an equivalence class on their own. If H has (one) simple factor of type E6 , E7 or E8 then any two unipotent characters with same uniform projection and with Frobenius eigenvalue either a third or sixth root of unity are equivalent, the two cuspidal unipotent characters of E7 (q) with Frobenius √ eigenvalue ± −1 are equivalent, as are the pairs of principal series characters of E7 (q) and E8 (q) lying in an exceptional family. We then let E˜ (HF , 1) be the set of sums over equivalence classes of unipotent characters of HF . Thus, in particular, all elements of E˜ (HF , 1) have norm either 1 or 2. Via the unique Jordan decomposition from Theorem 4.7.1 this also defines corresponding sets E˜ (LF , s), for L an F-stable Levi subgroup of a simple group of exceptional type with connected centre. We then have: Corollary 4.7.7 Let G be simple with connected centre, F : G → G a Steinberg map, and assume that the Mackey formula holds for GF . Then for all F-stable Levi subgroups L∗  M∗  G∗ with duals L  M  G and semisimple elements s ∈ L∗F the diagram ZE˜ (MF , s) . ⏐ RLM ⏐ ZE˜ (LF , s)

JsM

−−−→ JsL

−−−→

ZE˜ (CM∗ (s)F , 1) . C ∗ (s) ⏐ RCLM∗ (s) ⏐ ZE˜ (CL∗ (s)F , 1)

commutes, where Js• denotes the Jordan decomposition specified in Theorem 4.7.1. Proof This follows from the previous Theorem 4.7.5 except for the pairs of equivalent characters. But for those, Lusztig induction of the sum over an equivalence class is again uniquely determined by its norm and its uniform projection.  Corollary 4.7.8 Let G and s ∈ G∗F be as in Theorem 4.7.5. Then for any d  1 the relations d and *d agree on E (GF , s). In particular, ρ ∈ E (GF , s) is d-cuspidal if and only if ∗RG T (ρ) = 0 for all F-stable maximal tori T  G contained in some proper d-split Levi subgroup of G. The first claim follows from Corollary 4.7.7, using that it holds for unipotent characters by Theorem 4.6.20, and the second is a direct consequence of this by the discussion after Corollary 3.5.25. As already pointed out there, this corollary had

4.8 Disconnected Groups, Groups with Disconnected Centre

351

been obtained by Cabanes–Enguehard [CE99, Thm. 4.2] in a wide range of cases using block theoretic methods. Arguments as in the proof of Theorem 4.7.5 had been used by Kessar and Malle [KeMa13] and Hollenbach [Ho19] to obtain partial results for isolated series in exceptional groups. We end this section by stating another application of Digne and Michel’s unique Jordan decomposition for groups with connected centre, namely a result of Srinivasan and Vinroot [SrVi18, Thm. 5.1] on Galois action (see also the earlier [SrVi15] for the case of complex conjugation): Theorem 4.7.9 (Srinivasan–Vinroot) Let G be connected reductive with connected centre with a Frobenius endomorphism F such that the Mackey formula holds for GF . Then for any semisimple element s ∈ G∗F and field automorphism σ of K the diagram E (GF , s) ⏐ 4σ ⏐

JsG

−→ JsGr

E (GF , sr ) −→

Uch(CG∗ (s)F ) ⏐ 4σ ⏐ Uch(CG∗ (s)F )

commutes, where r ∈ N is such that σ(ζ) = ζ r for all |GF |th roots of unity ζ, JsG denotes the Jordan decomposition from Theorem 4.7.1 and the downward arrows mean conjugation by σ. Sketch of proof Let σ be as in the statement. By Proposition 3.3.15 the Lusztig series E (GF , s) is mapped to E (GF , sr ) by σ. As r is prime to the order of s by assumption, s and sr have the same centraliser in G∗ , so the diagram makes sense. Now consider the map J˜sG : E (GF , s) → Uch(CG∗ (s)F ),

ρ → σ −1 (JsG (σ(ρ))).

If we can show that J˜sG satisfies the same conditions as JsG in Theorem 4.7.1, then by the uniqueness statement we must have J˜sG = JsG , that is, σ◦JsG = JsG ◦σ, as claimed. For properties (1), (4) and (5) in Theorem 4.7.1 this follows from Corollary 3.3.14. For property (2) one can use Proposition 4.5.5, and for property (7) the claim is straightforward. We omit further details. 

4.8 Disconnected Groups, Groups with Disconnected Centre The notion of connected reductive groups is a very useful one in the character theory of finite groups of Lie type, but it has two important shortcomings. First, Lusztig’s Jordan decomposition of characters of groups with disconnected centre

352

Unipotent Characters

naturally leads to having to consider unipotent characters of groups that are no longer connected. And secondly, for many applications of character theory to problems that have a reduction to some sort of decorated finite simple groups one would need to know about representations of automorphism groups of the finite quasi-simple groups, and the latter do include extensions of simple groups of Lie type by their graph automorphisms. For both reasons it seems necessary to develop a character theory of finite disconnected reductive groups. Some important steps in this direction have been made, and Lusztig has started an investigation of character sheaves on disconnected groups (see [Lu09e] and the references there), but the theory is still far away from as complete a picture as, for example, given by Lusztig’s Jordan Decomposition Theorem 2.6.4 in the connected case. Here, we explain some of the available results, and indicate in particular their relevance to the first problem mentioned above: how to obtain the commutation of (some) Jordan decomposition map with Lusztig induction for groups with disconnected centre, at least in certain situations. In particular for type A, or more precisely, for G = SLn , which in some sense constitutes the worst case as its disconnected centre may become arbitrarily large when n increases, a definite result has been obtained by Bonnafé [Bo06] and Cabanes [Ca13]. 4.8.1 Let us start by establishing some notions from the structure theory of disconnected groups. Let G be a (not necessarily connected) reductive group. A parabolic subgroup of G is any closed subgroup P  G containing a Borel subgroup of G◦ . Its connected component of the identity P◦ is then a parabolic subgroup of G◦ and thus has a semidirect product decomposition P◦ = Ru (P)  L◦ with L◦ a Levi complement of P◦ . Then P = Ru (P)  L with L := NG (L◦ ) (see [Bo99, §6.1]). The subgroup L is called a quasi-Levi subgroup of G. A special class of parabolic subgroups are the quasi-Borel subgroups, the normalisers in G of Borel subgroups of G◦ , and their quasi-Levi subgroups are called maximal quasi-tori. Since all Borel subgroups of G◦ are G◦ -conjugate, all quasi-Borel subgroups of G are G-conjugate, and similarly all maximal quasi-tori of G are G-conjugate. Example 4.8.2 In the disconnected case quasi-tori may contain non-trivial unipotent elements. Indeed, let G = SL3 a in characteristic 2 with a inducing the transpose-inverse automorphism of order 2. Then the maximal torus T0 of G◦ of diagonal matrices is a-invariant, so the quasi-torus NG (T0 ) = T0 a contains the unipotent element a  1. This can only happen, though, for so-called quasi-central elements. Following Steinberg [St68, §9], an automorphism a of a connected reductive group G is called quasi-semisimple if it stabilises a pair T  B consisting of a maximal torus contained in a Borel subgroup of G, and quasi-central if it is

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353

quasi-semisimple and its centraliser CG (a) in G has maximal dimension among the quasi-semisimple elements in its Inn(G)-coset. Such a quasi-central element a naturally acts on the Weyl group W = NG (T)/T of G. Quasi-central elements behave somewhat as a replacement for the identity element in the non-trivial components. For example, a unipotent element can be quasi-semisimple only if it is quasi-central [Spa82a, Cor. II.2.21]. Example 4.8.3 An important class of examples of disconnected groups are the almost simple groups G obtained from a simple algebraic group G◦ by extension with a non-trivial graph automorphism a. The various cases are (up to isogeny): (1) for G◦ = SLn , n  3, with a inducing the transpose-inverse automorphism, any outer element with centraliser either Spn (when n is even) or one of GOn or Spn−1 × {±1} (if n is odd) is quasi-central; (2) for G◦ = SO2n with n  3 and G = GO2n the quasi-central elements in GO2n \ SO2n are those with centraliser GO2n−1 ; (3) for G◦ of simply connected or adjoint type D4 with a inducing triality, the quasi-central elements in the outer cosets have centraliser of type G2 ; (4) for G◦ of type E6 and a inducing the non-trivial graph automorphism of order 2, the outer quasi-central elements have centraliser F4 . See [DiMi94, pp. 356–357] for a further discussion of the quasi-central elements. Whenever convenient we will restrict ourselves to a situation when G/G◦ is cyclic, as in the above examples. This is certainly sufficient if one is just interested in character values, since any element a ∈ G lies in the cyclic extension G◦ a of G◦ . In this case parabolic subgroups and quasi-Levi subgroups are parametrised in terms of the centraliser of a quasi-central element (see [DiMi94, Cor. 1.25]): Proposition 4.8.4 Let G = G◦ a be reductive with a ∈ G quasi-central. (a) The map P → CP◦ (a) defines a bijection between parabolic subgroups of G containing a and parabolic subgroups of CG◦ (a). (b) The map L → CL◦ (a) defines a bijection between a-stable quasi-Levi subgroups of parabolic subgroups of G containing a and Levi subgroups of CG◦ (a). Its inverse is given by L → CG (Z◦ (L))a . We now consider reductive groups with a Frobenius map. Here the first basic result is as follows (see [Bo99, Lemme 6.2.1]): Lemma 4.8.5 Let G be reductive with a Frobenius map F : G → G. Then there exists an F-stable supplement A  G to G◦ (that is, we have G = G◦ A) such that A ∩ G◦ is central and all elements of A induce quasi-central automorphisms on G◦ .

354

Unipotent Characters

In particular, if G = G◦ a with a quasi-central then we may always assume that a is F-stable. The classification of F-stable maximal quasi-tori can be derived from Proposition 4.8.4 and is again in terms of the centraliser of a quasi-central element: Proposition 4.8.6 Let G = G◦ a be a reductive group with a Frobenius map F : G → G such that a ∈ GF is quasi-central. Then: (a) There exists an F, a -stable maximal torus T0 of G◦ . (b) Any F-stable maximal quasi-torus of G has a GF -conjugate containing a. (c) If T is an F-stable maximal quasi-torus of G containing a then CT◦ (a) is a maximal torus of CG◦ (a), and T → CT◦ (a) induces a bijection between GF classes of F-stable maximal quasi-tori of G and CG◦ (a)F -classes of F-stable maximal tori of CG◦ (a). Proof

See [DiMi94, 1.36(ii)] for (a), [DiMi94, 1.40] for (b) and (c).



The CG◦ (a)F -classes of F-stable maximal tori of CG◦ (a) are parametrised by the F-conjugacy classes of CW◦ (a), where W◦ := WG◦ (T0 ) (see, e.g., Remark 2.3.21). Then an F-stable maximal quasi-torus of G is said to be of type w ∈ CW◦ (a) with respect to T0 if it corresponds to a maximal torus of type w under the map in Proposition 4.8.6(c). Example 4.8.7 Let G be the extension of SLn , n  3, with the transpose-inverse automorphism. By Example 4.8.3 the Weyl group of the centraliser of a quasicentral outer element is of type Bm , where m = $n/2%, and thus by Proposition 4.8.6 the classes of F-stable maximal quasi-tori of G are in bijection with the conjugacy classes of this Weyl group, hence with pairs of partitions of m (see [GePf00, Prop. 3.4.7])). Note that there are fewer of these than there are partitions of n, unless n = 4. Definition 4.8.8 Let L be an F-stable quasi-Levi subgroup of the parabolic subgroup P of G. The unipotent radical Y := Ru (P) satisfies condition (∗) in 2.2.4 with respect to LF , and as in 3.3.2 we define (generalised) Lusztig induction (or twisted induction) G RLP : ZIrr(LF ) → ZIrr(GF )

by G RLP (λ)(g) :=



(−1)i Trace g ∗, Hic (L −1 (Y), Q )λ

for g ∈ GF

i0

Irr(LF )

for any λ ∈ (see [DiMi94, Def. 2.2], and [Bo99, 6.3]). We again denote by ∗RG the adjoint map, Lusztig restriction, sending characters of GF to virtual LP F characters of L .

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355

The definition shows that ◦

F

F

G ResG ◦ RLP = RLG◦ P◦ ◦ ResLL◦F , G◦F

so in the case of connected groups the above specialises to the ordinary Lusztig induction introduced in 3.3.2, and in fact many of the basic properties from that case can be shown to carry over to the more general setting: Proposition 4.8.9 Let Q  P be parabolic subgroups of G with F-stable quasiLevi subgroups M, L respectively, such that M  L. Then G L G RLP ◦ RML∩Q = RMQ

∗ L RML∩Q

and

∗ G ◦ ∗RG LP = RMQ .

This is shown in [Bo99, Prop. 6.3.3]. As a particular case this includes the situation when the quasi-Levi subgroup L contains G◦ and hence P = L, where we get F

G RLL = IndG LF

and

∗ G RLL

F

= ResG LF

(see [Bo99, Prop. 6.3.2]). As in Proposition 3.3.3 in the connected case, we have the following: Proposition 4.8.10 Let L be an F-stable quasi-Levi subgroup of an F-stable parabolic subgroup P of G. Then F

F

G RLP = IndG ◦ InflPL F . PF

That is, Lusztig induction specialises to (generalised) Harish-Chandra induction, see [Bo99, Prop. 6.3.4]. In fact, there is a generalisation of Harish-Chandra theory to disconnected groups with suitable attached Hecke algebras of extended Weyl groups similar to what we described in Section 3.2 but we will not go into this here (see [DiMi85] and [Ma91, §1] for details). There is also a character formula for RLG in the spirit of Theorem 3.3.12, see [DiMi94, Prop. 2.6], which in the case when p divides |GF : G◦F | involves a new 2-parameter Green function. On the identity element we obtain (see [DiMi94, Cor. 2.5]): Proposition 4.8.11 Let L be an F-stable quasi-Levi subgroup of G. Then RLG (λ)(1) = G◦ L◦

|GF : LF | λ(1) |G◦F : L◦F | p

for λ ∈ Irr(LF ).

The character formula also immediately yields the following extension of Proposition 3.3.16: Proposition 4.8.12 Let G = G◦ a and f ∈ CF(GF ) p be p-constant. Then for every F-stable quasi-Levi subgroup L containing a of some parabolic subgroup P of G we have:

356

Unipotent Characters



GF

G G π ⊗ ResL F ( f ) = RLP (a) RLP (π) ⊗ f for all π ∈ CF(LF ),

G ∗ G F (b) ∗RG LP (η) ⊗ ResL F ( f ) = RLP (η ⊗ f ) for all η ∈ CF(G ). F

G In particular, ∗RG LP ( f ) = ResL F ( f ). F

See [DiMi94, Prop. 2.11]. The Mackey formula is at present only proved in a few cases, see [DiMi94, Thms. 3.2 and 4.5]: Theorem 4.8.13 Let G be reductive with G/G◦ cyclic. Then the Mackey formula for Lusztig induction holds if (1) one of the quasi-Levi subgroups is a quasi-torus; or (2) both quasi-Levi subgroups are contained in F-stable parabolic subgroups. G In particular, in these cases RLP does not depend on the choice of P.

Here, the last assertion follows from the preceding one by the same formal argument as in the connected case. We will write RLG if this operator does not depend on the choice of parabolic subgroup. The Mackey formula can be used to derive the following extension of Proposition 4.8.11 to arbitrary quasi-central elements, see [DiMi94, Prop. 4.15]: Proposition 4.8.14 Let G = G◦ a be reductive with a quasi-central element a and L an F-stable quasi-Levi subgroup of G containing a. Then C

◦ (a)

RLG (λ)(a) = RC G◦ (a) (1CL◦ (a) ) λ(a) L

for all λ ∈ Irr(LF ).

Digne and Michel [DiMi94, Def. 3.10] also introduce a duality operator: Definition 4.8.15 Assume that G = G◦ a . The duality operator on class functions on GF is defined as

DG := CL◦ (a) RLG ◦ ∗RG L P⊇B

where B is a fixed F-stable quasi-Borel subgroup of G containing a and the sum runs over all F-stable parabolic subgroups P of G containing B with an F-stable Levi subgroup L containing a. (This will not depend on the choices for L.) This operator is a self-adjoint involution and satisfies properties analogous to those of the Alvis–Curtis–Kawanaka–Lusztig duality for connected groups (see [DiMi94, §3.1]). For example by [DiMi94, Prop. 3.13] we have the following analogue of Proposition 3.4.7:

4.8 Disconnected Groups, Groups with Disconnected Centre

357

Proposition 4.8.16 Assume that G = G◦ a . Then for any ρ ∈ Irr(GF ) restricting irreducibly to G◦F , the class function ψ on GF defined by F

ψ(g) := DG◦ a i (ResG (ρ))(g) G◦F a i

for g ∈ G◦F ai , i ∈ N,

is up to sign an irreducible character of GF . It is then natural to define (see [Ma93a, §2] and [DiMi94, Def. 3.16]): Definition 4.8.17 The irreducible character StG := DG (1G ), where 1G is the trivial character of GF , is called the Steinberg character of GF . The Steinberg character is an actual character; its values are given by a formula completely similar to the connected case: Proposition 4.8.18 Let G = G◦ a and g ∈ aGF . Then StG (g) =

εCG◦ (a) εCG◦ (g) |CG◦ (g)F | p

if g is quasi-semisimple,

0

else.

See [DiMi94, Prop. 3.18]. The Steinberg character can be used to prove an analogue for disconnected groups of Steinberg’s formula for the number of unipotent elements (see the remarks after Corollary 3.4.19). This is worked out in [LLS14, Thm. 1.1]. The character StG defined above is an extension of the Steinberg character of G◦F to the disconnected group GF . In Remark 2.6.26 the unipotent characters Uch(GF ) F of GF were defined as the constituents of IndG (ρ), for ρ ∈ Uch(G◦F ). We have G◦F the following alternative characterisation, which is far from obvious given that the RTG (1T ) are virtual characters: Proposition 4.8.19 Let G be reductive with a Frobenius map F. The unipotent characters of GF are exactly the irreducible constituents of the various RTG (1T ) where T ranges over the F-stable maximal quasi-tori of G. Proof

This can be deduced from the formula

F 1 ρ(1) IndG (ρ) = Rw (1) RTGw , ◦F G |W◦ | w ∈W◦ ◦F ρ∈Uch(G

)

which in turn follows from Proposition 2.3.24 together with Proposition 4.8.9, see [Bo99, Lemme 6.4.2].  It follows from this that as in the connected case, unipotent characters of GF are trivial on Z◦ (G)F [Bo99, Prop. 6.4.3].

358

Unipotent Characters

Example 4.8.20 The parametrisation of unipotent characters of disconnected groups is rather straightforward, given the result for the connected case, by using the knowledge of the action of automorphisms from Theorem 4.5.11 in conjunction with the extendability result in Theorem 4.5.12. For example, consider G = GO2n where G◦ = SO2n . If F is an untwisted Frobenius map, then the unipotent characters 1,0 of GF = GO+2n (q) are parametrised by equivalence classes of symbols in Xd,n of rank n and defect d ≡ 0 (mod 4). Here the symbols with two equal rows label those unipotent characters of GF whose restriction to SO+2n (q) is reducible. The decomposition of RTG (1T ) into unipotent characters is known. For G◦ simple either of exceptional type, or of classical type of small rank, it was determined in [Ma93a], which also gave a conjecture for the general case ([Ma93a, §7]). This was shown in [DiMi94, Thm. 5.11], based on results of Asai [As84c] on Shintani descent to deal with the case of classical groups. Lusztig [Lu12b, Thm. 2.4(ii)] gives a rather different proof in the general case in the spirit of his approach in the connected case. The first step is again a base change via the character table of a suitable Weyl group; here again σ denotes the automorphism of W◦ induced by F: Definition 4.8.21 Assume that G = G◦ a with a quasi-central element a. Let φ ∈ Irr(CW◦ (a))σ and φ˜ a σ-extension of φ. Then

1 ˜ φ(w) RTGw (1Tw ), RφG˜ := |CW◦ (a)| ◦ w ∈CW (a)

with Tw  G a quasi-torus of type w, is called the corresponding almost character. Let us point out the following subtlety: even if the restriction of such an almost character RφG˜ to any coset of G◦F in GF coincides with the restriction of an irre-

ducible character of GF , RφG˜ itself need not be an irreducible character. We refer to this as the gluing problem. As in the connected case, a class function on a G◦F -coset xG◦F in GF is called uniform if it is a linear combination of the restriction to xG◦F of characters RTG (θ), for F-stable maximal quasi-tori T and θ ∈ Irr(TF ). Again, the trivial character on any coset turns out to be uniform by [DiMi94, Prop. 4.12]. Example 4.8.22 A noteworthy difference to the connected case is that there can be unipotent characters of GF whose restriction to some coset of G◦F is orthogonal to all RTG (1T ), that is, its uniform projection is zero: (a) For G the extension of SLn , n  3, by the transpose-inverse automorphism, by [DiMi94, Thm. 5.2] the almost characters yield extensions of those unipotent characters of SLn (q) or SUn (q) labelled by partitions with a 2-core of size 0 or 1 (depending on the parity of n), that is, for those which lie in the principal 2-HarishChandra series of SLn (q) (respectively in the principal series for SUn (q)). All other

4.8 Disconnected Groups, Groups with Disconnected Centre

359

extensions of unipotent characters are orthogonal to the space of uniform functions, and no construction of these seems to be known. (b) More concretely, consider the case n = 3, when G is the extension of SL3 by the transpose-inverse graph automorphism a and F a Frobenius map commuting with a. The three unipotent characters of G◦F extend to GF by Theorem 4.5.11, but the restrictions of the extensions of the unipotent character ρ21 of G◦F to the coset G◦F a are orthogonal to the restrictions of all RTG (1T ) to this coset [Ma93a, Thm. 4] (or, to put it differently, the two extensions of ρ21 to GF have the same multiplicity in all RTG (1T )). For F such that G◦F = SU3 (q) this can also be deduced from the √ fact that the character field of these extensions contains −q (see [Ma90b, Tab. 2]) G while the RT (1T ) are rational valued. 4.8.23 When the index |G : G◦ | is divisible by the characteristic p of k, outer cosets of G◦ may contain non-trivial unipotent conjugacy classes. One can then define 1-parameter Green functions as in the connected case as the restriction of the RTG (θ) to the unipotent elements in a coset, see [Ma93a, Sect. 8–10]: Let Guni denote the set of unipotent elements of G and let a ∈ G be quasi-central. For T  G an F-stable maximal quasi-torus with a ∈ TF set G

F for u ∈ (aG)uni . QG T (u) := RT 1T (u) These Green functions have been computed for groups of exceptional type [Ma93b] and for groups of classical type and small rank [Ma93a]. Sorlin [So04] has defined a Springer correspondence which was then determined in [MaSo04] for groups of classical type. A generalised Springer correspondence in this setting was introduced by Lusztig [Lu04a] and determined completely for groups of classical type. In [Ma05] it is calculated for the groups of exceptional type and it is shown that the obvious generalisation of the Lusztig–Shoji algorithm (see 2.8.9) applied with this Springer correspondence yields the Green functions computed in [Ma93a, Ma93b]. It is to be expected that this holds in general. After this short excursion into the representation theory of disconnected groups, we now prove a generalisation of Theorem 4.7.2 to groups with cyclic centre. Theorem 4.8.24 Let G be a Levi subgroup of a simple group with cyclic centre, F : G → G a Frobenius endomorphism, and assume that the Mackey formula holds for GF . Then for all F-stable Levi subgroups L∗  G∗ with dual L  G and all semisimple s ∈ L∗F such that CL∗ (s) is connected and only has components of classical types A, B, C and D, there is a Jordan decomposition Js• as in Theorem 2.6.22

360

Unipotent Characters

for which the diagram ZE (GF , s) . ⏐ RLG ⏐ ZE (LF , s)

JsG

−−−→ JsL

−−−→

ZUch(CG∗ (s)F ) . C ∗ (s) ⏐ RCLG∗ (s) ⏐ ZUch(CL∗ (s)F )

commutes. Observe that here Lusztig induction on the right-hand upwards arrow is to a possibly disconnected group as introduced in Definition 4.8.8. ˜ be a regular embedding with dual epimorphism i ∗ : G ˜∗ → Proof Let i : G → G ∗ ∗ ∗ ∗ ˜ and L ˜ , dual to L. ˜ = LZ(G) ˜ the full preimage of L in G ˜ Let s˜ ∈ L ˜ ∗F G . Let L be a preimage of s. By Theorem 4.7.5 the middle square of the following diagram commutes: ˜

JG

Ind Ind s˜ ˜ F , s˜) −−→ ZUch(CG˜ ∗ (s˜)F ) −−→ ZUch(CG∗ (s)F ) ZE (GF , s) −−→ ZE (G . . . . C ˜ ∗ (s) ˜ ˜ C ∗ (s) ⏐ RLG˜ ⏐ RC G˜ ∗ (s) ⏐ RCLG∗ (s) ⏐ RLG L ˜ ⏐ ⏐ ⏐ ⏐ ˜ JsL˜ Ind ˜ F , s˜) −−→ ZE (LF , s) −−→ ZE (L ZUch(CL˜ ∗ (s˜)F ) = ZUch(CL∗ (s)F )

Let us explain the other parts of the diagram. The horizontal arrows on the left mean induction and then intersecting with the Lusztig series on the right. On the righthand side bottom row, since CL∗ (s) is connected, the unipotent characters of CL∗ (s)F are just the (deflations of the) unipotent characters of CL˜ ∗ (s˜)F . Similarly, on the top row, the unipotent characters of CG◦ ∗ (s)F are just the (deflations of the) unipotent characters of CG˜ ∗ (s˜)F , and the arrow is ordinary induction from CG◦ ∗ (s)F to CG∗ (s)F . Now the left-hand square commutes by Corollary 3.3.25, and the right-hand one by Proposition 4.8.10. Let λ ∈ E (LF , s). Since CL∗ (s) is connected, λ is L˜ F -invariant, so extends to ˜F F ˜ L (since by assumption Z(L)/Z◦ (L) is cyclic), and all constituents of IndLL F (λ) ˜ F : GF | distinct preimages lie in distinct Lusztig series corresponding to the | G F of s. So there is a unique constituent λ˜ in E (L˜ , s˜), occurring with multiplicity 1 by the Multiplicity-Freeness Theorem 1.7.15. Let ρ be a constituent of RLG (λ), and g ∈ L˜ F . Then RLG (λ) = RLG (λ g ) = RLG (λ)g , so all constituents of RLG (λ) in one ˜ F -orbit appear with the same multiplicity. G Since Z(G)/Z◦ (G) is cyclic, so is CG∗ (s)/CG◦ ∗ (s) and thus the induction from CG◦ ∗ (s)F to CG∗ (s)F is multiplicity-free. By Jordan decomposition 2.6.22, the num˜ ˜ is the same as the number of ber of characters in ZUch(CG∗ (s)F ) above JsG ˜ ( ρ) F ˜ G F ˜ constituents of ResG F ( ρ), ˜ for any ρ˜ ∈ E (G , s˜). Choosing any bijection between these two sets, the commutation of the above diagram then shows our claim. 

4.8 Disconnected Groups, Groups with Disconnected Centre

361

In the last step of this proof, one could make a more specific choice of bijection by fixing the image of one character, and then using the action of the ˜ F on the left-hand side, and of the isomorphic group ˜ F /GF Z(G) cyclic group G F ◦ F Irr(CG∗ (s) /CG∗ (s) ) on the right to assign the other images. Example 4.8.25 Let us consider the simplest example: assume that n is prime and GF = SLn (q) with n dividing q − 1. Then it is easily seen that there is a unique conjugacy class of semisimple elements in G∗ = PGLn with disconnected centraliser, namely the class of the image s in PGLn of s˜ = diag(1, ζ, . . . , ζ n−1 ) ∈ GLn , where ζ ∈ F×q has order n (see, e.g., [Bo05]). Here CG∗ (s)/CG◦ ∗ (s) is cyclic of order n and F acts trivially on it, so the G∗ -class of s splits into n distinct G∗F -classes, with representatives s0, . . . , sn−1 , with CG◦ ∗ (si )F a torus of order (q − 1)n−1 for i = 0 and of order (q n − 1)/(q − 1) when i > 0. Hence |Uch(CG∗ (si )F )| = n = |E (G, si )| by Jordan decomposition (Theorem 2.6.22). For any F-stable proper Levi subgroup L∗ < G∗ containing si we have that CL∗ (si ) is connected, hence we are in the situation of Theorem 4.8.24. Then any choice of bijection between the n elements of E (GF , si ) and of Uch(CG∗ (si )F ) will make Lusztig induction RLG commute with this Jordan decomposition. 4.8.26 The previous example has been vastly generalised by Bonnafé [Bo99] and subsequently Cabanes [Ca13]. For this let G = SLn with a Frobenius map F : G → G. Here the dual group is G∗ = PGLn . Since Z(SLn ) is cyclic, the centraliser H := CG∗ (s) of any semisimple element s ∈ G∗ = PGLn has the property that H/H◦ is cyclic of order dividing |Z(SLn )| = n p and thus H is a possibly disconnected group as considered above. Bonnafé works in a rather more general setting than ours here, in that he allows groups H such that H/H◦ acts on the simple factors of H◦ in a wreath fashion. We restrict ourselves to the case above where H = H◦  A, with A cyclic, permuting the simple factors of H◦ and no element of A inducing a non-trivial graph automorphism on any simple factor. For every a ∈ A and a-invariant unipotent character ρ of H◦F he defines an almost character as in Definition 4.8.21 on the stabiliser HρF , which turns out to be (up to sign) an irreducible character of HρF extending ρ, see [Bo99, Thm. 7.3.2] for SLn (q) and [Ca13, Thm. 3.6] for SUn (q). This implies in particular: Corollary 4.8.27 In the above situation, all unipotent characters of HF are uniform and thus the Mackey formula holds for HF on the subspace of unipotent class functions. This is as in the connected case, but in contrast to the situation when non-trivial graph automorphisms are involved, see Example 4.8.22. The explicit description of the unipotent characters in terms of characters RTH (1), together with transitivity,

362

Unipotent Characters

also allows one to work out the decomposition of all RLH (1) in this case, see [Bo99, Thm. 7.6.1] for SLn (q), and [Ca13, Thm. 4.4] for SUn (q). Theorem 4.8.28 (Bonnafé [Bo06, §27], Cabanes [Ca13, Thm. 4.9]) Let F be a Frobenius map on G = SLn and assume q is large enough. Then for every semisimple element s ∈ G∗F and any F-stable Levi subgroup L∗  G∗ containing s, with dual L, there exists a bijection JsL : E (LF , s) → Uch(CL∗ (s)F ) such that for all F-stable Levi subgroups L∗  M∗  G∗ the following diagram commutes: ZE (MF , s) . ⏐ RLM ⏐ ZE (LF , s)

JsM

−−−→ ZUch(CM∗ (s)F ) . C ∗ (s) ⏐ RCLM∗ (s) ⏐ JsL −−−→ ZUch(CL∗ (s)F )

The assumption on q is introduced by Lusztig’s result [Lu88, Thm. 1.14] which at present is only known under this condition. About the proof For the diagram to make sense one first needs to observe that CL∗ (s) is a quasi-Levi subgroup of CM∗ (s) in the sense introduced above, so that the Lusztig induction on the right-hand side is defined. Then, it should be noted that neither Lusztig induction in the statement mentions a parabolic subgroup containing the (quasi-)Levi subgroup. This is justified by the validity of the Mackey formula for the groups in question, see Theorem 3.3.7 and Corollary 4.8.27. The main task is then to define the maps JsL such that the commutation holds. They are obtained by identifying both sides with the set of irreducible characters of a suitable relative Weyl group.  The commutation remains open in general for groups with disconnected centre.

Appendix Further Reading and Open Questions

In this appendix, we loosely present, somewhat informally and in no particular order, topics and problems that were touched upon in the main text but, for one reason or another, could not be thoroughly dealt with. Thus, we mainly collect open ends and problems for which the theory is not complete or not even existent (yet), as well as questions it would be important to solve for various applications. We also mention related projects and topics, with indications for further reading.

A.1 Representatives of Conjugacy Classes As already mentioned in the preface, we do not discuss in this book any particular aspects of the problem of determining the conjugacy classes of GF . Nevertheless, it may be useful to indicate some critical issues related to this problem. On a general level, the Jordan decomposition of elements provides the framework in which the conjugacy classes are described: first one determines the classes of semisimple elements and then, for each semisimple class representative s ∈ GF , one determines the unipotent classes of CG (s)F . (Recall from 2.2.13 that CG◦ (s) is connected reductive, and that all unipotent elements of CG (s) are already contained in CG◦ (s).) See also [Spa82a], [Der84], [Ca85], [Hum95] for further information and detailed references. Thus, the problem of obtaining a classification of the conjugacy classes of GF is essentially under control. As a model case see [Miz77], where the conjugacy classes of GF = E6 (q) are determined. As briefly discussed in Remark 2.7.4, there is an issue of choosing class representatives g ∈ GF such that the action of F on A(g) = CG (g)/CG◦ (g) is as simple as possible. Specific choices of class representatives are needed, for example, in order to fix characteristic functions of F-stable character sheaves, or for the computation of Green functions (see Example 2.7.27 and Remark 2.8.8). More concretely, let us assume that G is a simple algebraic group. Then G = 363

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Further Reading and Open Questions

xα (t) | α ∈ Φ, t ∈ k , where Φ is the root system of G with respect to a maximally split torus T0 of G. Representatives for the unipotent classes are typically described by explicit expressions as products of elements xα (t), for various α and t. On the other hand, it is known that every semisimple element of G is conjugate to an element of T0 , but the conjugation takes place in the algebraic group G. In order to obtain representatives in GF , one could use Steinberg’s cross section; see [Hum95, §4.10 and §4.15]. This produces a set of representatives for the semisimple classes, but it is totally unrelated to T0 . Consequently, we will also have a problem for all mixed classes, that is, classes that are neither unipotent nor semisimple. For such classes, it appears to be very difficult – both conceptually and in concrete terms – to specify representatives with ‘good’ properties.

A.2 Green Functions for Type E8 and p = 2, 3, 5 There is a long tradition of work on Green functions; the principal ideas and methods are summarised in Shoji’s survey [Sho86]. At that point, the Green functions were known in all cases where q is a power of a good prime for G, or where q is arbitrary, G is of small rank and the whole table of unipotent character values is available (like for G2 , 3D4 , 2B2 , 2G2 ). A precursor of the full Lusztig–Shoji algorithm was employed in [Sho82], [BeSp84] to determine the Green functions for G of type F4 , E6 , E7 and E8 in good characteristic. Explicit results for the larger groups of exceptional type in characteristic 2 are obtained in [Ma90], [Ma93b]. Classical groups in characteristic 2 are dealt with in [Sho07]. In view of [Ge19c], the only open cases (at the time of this writing) are groups of type E8 in bad characteristic. There is some hope that extensions of the methods employed in [Ge19c] will lead to a solution of these cases as well. The methods in [Ge19c] rely on a theoretical result which also led to a solution of an old problem concerning the Glauberman correspondence in the general character theory of finite groups; see [Ge19b].

A.3 Character Tables of F4 (3) and E7 (2) One of the ultimate computational challenges is the determination of the ‘generic’ character tables of the large groups of exceptional type F4 , E6 , 2E 6 , E7 and E8 . (Complete tables for the smaller groups of exceptional type are already known and contained in the library of CHEVIE; see Table 2.4, p. 103.) Note that for each type of group, there will be several such tables according to the congruence classes of q modulo a certain integer that is determined by the underlying complete root datum.

A.4 Mackey Formula

365

(For example, in Section 2.1 we have seen that two such tables are required for GF = SL2 (q), one for q odd and one for q a power of 2; for type E8 one needs to distinguish congruence classes mod 60.) A considerable amount of information (on conjugacy classes, Green functions, . . .) is already available electronically, via Michel’s extensions [MiChv] of CHEVIE and Lübeck’s data webpage [Lue07]. However, a number of technically difficult problems related to the decomposition of the Lusztig induction functor RLG and the evaluation of characteristic functions of character sheaves remain to be addressed. For various reasons (see, e.g., [Ge19a], [Ge19c]) it is extremely useful to have explicit character tables for concrete, small values of q available; in particular, this includes the cases where q = p is a bad prime for G. The Cambridge ATLAS [CCNPW] contains the character tables of F4 (2) and of the non-abelian composition factor of 2E 6 (2)sc . In addition, the character table library of GAP (see [Bre18]) contains the character tables of E6 (2) and the whole group 2E 6 (2)sc . As far as the remaining exceptional groups are concerned, we find the following numbers using Lübeck’s online data [Lue07]: |Irr(F4 (3))| = 273,

|Irr(E6 (3))| = 1269,

|Irr(E7 (2))| = 531, |Irr(E8 (2))| = 1156,

|Irr(2E 6 (3))| = 1389,

|Irr(E7 (3)sc )| = 5052,

|Irr(E8 (3))| = 12825,

|Irr(E8 (5))| = 519071.

Thus, it should be within reach to determine the individual character tables of F4 (3), E7 (2) and, perhaps, even of E8 (2). The knowledge of these tables would be very useful for solving the open problems on characteristic functions of cuspidal character sheaves mentioned in Example 2.7.27.

A.4 Mackey Formula The Mackey formula for Lusztig induction is known in most cases (see Theorem 3.3.7) but its proof is far from satisfactory and requires difficult and tricky case-by-case arguments. Knowledge of the Mackey formula implies in particular independence of Lusztig induction from the choice of parabolic subgroup, a result which would be very important to have even independently.

A.5 Jordan Decomposition For connected reductive groups with connected centre, Digne and Michel defined a unique Jordan decomposition (see Theorem 4.7.1). Yet, this Jordan decomposition is not constructed (or proved) by an intrinsic method, but by some ad hoc assumptions

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Further Reading and Open Questions

and conventions. It would be highly desirable to find a general approach to a unique Jordan decomposition. This would most likely also lead to a general result on commutation of Jordan decomposition with Lusztig induction, which we could here only arrive at by some detailed case-by-case discussion, and even then not in complete generality (see Theorem 4.7.5). In the general case of groups with not necessarily connected centre, there has so far not been a method to define a unique Jordan decomposition except for groups of type A (see Theorem 4.8.28), and consequently the very important commutation problem is widely open. It might be hoped that Lusztig’s approach via categorical centres [Lu16] could give a conceptual solution. A first important open question here is already whether Jordan decomposition can be chosen such that it commutes with Harish-Chandra induction (see Corollary 4.7.6 for the case of connected centre). This is also related to the much deeper question on existence of Morita equivalences for Brauer -blocks of characters in Lusztig series corresponding under Jordan decomposition as discussed in [BoRo93, BDR15].

A.6 d-Harish-Chandra Theory and Cyclotomic Hecke Algebras The d-Harish-Chandra theory presented in Section 3.5 should be explained by the so-called cyclotomic Hecke algebras, certain deformations of group algebras of complex reflection groups, that conjecturally occur as endomorphism algebras of Lusztig induced d-cuspidal modules, see the conjectures in Broué–Malle [BrMa93]. That had first been shown (not under this point of view) by Lusztig [Lu76c] in certain cyclic cases, and in some further particular situations by Digne, Michel and Rouquier [DMR07], and Digne–Michel [DiMi14]. It should lead to a conceptual proof of the Comparison Theorem 4.6.21 and the analogue of Howlett–Lehrer–Lusztig theory stated in Theorem 4.6.24. Moreover this would give a unified approach to not only proving but also ‘understanding’ Broué’s abelian defect group conjecture in modular representation theory for groups of Lie type (see [BrMa93]). This is closely connected to the so-called spetses programme which strives to construct analogues of finite reductive groups whose Weyl group is a ‘spetsial’ complex reflection group (see [Ma98]), begun in [Ma95, BMM99, BMM14].

A.7 Disconnected Groups The representation theory of disconnected groups is still at its beginnings; we sketched some results of Digne–Michel [DiMi94] and Bonnafé [Bo99] in Section 4.8. Probably one would need a continuation of Lusztig’s approach via charac-

A.8 Shintani Descent

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ter sheaves on disconnected groups [Lu09e], leading, for example, to an algorithm for the computation of Green functions extending the Lusztig–Shoji algorithm discussed in Section 2.8, to a construction of those unipotent characters with zero uniform projection (see Example 4.8.22), and more generally to an analogue of Jordan decomposition for characters on a coset.

A.8 Shintani Descent Another interesting direction, which we did not touch upon in this book, is the character theory of finite groups obtained by extending a finite reductive group (connected or not) by a group of field automorphisms. This is one of the themes of Shintani descent, originating in [Shi76] (see also the survey [Di87]). Shintani descent is an important tool for studying various powers of a Frobenius map F : G → G at the same time; see [Lu84a, Chap. 2], [DiMi85, Chap. III], [Kaw87, §1]. The ‘Shintani descent identity’, due independently to Asai, Digne–Michel and Lusztig, connects zeta functions of Deligne–Lusztig varieties to Hecke algebras; see Curtis’ survey [Cu88, §2] and further references there. Shintani descent also plays an important role in determining the Lusztig induction functor RLG ; see Asai’s papers (as discussed in Section 4.6) as well as [Sho85], [Sho87].

A.9 Automorphisms and Galois Action Both of the latter questions, on characters of disconnected groups as well as on extensions by field automorphisms, are closely related to the more general question of the action of outer automorphisms on the irreducible characters of finite reductive groups. While this is solved for groups with connected centre (see e.g. Theorems 4.5.11 and 4.5.12), the question is open in the general case. See Cabanes– Späth [CaSp17] and Taylor [Ta18b] for the symplectic groups and [Ma17b] for partial results for cuspidal characters in quasi-isolated series. For example it would be desirable to have a Jordan decomposition equivariant with respect to outer automorphisms, and, in fact, with respect to field automorphisms (see Theorem 4.7.9 for connected centre groups).

A.10 Some Applications Many applications of the character theory of finite groups are related to the following two purely group-theoretical problems. Let Γ be a finite group.

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Further Reading and Open Questions

• If C1, C2, C3 are conjugacy classes of Γ (not necessarily distinct), then compute the number of pairs (x, y) ∈ C1 × C2 with x y ∈ C3 . • If Γ   Γ is a subgroup and C is a conjugacy class of Γ, then compute the cardinality |C ∩ Γ  | or, more generally, |C ∩ Γ  gΓ  | for any g ∈ Γ. It is well known that both problems can be solved efficiently, once sufficient information about the character table of Γ is known. (See the exercises in [Is76, Chap. 3] for the first problem, and [CuRe81, §11.D] for the second problem.) In the case where Γ = GF is a finite group of Lie type, this character-theoretic approach to group-theoretical problems has been applied successfully in a variety of situations: to mention but a few, see Malle–Matzat [MaMa18] (realisation of finite groups as Galois groups); Malle–Saxl–Weigel [MSW94] (generation of finite classical groups); Lusztig [Lu03a] (subgroups isomorphic to the alternating group A5 inside groups of Lie type of type E8 ); Lusztig [Lu11] (construction of a surjective map from conjugacy classes in the Weyl group W to unipotent classes of the underlying algebraic group G); the solution of a conjecture of P. Neumann on the existence of elements with small fixed spaces in linear groups by Guralnick–Malle [GM12a]; their construction of Beauville surfaces for all but one finite non-abelian simple group [GM12b]; the resolution of cases of the Arad–Herzog conjecture on products of conjugacy classes by Guralnick–Malle–Tiep [GMT13]; and the solution of the Ore conjecture on commutators in simple groups by Liebeck–O’Brien–Shalev–Tiep (see [Ma14], which also describes further applications to images of word maps). For further applications of a somewhat different nature, see Liebeck–Shalev [LiSh05] (character degrees and random walks on finite groups), Lusztig [Lu00] 2 (existence of GF -invariant vectors in representations of GF , a problem which has previously been studied in the context of Lie groups and p-adic groups), Reeder [Re07] (study of the restriction of Deligne–Lusztig characters, motivated by a problem for finite orthogonal groups).

A.11 Related Topics and Projects In this book, we are exclusively dealing with algebraic groups defined over finite fields, and representations of these groups over fields of characteristic 0, following the approach initiated by Deligne and Lusztig in the 1970s. One of the features of this theory is that it can be, and has been, distilled into finite combinatorial terms (root data, Hecke algebras, Lusztig’s Fourier matrices and so on), that lead to efficient algorithms and implementations on a computer: see the CHEVIE project. These programs have turned out to be highly useful, both in a number of applications, and in

A.11 Related Topics and Projects

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the explicit verification of certain general properties for groups of exceptional type. (An example is mentioned in the remarks concerning the proof of Theorem 2.7.25.) There is an analogous, multi-author project for real reductive groups and unitary representations, under the title ‘Atlas of Lie groups and representations’; see www. liegroups.org/. The article by Adams et al. [ALTV] provides a good overview and also gives a detailed description of the atlas algorithms for computing unitary representations. Again, the general theory is translated into finite combinatorial terms and efficient computer implementations. One of the major successes of this project attracted a huge interest in the mathematical community and beyond; see [Vo07]. In another direction, the Harish-Chandra philosophy discussed in Chapter 3 has been appearing in a number of different contexts which, although sometimes based on entirely different theoretical foundations, eventually lead to similar combinatorial patterns. A recent example is the work of Achar et al. [AHJR], where the Springer correspondence (which we encountered in this book in Section 2.8, in connection with the problem of computing Green functions) has been generalised to a modular setting where one considers sheaves with coefficients in a field of characteristic  > 0 (and not just in Q ). There are results about the classification of cuspidal objects and the parametrisation of Harish-Chandra series which are highly reminiscent of results in [DD97], [GHM94], [GHM96], [GeHi97] (where -modular representations of finite groups of Lie type are considered). Compare, for example, the descriptions of modular Harish-Chandra series for GLn in [AHJR, Example 6.1] and in [DD97, §4]; or the characterisation of a modular Steinberg object in [AHJR, Theorem 7.1] and in [GHM94, Theorem 4.2].

Index

on semisimple elements, 113 characteristic exponent, 36 characteristic function, 193 class fusion, 94, 113 co-character group, 11 cohook of a symbol, 304 coinvariant algebra, 276 comparison theorem, 229, 340 complete root datum, 75 complex reflection group spetsial, 282 complex reflection groups, 260 conformal group, 81 connected, 4 content of a symbol, 302 core of a symbol, 303 cuspidal module, 217 pair, 217 symbol, 303 cuspidal pair, 217 cuspidal unipotent characters in classical types, 314

1-split Levi subgroup, 213 2-parameter Green function, 241 a-function, 278 a-invariant, 233, 278 adding a cohook, 305 adding a hook, 304 addition of symbols, 303 adjoint quotient, 62 adjoint representation, 12 adjointness of R G L , 216 affine algebraic group, 3 affine variety, 2 algebraic B N -pair, 14 almost character for disconnected groups, 358 almost characters, 193 Alvis–Curtis–Kawanaka–Lusztig duality, 249 Asai Levi subgroup, 336 average value, 183 b-invariant, 272 base, 18 β-set, 277 B N -pair, 13 of rank 1, 224 Bruhat cells, 14 Bruhat decomposition, 14

d-cocore, 305 of a symbol, 305 d-core, 304 of a symbol, 305 d-cuspidal character, 266 d-cuspidal pair, 266 d-Harish-Chandra series, 267 d-Harish-Chandra theory, 340 d-Jordan cuspidal character, 267 d-principal series, 341 d-regular element, 259 d-split Levi subgroup, 258 d-split Levi subgroups

Cartan matrix, 19 Cartan type, 19 central isogeny, 39 central isotypy, 39 character formula for RLG , 242 character group, 10 character of -defect zero, 298 character sheaves, 181, 192 character values

390

INDEX in GL n , 262 in classical types, 263 in exceptional types, 264 d-torus, 257 decomposition of RLG in classical types, 330, 336 in type A, 328 decomposition theorem, 292 defect of a symbol, 302 defined over F q , 40 degenerate symbol, 303 degree formula, 233 degree polynomial, 130 degree polynomials of unipotent characters, 324 Deligne–Lusztig character, 109 diagonal automorphisms, 62 distinguished symbol, 302 Drinfeld double, 290 dual complete root datum, 79 dual root datum, 16 duality, 64 duality operator, 249 for disconnected groups, 356 Dynkin diagram, 19 e-symbols, 305 Ennola dual, 78, 206 Ennola duality, 300 exceptional characters, 221 exceptional family, 281 extended Coxeter group, 222 F-simple, 63 fake degree, 150, 276, 283 fake degrees for type A n , 277 for type B n , 277 for type D n , 277 for type 2D n , 284 families in Irr(W ), 280 in type A n , 282 in type B n , 282 in type D n , 282 families of unipotent characters, 287 in classical types, 308 finite group of Lie type, 42 finite reductive group, 42 Fourier matrix, 293 for 2B2 , 319 for 2F4 , 321 for 2 G2 , 319 for classical types, 309

391 of S2 and S3 , 294 Frobenius eigenvalues, 295 in classical types, 316 in exceptional types, 318 Frobenius map, 40 fundamental group, 19, 63 Galois action and RLG , 243 and Lusztig series, 243 on unipotent characters, 323 Gelfand–Graev character, 254 general orthogonal group, 5 general position, 110 general unitary group, 49 generalised decomposition map, 244 generalised Gelfand–Graev characters, 254 generic character table, 102, 181 generic degree, 231 generic finite reductive group, 75 generic Iwahori–Hecke algebra, 219 generic twisted extended Iwahori–Hecke algebra, 222 generic unipotent characters, 323 genus of a symbol, 302 geometric conjugacy classes of characters, 121 geometric series of characters, 121 geometrically conjugate, 121, 145 good prime, 188 Green function, 112, 241 Green functions, 359 Harish-Chandra induction, 110, 214 Harish-Chandra restriction, 214 Harish-Chandra series, 217 in classical groups, 315 in exceptional groups, 318 in unitary groups, 300 Hecke algebra, 218, 223 Hom-functor, 218 homomorphism of root data, 16 hook formula for classical types, 307 for type A, 298 hook length, 304 hook of a symbol, 304 indecomposable Cartan matrix, 19 index representation, 221 indivisible, 139 inflation, 94 irreducible complex reflection groups, 260 isogeny, 36

392 isogeny theorem, 35 isotypy, 39 Iwahori–Hecke algebra, 220 j-induction, 273 Jordan decomposition for unipotent characters, 293 Jordan decomposition of characters, 134, 167, 168, 178 and Harish-Chandra series, 246, 349 commutes with RLG , 345, 347, 350, 359 given by RLG , 246 uniqueness of, 343 Jordan decomposition of elements, 8

-adic cohomology groups with compact support, 106 Lefschetz number, 107 leg length of a hook, 327 length function, 13 Levi complement, 212 Levi decomposition, 212 Levi subgroup, 212 Φ-split, 258 d-split, 258 1-split, 213 of type (I, w), 237 split, 213 linear algebraic group, 4 linear characters, 96 Lusztig induction, 237 and regular embeddings, 248 and restriction of scalars, 248 in disconnected groups, 354 is generic, 324 Lusztig restriction, 238 Lusztig series Galois action, 243 Lusztig series of characters, 167 Mackey formula for R G L , 216 for RLG , 240 for disconnected groups, 356 maximal quasi-torus, 352 maximal toric sub-datum, 79 maximally split, 44, 64 Mellin transform, 291 modified Green functions, 200 non-abelian Fourier transform, 290 non-abelian Fourier transform matrix, 293 norm map, 120 one-parameter Hecke algebra, 220

INDEX order polynomial, 75 ordinary automorphism, 69, 283 orthogonality relations for Green functions, 195 p-constant class function, 243 p-isogeny of root data, 20 parabolic subgroup, 212, 261, 352 parabolic subgroups of GL n (q), 214 of GU n (q), 214 perfect pairing, 11 Φ-split Levi subgroup, 258 Φ-torus, 258 Poincaré polynomial, 231 positive roots, 18 preferred σ-extension, 285 preservation results, 35 principal series, 218 projective general linear group, 34 property (U d ), 266 quasi-Borel subgroup, 352 quasi-central automorphism, 352 quasi-central elements in almost simple groups, 353 quasi-isolated conjugacy class, 246 quasi-Levi subgroup, 352 quasi-semisimple automorphism, 352 rank, 19, 111, 212 of a symbol, 302 rational series of characters, 167 rational structure, 40 reduced expression, 14, 219 reduced symbol, 308 reductive, 10 reductive B N -pair, 14, 29 regular character, 252 regular elements, 123 regular embedding, 80 and Lusztig induction, 248 regular functions, 2 regular number, 259 regular unipotent class, 115, 198 regular unipotent elements, 115 regular vector, 259 relation 1 , 217 relation  d , 267 relation * d , 267 relation +, 280 relative F-rank, 111 relative rank, 226

INDEX relative Weyl group, 223, 259 in classical types, 269 in exceptional types, 261 removing a cohook, 304 removing a hook, 304 representation, 12 root datum, 15 dual, 16 of adjoint type, 24 of simply connected type, 24 root exponents, 21, 36 root space decomposition, 12 root subgroup, 28 roots, 12 Schur element, 231 Schur elements in type A n , 233 Schur index, 323 semidirect product (of algebraic groups), 9 semisimple, 13 semisimple character, 171 semisimple element, 8 semisimple group of adjoint type, 55 semisimple group of simply connected type, 55 semisimple rank, 155, 170 series of finite groups of Lie type, 76 Shephard–Todd classification, 260 shift operation on symbols, 302 σ-character table, 99 σ-conjugate, 96 σ-extension, 97 σ-class function, 96 sign representation, 221 similarity of symbols, 302 simple algebraic group, 10 simply connected covering, 62 Singer cycle, 71 special character, 281 special linear group, 33 special orthogonal group, 5 special unipotent character, 289 specialisation, 220 spetses, 323 spetsial reflection groups, 282 split Levi subgroup, 213 split torus, 213 Springer correspondence, 196 standard Frobenius map, 40 standard Levi subgroup, 212 standard parabolic subgroup, 212

393 Steinberg character, 251 for disconnected groups, 357 Steinberg map, 42 Sylow d-torus, 259 symbol, 301 addition, 303 cohook, 304 cohook removal, 304 content, 302 core, 303 cuspidal, 303 d-cocore, 305 d-core, 304 defect, 302 degenerate, 303 distinguished, 302 genus, 302 hook, 304 hook removal, 304 rank, 302 reduced, 308 shift operation, 302 similarity, 302 symplectic group, 5 tangent space, 6 Tits system, 13 toric datum, 79 torus, 8 torus of type w, 70, 80, 127 transitivity of R G L , 215 of RLG , 239 twisted induction, 238 twisting operator, 296 uniform almost character, 142 uniform criterion, 226, 266 uniform function, 117 on disconnected groups, 358 uniform pairs, 196 uniform projection, 118 unipotent almost characters, 293 unipotent character, 122 unipotent characters rationality of, 311, 323 and automorphisms, 325 families of, 287 in classical types, 305 in exceptional types, 317 in type A, 297 of -defect 0, 298, 308 of GO±2n (q), 358

394 of disconnected groups, 357 special, 289 unipotent classes, 114 unipotent element, 8 unipotent principal series, 228 unipotent radical, 9 unipotent support, 189

INDEX unipotent uniform almost characters, 150 virtual character, 93 weight spaces, 12 weights, 12 Weyl group, 13, 15 Zariski topology, 2