Hecke Algebras with Unequal Parameters [New ed.] 0821833561, 9780821833568

Table of contents :
[CRMMS18] G. Lusztig, Hecke Algebras with Unequal Parameters, 2003
Contents
Introduction
Chapter 1. Coxeter Groups
Chapter 2. Partial Order on W
Chapter 3. The Algebra H
Chapter 4. The Bar Operator
Chapter 5. The Elements c_w
Chapter 6. Left or Right Multiplication by c_s
Chapter 7. Dihedral Groups
Chapter 8. Cells
Chapter 9. Cosets of Parabolic Subgroups
Chapter 10. Inversion
Chapter 11. The Longest Element for a Finite W
Chapter 12. Examples of Elements D_w
Chapter 13. The Function a
Chapter 14. Conjectures
Chapter 15. Example: The Split Case
Chapter 16. Example: The Quasisplit Case
Chapter 17. Example: The Infinite Dihedral Case
Chapter 18. The Ring J
Chapter 19. Algebras with Trace Form
Chapter 20. The Function a_E
Chapter 21. Study of a Left Cell
Chapter 22. Constructible Representations
Chapter 23. Two-Sided Cells
Chapter 24. Virtual Cells
Chapter 25. Relative Coxeter Groups
Chapter 26. Representations
Chapter 27. A New Realization of Hecke Algebras
Bibliography
[Bé] R. Bédard, Cells for two Coxeter groups, Comm. Algebra 14 (1986), 1253-1286.
[Br] K. Bremke, On generalized cells in affine Weyl groups, J. Algebra 191 (1997), 149-173.
[BM] K. Bremke and G. Malle, Reduced words and a length function for G(e,1,n), Indag. Math. (N.S.) 8 (1997), 453-469.
[DL] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), 103-161.
[Ge] M. Geck, Constructible characters, leading coefficients and left cells for finite Coxeter groups with unequal parameters, Represent. Theory 6 (2002), 1-30.
1. Introduction
2. On the generalized left cells of W
3. Generalized a-invariants and leading coefficients
4. Orthogonal representations and leading coefficients
5. Constructible characters and leading coefficients in type Bn
6. Cells and leading coefficients in type I2(m)
7. Cells and leading coefficients in type F4
References
[Ho] P. N. Hoefsmit, Representations of Hecke algebras of finite groups with BN-pairs of classical type, Ph.D. Thesis, Univ. of British Columbia, Vancouver, BC, 1974.
[Iw] N Iwahori, On the structure of the Hecke ring of a Chevalley group over a finite field, J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), 215-236.
[IM] N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke ring of p-adic Chevalley groups, Inst. Hautes Etudes Sci. Publ. Math. 25 (1965), 5-48.
[KL1] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184.
[KL2] D. Kazhdan and G. Lusztig, Schubert varieties and Poincare duality, Geometry of the Laplace Operator (Honolulu, 1979), Proc. Symp. Pure Math., vol. 36, Amer. Math. Soc., Providence, RI, 1980, pp. 185-203.
[Lu1] G. Lusztig, Coxeter orbits and eigenspaces of Frobenius, Invent. Math. 28 (1976), 101-159.
[Lu2] G. Lusztig, Irreducible representations of finite classical groups, Invent. Math. 43 (1977), 125-175.
[Lu3] G. Lusztig, Left cells in Weyl groups, Lie Group Representations. I, Lecture Notes in Math., vol. 1024, Springer, 1983, pp. 99-111.
[Lu4] G. Lusztig, Some examples of square integrable representations of semisimple p-adic groups, Trans. Amer. Math. Soc. 227 (1983), 623-653.
[Lu6] G. Lusztig, Cells in affine Weyl groups, Algebraic Groups and Related Topics (Kyoto Nagoya, 1983), Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 255-287.
[Lu8] G. Lusztig, Cells in affine Weyl groups. II, J. Algebra 109 (1987), 536-548.
[Lu9] G. Lusztig, Introduction to character sheaves, The Arcata Conference on Representations of Finite Groups (Arcata, CA, 1986), Proc. Symp. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 165-180.
[Lu10] G. Lusztig, Intersection cohomology methods in representation theory, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, pp. 155-174.
[Lu11] G. Lusztig, Classification of unipotent representations of simple p-adic groups, Internat. Math. Res. Notices (1995), 517-589.
[Lu12] G. Lusztig, Lectures on Hecke algebras with unequal parameters, MIT Lectures, 1999; math.RT 0108172.
[Lu13] G. Lusztig, Classification of unipotent representations of simple p-adic groups. II, Represent. Theory 6 (2002), 243-289; math.RT 0111248.

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Hecke Algebras with Unequal Parameters

Volume 18

CRM MONOGRAPH SERIES Centre de Recherches Mathematiques Universite de Montreal

Hecke Algebras with Unequal Parameters G. Lusztig The Centre de Recherches Mathematiques (CRM) of the Universite de Montreal was created in 1968 to promote research in pure and applied mathematics and related disciplines. Among its activities are special theme years, summer schools, workshops, postdoctoral programs, and publishing. The CRM is supported by the Universite de Montreal, the Province of Quebec (FCAR}, and the Natural Sciences and Engineering Research Council of Canada. It is affiliated with the Institut des Sciences Mathematiques (ISM) of Montreal, whose constituent members are Concordia University, McGill University, the Universite de Montreal, the Universite du Quebec a Montreal, and the Ecole Polytechnique. The CRM may be reached on the Web at www.crm.umontreal.ca.

American Mathematical Society Providence, Rhode Island USA

The production of this volume was supported in part by the Fonds quebecois de la recherche sur la nature et les technologies (FQRNT) and the Natural Sciences and Engineering Research Council of Canada (NSERC). The work of G. Lusztig was supported by the National Science Foundation.

2000 Mathematics Subject Classification. Primary 20C08.

Library of Congress Cataloging-in-Publication Data Lusztig, George, 1946Hecke algebras with unequal parameters / George Lusztig. p. cm. - (CRM monograph series, ISSN 1065-8599; v. 18) ISBN 0-8218-3356-1 (alk. paper) Includes bibliographical references. 1. Hecke algebras. 2. Linear algebraic groups. 3. Representations of groups. II. Series. Q,A.179.L88 2003 512'.55-dc21

I. Title.

2002038468

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissien©ams. erg.

© 2003 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America.

I§ The paper

used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. This volume was submitted to the American Mathematical Society in camera ready form by the Centre de Recherches Mathematiques. Visit the AMS home page at http://www. ams. erg/ 10987654321

08 07 06 05 04 03

Contents Introduction

1

Chapter 1.

Coxeter Groups

Chapter 2.

Partial Order on W

19

Chapter 3.

The Algebra

23

Chapter 4.

The Bar Operator

27

Chapter 5.

The Elements

31

Chapter 6.

Left or Right Multiplication by

Chapter 7.

Dihedral Groups

37

Chapter 8.

Cells

43

Chapter 9.

Cosets of Parabolic Subgroups

47

9

7-{

Cw C8

35

Chapter 10.

Inversion

53

Chapter 11.

The Longest Element for a Finite W

57

Chapter 12.

Examples of Elements

61

Chapter 13.

The Function a

63

Chapter 14.

Conjectures

67

Chapter 15.

Example: The Split Case

73

Chapter 16.

Example: The Quasisplit Case

77

Chapter 17.

Example: The Infinite Dihedral Case

81

Chapter 18.

The Ring J

85

Chapter 19.

Algebras with Trace Form

91

Chapter 20.

The Function aE

93

Chapter 21.

Study of a Left Cell

101

Chapter 22.

Constructible Representations

105

Chapter 23.

Two-Sided Cells

117

Dw

V

CONTENTS

vi

Chapter 24.

Virtual Cells

121

Chapter 25.

Relative Coxeter Groups

125

Chapter 26.

Representations

127

Chapter 27.

A New Realization of Hecke Algebras

129

Bibliography

135

Introduction These notes are an expanded version of the Aisenstadt lectures given at the CRM, Universite de Montreal, in May/ June 2002; they also include material from lectures given at MIT during the Fall of 1999 [Lu12]. I wish to thank Jacques Hurtubise for inviting me to give the Aisenstadt lectures. Hecke algebras arise as endomorphism algebras of representations of groups induced by representations of subgroups. In these notes we are mainly interested in a particular kind of Hecke algebras, which arise in the representation theory of reductive algebraic groups over finite or p-adic fields (see 0.3, 0.6). These Hecke algebras are specializations of certain algebras (known as Iwahori-Hecke algebras) which can be defined without reference to algebraic groups, namely by explicit generators and relations (see 3.2) in terms of a Coxeter group W (see 3.1) and a weight function L: W --+ Z (see 3.1), that is, a weighted Coxeter group. An Iwahori-Hecke algebra is completely specified by a weighted Coxeter graph, that is, the Coxeter graph of W (see 1.1) where for each vertex we specify the value of L at the corresponding simple reflection. A particularly simple kind of Iwahori-Hecke algebras corresponds to the case where the weight function is constant on the set of simple reflections (equal parameter case). In this case one has the theory of the "new basis" [KLl] and cells [KLl, Lu6, Lu8]. The main goal of these notes is to try to extend as much as possible the theory of the new basis to the general case (of not necessarily equal parameters). We give a number of conjectures for what should happen in the general case and we present some evidence for these conjectures. We now review the contents of these notes. Chapter 1 introduces Coxeter groups following [Bo]. We also give a realization of the classical affine Weyl groups as periodic permutations of Z following an idea of [Lu4]. Chapter 2 contains some standard results on the partial order on a Coxeter group. In Chapter 3 we introduce the Iwahori-Hecke algebra attached to a weighted Coxeter group. Useful references for this are [Bo, GP]. In Chapter 4 we define the bar operator following [KL 1]. This is used in Chapter 5 to define the "new basis" (cw) of an Iwahori-Hecke algebra following [KLl] for equal parameters and [Lu3] in general. In Chapter 6 we study some multiplicative properties of the new basis, following [KLl] and [Lu3]. In Chapter 7 we compute explicitly the "new basis" in the case of dihedral groups. In Chapter 8 we define left, right and two-sided cells. In Chapter 9 we study the behavior of the new basis in relation to a given parabolic subgroup. In Chapter 10, Chapter 12 we study a "basis" dual to the new basis. In Chapter 11 we consider the case of finite Coxeter groups. In Chapter 13 we study the function a on certain weighted Coxeter groups following an idea from [Lu6]. In Chapter 14 we present a list of conjectures concerning cells and the function a and we show that they can be deduced from a much shorter list

INTRODUCTION

2

of conjectures. These conjectures are established in a "split case" in Chapter 15 (following [Lu8]), in a "quasisplit case" in Chapter 16 and for an infinite dihedral group in Chapter 17. Note that in the first two cases the proof requires arguments from intersection cohomology while in the third case the argument is computational. In Chapter 18, assuming the truth of these conjectures we develop the theory of "asymptotic Hecke algebras" J in the weighted case, following an idea from [Lu8]. Chapter 19, Chapter 20, Chapter 21 (where W is assumed to be a Weyl group) are in preparation for Chapter 22 where the class of constructible representations of W is introduced and studied in the weighted case (conjecturally these are the representations of W carried by left cells), for Chapter 23 where two-sided cells are discussed and for Chapter 24 where certain virtual representations of W ( "virtual cells") are discussed. In Chapter 25 we discuss the weighted Coxeter groups which arise in the examples 0.3 and 0.6. We formulate a conjecture (25.3) which relates the two-sided cells of such a weighted Coxeter group to the two-sided cells of a larger Coxeter group with the weight function given by the length. In Chapter 26 we state (following [Lul3]) the classification of irreducible representations of Hecke algebras of the type discussed in 0.6 in terms of the geometry of the dual group. In Chapter 27 we give a new realization of a Hecke algebra as in 0.3 or 0.6 as a space of functions on the rational points of an algebraic variety defined over lF q. This leads us to a (partly conjectural) geometrical interpretation of the coefficients Py,w of the new basis of the Hecke algebra in terms of intersection cohomology, generalizing the results of [KL2]. We expect that this geometrical interpretation should play a role in the proof of the conjectures in Chapter 14 in the cases arising from algebraic groups as in 0.3, 0.6. 0.1. In 0.1-0.8 we give a survey of the theory of Hecke algebras arising from red.uctive groups. Let r be a group acting transitively on a set X. If E is a f-equivariant Cvector bundle over X (with discrete topology) then the fibre Ex of E at x E X is naturally a representation of r x = {g E r; gx = x}. Moreover, for x E X, E f---+ Ex is an equivalence from the category of f-equivariant vector bundles on X of finite dimension and that of finite-dimensional r x-modules over (C. Let E be a f-equivariant C-vector bundle of finite dimension over X. Then r acts naturally on the vector space ffixEX Ex. (This is the representation of r induced by the representation of r x on Ex, for any x EX.) The C-algebra H

= H(r,X,E) = Endr ( EB Ex) xEX

is called the Hecke algebra. The image of the obvious embedding

H

IJHom(Ex,Ex,),

C

f---+

(¢~,)(x,x')EXxX

(x,x')EXxX consists of all (!!,) E TI(x,x')EXxX Hom(Ex, Ex') such that • for any x E X we have

J;, =

0 for all but finitely many x' E X;

• for any g Er and any (x, x') EX x X, the compositions Ex

Ex .!!..., E 9

1::, x ---, Egx'

coincide.

~

Ex' .!!..., Egx',

INTRODUCTION

3

For any r-orbit C in Xx X we set

He= H(r,X,E)e = {¢ E H;q'>~,-/- 0 Then He

*

(x',x) EC}.

= 0 unless C is finitary in the following sense:

• for some (or any) x EX, the set {x' EX; (x',x) EC} is finite, in which case He-"'--+ Homrxnrx,(Ex,Ex,), ¢ i--+ ¢~, for (x', x) EC. Moreover,

(a)

H=

E9He. e finitary

0.2. To explain how Hecke algebras arise from reductive algebraic groups we need the notion of "unipotent cuspidal representation." Let p be a prime number and let lF be an algebraic closure of the finite field with p elements. Let q be a power of p and let lF q be the subfield of lF with q elements. Let G be a connected reductive algebraic group over lF with a fixed lF q structure and let F: G --+ G be the corresponding Frobenius map. We refer to [DL] for the notion of unipotent cuspidal representation of the finite group GF = G (lF q). We will only give here the definition assuming that q is sufficiently large. Let E be an irreducible representation over 2. We say that W is irreducible if this graph is connected. It is easy to see that in general, W is naturally a product of irreducible Coxeter groups, corresponding to the connected components of S. In the setup of 1.3, let ( , ) : E x E --+ JR. be the symmetric JR.-bilinear form given by (e 8 ,e8 ,) = -cos(1rjm 8 , 8 1). Then cr(w): E--+ E preserves (,) for any wEW.

We say that W is tame if ( e, e) 2: 0 for any e E E. It is easy to see that, if W is finite then W is tame. We say that Wis integral if, for any s =f:. s' in S, we have 4cos 2 (1r/ms,s') EN (or equivalently ms,s' E {2, 3, 4, 6, oo}). We will be mainly interested in the case where Wis tame. The tame, irreducible W are of three kinds: (a) finite, integral; (b) finite, nonintegral; (c) tame, infinite (and automatically integral). 1.12. Fork E Z define Pk: Z--+ Z by pk(z) = z + k. Let n 2'. 2. Let W be the group of all permutations er: Z --+ Z such that crpn = PnCT. Define x: W --+ Z by x(cr) = ~kEX(cr(k) - k) where Xis a set of representatives for the residue classes mod n in Z. One checks that x does not depend on the choice of X and x is a group homomorphism with image nZ. Now W' = ker(x) is generated by { sm; m E Z/nZ} where Sm: Z --+ Z is defined by

sm(z)=z+l

ifz=mmodn,

sm(z) = z - 1

if z = m

sm(z) = z

for all other z E Z.

+ 1 mod n,

It is a Coxeter group on these generators, said to be of type An-1· (This description of W' appears in [Lu4].) For n 2'. 3, m, m' E Z/nZ are joined by a single edge in the Coxeter graph if m - m' = 1 mod n and are not joined otherwise. For n = 2, 0, 1 E Z/2Z are joined by a quadruple edge in the Coxeter graph. The length function on W' is given by

1. COXETER GROUPS

14

where, for a E W',

Ya= {(i,j) E Z x Z;i < j,a(i) > a(j)} and Ya/Tn is the (finite) set of orbits of Tn: Ya-+ Ya, (i,j)' I--+ (i + n,j + n). 1.13. Assume now that n = 2p?: 4, where p EN. Let W be the subgroup of W consisting of all a E W that commute with the involution Z -+ Z, z 1--+ 1 - z. We compute x(a) for a E W, taking X = {-(p -1), ... , -1, 0, 1, 2, ... ,p}:

x(a)

L

=

(a(k) - k)

L

=

L

+

kE[l,p]

(a(l - k) - (1- k))

kE(l,p]

L

(a(k)-k)+

kE(l,p]

(1-a(k)-(1-k))=O.

kE[l,p]

Thus, Wis a subgroup of W'. Now Wis generated bys~, s~, ... , s~ where

It is a Coxeter group on these generators, said to be of type Cp. The Coxeter graph is

•=•-•-···-•=• with vertices corresponding to 0, 1, 2, ... , p - 1, p. Let a E W. We have a partition Ya= Y,;1 LJ Y; where

Y1 = {(i,j) E Z x Z;i < j,a(i) > a(j),i+ j

# 1 mod 2p},

Y; = {(i,j) E Z x Z;i < j,a(i) > a(j),i+j = 1 mod 2p}. Now Y; /rn is the fixed point set of the involution of Ya/Tn induced by the involution (i,j) 1--+ (1 - j, 1 - i) of Ya. Hence we have ~(Ya/Tn) = 2l 0 (a) + l 1 (a) where zo (a) = U(Y,;1 /rn) /2, l 1 ( a) = U(Y; /rn) are integers. Now (i, j) 1--+ ( i, (i + j - 1) /2p) is a bijection of with

Y;

{(i, h)

E

Z x Z; 2i < 1 + 2ph, 2a(i) > 1 + 2ph} = {(i, h)

It follows that z1 (a)

E

Z x Z; i S:. ph < a(i)}.

I: J(i)

=

iE[l-p,p] i l(sy). Then sy < z. We can assume that sy < y (otherwise the result is trivial). We can find t ET such that sy < tsy :::; z 19

2. PARTIAL ORDER ON W

20

and l(tsy) = l(sy) + 1. If t = s, then y::; z and we are done. Hence we may assume that t # s. We show that y < stsy.

(a)

Assume that (a) does not hold. Then y, tsy, sy, stsy have lengths q + 1, q + 1, q, q. We can find a reduced expression y = ss 1s 2 ···Sq- Since l(stsy) < l(y), we see from 2.2 that either sts = ss 1 ···Si··· s 1s for some i E [1, q] or sts = s. (This last case has been excluded.)· It follows that tsy

=

S1

···Si··· S1SSS1S2 ···Sq

=

S1 · · · Si-lSi+l

···Sq.

Thus, l(tsy) ::; q - 1, a contradiction. Thus, (a) holds. Let y' = stsy. We have sy' ::; z ::; sz and Z( z) - l ( sy') < l ( z) - l ( sy). By the induction hypothesis, y' ::; sz. We have y < y' by (a), hence y::; sz. The lemma is proved. D 2.4. PROPOSITION. The following three conditions on y, w E W are equivalent:

(i) y ::; w; (ii) for any reduced expression w = s 1 s 2 · · · Sq there exists a subsequence i 1 < i 2 < · · · < ir of 1, 2, ... , q such that y = Si 1 Si 2 ···Sir, r = l(y); (iii) there exists a reduced expression w = s 1s 2 ···sq and a subsequence i 1 < i 2 < · · · < ir of 1, 2, ... , q such that Y = Si 1 Si 2 ···Sir. Proof of (i) ===} (ii). We may assume that y < w. Let y = Yo, Y1, Y2, ... , Yn = w be as in 2.1. Let w = s 1 s 2 ···Sq be a reduced expression. Since Yn-1Y;,, 1 E T, l(Yn-d = l(yn) - 1, we see from 2.2 that there exists i E [1, q] such that Yn-1Y;,, 1 = s1s2 ···Si··· s2s1 hence Yn-1 = s1s2 · · · Si-lsi+l ···sq (a reduced expression). Similarly, since Yn-2Y;} 1 E T, l(Yn-2) = l(Yn-1) - 1, we see from 2.2 (applied to Yn-1) that there exists j E [1, q] - {i} such that Yn-2 equals

(depending on whether i < j or i > j). Continuing in this way we see that y is of the required form. The proof of (ii) ===} (iii) is trivial. Proof of (iii) ===} (i). Assume that w = s 1s 2 ···sq (reduced expression) and y = Si 1 Si 2 ···Sir where i1 < i2 < · · · < ir is a subsequence of 1, 2, ... , q. We argue by induction on q. If q = 0 there is nothing to prove. Now assume q > 0. If i 1 > 1, then the induction hypothesis is applicable to y, w' = s2 .•. Sq and yields y ::; w'. But w' ::; w hence y ::; w. If i 1 = 1 then the induction hypothesis is applicable to y' = Si 2 • • • sir, w' = s2 · · · sq and yields y' ::; w'. Thus, s 1 y::; s 1 w < w. By 2.3 we then have y::; w. The proposition is proved. D 2.5. COROLLARY. Let y, z E W and lets E 8.

(a) Assume that sz < z. Then y::; z (b) Assume that y < sy. Then y ::; z

ht such that w E W.

T} = -Ts-l for any s ES. We have TJ = sgn(w)T,:-! 1 for any

CHAPTER 4

The Bar Operator 4.1. = Ts s1s2 ···sq Let-:

Ts-l

We preserve the setup of 3.1. Fors ES, the element Ts E 7-l is invertible: - (Vs - v; 1). It follows that Tw is invertible for each W E W; if W = is a reduced expression in W, then T;;; 1 = Ts~ 1 · · · Ts-;, 1 Ts-,_ 1 . A--, A be the ring involution which takes vn to v-n for any n E Z.

4.2. LEMMA. (a) There is a unique ring homomorphism-= 7-l --, 7-l which is A-semilinear with respect to-= A--, A and satisfies Ts = Ts-l for any s ES. (b) This homomorphism is involutive. It takes Tw to T;;), for any w E W. The following identities can be deduced easily from 3.2(a), (b): (Ts-l - v_;- 1)(Ts-l +Vs)= 0

T-1T-1T-1 ... s

s1

s

=

for SES, T-1T-1T-1 ... s'

s

s'

(both products have ms,s' factors) for any s -/=- s' in S such that ms,s' < oo; (a) follows. We prove (b). Let s E S. Applying - to T/fs = l gives 7\Ts = l. We have also TsTs = l hence Ts = Ts. It follows that the square of - is 1. The second assertion of (b) is immediate. The lemma is proved. D 4.3.

For any w E W we can write uniquely T w = LyEW ry,wTy where ry,w E

A are zero for all but finitely many y. 4.4. LEMMA. Let w E W ands ES be such that w > sw. For y E W we have ry,w ry,w

= =

rsy,sw rsy,sw

if sy

+ (vs

- v; 1 )ry,sw

if sy

< >

Y, y.

We have Tw

= Ts- 1 Tsw =

(Ts - (Vs -

v,;- 1 )) L ry,swTy y

=L y

=

ry,swTsy - L(vs - v; 1 )ry,swTy y

L Tsy,swTy - L (Vs y

L

+

(vs -v; 1 )ry,swTy

y;syy

The lemma follows.

D

4.5. LEMMA. For any y, w we have ry,w

= sgn(yw)ry,w·

We argue by induction on l(w). If l(w) = 0, then w = l and the result is obvious. Assume now that l(w) ~ 1. We can finds ES such that w > sw. Assume 27

4. THE BAR OPERATOR

28

first that sy < y. From 4.4 we see, using the induction hypothesis, that ry,w

= fsy,sw = sgn(sysw)rsy,sw = sgn(yw)ry,w

Assume next that sy > y. From 4.4 we see, using the induction hypothesis, that fy,w = fsy,sw

+ (v_;- 1 -

Vs)ry,sw

= sgn(sysw)rsy,sw + (v_;- 1 - Vs) sgn(ysw)ry,sw = sgn(yw)(rsy,sw + (vs - v; 1 )ry,sw) = sgn(yw)ry,w· D

The lemma is proved. 4.6.

LEMMA.

For any x, z E W we have I:y rx,yry,z = Ox,z·

Using the fact that - is an involution, we have Tz

= 1\ =

Ly ry,zTy = Ly ry,/f°y =LL ry,zfx,yTx. y X

We now compare the coefficients of Tx on both sides. The lemma follows.

D

4.7. PROPOSITION. Let y,w E w. (a) If ry,w -=/- 0, then y :S; w. (b) Assume that L(s) > 0 for alls ES. If y :S; w, then r y,w = VL(w)-L(y) mod VL(w)-L(y)-lz[v-l] , ry,w = sgn(yw)v-L(w)+L(y) mod V-L(w)+L(y)+lz[v].

(c) Without assumption on L , r y,w E vL(w)-L(y)z[v 2 , v- 2 ] . We prove (a) by induction on l (w). If l (w) = 0 then w = 1 and the result is obvious. Assume now that l(w) ?: l. We can find s E S such that w > sw. Assume first that sy < y. From 4.4 we see that rsy,sw -=/- 0 hence, by the induction hypothesis, sy :S; sw. Thus sy :S; sw < w and, by 2.3, we deduce y :S; w. Assume next that sy > y. From 4.4 we see that either rsy,sw -=/- 0 or ry,sw -=/- 0 hence, by the induction hypothesis, sy :S; sw or y :S; sw. Combining this with y < sy and sw < w we see that y :S; w. This proves (a). We prove the first assertion of (b) by induction on l(w). If l(w) = 0 then w = 1 and the result is obvious. Assume now that l(w)?: l. We can finds ES such that w > sw. Assume first that sy < y. Then we have also sy y. From y < sy, y :S; w we deduce using 2.5(b) that y :S; sw. By the induction hypothesis, we have ry,sw = vL(sw)-L(y) + strictly lower powers. Hence 1 )r = vL(s)vL(sw)-L(y) (v s - vs y,sw

+ strictly lower

= vL(w)-L(y) + strictly lower

powers

powers.

On the other hand, if sy :S; sw, then by the induction hypothesis,

+ strictly lower powers = vL(w)-L(y)- 2 L(s) + strictly lower powers

r 8 y,sw = vL(sw)-L(sy)

4. THE BAR OPERATOR

29

while if sy 1:. sw then rsy,sw = 0 by (a). Thus, in ry,w = T8 y,sw+(v 8 -v; 1 )ry,sw, the term r sy,sw contributes only powers of v which are strictly smaller than L( w) - L(y) hence ry,w = vL(w)-L(y) + strictly lower powers. This proves the first assertion of (b). The second assertion of (b) follows from the first using 4.5. We prove (c) by induction on l(w). If l(w) = 0 then w = 1 and the result is obvious. Assume now that l(w) 2 1. We can finds ES such that w > sw. Assume first that sy < y. By the induction hypothesis, Ty,w = Tsy,sw E VL(sw)-L(sy)z[v 2,v- 2] = VL(w)-L(y)z[v 2,v- 2] as required. Assume next that sy r y,w

= r sy,sw + (v s -v-l)ry,sw 8

> y. By the induction hypothesis,

E VL(sw)-L(sy)z[v2 , V-2] +vL(s)VL(sw)-L(y)z[v2 , V-2]

= VL(w)-L(y)z[v2, V-2], as required. The proposition is proved. 4.8.

PROPOSITION

D

(D. N. Verma). For x < z in W we have Ly;x:S:y:S:z sgn(y) =

0. Using 4.5 we can rewrite 4.6 (in our case) in the form (a)

Lsgn(xy)rx,yry,z

-"

= 0.

y

Here we may restrict the summation to y such that x :S: y :S: z. In the rest of the proof we shall take L = l. Then 4.7(b) holds and we see that if x :S: y :S: z, then r x,y r y,z = vl(y)-l(x)vl(z)-l(y)

+ strictly lower powers of v.

Hence (a) states that

L

sgn(xy)vl(z)-l(x)

+ strictly lower powers of v

is 0.

y;x:S:y:S:z

In particular Ly;x:S:y:S:z sgn(xy) 4.9.

(a)

= 0.

The proposition is proved.

Now-= 1{---+ 1{ commutes with hf----+ hb. Hence ry-1,w-1

= ry,w

On the other hand, it is clear that - : 1{

for any y,w E W. ---+ 1{

and t : 1{

---+ 1{

commute.

D

CHAPTER 5

The Elements 5.1.

Cw

We preserve the setup of 3.1. For any n E Z let

E9zvm,

A:::;n=

A:c:n

=

m;m:'.Sn

E9zvm,

An =

m;m 0. Since Pz' = 'i:;x' hs,x,x' hx, ,y,z', there exists x' such that Kn(hs,x,x'hx',y,z')-/=- 0. Since sx > x, we see from 6.5 that hs,x,x' E Z hence Kn(hs x x1hx 1 y z = hs x x1Kn(hx 1 y z ' ' ' ' ' ' '' Thus we have nn(hx',y,z')-/=- 0. This proves (P) in our case. 1 )

1 ).

D

15.6. Since (P) and Pl-P3 are known, we see that Pl-P11 and P13, P14 hold in our case (see Chapter 14). The same arguments can be applied to W 1 where IC S, hence Pl-P11 and P13, P14 hold for W 1 . By 14.12, P12 holds for W. Thus, Pl-P14 hold for W.

15. EXAMPLE: THE SPLIT CASE

75

15.7. Proof of Pl5. By 14.15, we see that it is enough to prove 14.15(a). Let y, w, s, s' be as in 14.15(a). In our case, by 6.5, the equation in 14.15(a) involves only integers, hence it is enough to prove it after specializing v = v'. If in 14.15 we specialize v = v', then the left and right module structures in 14.15 clearly commute, since the left and right regular representations of 'H commute. Hence the coefficient of ey in ( (C 8 ew )cs' - c8 (ewes,)) v=v' is 0. By the computation in 14.15, this coefficient is

L

hw,s' ,y' hs,y' ,y -

y' ;y' s' 0. Hence tu,txtuty E aty-ity+J+. Since ty-dy has a coefficient 1 and the other coefficients are 2: 0, it follows that tu,txtuty -=/- 0. Thus, txtuty -=/- 0. We see that the first and second conditions are equivalent. The proposition is proved. D 18.5. Assume now that we are in the setup of 7.1 with m = oo and L 2 > L 1 . From the formulas in 17.3, 17.4 we can determine the multiplication table of J. We find t22k+l t22k'+1

=

L L

t22k+2k'+l-4u,

t12k+3 ti2k' +3

=

uE[O,k] t22k+1 t22k'+2

=

t22k+2k'+2-4u,

t12k+3 t12k' +2

=

uE[O,k]

t22k+2st l2k 1 +s

=

L L

=

t22k+2k 1 +2-4u'

t22k+2 t12k' +2

=

t12k+2k' +2-4u,

=

t22k+2k' +1-4-u,

uE[O,k]

t12k+2k' +2-4u,

t12k+2 t22k' +2

uE[O,k] h1h1

t12k+2k' +3-4u,

uE[O,k]

uE[O,k] t12k+2 t22k' +1

L L L L

uE[O,k]

=

t12k+2k' +3-4u,

uE[O,k]

titi=ti;

h1,

here k, k' 2: 0 and k = min(k, k'). All other products are 0. Let R be the free Abelian group with basis (bk)kEN· We regard Ras a commutative ring with multiplication

uE[O,min(k,k')]

Let Jo = ~wEW-{l,li} 7!..tw. The formulas above show that J = Jo EB 7l..t1 EB 7!..ti 1 (direct sum of rings) and that the ring J 0 is isomorphic to the ring of 2 x 2 matrices with entries in R, via the isomorphism defined by:

Note that R is canonically isomorphic to the representation ring of SL 2 ((C) with its canonical basis consisting of irreducible representations. 18.6. Assume that we are in the setup of 7.1 with m = oo and L 2 methods similar (but simpler) to those of §17 and 18.5, we find t22k+1 t22k'+1

=

L

= L1.

By

t22k+2k'+1-2u.

uE[0,2 min(k,k')]

Let J1 be the subring of J generated by t 22 k+i, k E N. While, in 18.5, the analogue of J 1 was isomorphic to R as a ring with basis, in the present case, J 1 is canonically isomorphic to R', the subgroup of R generated by bk with k even. (Note that R' is a subring of R, naturally isomorphic to the representation ring of PGL 2 ((C).)

18. THE RING J

88

18. 7. In the setup of 7.1 with m = 4 and £ 2 = 2, £ 1 = 1 (a special case of the situation in Chapter 16), we have

(direct sum of rings) where Jo is the subgroup of J generated by t2 1 , t2 2 , ti 2 , t1 3 • The ring J0 is isomorphic to the ring of 2 x 2 matrices with entries in Z, via the isomorphism defined by:

Moreover, ti, ti 1 , t2 4 are idempotent. On the other hand,

Notice the minus sign! (It is a special case of the computation in 7.8.) 18.8. For any z E W we set nz = nd where dis the unique element of 'D such that d rv .c z- 1 and nd = ±1 is as in 14.l(a), see P5. Note that z f-> nz is constant on right cells.

18.9.

THEOREM.

The A-linear map¢: 1i--+ JA =A® J given by

L

¢(cl)=

hx,d,znztz

(x E W)

zEW

dE'D a(d)=a(z)

is a homomorphism of A-algebras with l. Consider the equality

(a) w

w

(see P15) with a(x 2 ) = a(y) = a. In the left hand side we may assume that ~n w ~.c x 2 hence (by P4) a(y) ;::: a(w) ;::: a(x 2 ), hence a(w) = a. Similarly in the right-hand side we may assume that a(w) = a. Picking the coefficient of v'a in both sides of (a) gives y

(b) w

w

Let x, x' E W. The desired identity ¢(clc!,) = ¢(cl)¢(c!,) is equivalent to

~

,L....,;

hx x' whw dunu

wEW

''

''

=

dE'D

d,d'E'D a(d)=a(z) a(d')=a(z')

a(d)=a'

for any u E W such that a( u)

~ hx 'd' zhx 1 'd'' z'""z ,L....,; 1 ,z', u-1 nznz' z,z'EW

= a'.

In the right hand we may assume that

a(d) = a(z) = a(d') = a(z') = a'

and

nz

=

nu

18. THE RING J

89

(by P8, P4). Hence the right-hand side can be rewritten (using (b)):

L

L

hx 1 ,d',z' hx,d,z"/z,z',u- 1'nu'nz 1 z' EW,d,d' E'D z;a(z)=a' a(d)=a(d')=a(z')=a'

L

L

hx 1 ,d',z' hx,w,u"/d,z',w-l'nu'nz'· z'EW,d,d'E'D w;a(w)=a' a(d)=a(d')=a(z')=a'

By P2, P3, P5, this equals '"""' hx 1 d' z'hx z' u'nu L.....it ' ' ' ' z'EW,d'E'D a(d')=a(z')=a'

'"""' hx x' whw d' u'nu, ~ '' '' wEW,d'E'D a(d')=a'

Thus cp is compatible with multiplication. Next we show that¢ is compatible with the unit elements of the two algebras. An equivalent statement is that for any z E W such that a(z) = a, the sum I:dE'D;a(d)=a h1,d,z'nz equals r'iz if z E 'D and is O if z €/. 'D. This is clear since h1,d,z

= bz,d·

cL

18.10. If we identify the A-modules H and JA via 1--+ nwtw, the obvious left JA-module structure on JA becomes the left JA-module structure on H given by tx

*

ct = L 'Yx,w,z- 1'nw'nzC! zEW

Let Ha = ffiw;a(w)=a Act, H?.a = ffiw;a(w)?.a x, w. For any h E H, w E W we have

(a)

hct

= c/>(h) * ct

Indeed, we may assume that h

(cl)* ct=

= 4.

L

hx,d,z'nztz dE'D,z a(d)=a(z)

Act,

mod

ct E Ha(w) for all

H?.a(w)+l·

dE'D,z,u a(d)=a(w)=a(u) A

as required.

*

* ct

'"""' L...J h x,t,u"/d,w,t-1 nwCut dE'D,t,u a(d)=a(w)=a(u)

L

tx

Using 18.9(b), we have

dE'D,z,u a(d)=a(z)

hx,w,uct u;a(w)=a(u)

We have

=

L

hx,w,u'Yd,w,w-l 'nwCt dE'D,u a(d)=a(w)=a(u)

= clct mod H?.a(w)+l,

18. THE RING J

90

18.11. Let A---+ R be a ring homomorphism of A into a commutative ring R with l. Let 7-lR = R @A 7-l, JR= R 12?A (JA) = R@ J, 7-lR,?_a = R@A 7-l-:::_a, Then extends to a homomorphism of R-algebras c/>R : 7-lR ---+ JR, The JA-module in 18.10 extends to a JA-module structure on 7-{R denoted again by*· From 18.lO(a) we deduce

heL = c/>R(h) * eL

(a)

mod 7-lR,?_a(w)+l

for any h E 7-lR, w E W.

18.12. PROPOSITION. (a) If N is a bound for w, L, then (ker R)N+l = 0. (b) If R = R 0[v,v- 1] where Ro is a commutative ring with 1, vis an indeterminate and A---+ R is the obvious ring homomorphism, then ker c/>R = 0. We prove (a). If h E kerc/>R then by 18.ll(a), we have h1iR,?_a C 7-lR,?_a+l for any a 2 0. Applying this repeatedly, we see that, if h 1 , h 2 , ... , hN+l E 7-l, we have h1h2 · · · hN+l E 7-lR,?_N+l = 0. This proves (a). We prove (b). Leth= LxPxcl E kerc/>R where Px ER. Assume that h-=/=- 0. Then Px -=/=- 0 for some x. We can find a 2 0 such that Px -=/=- 0 ===} a(x) 2 a and X = {x E W;px-=/=- O,a(x) = a} is nonempty. We can find b E Z such that Px E vbz[v- 1] for all x EX and such that X' = {x EX; 7rb(Px)-=/=- O} is nonempty. Let x 0 EX'. We can find d ED such that I XO, d,Xo-1 = 1 X 0-1 ,Xo, d-=/=- 0. We have hcdt =

LxPxclet If a(x) > a, then cle~ E 7-lR,?_a+l· Hence he~ = LxEX Pxcle~ mod 7-lR,?_a+l· Since c/>R(h) = 0, from 18.ll(a) we have he~ E mod7-lR,?_a+l· It follows that LxEX Pxele~ E 7-lR,?_a+l· In particular the coefficient of ela in LxEX Pxelc~ is 0. In other words, LxEX Pxhx,d,xo = 0. The coefficient of va+b in the last sum is

~ 7rb(Pxhx 'd' x-0 1 = 7rb(Pxohxo 'd' x-0 1 ,L...it xEX and this is on the one hand O and on the other hand is nonzero since 7rb (Pxo) -=/=- 0 and rx d x-1 -=/=- 0, by the choice of Xo, d. This contradiction completes the proof. D O,

'

0

CHAPTER 19

Algebras with. Trace Form 19.1. Let R be a field and let A be an associative R-algebra with 1 of finite dimension over R. We assume that A is semisimple and split over R and that we are given a trace form on A that is, an R-linear map T: A -----t. R such that (a, a')= T(aa') = T(a'a) is a nondegenerate (symmetric) R-bilinear form (, ) : Ax A -----t R. Note that ( aa', a") = (a, a' a") for all a, a', a" in A. Let Mod A be the category whose objects are left A-modules of finite dimension over R. We write E E Irr A for "E is a simple object of Mod A". Let (ai)iEI be an R-basis of A. Define an R-basis (a~)iEJ of A by (ai, aj) = t5ij· Then

(a) I:i ai 0 a~ EA 0 A is independent of the choice of (ai).

z:=i

19.2. PROPOSITION. (a) We have T(ai)a~ = 1. (b) If EE Irr A, then Li tr(ai, E)a~ is in the center of A. It acts on E as a scalar fe E R times the identity and on E' E Irr A, not isomorphic to E, as zero. Moreover, fe does not depend on the choice of (ai). (c) One can attach uniquely to each E E Irr A a scalar ge E R ( depending only on the isomorphism class of E), so that

Lgetr(a,E)

= T(a) for all a EA,

E

where the sum is taken over all E E Irr A up to isomorphism. (d) For any EE Irr A we have fege = 1. In particular, fe i= 0, ge i= 0. (e) If E, E' E Irr A, then ~i tr(ai, E) tr(a~, E') is fe dimE if E, E' are isomorphic and is 0, otherwise. Let A = ffi~=l An be the decomposition of A as a sum of simple algebras. Let Tn: An -----t R be the restriction of T. Then Tn is a trace form for An, whose associated form is the restriction of ( , ) and (An, An') = 0 for n i= n'. Hence we can choose (ai) so that each ai is contained in some An and then a~ will be contained in the same An as a~. We prove (a). From 19.l(a) we see that I:i T(ai)a~ is independent of the choice of (ai). Hence we may choose (ai) as in the first paragraph of the proof. We are thus reduced to the case where A is simple. In that case the assertion is easily verified. We prove (b). From 19.l(a) we see that I:i tr(ai, E)a~ is independent of the choice of (ai). Hence we may choose (ai) as in the first paragraph of the proof. We are thus reduced to the case where A is simple. In that case the assertion is easily verified. We prove (c). It is enough to note that a f---> tr(a, E) form a basis of the space of R-linear functions A -----t R which vanish on all aa' - a' a and T is such a function. 91

92

19. ALGEBRAS WITH TRACE FORM

We prove (d). We consider the equation in (c) for a sides by a~ and sum over i. Using (a), we obtain

LL gE tr( ai, E)a~ = i

= ai,

we multiply both

LT( ai)a~ = l.

E

Hence LE gE Li tr(ai, E)a~ = l. By (b), the left-hand side acts on a E' E Irr A as a scalar gE, fa, times the identity. This proves (d). (e) follows immediately from (b). The proposition is proved. D 19.3. Now let A' be a semisimple subalgebra of A such that T 1, the restriction of T to A' is a trace form of A'. (We do not assume that the unit element lA, of A' coincides to the unit element 1 of A.) If EE Mod A then lA'E is naturally an object of Mod A'. Hence if E' E Irr A', then the multiplicity [E':lA,E] of E' in lA, E' is well-defined. Note that, if a' EA', then tr(a', lA,E) = tr(a', E).

19.4. LEMMA. Let E' E Irr A'. We have gE, E E Irr A ( up to isomorphism).

= I:E[E':lA,E]gE, sum over all

By the definition of gE,, it is enough to show that (a)

LL[E':l'E]gEtr(a',E') E'

= T(a')

for any a' EA'.

E

Here E' (resp. E) runs over the isomorphism classes of simple objects of ModA' (resp. ModA). The left hand of (a) is

LgE L[E':l'E] tr(a',E') E E' This completes the proof.

= LgEtr(a', l'E) = LgEtr(a',E) = T(a'). E

E D

CHAPTER 20

The Function

aE

20.1. In this section we assume that the assumptions of 18.1 hold and that Wis finite. The results of Chapter 19 will be applied in the following cases.

(a) A = He, R = C. Here A-+ 0. We also assume that a, b are such that W, L satisfies the assumptions of 18.l. Then f E°',/3 is defined in terms of this L. v 22.12. LEMMA (Hoefsmit [Ho]). We have f E;;,/3

= Ha( V2a)H13( V2a)Ga,f3( V2a' V2b)G13,a( V2a' V-2b).

We will rewrite the expression above using the following result. 22.13. LEMMA. Let N be a large integer. We have

H (q) =

'°'·

qL..,,E[l,N-1]

(') 2

Ga,{3(q, y)G13,a(q, Y-1)

=

TI!1 IJhE[l,aN-;+l+i-l](qh -1)/(q -1) Til::C:iN-i+l +i-1] (qhy l) Tif=l I1hE[l,f3N-J+l +J-1] (qhy-l 1) x---------------------~---TI . (q"N-;+1+i-l !y+qf3N-J+l+j-1 1y-l) i,JE[l,N] V 17 V 17

The proof is by induction on n. We omit it. 22.14. PROPOSITION. (a) Ifb' = 0 then f[Al• is equal to 2d where 2d+r is the number of singles in A. If b' > 0 then f[Al• = 1. (b) We have a[A] = AN - EN where iE[l,N+r] jE[l,N]

EN=

1::C:i g[A'l• + g[AIIJ•i the contribution of the other Eis:::> 0 (see 20.13(b)). Hence (d) forces [[A']:[AIJ] = [[A']:[AIIJ] = 1 and [[A']:E] = 0 for all other E in the sum. The lemma follows. D 22.18. LEMMA. [A]® sgn = [A]. (Notation of 22.14.) This can be reduced to a known statement about the symmetric group. We omit the details. 22.19. Let Z be a totally ordered finite set z 1 < z 2 < · · · < ZM. For any r E [O, M] such that r = M mod 2 let Zr be the set of subsets of Z of cardinal (M -r)/2. An involution l: Z---+ Z is said to be r-admissible if the following hold:

(a) l has exactly r fixed points; (b) if M = r, there is no further condition; if M > r, there exist two consecutive elements z, z' of Z such that l( z) = z', l( z') = z and the induced involution of Z - {z,z'} is r-admissible.

22.

CONSTRUCTIBLE REPRESENTATIONS

113

Let Invr(Z) be the set of r-admissible involutions of Z. Toi E Invr(Z) we associate a subset Si of Zr as follows: a subset Y C Z is in Si if it contains exactly one element in each nontrivial i-orbit. Clearly, ~(Si)= 2P 0 where p 0 = (M - r)/2. (In fact, Si is naturally an affine space over the field lF 2 .) 22.20. LEMMA. Assume that Po > 0. Let YE Zr· (a) We can find two consecutive elements z, z' of Z such that exactly one of z, z' is in Y. (b) There exists i E Inv r ( Z) such that Y E Si. (c) Assume that for some k E [O, Po - l], z1 , z 2 , ... , Zk belong to Y but Zk+l (f. Y. Let l be the smallest number such that l > k and zz E Y. There exists i E Invr(Z) such that YES, and i(zz) = Zl-l· We prove (a). Let zk be the smallest element of Y. If k > l then we can take (z, z') = (zk-l, zk)- Hence we may assume that z 1 E Y. Let Zk' be the next smallest element of Y. If k' > 2 then we can take ( z, z') = (Zk' - l , Zk'). Continuing like this we see that we may assume that Y = {z1 , z2 , ... , zp 0 } . Since p 0 < M, we may take (z,z') = (zp 0 ,Zp 0 +1)We prove (b). Let z, z' be as in (a). Let Z' = Z - {z, z'} with the induced order. Let Y' = Y n Z'. If p 0 ::::, 2 then by induction on p0 we may assume that there exists i' E Inv r ( Z') such that Y' E Si'. Extend i' to an involution i of Z by z 1-+ z', z' 1-+ z. Then i E Invr(Z) and YE Sl. If Po= l, define i: Z-+ Z so that z 1-+ z', z' 1-+ z and i = l on Z - {z, z'}. Then i E Invr(Z) and YE Si. We prove (c). We have l ::::, k + 2. Hence zz_ 1 tf. Y. Let (z, z') = (zz-1, zz). We continue as in the proof of (b), except that instead of invoking an induction hypothesis, we invoke (b) itself. D 22.21. Assume that M > r. We consider the graph whose set of vertices is and in which two vertices Y # Y' are joined if there exists i E Inv r (Z) such that Y E Si, Y' E Sl.

Zr

22.22. LEMMA. This graph is connected. We show that any vertex Y = {Zi 1 , Zi 2 , . • • , ziPo} is in the same connected component as Y0 = {z1, z 2 , ... , Zp 0 } . We argue by induction on my = i1 + i2 + · · · + ip 0 • If my = 1 + 2 + · · · + p 0 then Y = Y 0 and there is nothing to prove. Assume now that m > 1 + 2 + · · · + p 0 so that Y # Y0 . Then the assumption of Lemma 22.20(c) is satisfied. Hence we can find I such that z 1 E Y, zz_ 1 (/:. Y and i E Invr(Z) such that YE Si and i(zz) = zz_ 1 . Let Y' = (Y - {zz}) U {zz-1}. Then Y' E Sl hence Y, Y' are joined in our graph. We have my, = my - 1 hence by the induction hypothesis Y', Y0 are in the same connected component. It follows that Y, Y0 are in the same connected component. The lemma is proved. D 22.23.

Assume that b' = 0. Let

Z E M~b;n·

Let Z be the set of singles of Z.

Each set Y E Zr gives rise to a symbol Ay in :'.J\;.1 (Z): the first row of Ay consists of Z - Y and one element in each double of Z; the second row consists of Y and one element in each double of Z. For any i E Invr(Z) we set

c(Z,i) = EBYEs,[Ay] E ModW. 22.24. PROPOSITION. (a) In the setup of 22.23, let i E Invr(Z). Then c(Z, i) E Con(W).

22. CONSTRUCTIBLE REPRESENTATIONS

114

(b) All constructible representations of W are obtained as in (a). We prove (a) by induction on n. If n = 0 the result is clear. Assume that 73:_,?. l. We may assume that O is not a double of Z. Let at be the largest entry of Z. (A) Assume that there exists i, 0 ::; i < t, such that ai does not appear in Z. Then Z is obtained from Z' E M;;b·n-k with n - k < n by increasing each of the k largest entries by a and this set of l~rgest entries is unambiguously defined. The set Z' of singles of Z' is naturally in order preserving bijection with Z. Let L' correspond to l under this bijection. By the induction hypothesis, c(Z', L1 ) E Con(Wn-k)Since, by 22.5, the sign representation sgnk of 6k is constructible, it follows that sgnk~c(Z1 ,L 1 ) E Con(6k x Wn-k). Using 22.17, we have

j~txwn-k (sgnk ~c(Z', L')) = c(Z, l) hence c(Z, L) E Con(W). (B) Assume that there exists i, 0 < i ::; t such that ai is a double of be as in 22.8 (with respect to our t). Then O is not a double of

Z and

Z.

Let Z

the largest

entry of Z is at. Let Z be the set of singles of Z. We have Z = { at- z; z E Z}. Thus Z, Z are naturally in (order reversing) bijection under j r--+ at-j. Let L1 E Invr(Z) correspond to_l under th~ bijection. Since at - ai does not :ppear in

Z,

(A) is

applicabletoZ. Hencec(Z,L') E Con(W). By22.18wehavec(Z,L')®sgn=c(Z,L) hence c(Z, L) E Con(W). (C) Assume that we are not in case (A) and not in case (B). Then Z = {O, a, 2a, ... , ta} = z. We can find ia, (i + l)a in Z such that L interchanges ia, (i + 1)a and induces on Z - {ia, (i + 1)a} an r-admissible involution L1 . We have

Z' = {O, a, 2a, ... , ia, ia, (i + l)a, (i + 2)a, ... , (t - l)a} E M~b;n-k with n - k < n. The set of singles of Z' is Z' = {O, a, 2a, ... , (i - l)a, (i + l)a, ... , (t - l)a}. It is in natural (order preserving) bijection with Z -{ia, (i+ l)a}. Hence Li induces L' E Invr(Z'). By the induction hypothesis we have c(Z', l 1 ) E Con(Wn-k). Hence sgnk ~c(Z', L1 ) E Con(6k x Wn-k) where sgnk is as in (A). By 22.17 we have jltxwn-k (sgnk ~c(Z', L')) = c(Z, l) hence c(Z, L) E Con(W). This proves (a). We prove (b) by induction on n. If n = 0 the result is clear. Assume now that n ?. l. By an argument like the ones used in (B) we see that the class of representations of W obtained in (a) is closed under ® sgn. Therefore, to show that CE Con(W) is obtained in (a), we may assume that C = jtxwn-k (C') for some k > 0 and some C' E Con(6k x Wn-k)- By 22.5 we have C' = E ~ C" where E is a simple 6k-module and C" E Con(Wn-k). Using 22.5(a) we have E = xsk" (sgn ~E') where k' + k" = k, k' > 0 and E' is a simple 6k11-module.

j~:,

Let

C=

j:;,~;'wn-k (E' ® C') E Con(Wn_k'). Then C =

the induction hypothesis,

C is of the form

jK, xwn-k' (sgnk' ®C).

By

described in (a). Using an argument as

22. CONSTRUCTIBLE REPRESENTATIONS

115

in (A) or (C) we deduce that C is of the form described in (a). The proposition is D proved. 22.25.

PROPOSITION.

Assume that b' > 0.

(a) Let EE Irr W. Then EE Con(W). (b) All constructible representations of W are obtained as in (a). We prove (a). We may assume that E = [A] where A E Sy;;b·n does not contain both O and b'. We argue by induction on n. If n = 0 the resi'.ilt is clear. Assume now that n 2': 1. (A) Assume that either ( 1) there exist two entries z, z' of A such that z' - z > a and there is no entry z" of A such that z < z" < z', or (2) there exists an entry z' of A such that z' 2': a and there is no entry z" of A such that z" < z'. Let A' be the symbol obtained from A by subtracting a from each entry z of A such that z 2': z' and leaving the other entries of A unchanged. Then A' E Sy;;b·n-k with n - k < n. By the induction hypothesis, [A'] E Con(Wn-k)- Since, by 22'.5, the sign representation sgnk of lfh is constructible, it follows that sgnk iz;J[A'] E Con(6k x Wn-k)- Using 22.17, we have xwn_Jsgnk iz;J[A']) = [A] hence [A] E Con(W). (B) Assume that there exist two entries z, z' of A such that O < z' - z < a. Lett be the smallest integer such that at+ b' 2': Ai for all i E [1, N + r] and at 2': µj for all j E [1, NJ. Let A E Sy~;i,~,;_-N-r be as in 22.8 with respect to this t. Then A does not contain both O and b'. Now (A) is applicable to A. Hence [A] E Con(W). By 22.18 we have [A]® sgn = [A] hence [A] E Con(W). (C) Assume that we are not in case (A) and not in case (B). Then the entries of A are either 0, a, 2a, ... , ta or b', a+ b', 2a + b', . .. , ta+ b'. This cannot happen for n 2': 1. This proves (a).

jt

The proof of (b) is entirely similar to that of 22.24(b ). The proposition is proved. D

22.26. We now assume that n 2': 2 and that W' = W~ is the kernel of Xn: Wn ----+ ±1 in 22.10. We regard W~ as a Coxeter group with generators s 1 , s 2 , ... , Sn-las in 22.9 ands~= (n -1,n')((n -1)',n) (product of transpositions). Let L: W' ----+ N be the weight function given by L(w) = al(w) for all w. Here a> 0. For A E Sy:;0 we denote by Atr the symbol whose first (resp. second) row is the second (resp.' first) row of A. We then have Atr E Sy:;0 • Z,From the definitions we see that the simple Wn-modules [A], [Atr] have the sa~e restriction to W'; this restriction is a simple W'-module [c!l] if A# Atr and is a direct sum of two nonisomorphic simple W'-modules [Ic!l], [IIc!l] if A= Atr_ In this way we see that the simple W' -modules are naturally in bijection with the set of orbits of the involution of Sy a O·n induced by A f---+ A tr except that each fixed point of this involution corresponds to' two simple W' -modules. Let Z E M!:o·n· Let Z be the set of singles of Z. Assume first that Z # 0. Each set Y E .Z,0 '~ives rise to a symbol Ay in Sy:o;n: the first row of Ay consists of Z - Y and one element in each double of Z; the second row consists of Y and one element in each double of Z. For any i E Inv 0 (Z) we define c(Z, i) E Mod W

116

22. CONSTRUCTIBLE REPRESENTATIONS

by

c(Z, l) EB c(Z, l) =

EB [Ay] E Mod W YES,

Note that Y and Z - Y have the same contribution to the sum. A proof entirely similar to that of 22.24 shows that c(Z, l) E Con(W). Moreover, if Z = 0 and A= Nr E Sy:l;n is defined by 7rN(A) = Z, then [1c!l] E Con(W) and [IIc!l] E Con(W). All constructible representations of W are obtained in this way. 22.27. Assume that Wis of type F 4 and that the values of L: W-----+ Non S are a, a, b, b where a > b > 0. Case l. Assume that a = 2b. There are four simple W-modules p 1 , P2, Ps, pg (subscript equals dimension) with a= 3b. Then P1 EB P2,

P1 EB Ps,

P2 EB pg,

Ps EB pg E Con(W).

(They are obtained by j from the Wr of type B 3 with parameters a, b, b.) The simple W-modules pl, pi have a= 15b and

Pt Pt Pi EB Pt Pi EB Pt Pt EB Pt Pi EB Pi ECon(W).

There are five simple W-modules P12, P16, p5, p~, p4 (subscript equals dimension) with a = 7b. Then p4 EB P16,

P12 EB Pl6 EB P6,

P12 EB Pl6 EB p~ E Con(W).

All 12 simple W-modules other than the 13 listed above, are constructible. All constructible representations of W are thus obtained. Case 2. Assume that a t/:. {b, 2b}. The simple W-modules P12, P16, P6, p~, p4 in Case 1 now have a = 3a + b and p4 EB P16,

P12 EB Pl6 EB P6,

P12 EB Pl6 EB p~ E Con(W).

All 20 simple W-modules other than the 5 listed above, are constructible. All constructible representations of W are thus obtained. 22.28. Assume that W is of type G 2 and that the values of L: W -----+ N on S are a, b where a> b > 0. Let p 2 , p; be the two 2-dimensional simple W-modules. They have a= a and p 2 EB p; is constructible. All 4 simple W-modules other than the 2 listed above, are constructible. All constructible representations of W are thus obtained. 22.29. Let£, be the set of all weight functions L: W-----+ N such that L(s) > 0 for all s E S. We assume that Pl-P15 in Chapter 14 hold for any L E £,. For L, L' E £, we write L rv L' if the constructible representations of W with respect to L are the same as those with respect to L'. This is an equivalence relation on £. From the results in this section we see that any equivalence class for rv contains some L which is attached to some (G, F, P, E) as in 0.3. We expect that the constructible representations of W are exactly the representations of W carried by the left cells of W (for fixed L E £). (For L = l this holds by [Lu7]. For W of type F 4 and general L this holds by [Ge].) This would imply that for L, L' E £, we have L rv L' if and only if the representations of W carried by the left cells of W with respect to L are the same as those with respect to L'.

CHAPTER 23

Two-Sided Cells 23.1. We define a graph 9w as follows. The vertices of 9w are the simple Wmodules up to isomorphism. Two nonisomorphic simple W-modules are joined in 9w if they both appear as components of some constructible representation of W. Let 9w / rv be the set of connected components of 9w. The connected components of 9w are determined explicitly by the results in Chapter 22 for W irreducible. For example, in the setup of 22.4, 22.5 we have 9w = 9w / rv. In the setup of 22.24, 9w/rv is naturally in bijection with Ma,b;n· (Here, 22.22 is used). In the setup of 22.25, we have 9w = 9w/rv. We show that:

(a) if E, E' are in the same connected component of 9w then E '""£R E'. We may assume that both E, E' appear in some constructible representation of W. By 22.2, there exists a left cell r such that [E:[rl] # 0, [E':[rl] # 0. By 21.2, we have [E.:Jit'] # 0, [E.:Jf] # 0. Hence E rv£R E', as desired. 23.2. Let cw be the set of two-sided cells of W, L. Consider the (surjective) map Irr W -----> cw which to E associates the two-sided cell c such that E '""£R x for x E c. By 23.1 this induces a (surjective) map

ww: 9w/rv ___, cw.

(a)

We conjecture that ww is a bijection. This is made plausible by: 23.3.

PROPOSITION.

Assume that W, L is split. Then ww is a bijection.

Let E, E' E Irr W be such that E rv £R E'. By 22.3, we can find constructible representations C, C' such that [E:C] # 0, [E':C'] # 0. By 22.2, we can find left cells r, I'' such that C = [I'], C' = [I'']. Then [E:[rl] # 0, [E':[r'l] # 0. Let d E V n I', d' E V n I''. Since rd = [I'] and [E:[I'l] # 0, we have E '""1:,n d. Similarly, E' rv £R d'. Hence d' rv £R d'. By 18.4( c), there exists u E W such that tdtutd' # 0. (Here we use the splitness assumption.) Note that j r-+ jtutd' is a Jclinear map Jf -----> Jf. This map is nonzero since it takes td to tdtutd, # 0. Thus, HomJc(Jit', Jf) # 0. Using 21.2, we deduce that Homw([r], [I''])# 0. Hence there exists E E Irr W such that E is a component of both [I'] = C and [I''] = C'. Thus, both E, E appear in C and both E, E' appear in C'. Hence E, E' are in the same connected component of 9w. The proposition is proved. D 23.4. Assume now that W, S, L, W, u are as in 16.1, 16.3 and W is an irreducible Weyl group. Let be the set of all u-stable two-sided cells of W. Let c'fy be the set of all

cw

two-sided cells of W which meet W. We have c'fy C 117

cw cw. C

Let f: cw

----->

c'fy

118

23. TWO-SIDED CELLS

be the map which attaches to a two-sided cell of W the unique two-sided cell of W containing it; this map is well-defined by 16.13(a) and is obviously surjective. 23.5. PROPOSITION. In the setup of 23.4, ww is a bijection and is a bijection.

f:

cw-> cw

Since ww, f are surjective, the composition f ww: 9w /"' -> cw is surjective. Hence it is enough to show that this composition is injective. For this it suffices to check one of the two statements below:

(a) tt(Qw/rv) = tt(c~); (b) the composition 9w/"' for x E c) is injective.

~ c~ C cw

L

NEBN (where f'(c)

= (a(x), a(xwo))

Note that the value of the composition (b) at Eis (aE, aEt ).

Case l. W is of type G 2 and W is of type D 4. Then (b) holds: the composition (b) takes distinct values (0, 12), (1, 7), (3, 3), (7, 1), (12, 0) on the 5 elements of 9w/rv. Case 2. Wis of type F4 and Wis of type E 6 • Then again (b) holds. Case 3. W is of type Bn with n ~ 2 and W is of type A2n or A2n+l· Then u is conjugation by the longest element w0 of W. We show that (a) holds. Let Y be the set of all EE Irr W (up to isomorphism) such that tr(w0 , E) =I= 0. Let Y' be the set of all E' E IrrW (up to isomorphism). By 23.4 and 23.1 we h~e a natural bijection between cw and the set of isomorphism classes of E E Irr W. If c E cw corresponds to E, then the number of fixed points of u on c is clearly ± dim(E) tr(wo, E). Hence tt(cw) = tt Y. From 23.1 we have tt(Qw/"') = tt Y. Hence to show (a) it suffices to show that ttY = ttY'. But this is shown in [Lu3].

Case 4. Assume that W is of type Dn and W is of type Bn-l with n ~ 3. We will show that (a) holds. We change notation and write W' instead of W, W'u instead of W. Then W' is as in 22.26 and we may assume that ucolonW'-> W' is conjugation by Sn (as in 22.26). Let Mf{n be the set of all elements in Mfo-n whose set of singles is nonempty. Let ' ' ' '

M'l,O;n --

r

N~oo

MN,! l,O;n·

By 22.26 and 23.3, c\v, is naturally in bijection with Mi O·n· By 23.1, Qw,u/rv is naturally in bijection with M 1,2;n-l · The identity map 'is clearly a bijection ~ MN+l' . duces a b"" . M 1,2;n-1 --+ ~ M'i,o;n· H ence to prove M 1N,2;n-l --+ l,O;n,.. I t 1n 1Ject10n that tt(Qw,,,,/rv) :S: tt(cw,) it suffices to prove that tt(Mi,o;n) = tt(Mi,o;n). In other words, we must show that (c) any u-stable two-sided cell of W' meets W'u. Now 22.26 and 23.3 provide an inductive procedure to obtain any u-stable two-sided cell of W'. Namely such a cell is obtained by one of two procedures: (i) we consider au-stable two-sided cell in a parabolic subgroup of type 6 k x Dn-k (where n - k E [2, n - l]) and we attach to it the unique two-sided cell of W' that contains it; (ii) we take a two-sided cell obtained in (i) and multiply it on the right by the longest element of W'.

23. TWO-SIDED CELLS

119

Since we may assume that (c) holds when n is replaced by n - k E [2, n - 1], we see that the procedures (i) and (ii) yield only two-sided cells that contain u-fixed elements. This proves (c). The proposition is proved. D

CHAPTER 24

Virtual Cells 24.1. In this section we preserve the setup of 20.1. A virtual cell of W (with respect to L: W----+ N) is an element of K(W) of the form 'Yx (see 20.16) for some x E W.

24.2. LEMMA. Let x E W and let

r

be the left cell containing x.

(a) If 'Yx -=/- 0 then x E f n r(b) 'Yx is a C-linear combination of E E Irr W such that [E: [r]] -=/- 0. 1.

Assume that 'Yx -=/- 0. Then there exists EE Irr Jc such that tr(tx, £) -=/- 0. We have E = ffidED tdE and tx: E ----+ E maps the summand tdE (where x "'.c d) into td,, where d' "'.c x- 1 and all other summands to 0. Since tr(tx, E) -=/- 0, we must have tdE = td,E-=/- 0 hence d = d' and x "'.c x- 1 . This proves (a). We prove (b). Let d E 'D n r. Assume that E E Irr W appears with -=/- 0 coefficient in 'Yx· Then tr(tx, E•) -=/- 0. As we have seen in the proof of (a), we have tdE• -=/- 0. Using 21.3, 21.2, we deduce [E.:Jif'] -=/- 0 and [E.:[rJ•l -=/- 0. Hence [E: [rl] -=/- 0. The lemma is proved. D 24.3. In the remainder of this section we will give a number of explicit computations of virtual cells.

24.4. LEMMA. In the setup of 22.10, w 0 acts on [A] as multiplication by E[A]

=

(-l)I:i(a-lµi-j+l)_

Using the definitions we are reduced to the case where k =nor l = n. If k = n we have E[A] = 1 since [A] factors through 6n and the longest element of Wn is in the kernel of Wn----+ 6n. Similarly, if l = n we have E[A] = Exn = (-1r. The lemma is proved. D 24.5. PROPOSITION. Assume that we are in the setup of 22.23. Let l E Invr(Z) and let K: S, ----+ IF2 be an affine-linear function. Let

c(.Z,l,K) =

L (-lt(Yl[Ay] E K(W). YES,

There exists x E W such that 'Yx

= ±c(Z, l, r;,).

To some extent the proof is a repetition of the proof of 22.24(a), but we have to keep track of K, a complicating factor. We argue by induction on the rank n of Z. If n = 0 the3esult is clear. Assume now that3 2:: 1. We may assume that O is not a double of Z. Let at be the largest entry of Z. 121

122

24. VIRTUAL CELLS

(A) Assume that there exists i, 0 ~ i < t, such that ai does not appear in Z. Then Z is obtained from a multiset Z' of rank n - k < n by increasing each of the k largest entries by a and this set of largest entries is unambiguously defined. The set Z' of singles of Z' is naturally in bijection with Z. Let i', r,, 1 correspond to i, 1o, under this bijection. By the induction hypothesis, there exists x' E Wn-k such that '"Y::n-k = ±c(Z', i', r,, 1 ) E K(Wn-k), Let Wo,k be the longest element of 6 k. Then

(a) and

(b) W lwo,kX 1

•W

( SkXWn-k)

= Jsk XWn-k lwo,kX' = j~txwn_Jsgnk IZlr:n-k) =

±j:kXWn-k (sgnk IZlc(Z', l 1, 1o, 1 ))

= ±c(Z, l, 1o,),

as required. (B) Assume that there exists i, 0 < i ~ t such that ai is a double of

Z.

Let

Z

be as in 22.8 (with respect to our t). Then O is not a double of Zand the largest -

-

entry of Z is at. Let Z be the set of singles of Z. We have Z = {at-z; z E Z}. Thus Z, Z are naturally in (order reversing) bijection under j c-+ at - j. Let i' E Inv r (Z) correspond to i under this bijection and let r,, 1 : S,, --+ IF 2 correspond to 1o, under this bijection. Define r,, 11 : S,, --+ IF2 by 1o,"(Y) = 1o,'(Y) + I:;yEY a- 1 y (an affine-linear function). Since at - ai does not appear in exists x' E W such that rx' rx'wo

Z,

(A) is applicable to

Z.

Hence there

= ±c(Z, i', 1o,"). By 20.23, 22.18, 24.4, we have

= (-1)/(x')((rx') = ±((c(Z, i', r,, 11 )) ® sgn = ±c(Z, i', 1o, 1) ® sgn = ±c(Z, i, 1o,),

as desired. (C) Assume that we are not in case (A) and not in case (B). Then Z = {O, a, 2a, ... , ta}= Z. We can find ia, (i + l)a in Z such that i interchanges ia, (i + 1 )a and induces on Z - {ia, (i + 1)a} an r-admissible involution i 1 .

(Cl) Assume first that 1o,(Y) = 1o,(Y * {ia, (i symmetric difference.) Let

+ l)a})

for any Y E S,.

(* is

Z' = {O, a, 2a, ... , ia, ia, (i + l)a, (i + 2)a, ... , (t - l)a}. This has rank n - k < n. The set of singles of Z' is Z' = {O, a, 2a, ... , (i - l)a, (i + l)a, ... , (t - l)a}. It is in natural (order preserving) bijection with Z -{ia, (i+ l)a}. Hence i 1 induces i' E Invr(Z'). We have an obvious surjective map of affine spaces Jr: S,--+ S,, and 1o, is constant on the fibres of this map. Hence there is an affine-linear map r,, 1 : S,, --+ IF 2 such that 1o, = r,, 1Jr. By the induction hypothesis, there exists x' E Wn-k such that r:n-k = ±c(Z', i', r,, 1) E K(Wn-k), Let Wo,k be the longest element of 6k. Then (a), (b) hold and we are done. ( C2) Assume next that 1o,(Y) i- 1o,(Y * {ia, (i + 1)a}) for some (or equivalently any) Y E S,. We have

Z = {O, 0, a, a, 2a, 2a, ... , ta, ta} -

{at - 0, at - a, ... , at - at} =

Z=

Z.

123

24. VIRTUAL CELLS

Let l' E Inv r ( Z) correspond to l under the bijection z f-+ ta - z of Z with itself; let t,, 1 : Si' -+ IF 2 correspond to t,, under this bijection. Let t,,11 : Si' -+ IF 2 be given by t,, 11 (Y) = t,, 1 (Y) + I:yEY a- 1 y (an affine-linear function). Note that l 1 interchanges (t - i - l)a, (t - i)a and induces on Z - {(t - i - l)a, (t - i)a} an r-admissible involution. We show that for any Y E Si' we have t,, 11 (Y) = t,, 11 (Y * {(t - i - l)a, (t - i)a} ), or equivalently t,, 1 (Y) = t,, 1 (Y * {(t- i - l)a, (t- i)a})

+ l.

This follows from our assumption t,,(Y) = t,,(Y * {ia, (i + l)a}) + 1 for any YE Si. We see that case (Cl) applies to l', t,, 11 so that there exists x' E W with rx' = ±c(Z, l', t,, 11 ). By 20.23, 22.18, 24.4, we have rx'wo

= (-l)l(x')((rx,) = ±((c(.Z,/,t,,")) ®sgn = ±c(.Z,/,t,,') ®sgn = ±c(Z,l,t,,),

as desired. The proposition is proved. 24.6.

D

Assume that we are in the setup of 22.27. By 22.27,

P4

+ P16,

P12

+ Pl6 + P6,

P12

+ P16 + P~

are constructible, hence (by 22.2, 21.4) are of the form ndrd for suitable d E V, hence are ± virtual cells. Let d E V be such that ndrd = P12 + P16 + p5. Let r be the left cell that contains d. Recall (21.4) that [r] = A EBB EB C where A= P12, B = P16, C = p5. By the discussion in 21.10 we see that J[;nr-i has exactly three simple modules (up to isomorphism), namely tdA•, tdB•, tdC•, and these are I-dimensional. Since Jrnr-i is a semisimple algebra (21.9), it follows that it is commutative of dimension 3. Hence r n r- 1 consists of three elements d, x, y. Let PA, PB, pc denote the traces of tx on A•, B •, C• respectively. Let qA, qB, qc denote the traces of ty on A•, B•, C• respectively. By 20.24, PA, PB, Pc, qA, qB, qc are integers. Recall that the traces of ndtd on A•, B •, C • are 1, 1, 1 respectively. Since f A11a, f B11a, Jc. are 6, 2, 3 we see that the orthogonality formula 21.10 gives 1 + p~

+ q~ = 6,

1 +PAPE+ qAqB = 0,

1 + p~

l

+ q~ = 2, 1 + Pt + qt = 3, + PAPC + qAqc = 0, l + PBPC + qBqc =

Solving these equations with integer unknowns we see that there exist so that (up to interchanging x, y) we have (PA,qA)

= (21:,1: 1),

(PB,qB)

ry =

P12 - Pl6

= (0,-1: 1),

(pc,qc)

E, E1

0.

E {1, -1}

= (-1:,E').

+ P6· Hence 2p12 - P6, P12 - P16 + P6 are ± virtual cells. argument shows that 2p 12 - P6', p 12 - p15 + P6' are

Then Erx = 2p12 - P6,

E1

The same ±virtual cells. A similar (but simpler) argument shows that p4 - p16 is ±a virtual cell. Assume now that we are in the setup of 22.27 (Case 1). By 22.27, P1

+ P2,

P1

+ Ps,

P2

+ pg,

Ps

+ pg,

Pt

+ Pt

Pt

+ Pt

P!

+ Pt

Pl

+ Pt

are constructible, hence by 22.2, 21.4 are of the form ndrd for suitable d E V, hence are ±virtual cells. By an argument similar to that above (but simpler) we see that P1 - P2,

P1 - Ps,

P2 - pg,

Ps - pg,

t P1t - P2,

t P1t - Ps,

t P2t - Pg,

t Pst - Pg,

124

24. VIRTUAL CELLS

are ±virtual cells. 24. 7. Assume that we are in the setup of 22.29. By 22.29, p2 + p; is constructible, hence by 22.2, 21.4, is of the form na"(d for some d E 'D, hence is ±a virtual cell. As in 24.6, we see that p 2 - p; is ±a virtual cell.

CHAPTER 25

Relative Coxeter Groups 25.1. Let W, S be a Coxeter group and let u E Aw (see 1.17). We assume that W is a Weyl group or an affine Weyl group. Let J be au-stable subset of S such that WJ is finite (that is, J-/- S when Wis infinite). Let U: W----+ {permutations of R} be as in 1.5. Let W be the set of all w E W such that U(w) carries {(1, s); s E J} onto itself. (A subgroup of W.) Alternatively,

W = {w E W;wWJ = WJw,w has minimal length in wWJ = WJw}. Let K be the set of all u-orbits k on S - J such that WJuk is finite. (In the case where W is infinite, K consists of all u-orbits on S - J if tt (u \ ( S - J)) ~ 2 and K = 0 if tt(u\(S - J)) = 1.) We assume that J is u-excellent in the following sense: for any k EK we have wi{UkJwi{uk = J. For k E K we have wi{ukwi{ wi{uk = wi{ hence WJUkWJ - WJWJUk Tk ·.O O 0 0

satisfies Tf = 1. If k EK then U(wi{uk) maps {(1, s); s E JU k} onto {(-1, s); s E JU k}. It also maps {(±1,s);s E J} onto {(±1,s);s E J}. Hence it maps {(1,s);s E J} onto {(-1,s);s E J}. Similarly, U(wi{) maps {(-1,s);s E J} onto {(1,s);s E J}. Hence U(Tk) = U(wi{)U(wi{uk) maps {(1, s); s E J} onto {(1, s); s E J}. Thus, Tk E W. More precisely, Tk E wu, the fixed point set of u: W ----+ W. The following result is proved in [Lul] assuming that W is a Weyl group (see [Lull] for the case where Wis an affine Weyl group).

(a)

wu

is a Coxeter group on the generators {Tk; k E K}. Moreover, if W is a Weyl group then is a Weyl group; if W is an affine Weyl group and tt(u\(S - J)) ~ 2 then is an affine Weyl group; if W is an affine Weyl group and tt(u\(S - J)) = l then = {1}.

wu

wu

wu

25.2. We now strengthen our assumption on J by assuming that there exists an adjoint reductive group GJ defined over lFq whose Coxeter graph is J (a full subgraph of the Coxeter graph of W), such that u: J ----+ J is induced by the Frobenius map of G J and that G J (JF q) admits a unipotent cuspidal representation E; let c 0 be the two-sided cell of WJ (with the weight function given by length) corresponding to this unipotent representation in the classification [Lu5]. The function a: W----+ N (see 13.6) (defined in terms of the weight function given by the length) takes a constant value a on c 0 and a constant value ak on coTk fork EK (see 9.13, P11, 15.6). The function {Tki k E K} ----+ Z given by Tk f----, ak - a takes equal values at two elements Tk, Tk' that are conjugate in wu (case by case check) hence it is the restriction of a weight function L: wu ----+ Z. This weight function takes > 0 values on {Tki k E K}. Let aL: wu ----+ N be the function defined like 125

126

25. RELATIVE COXETER GROUPS

wu (instead of W) and the weight function just defined. Define a' : wu -+ N by a' (x) = a(yx) where y is any element of c 0 . This is independent of the choice of y, by 9.13, Pll, 15.6.

a: W-+ N (see 13.6) in terms of

25.3. CONJECTURE. (a) aL = a'. (b) Let c be a two-sided cell of wu (relative to the weight function L) as in 25.2. There exists a ( necessarily unique) two-sided cell c of W ( relative to the weight function given by length) such that yx E c for any y Eco, x E c. Moreover the map c f-+ c is injective. This would reduce the problem of computing the two-sided cells of wu (relative to the weight function L) to the analogous problem for W (relative to the weight function given by length).

CHAPTER 26

Representations 26.1. Let W, S be an affine Weyl group and let u E Aw (see 1.17). Let J be a u-:stable subset of S with J-/=- S. Let U(J) be the set of isomorphism classes of unipotent cuspidal representations of GJ(lFq) (as in 25.2). Note that U(J) is independent of the choice of G J. Let E E U ( J). Let 1i (W, J, E) be the I wahoriHecke algebra attached to (defined as in 25.1 in terms of W, S, J) and to the weight function L: N (defined as in 25.2). Let n be as in 1.18. Let

wu-,

wu

nu= {a E n;ua

= au},

nu,J

= {a

E nu;a(J)

= J}.

If a E nu,J then a : W -, W restricts to an automorphism of wu as a Coxeter group; this automorphism is compatible with the weight function L: wu -, N hence it induces an automorphism of the algebra H(W, J, E). Hence we may form a semidirect product algebra H(W, J, E) ©A A[nu,J] where A[nu,J] is the group algebra of nu,J over A. Let v 0 E C* be such that v 0 = 1 or v 0 is not a root of 1. Let (H(W, J, E) ©A A[nu,JDvo

be the C-algebra obtained from H(W, J, E) ©A A[nu,J] by the change of scalars A-t C, v 1-----t vo. Let I=

LJ Irr(H(W, J, E) ©A A[nu,JDvo

where Irr stands for the set of isomorphism classes of simple modules of an algebra and the disjoint union is taken over all ( J, E) as above modulo the action of nu. On the other hand, let g be a connected, simply connected almost simple reductive group over C, of type "dual" to that of W. Let A(Q) be the group of automorphisms of g modulo the group of inner automorphisms of g. There is a natural action of A(Q) on g (well-defined up to conjugacy) and we form the semidirect product g of g and A(Q) via this action. Note that g may be identified with the identity comp~nent of Q. Let J be the set of all pairs (C, £) where C is a 9-conjugacy class in g and £ is an irreducible 9-equivariant local system on C. 26.2.

THEOREM.

There is a natural bijection I+-+

J.

This is shown in [Lull, Lu13]. Using this bijection we may transfer the partition of I into pieces indexed by the various ( J, E) into a partition of J into pieces again indexed by the various ( J, E). This partition can be described purely in terms of the geometry of g (see [Lul3]).

127

CHAPTER 27

A New Realization of Hecke Algebras 27.1. Let G, F, P, E, w, s, J, W, u, ... be as in 0.3. Let H = H( cF, pF, E). In this section we give a new realization of the Hecke algebra Has a function space. We will identify );~: Ep2 ----+ Ep, is the composition

E

where g1, g2 E GF, g2Pog 2 1 = P2, g1Pog1 1 = Pi; if (A, P2) !:/. Ow then (we/>);~: Ep2 ----+ Ep, is 0. ((we/>)~ is independent of the choices of g1, g2.) 27.3.

For was in 27.2 we have Ow-'= {(A,A) E P x P; (A,A) E Ow}-

Let

129

27. A NEW REALIZATION OF HECKE ALGEBRAS

130

Then ~U = ql(w) where l is length in W. The composition (w¢)(w- 1 ¢) has as (Po, P0 )-component the sum over all Pi EU of the compositions

E Po

i(a91,1) ---+

E

Po

91

--+

E

91 1 P1 - - - t

E

Po

i(a1,91) ---->

Epo

where g1 E GF, g1Pog 11 = Pi, that is ql(w) times the identity map of Ep0 • Thus, (a)

(w¢)(w- 1 ¢)

= i(wl(1¢) + linear combination of w' ¢ with w' i=

27.4. Let w,w' E wu be such that l(ww') in W.) Then

1.

= l(w) + l(w'). (Here l is length

(a) (Pi,P2)EOw,(P2,P3)EOw =? (Pi,P3)EOww', (b) if (Pi, P3) E Oww' then there is a unique P2 E P such that (Pi, P2) E Ow, (P2,P3) E Ow'· If Pi, P 2 , P3 are as in (a) we have = Ep3 ---+ Ep1 . From the definitions we see that 1

'lj;:: 'lj;::'l/1:!:

(c) 27.5. For w E Wu, w

.*(x(lo)w) where X(lo)w: Mt',--+

I['.

is the character of lo: MF--+ GL(Ep0 ) restricted to Mt',.

27.11. The obvious homomorphism Aut(L)--+ Aut(Lad) defines for any w E an isomorphism of Mw = Aut(L)w with a connected component of the reductive algebraic group Aut(Lad) with identity component Lad· Hence we have the notion of character sheaf on Mw (see [Lu9]). Let Mw be the set of isomorphism classes of character sheaves on Mw. Let A E Mw. Then A is Lad-equivariant for the conjugation action of Mw. Since >. is smooth with connected fibres of fixed dimension, a suitable shift of >. * (A) is a simple perverse sheaf A on Zw. Let Att be the unique simple perverse sheaf on Z, whose support is the closure in Z of the support of A and which satisfies Att lzw = A. Let M; be the set of all A E Mw such that F* A ~ A. For any A E M; we choose an isomorphism : F* A ~ A. There are induced isomorphisms ¢: F* A ~ A,¢: F* Att ~ A}. Let

wu

F XA,¢: MW --+ Qz,

XA,¢: Zw

F

-

--+ Qz,

XArt,¢:

z F --+ Qz

be the corresponding characteristic functions (alternating sums of traces of Frobenius at stalks of cohomology sheaves at various F-fixed points). We have cA,¢ = (-l)N>.*(cA,¢) where N = dimZw - dimMw and cA,¢ = CArt,¢1Z!·

It is known [Lu9] that the functions XA,¢ (where A runs through M;) form a basis for the vector space of functions --+ I['. that are constant on the orbits of M°F (acting on M; by conjugation). Hence

M;

L

X(lo)w =

(AXA,¢

AEM! where (A E I('. are uniquely determined. Applying >. * to both sides we deduce

ql(w)/2j~ = L(A(-l)NXA,¢· A

27. A NEW REALIZATION OF HECKE ALGEBRAS

134

Hence

f~ = q-l(w)l 2(-l)w~=~AXA1,q,IZ!· A

The following conjecture provides a geometric interpretation of the polynomials Py,w (see 5.3) attached to the Coxeter group wu with its weight function L: wu -----t N. 27.12.

CONJECTURE.

Assume that y

E

wu.

We have

q-l(w)/ 2(-l)N L~AXAi,q,lz: = Py,wlv=.,fofi, A

L~AXAi,q,lzF-uyEWuz: = 0. A

27.13. We now consider the special case where P is the set of Borel subgroups of G and Eis the trivial vector bundle C. Then W = W. In this case the homomorphism lo is trivial. For w E wu = wu and (P1, P2, gU~) E Yo we have

fw(Pi,P2,gU$J = 0,

if (A,P2) (j_ Ow, 2 fw(Pi, P2, gU$J = q-l(w)/ , if (A, P2) E Ow,

In particular, the functions in ""(H) do not depend on the third coordinate gU$1 which can therefore be omitted. For f', f" in ""(H) we have

(!' * f")(P,P') = L

f'(P,F)f"(F,P').

FEPF

In the present case, Conjecture 27.12 states that Py,wlv=.,fo is (up to normalization) the restriction to of the characteristic function of the intersection cohomology sheaf of the closure of Zw in Z. Equivalently, if for w E wu we set

z:

Pw = {P' E P; (Po, P') E Ow}, then Py,w lv=.,fo is (up to normalization) the restriction to Pt' of the characteristic function of the intersection cohomology sheaf of the closure of Pw in P. This property is known to be true; it is proved in [KL2] in the case where u = l on W and is stated in the general case in [Lu3]. 27.14. Let G, F, P, E, W, S, J, W, u, . . . be as in 0.6. Let H = H(GF,pF,E). Everything in 27.1-27.13 extends to this case (we replace G by G throughout) with the following modifications. In the definition of SB (see 27.7) we must now restrict ourselves to functions f : Yo -----t C such that

{(P,P')

E

pF x PF;f(P,P',gU$) =f. 0 for some g E GF}

is contained in the union of finitely many G-orbits on P x P. Also, when defining the multiplication* in 27.7 only the definition 27.7(a) makes now sense (in 27.7(b) the quantity ~(U$) is infinite hence does not make sense). In 27.13 one should use Iwahori subgroups instead of Borel subgroups.

Bibliography [Be] [Bo] [Br] [BM] [DL] [Ge] [GP] [Ho] [Iw] [IM]

[KLl] [KL2] [Lul] [Lu2]

[Lu3] [Lu4]

[Lu5] [Lu6]

[Lu7] [Lu8]

[Lu9] [LulO]

[Lull]

R. Bedard, Cells in two Coxeter groups, Comm. Algebra 14 (1986), 1253-1286. N. Bourbaki, Groupes et algebres de Lie, Chapters 4, 5, 6, Actualites Sci. Indust., vol. 1337, Hermann, Paris, 1968. K. Bremke, On generalized cells in affine Weyl groups, J. Algebra 191 (1997), 149-173. K. Bremke and G. Malle, Reduced words and a length function for G(e, l, n), Indag. Math. (N.S.) 8 (1997), 453-469. P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), 103-161. M. Geck, Constructible characters, leading coefficients and left cells for finite Coxeter groups with unequal parameters, Represent. Theory 6 (2002), 1-30. M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Math. Soc. Monogr. (N.S.), vol. 21, Clarendon Press, New York, 2000. P. N. Hoefsmit, Representations of Hecke algebras of finite groups with EN-pairs of classical type, Ph.D. Thesis, Univ. of British Columbia, Vancouver, BC, 1974. N Iwahori, On the structure of the Hecke ring of a Chevalley group over a finite field, J. Fae. Sci. Univ. Tokyo Sect. I 10 (1964), 215-236. N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke ring of p-adic Chevalley groups, Inst. Hautes Etudes Sci. Pub!. Math. 25 (1965), 5-48. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. ___ , Schubert varieties and Poincare duality, Geometry of the Laplace Operator (Honolulu, 1979), Proc. Symp. Pure Math., vol. 36, Amer. Math. Soc., Providence, RI, 1980, pp. 185-203. G. Lusztig, Coxeter orbits and eigenspaces of Jilrobenius, Invent. Math. 28 (1976), 101-159. ___ , Irreducible representations of finite classical groups, Invent. Math. 43 (1977), 125-175. - - - , Left cells in Weyl groups, Lie Group Representations. I, Lecture Notes in Math., vol. 1024, Springer, 1983, pp. 99-111. ___ , Some examples of square integrable representations of semisimple p-adic groups, Trans. Amer. Math. Soc. 227 (1983), 623-653. - - - , Characters of reductive groups over a finite field, Ann. of Math. Stud., vol. 107, Princeton Univ. Press, Princeton, NJ, 1984. ___ , Cells in affine Weyl groups, Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 255-287. ___ , Sur les cellules gauches des groupes de Weyl, C. R. Acad. Sci. Paris Ser. I Math. 302 (1986), 5-8. - - - , Cells in affine Weyl groups. II, J. Algebra"109 (1987), 536-548. ___ , Introduction to character sheaves, The Arcata Conference on Representations of Finite Groups (Arcata, CA, 1986), Proc. Symp. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 165-180. ___ , Intersection cohomology methods in representation theory, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, pp. 155-174. ___ , Classification of unipotent representations of simple p-adic groups, Internat. Math. Res. Notices (1995), 517-589.

135

136

BIBLIOGRAPHY

[Lu12] - - - , Lectures on Hecke algebras with unequal parameters, MIT Lectures, 1999; math.RT /0108172. [Lu13] _ _ _ , Classification of unipotent representations of simple p-adic groups. II, Represent. Theory 6 (2002), 243-289; math.RT /0111248. [Xi] N. Xi, Representations of affine Hecke algebras, Lecture Notes in Math. vol. 1587, Springer, Berlin, 1994.

Titles in This Series 18 G. Lusztig, Hecke algebras with unequal parameters, 2003 17 Michael Barr, Acyclic models, 2002 16 Joel Feldman, Horst Knorrer, and Eugene Trubowitz, Fermionic functional integrals and the renormalization group, 2002 15 Jose I. Burgos Gil, The regulators of Beilinson and Borel, 2002 14 Eyal Z. Goren, Lectures on Hilbert modular varieties and modular forms, 2002 13 Michael Baake and Robert V. Moody, Editors, Directions in mathematical quasicrystals, 2000 12 Masayoshi Miyanishi, Open algebraic surfaces, 2001 11 Spencer J. Bloch, Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, 2000 10 James D. Lewis, A survey of the Hodge conjecture, Second Edition, 1999 9 Yves Meyer, Wavelets, vibrations and scaling, 1998 8 Ioannis Karatzas, Lectures on the mathematics of finance, 1996 7 John Milton, Dynamics of small neural populations, 1996 6 Eugene B. Dynkin, An introduction to branching measure-valued processes, 1994 5 Andrew Bruckner, Differentiation of real functions, 1994 4 David Ruelle, Dynamical zeta functions for piecewise monotone maps of the interval, 1994 3 V. Kumar Murty, Introduction to Abelian varieties, 1993 2 M. Ya. Antimirov, A. A. Kolyshkin, and Remi Vaillancourt, Applied integral transforms, 1993 1 D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, 1992

COMMUNICATIONS IN ALGEBRA, 1 4 ( 7 ) , 1 2 5 3 - 1 2 8 6 ( 1 9 8 6 )

CELLS FOR TWO COXETER GROUPS Robert B6dard Universite dtOttawa Dgpartement de math6matiques Ottawa, Ontario, Canada KIN 9B4

0.

Introduction In this article, we will discuss some results

about left and two-sided cells in Coxeter groups as defined by Kaszdan and Lusztig in (2). After recalling part of the work of Lusztig in this area, we will describe the decomposition of the affine Weyl group W (C ) a 3 in left and two-sided cells. We will also verify for this group some of the conjectures of G.Lusztig about these cells. For example, there are conjecturally only finitely many left cells in affine Weyl groups. Secondly, we will consider the compact hyperbolic Coxeter group (W,S) generated by S={si

/

i=1,2,3} such that

each si in S is of order 2 and also s1s2

,

s1s3 and

s2s3 are respectively of order 3,3 and 4. For this group we will show there are an infinite number of left cells contrarily to what is conjectured for affine Weyl groups

Copyright O 1986 by Marcel Dekker, Inc.

0092-7872/86/1407-1253$3.50/0

BEDARD

1254

The d e c o m p o s i t i o n i n l e f t a n d t w o - s i d e d

cells i s

known f o r t h e f o l l o w i n g a f f i n e Weyl g r o u p s : W a ( A n ) see ( 3 ) ,

(4) a n d

0, Wa(B2)

n22

= W a ( C 2 ) , Wa(G2) s e e (3).

The k n o w l e d g e o f t h e c e l l d e c o m p o s i t i o n o f W ( C ) m i g h t a 3 be h e l p f u l i n t h e s e a r c h of algorithms t o d e s c r i b e t h e c e l l s i n t h e g e n e r a l c a s e : Wa(C,)

for a l l n

2

2 . The

above example o f t h e h y p e r b o l i c C o x e t e r group l e a d s u s t o s u s p e c t t h a t t h e c o n , j e c t u r a l l y f i n i t e number o f l e f t c e l l s i n t h e c a s e o f t h e a f f i n e \Jeyl g r o u p s s h o u l d b e deeply linked t o t h e s t r u c t u r e of t h e s e groups. One o f t h e r e a s o n s f o r s t u d y i n g t h e s t r u c t u r e o f l e f t and two-sided

c e l l s f o r a f f i n e Weyl g r o u p s i s t h e

hope i t w i l l be h e l p f u l i n t h e u n d e r s t a n d i n g o f unramif l e d r e p r e s e n t a t i o n s of s p l i t p-adic

semisimple groups.

F o r a n y C o x e t e r g r o u p (IrJ,S), we u s e t h e d e f i n i t i o n s o f l e f t and two-sided

c e l l s and n o t a t i o n s : x

x L3 y a n d x L R y f o r x , y

x

T,R

y, x y y,

W a s d e s c r i b e d i n (2).

x L y ) mean:; x a n d y a r c i n the ( r c s p e c L i v c l y l e f t ) c e l 1. We a l s o w r i t e

!: ( r e ; ; p e i : t i . v ~ l y

two-sided

E

w(i4-1)

when i= 1,2,

w(n) > n

when i= n

. . . , n-1

Proof: It i s not t o o d i f f i c u l t t o prove t h a t

1 o E R , m 2 1) U { a o y a ! a E R 1 . We w a n t t o Ra= { a m,a c o m p u t e w - a i a n d f i n d when i t i s a n e g a t i v e a f f i n e r o o t We w i l l p r o v e t h e lemma when i= 1 , 2 ,

. . . ,n-1.

The o t h e r

two c a s e s i= 0 , n a r e d o n e i n t h e same s p i r i t w i t h m i n o r changes . We c a n a l w a y s w r i t e w= t ( y V ) . w l w h e r e w 1

E

w(Cn),

yV i s i n t h e c o r o o t l a t t i c e . Then w 0 am,a = a m ' , w l ( a ) wher e m T = m-(yV / w l ( a ) )

.

Write w ( i ) = w l ( i )

+ jN,

w ( i + l ) = w l ( i + l ) + j ' N . We

h a v e t h e f o l l o w i n g c a s e s : ( 1 ) i f I. 5- w 1( i ) 1 5 wl(i+l)

2

n , t h e n w*aj- a

(j'-,j ),wl(ai)

5

ri

and

and i t i s a

n e g a t i v c a r f i n e r o o t i f a n ? o n l y i f (*j > ,j ' ) o r (J- 3 a n d w 1 ( o1. )

E

R-). (2) if 1 5 w ( i ) 5 - n and 1

a q e g a t i v e a f f i n e r o o t i f and o n l y i f j > j ' .

( 3 ) i f n < w 1( i ) 5 - 2n a n d

'

TWO COXETER GROUPS

a n e g a t i v e a f f i n e r o o t i f and only i f j

n < w ( i t l ) 5 2n, t h e n waai- a ( j l

-

,w,

a n e g a t i v e a f f i n e r o o t i f and only i f and wl(ai)

, -

j'.

and it i s

( ai ) ( j > j' ) o r ( j = j'

R-).

E

From a l l t h i s d i s c u s s i o n , w e g e t w.a

i

is negative

i f arid o n l y i f w ( i ) > w ( i + l ) . We c a n d e s c r i b e t h e u n i q u e i n v o l u t i v f a u t o m o r p h i s m

2.4 $:\J

a

(C,)

+

Wa(Cn) s u c h t h a t $ ( s i ) = s

n-

.

1

f o r a l l i= 0 , 1 , . .

n a s follows: consider t h e unique permutation such t h a t O(i)= i i f i:l,2,

. . . ,n

i E O

a:

,

+

(mod N), f 3 ( i ) = i - ( n + l ) i f

(mod N ) a n d @ ( i ) =i - n i f i:-1,-2,

. . . , - n(mod

t h e n i t i s n o t d i f f i c u l t t o c h e c k t h a t $(w)= f3.w.8-l any w

2.5

E

T;Ja ( C n )

,(

for

.

F o r t h e r e s t o f t h i s s e c t i o n , PJ w i l l d e n o t e Wa(C3)

and w e u s e t h e d e s c r i p t i o n o f W i n 2 . 1 . F o r any s u b s e t J J of { 0 , 1 , 2 , 3 } , we d e n o t e b y W : t h e s e t o f a l l w E W s u c h t h a t R(w)=

CIS,

1

j

E

J}.

N)

1264

BEDARD

(2)= (Ci;))

12

*

where ( )

*

i s defined r e l a t i v e t o

BEDARD

n

=

(Fil))*

where ( )

=

(Fhl))*

where ( )

I

7 {

n

i s defined r e l a t i v e t o {sO,sl}, is defined r e l a t i v e t o {s2,s3},

w h c ~ ec i s t h e i d i n t i 1 . y c l e m e n t o f Wa(C

2.7

t h e di::tinct

l e f t c e l l s o f Wn(C3).

and i , an i n d e x , t h e n t h e l e f t c e l l in W

).

3 Theorem. 1) The s u b s e t s d e s c r i b e d i n 2 . 6 a r e all

I

x:~)

i s contained

. 3 ) The i n v o l u t i v e a u t o m o r p h i s m $ d e f i n e d

in 2 . 4 sends t h e l e f t c e l l where

Xii)

t o the l e f t c e l l $(xii))

1267

TWO COXETER G R O U P S

and

X

E

{A,B,CI

xii-l) if -. $({sj jcI})= {s. J and X e {A,R,C}

x!~)

if s and

.

J

X

.

I E

J

j ~ I i ,150 (nod 2)

~ E T I

{D,E,GI

4) The function a deflned In 1.4 takes the following values: a( A?))=

5, 9, a( B T(I))=

containing respectively sos2s3s2s3, S3sOS1S0S1, S2S3S2S3, S S S S

0 1 0 1' s1s2s1y S0S3y

sls3' s0s2, si for i= 0,1,2,3.

Proof: We will sketch a proof. By proposition 1.9,

(l)- w~~~ is a left cell such that a( A (1)) = 9. By symA123123 (1 ~ ) = AOl2 is also a left cell such metry $ ( w ~ ~ w012= that a( A : : ; ) =

9. We could describe these more yrecise-

ly using 2.1 and lemma 2.3:

BEDARD

1268

By applying proposition 1.14 to the left cell - (1)-A(l) r= A123 and S T = {so,sl}, we have 13 123"oS1 (1) (1) is a left cell and A123.s0 U ~/., , I,~ is rC)

(1)

(A!:)*=

\

A'"123"0

a union of at most 2 left cells. Thus A:;=

is

a left cell. We could describe these sets: (2) > w(3) > 3, 0 > w(l)

+ w(2))

By lemma 2.3, we get

and theorem 1.5, we have the same 2-sided cell as A(') 123

We can repeat this process to the new left cells and constructed this way many more left cells. By proposition 1.14 applied to the left cell

r=

s ) and by lemma 2.3, then both 2) 3 ( 2 L~ ( 1 ) are left cells conA 13 s and A13 2 13 "2'3 tained respectively i n W12-and Wi3 such that a( A!;))= A(') 13

and S T = {s

9. By pr'oposition 1.111 applied to the left a ( A"))= 13 cell r= A(1) and S T = {s s and by lemma 2.3, then 12 OJ 1 (3)- A12 (1) .sOsl are both left cells ~ ( 3 d12 )(1)~*so and A12 02 contained respectively in wo2 and such that

d2

a( A;:))=

(3)) = 9 . By proposition 1.14 applied to a( \2

the left cell

r= q3 ( ) and

4)

S T = is 0 ,sl} and by lemma 2.3, .s0s1 are both left

4;)=4;)

then A = s o and 03 cells contained respectively in

wo3 and

w13 such that

~

~

~

TWO COXETER GROUPS

(1) )=a( a( ,Ao3 =

4)

\3(4

)=

9. By proposition 1.14 applied to

{sl,s2 1 and by lemma 2.3, then A (1)23 .s,, is a left cell contained in w~~ such that and S f

=

4:) 1 2

a( A :) and S f =

9. By proposition 1.14 applied to r= A0( 23 ) 2 ) 3 is {s s2} and by lemma 2.3, then AOl3= 02 1' 2

=

a left cell contained in w013 such that a( Consider S f = {s2,s3 1 and the left cell For any w

E.

(4), 43

h(2) 1 3 ) = 9.

(4

q3

r=

. w-l is in the middle of the string

a(w-l) relative to S f and we have (,(!I)* is equal to 4'4) 3 .s3 U cells. But

(4) *s2 and q3

44)

it is a union of at most 2 left 12 -s3 is already a left cell in W

4:) 4:) =

4') 4;) 4'))

*s2 is contained in w2. Then 3 a left cell contained in w2 such that a( and

all left cells such that, for I

=

=

{0,1,2,3} and (i) index, each Aii) is contained in 'W and a( AI (3)- A(l).s Consider A02303 2

C

*s2 is

9.

1270

BEDARD

B e c a u s e s 2 k R(w) a n d a ( w ) = 9 , t h e number v o f p o s i t i v e f o r a n y w E A") 3' 03 ' 9 by lemma 1 . 7 . D e n o t e b y w o , t h e l o n g e s t

r o o t s i n t h e f i n i t e r o o t system C t h e n a ( A:::)=

e l e m e n t o f t h e f i n i t e Weyl g r o u p W(C ) , t h e s u b g r o u p 3 g e n e r a t e d b y { s l , s 2 , s 3 } . Then w s s s s s s i s t h e 0 0 1 2 3 0 2 unique element of minimal l e n g t h i n w c

023

ment w '

s

E

and f o r any 023 s u c h t h a t w i w 0 s 0 s l ~ 2 ~ 3 ~ 0t hs e2r,e i s a n e l e -

E

A 0(233 ) s u c h t h a t l ( w l ) = l ( w ) - 1 , w'= s w f o r some

S and w

w l . B e c a u s e a ( w ) = a ( w r ) = 9 , t h e number v

of p o s i t i v e r o o t s of t h e f i n i t e r o o t system C c o r o l l a r y 8.4 i n

0, we h a v e

w - wL 1

3

and

m w s s s s s s L 0 0 1 2 3 0 2

a n d A ( ~ )i s c o n t a i n e d i n a l e f t c e l l . 023 C o n s i d e r 8::$=

bJ023-

(6 i= l

A(i)) 023

Using t h e technique of s t r c a n p r o v e t h a t ~ ( li s) c o n t a i n e d i n a l e f t c e l l . We c a n 023 e a s i l y c h e c k t h a t s s s s s E 8 0i l2 l3 . By p r o p o s i t i o n 1 . 8 , a ( s 0 s 2 s 3 s 2 s 3 ) (5. T h u s we g e t a ( s 0 s 2 s 3 s 2 s 3 ) = 5 a n d (1) (3) a ( 8 0 2 3 ) = 5 . F o r a n y w E 8") and w' 023 AO232 w b e c a u s e a ( w ) = 5 , a ( w l ) = 9. So we c a n c o n c l u d e t h a t b o t h -

ewl

a n d ~ ( la r)e l e f t c e l l s c o n t a i n e d i n 023 023 ( that a ( A 0 233 ))= 9 a n d a( B(012 )3 ) = 5 .

w~~~

such

1271

TWO COXETER GROUPS

~ y s y m m e t r y , A (O3l) 3- A a(32 ) . s l = ( l ) = y013- A B01 3

B::;)

)

and 023 a r e b o t h l e f t c e l l s con$(

(3) (1) t a i n e d i n w 0 1 3 s u c h t h a t a ( AOl3)= 9 a n d a ( B ~

~ 5 .~

By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o t h e l e f t c e l l

r=

A"'

023

a n d S T = { s O , s l } a n d b y lemma 2 . 3 ,

then both

(5)=

a n d A ( 3 ) = A (032) 3 ~ ~ l ~a 0r e l e f t c e l l s con03 a n d wo3 s u c h t h a t a ( 1 3 tained respectively i n

A13

d3

a( A(033 ) ) = 9 . By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o t h e l e f t c e l l r= A l 2 ) a n d S T = {s,,s,} a n d b y lemma 2 . 3 , t h e n J

A::'=A::)

'"= A::). i

.s2 and

A1 3

s2s3 a r e both l e f t c e l l s such t h a t

contained r e s p e c t i v e l y i n w12 and

) = 9 . By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o

the l e f t c e l l

r=

( 3 ) and

A 03

S T = { s 2 , s } a n d b y lemma 2 . 3 ,

c e l l s contained respectively i n

3

wo2

and w13 such t h a t

9. By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o (7) t h e l e f t c e l l s A a 2 , A13( 6 ) a n d S T = { s 1 , s 2 } a n d b y lemma 2 . 3 , t h e n A12 = A O(7) 2 * s l y A$:)= A : ~ ) . S1 a r e b o t h l e f t a ( A;:))=

a ( A::))=

c e l l s c o n t a i n e d r e s p e c t i v e l y i n W' a ( A!'))=

a(

and

w~~

such t h a t

(2)) = 9 .

C o n s i d e r t h e p a i r S T = {s0,s1} and t h e l e f t c e l l w-l i s i n t h e middle of t h e (5). F o r a n y w E A03 03 r e l a t i v e t o { s ,- , s , -} a n d we h a v e s t r i n g a(w-')

(A~;')*=

A;;'. " so

A O( 35 ) - s 1 i s t h e u n i o n o f a t m o s t

2 l e f t c e l l s . Because c e l l i n w 1 3 a n d A:!)=

is already a l e f t A Os3 0 13 ~ ( ~ ) i .s sc o~n t a i n e d i n dis03

d3,

)

=

BEDARD

1272

i s a l e f t c e l l contained i n t h e n A:;) 13 w 1 3 s u c h t h a t a ( A:!))= 9.

j o i n t from

By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o t h e l e f t c e l l

r=

A!):

-

a n d S t = { s 2 , s ? } a n d by lemma 2 . 3 ,

-

then

A13( 8 ) - s 2 s 3 a r e b o t h l e f t c e l l s contained respectively i n

w2

a r e a l l l e f t cel1.s s u c h t h a t , i, a n index,

each

a n d bJ3 s u c h t h a t a ( A L 3 ) ) =

f o r I C { 0 , 1 , 2 , 3 1 and

i s c o n t a i n e d i n W'

and a l s o

By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o t h e l e f t c e l l

r=

8 ( l ) a n d S t = {",s1} a n d b y lemma 2 . 3 , t h e n 023 B::)= B ~ ~ ~ a.n ds lB ( ~ ) =~( 1 ) ~ ~a r e ~b o t h. l e fst c e~l l s s 03 c o n t a i n e d r e s p e c t i v e l y i n w 1 3 a n d wo3 s u c h t h a t ( 1 ) = a ( B O( 3I ) ) = 5; t h i s i s b e c a u s e t h e l e f t c e l l s a ( B13 ) y 8;;) 8 0( 12 3

a n d 8 ( l ) a r e a l l c o n t a i n e d i n t h e same 2 - s i 03 d e d c e l l . By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o t h e l e f t c e l l

r=

1 3 ' ~ ) a n d S t = { s 2 , s ) a n d b y lemma 2 . 3 , t h e n 13 3 (l)= a n d B13( 2 ) = * s 2 s 3a r e b o t h l e f t c e l l s %2 B13 2 13 c o n t a i n e d r e s p e c t i v e l y i n w 1 2 and w13 s u c h t h a t

~

TWO COXETER GROUPS

(1 ) = a( 8 : : ) ) = a( B12

5. By proposition 1.14 applied to

the left cells B12

, )::B

and S f = {so,sl) and by lem-

(1) (1)- (1) ma 2.3, then = ):B B12 B1 %2 .SoS1' (2) (4)= B13 (2)-sOs1 are all left cells conB13 "0' '13 tained respectively in wo2, wl, wo3 and w13 such that

A)=

a( B;:))=

a( B~(l))= a( B::))=

(4))= 5. ~y proposia( B~~

tion 1.14 applied to the left cell T= B ( ~ ) and 13 S 1 = {sl,s2}, then B$:)= a(

~6:))-

(Bi:)*

is a left cell and

5. We can describe 8')2 3

as follows:

w(2)>w(1)>3, w(2)>w(3)>3,

By lemma 2.3, Bi;)

is contained in W 2 3

w(2)>7

.

BY proposition 1.14 applied to the left cell 8(4) and S T = {s,,s 1 and by lemma 2.3, then 13 3 B(~)= 2 ~13 ( ~ ) .and s ~B 3 (I)= ~13 ( ~ ) . sare ~ sboth ~ left cells

='l

contained respectively in

w2

and

w3

(a)=

such that a( B2

1274

BEDARD

a r e a l l l e f t c e l l s such t h a t ,

f o r I C {0,1,2,3} and

e a c h BI( i ) i s c o n t a i n e d i n W'

i, an index,

and

a ( BI$ ~ ) ) =5 . Denote

(A$;)

$3C 2(1)= 3

and c ( ~ ) = Wol01

(AA:)

"

A::)

A$;)

u

B;;))

BO1

We c a n e a s i l y c h e c k t h a t s s s s E c ( ~ a) n d a l s o 2 3 2 3 23 s s s s E Ci:). Using t h e techniques of s t r i n g s desc r i b e d i n 1 . 1 2 , we c a n p r o v e d t h a t C (') i s c o n t a i n e d i n 23 a l e f t c e l l . Thus c ( ~ i) s t h e l e f t c e l l c o n t a i n i n g 23 s 2 s 3 s 2 s 3 . BY p r o p o s i t i o n 1 . 8 , a ( s s s s ) 5 4 . T h u s we 2 3 2 3 g e t a ( z 2 s 3 s 2 s 3)= 4 a n d a ( c ( ~ ) ) =4 . By s y m m e t r y , 23 (1)C O , - $ ( c;:)) i s t h e l e f t c e l l c o n t a i n i n g sOslsOsl and

By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o t h e l e f t c e l l

r=

( I ) (c$:))* is a left cell c ( ~ a) n d S t = { s 23 l y s 2 } ' C13 s u c h t h a t a ( c ( ~ ) ) =4 . We c o u l d d e s c r i b e C ( 1 ) a s f o l l o w s : 13 13

By lemma 2 . 3 ,

c:;)

i s c o n t a i n e d i n W1 3

.

TWO COXETER GROUPS

By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o t h e l e f t c e l l

r=

C ' l ) a n d S f = ( s s } a n d b y lemma 2 . 3 , t h e n 13 0' 1 c ( l i = ( l ) . s O a n d C13( 2 ) = C13(1). s O s l a r e b o t h l e f t C e l l s 03 '13 contained r e s p e c t i v e l y i n wo3 and i n w13 such t h a t

a ( C O( 13 ) ) = a ( c::))=

-

)

c

and

4 . By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o

c!;)=

*

(

i s a u n i o n o f a t most

two l e f t c e l l s .

\

By t h e t e c h n i q u e o f s t r i n g s , we c a n p r o v e t h a t a l e f t c e l l . I t f o l l o w s t h a t a ( c:;))= 12 ma 2 . 3 , c::) i s contained i n W

4 and

c:;)

is

b y lem-

.

By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o t h e l e f t c e l l

r= C 13 kL'

c::)

cil)=

a n d S T = { s 2 , s 3 ) a n d by lemma 2 . 3 , t h e n as

2

a n d c ( l ) = C13( 2 ) "2'3 3

are both l e f t c e l l s

c o n t a i n e d r e s p e c t i v e l y i n W2 a n d a ( c!'))=

3

r=

cell

(3):

4. By p r o p o s i t i o n ~ 1 (2 l a) n d S 1 -

such t h a t a (

cil))-

1.111 applied t o the l e f t

{ S ~ , S a~n )d by lemma 2 . 3 , t h e n

( l ) .so and C i 2 ) = 5 2

02

w3

(l) a r e both l e f t c e l l s C12 . s o s l

contained respectively i n

wo2

a n d W'

(3))-

such t h a t a ( Co2

I

By s y m m e t r y , t h e n ?4

()

c;:)=

(c;:))*=

$ ( c::))

i s defined r e l a t i v e t o t h e p a i r {sl,s2},

where

BEDARD

1276

2 ) ( c i ; ) ) * = g ( c!:)) where ( ) * i s d e f i n e d r e l a t i v e 5 2 t o t h e p a i r { s o , s l } , c:')== C O( 22 1 a s 1 = I$( c k l ) ) ,

for I C(0,1,2,3}

a n d i , a n i n d e x , e a c h C I( i ) i s c o n t a i I ,

T

n e d i n W'

\

Consider

~$2'-

"

4.

a n d a ( c:")= W"-

(6

iI

2

A!:)

8): i=l

i=l

u

(i,) C12

i=l

U s i n g t h e t e c h n i q u e o f s t r i n g s , we c a n p r o v e t h a t D12( 1 i s c o n t a i n e d i n a l e f t c e l l . T h u s 17:)

It i s e a s y t o check t h a t sls2sl t i o n 1.8, a(sls2sl)

E

D!:).

is a left cell. By p r o p o s i -

5 3 . T h u s we g e t a ( s l s 2 s l

)= 3 and

3 . B e c a u s e $ ( ~ ( l ) i) s t h e l e f t c e l 1 c o n t a i 12 n i n g g ( s l s 2 s l ) = sls2s1, we h a v e I$( D12( 1 ) ) = Dl*( 1 1

a ( D!:))=

By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o t h e l e f t c e l l

r= D):!

Dbi)-

and S f - (sO,sl}

a n d by lemma 2 . 3 , t h e n b o t h

-

p:?j).so

and D i l ) =

vl2( 1 ) . s O s l a r e

t a i n e d r e s p e o t i v c l y i n W o 2 a n d W1

l e f t c e l l s con-

s u c h t h a t a ( D O(1) 2 )=

a ( D i l ) ) = 3 . By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o t h e l e f t

r=

D ( l ) a n d S T = { s 2 , s ? } a n d by lemma 2 . 3 , t h e n 02 (1)( 1 D O 3 - D O 2 ) . s 3 a n d D i g ) = D Oc 7) 2 -s3s2 a r e both l e f t c e l l s cell

contained r e s p e c t i v e l y i n Wo3 a ( D O( 31 ) = a ( D;:))=

and Wo2

such t h a t

3 . By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o

t h e l e f t c e l l T= D i g ) a n d S T = { s o , s l } a n d b y lemma 2 . 3 0 i 2 ) = D i g ) - s l a n d DO(1)- D o( 22 ) -s1sO a r e b o t h l e f t c e l l s

1277

TWO COXETER GROUPS

contained respectively in a( v;l')=

w1

and WO such that a( 0i2) ) =

3.

By symmetry, then D ! : ) =

(1) *S3= D12

+ ( D;;)),

2 (1).s0s1= + ( D;:)), (l)= ~ ( ~ ) - s +~( sDil)), ~ = D13 D13 2 12 (l)) are *s2s3= q( Do .s2= q,( 0i2)) , D(l)= ):!D 3 all left cells such that, for I C {O,l,2,3} and i, an

W index, Dii) is contained in ' notice D,$:'=

and a( D:i)

D ~ ~ ) . S ~D S12 (1) ~ ="3'0=

) = 3. We should

'13(1) .so= $( ):D

)

.

(1) Using the technique of strings, we can prove that E 0 3 is contained in a left cell. Thus ~ l i ) i sa left cell. We can easily check that s0s3

E

E(~). By proposition I.8 03

a(s s ) 5 2. Thus we get a(sOs3)= 2 and a( E::))= 2. 0 3 Because @( is the left cell containing +(sOs3)==

EA:)

we have I ) ( E('))= E (1) s s 0 3' 03 03 ' By proposition 1.14 applied to the left cell

union of at most 2 left cells.

1278

BEDARD

U s i n g t h e t e c h n i q u e o f s t r i n g s a n d by lemma 2 . 3 , we c a n show t h a t E ( ~ i )s a l e f t c e l l c o n t a i n e d i n w 1 3 a n d 13 (1 s u c h t h a t a ( E ( l ) ) = 2 . E a s i l y , we g e t t h a t s1s3 E E l j 13 By p r o p o s i t i o n 1 . 1 4 a p p l i e d t o t h e l e f t c e l l

r=

.

1 a n d b y lemma 2 . 3 , t h e n b o t h 13 3 E $ l ) = E ( 1 ) . s 2 a n d E ( X E ( 1 ) - s 2 s 3 a r e l e f t c e l l s con13 3 13 t a i n e d r e s p e c t i v e l y i n w2 a n d w3 s u c h t h a t a ( E i l ) ) = ~ ( la n)d S T = { s 2 , s

$( )

*

Ei?j')

where

i s d e f i n e d r e l a t i v e t o { s 2 , s 3 } , E1 ( l ) = E0(2l l .sl=+( E 2( 1 ) )

E : ~ ) = E0 2( ~ ) * ~ $ ~( E:')) S ~ = a r e a l l l e f t c e l l s contained r e s p e c t i v e l y i n Ido2,

W

1

a n d 'do s u c h t h a t a ( E::))=

(1) U s i n g t h e t e c h n i q u e o f s t r , i n g s , we c a n p r o v e t h a t F 1 i s c o n t a i n e d i n a l e f t c e l l . T h u s F!') Wc c a n e a s i l y c h e c k that,

- ii1). "1

c

a(sl) 2 1 . It, f o l l ~ w st h a t a ( s l ) -

is a l e f t c e l l .

~y p r o p o s i t i o n 1 . 8 ,

1 a n d a ( F:'))=

1.

By p r o p o s i t i o n 1 . 1 4 a o p l i e d t o t h e l e f t c e l l

r=

F!~)

and

s ' = { s O , s l } , t h e n FA')=

u n i o n of a t most 2 l e f t c e l l s . Here

(F('))~ F1

is the

1279

TWO COXETER GROUPS

By t h e t e c h n i q u e o f s t r i n g s a n d b y lemma 2 . 3 , we c a n

FA^)

prove t h a t

i s a l e f t c e l l c o n t a i n e d i n W'

t h a t a ( ~ ; l ) ) =1. E a s i l y we g e t t h a t s By symmetry, ( I ) ='d2-

2

(d

2 A $ ~ )

i=l

8ii)

"

C,

u

i=l

i=l

(1)

o E F o 2

(i)

such

u

Vii)

i =l

.

" E2

f i n e d r e l a t i v e t o { s , , ~ ~ )a,r e b o t h l e f t c e l l s c o n t a i ned r e s p e c t i v e l y i n W2 a n d bJ3 s u c h t h a t s , s3

F'l)

E

3

E

F 2( 1 ) ,

a ( F 3(1)) = 1.

and a ( F;'))=

F i n a l l y G ( ~ ) ={ e l i s t r i v i a l l y a l e f t c e l l s u c h t h a t a ( G ( ' ) ) = 0. To f i n i s h t h e p r o o f , we c a n c h e c k t h a t t h e l e f t c e l l s l i s t e d a r e a l l d l s t i n t and t h a t t h e r e u n i o n o f t h e l e f t c e l l s c o n t a i n e d i n bJ i s equal t o

w'.

I

,

foor a n y I C { 0 , 1 , 2 , 3 ~ ,

For t h e proof of t h i s l a s t p o i n t ,

it i s

e n o u g h t o c h e c k i t o n l y f o r I = { 0 , 2 } a n d {o}; f o r t h e o t h e r I ' s , i t Is a l r e a d y v e r i f i e d . 2.8

Theorem. C o n s i d e r t h e u n i o n X-

.U

Xii)

over a l l

1 3 1

l e f t c e l l s i n W ( C ) whose name c o n t a i n s a f i x e d c a p i a 3 t a l l e t t e r , w h e r e X E { A , 8 , C , D , E , F, G 3. Then W,(C

3

)= A

u 8 u C uV u E u F

i n two-sided

LJ

G i s t h e p a r t i t i o n of Wa(C

3

)

cells.

P r o o f : We c a n p r o v e e a c h X i s c o n t a i n e d i n a 2 - s i ded c e l l , f o r X

E

{

A,

8, C ,

D,

E,

F, G 1

.

To f i n i s h

1280

BEDARD

the proof, it suffises to notice that the value a(X) is distinct for each X. Because a is constant on the 2-sided cells, the result follows. 2.9

Consider the unipotent classes in SO ( k ) ( k is an

7

algebraically closed field such that char(k)$

2 ) , they

are in 1-1 correspondence with the set X; where

This correspondance is such that i

j

is the number of

Jordan cells of size j of a unipotent element. 2.10 There is a bijection U between the set of 2-sided cells in W (C ) and the set of unipotent classes in a 3 SO (k) ( or equivalently X+ ) given by:

7

U(A)

=

1+1+1+1+1+1+1, U(8)

U(V)

=

2+2t3, U(E)

=

=

1+1+1+2+2, U(C)=171+1+lt3

1+3+3, U(F)

=

1+1+5, U(G)

2.11 We have the following table I:

=

7.

TWO COXETER GROUPS

1281

where, in the first column, X runs thru the set of 2-sided cells, and in the fourth column, u is a unipotent element in U(X) and

$

is the variety of Borel sub-

groups in SO (k) containing u

7

2.12 Corollary. 1) U is the one-to-one correspondance between the set of two-sided cells in W and the unipotent classes in SO (k) of the conjecture D in

7

0.

2) Each two-sided cell of Wa(C ) has a 3 non-empty intersection with some finite parabolic subgroup of Wa(C ) . 3 3) There are only finitely many left cells in Wa(C

3

).

4) For any w

Wa(C3), w m $ ( w ) . 5) Each two-sided cell meets W 0 u {el E

in only one left cell. Proof: 1) By studying the left cells in hJO we can prove that A

5 6 6 C ifk

D

6E 6F

$

U

G

,

{el

.

It

-

follows from (7) : given X 1' X 2 both two-sided cells, then X1 X2 iff U ( X 2 ) C U(XI). This is part of the

&

conjecture D. The rest of the conjecture follows from table I 2), 3), 4) and 5) follow from theorems 2.7 and 2.8 3.

Left Cells for a Compact Hyperbolic Coxeter Group

3.1

In this section, denote by (W,S): the compact hy-

perbolic Coxeter group generated by S= {sl,s2,s 3) such

1282

BEDARD

2 t h a t si= e ( e i s t h e i d e n t i t y o f W ) f o r i= 1, 2 , 3 and s1s2,

s1s3"

s 2 s 3 have r e s p e c t i v e l y o r d e r 3 , 3 and

I t s Coxeter diagram i s

4.

ST

We w i l l p r o v e t h e r e a r e a n i n f i n i t e number o f l e f t c e l l s f o r t h i s group. 3.2

D e n o t e by { esl

s

E

S 1 : t h e standard b a s i s of W

S

a n d b y B: t h e c a n o n i c a l s y m m e t r i c b i l i n e a r f o r m o n R' a s s o c i a t e d t o (W,S) (1). We h a v e B ( e s , e s ) = 1 f o r s

,e S i ) =

B(es

,es

B(es i

1

)= - c o s ( n / 3 ) 1

E

S,

f o r i- 2 , 3 a n d

-C O S ( T / ~ ) .

B ( e s 2 , eS3)= B(es , e s 2 )=

3

It i s not d i f f i c u l t t o v e r i f y t h a t B i s non-degenerate with signature ( 2 , l )

3.3

D e n o t e b y { e;!

s

E

S 1 : the dual basis relative S

t o B of t h e s t a n d a r d b a s i s of W

,

i . e . B(es,e;)=

s=t a n d B ( es ye tx ) = 0 when s+t f o r s , t C= { x

A s= { x

3.4

iR

E

S

I

B(es,x)

> 0 for a l l s

R '

S . Denote by

81

x-

C

a s m es* a n d as>O} s&Ss T h e r e i s a n a t u r a l a c t i o n o f W on R w h o r e x i n E

iRS B(e,,x)>O)= { x

E

1 when

i:

R~ i s s e n t t o w ( x ) e R S by w E W a n d s u c h t h a t we h a v e

s ( x ) = x-

3.5

2 . B ( x , e s ) * e s f o r any s E S .

Lemma. F o r w

E

W a n d s E S , we h a v e :

1 ) i f ~ " ( c ) c A ~ ,t h e n s

d R(w)

2 ) i f W - ~ ( C ) C S ( A ~t )h ,e n s

E

R(w)

1283

TWO COXETER GROUPS

Proof: See th6orSme 1, chap.V 54.4 in ( I) 3.6

Denote by v= s2s3s1s2s3s2s1s3

sequence {wn/ n

E

IN} of elements in

'{sl,s

3

W. We define a

E

k

~ s1s3sl*v ~ = ,

W: w

1 if n= O

Cs,}

if n>O, n~ 2, 5 (mod 6)

Is2}

if n>O, n~ 3,

R(wn) is equal tod

4 (mod 6)

2) For k non-negative integer, we have

R(W~~-")=

Fs2} and R ( w ~ ~ + ~ . s ~{s31. )=

Proof: We sketch the proof. We must compute the action of w-l n on the dual basis { e : s r S 1 . So we can write v-I in matrix form relative to this basis. 2

Its characteristic polynomial is (A-1) ( A - ( 9 + 6 6 ) A + l ) and we can find matrices P and D where P is invertlble,

D is diagonale and v-l= PDP-'.

In this way, we can com-

non-negative integers

q

S, w-l(ef ) = a n t E S t, s,n can b e explicitely computed. By lem-

s 0 For n-~ and

E

where the a t,s,n ma 3.5 and the values of these coefficients a's, we can compute R(wn) and the result follows.

3.8

Proposition. 1) For all m,n

E

N, if mfn, then

BEDARD

1284

2) We have the following description CV

of the different ()-operation as defined in 1.13 on the

- 1 is at one end of a {s2,s }-string 3

w n : i) if n=O, wo Y

.CI

o(wgl) and (w0 ) = w 1 where ( ) is defined relative to Cs,,s33 ii) if n>O, n: of a (s

- 1 is at one end

0, 1 (mod 6), then: wn k

s } - string o(w;l) 2' 3

ntO (mod6)

and (w ) = i

f 1

(mod6)

" where ( ) is defined relative to {s2,s3}. Also wil be-

longs to a {sl,s3}-string and

^

if nzO (mod61

w

=

\wzitif n:l

(mod6)

N

where ( ) is defined relative to {sl,s3}.

-1

iii) if n>O, n= 2, 5 (mod 6), then: w n belongs to N if nE2 (mod6) a {sl,s2}-string and (w ) = w l w : ~ ~if 5

(mod6)

w

where ( ) is defined relative to {sl,s2}. Also wil beC

if n ~ 2(mod6)

longs to {sl,s }-string and (wn)= 3

if n ~ 5(mod6) V

where ( ) is defined relative to Is1, iv) if n>O, n:

S31 -1 3, 4 (mod 6), then: w n is at one end N

if n53 (mod 6)

of a {s2,s3}-string o(wil) and (w )= w [

i 4

m o d 6)

5.

where ( ) is defined relative to is2, s3}. Also wil belongs to {sl,s2)-string

and

if n ~ 3(mod 6) if nz4 (mod 6)

C

where ( ) is defined relative to {s1,s2}.

TWO COXETER GROUPS

1285

Proof: It follows easily from the construction of the wn and the lemma 3.7. 3.9

Proposition. For all m y n c N, mf. n, then w m 4 w

( See 1.15 for the definition of -and

+.

n.

)

Proof: We prove by induction on m that wm

wn

for all n > m. If m=O, R(wo)= {sl,s3}, while {slys3} for all n>O. Thus wo rlL wn for n>O.

R(wn)f.

If m>O and assume there exists n>m such that

.

wm -w n By proposition 3.8, we can always choose a pair {s,t}CS such that wil and wil belongs respectively to the {s,t}-strings u(wil) and w *

V

h

such that ( w ~ ) = w ~ - ~

-

and (wn ) = wnkl where ( ) is defined relative to {s,t}. CCI N then w ~ - ~(wm) = (wn)= wnklY a contradicIf wm NW,, tion. The result follows. 3.10 Corollary. There are an infinite number of left cells in W. Proof: It follows easily from proposition 3.8, 3.9 and lemma 1.16. ACKNOWLEDGMENTS The author wishes to thank George Lusztig for valuable discussions and suggestions on this subject and also for providing the preprint of the article:" Cells in Affine Weyl Groups."

REFERENCE 1.

N.Bourbaki, Groupes et Algsbres de Lie, Chap.IV,

V, VI, Hermann, Paris (1968).

BEDARD

1286

2.

D.Kazhdan and G.Lusztig, Representations of Coxeter Groups and Hecke Algebras, Inventiones math. 53 (1979) pp. 165-184. -

3.

G.Lusztig, Cells in Affine Weyl Groups, preprint.

4.

G.Lusztig, Some Examples of Square Integrable Representations of Semisimple p-adic Groups, Trans. Amer. Math. Soc.

5.

277 (1983) #2 pp. 623-653.

I.G.Macdonald, Affine Root Systems and Dedekind's n-Functions, Inventiones math.

6.

15 (1972)

pp.91-143.

J.Y.Shi, The Decomposition into Cells of the Affine Weyl Groups of Type A, Ph.D. Thesis, Mathematics Institute, Univ. of Warwick, Coventry, (1984)

7.

N.Spaltenstein, Classes Unipotentes et Sous-groupes de Borel, Lect. Notes in Math. 946, Springer-Verlag, Berlin-Heidelberg-New York (1982). Received:

May 1985

191, 149]173 Ž1997. JA966889

JOURNAL OF ALGEBRA ARTICLE NO.

On Generalized Cells in Affine Weyl Groups Kirsten Bremke* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139 Communicated by Walter Feit Received June 20, 1996

We determine the lowest generalized two-sided cell for affine Weyl groups. We show that it consists of at most < W0 < generalized left cells, where W0 denotes the corresponding finite Weyl group. For parameters coming from graph automorphisms, we prove that this bound is exact. For such parameters, we also characterize all generalized left cells for finite and affine Weyl groups. Q 1997 Academic Press

1. INTRODUCTION The concept of cells for an arbitrary Coxeter system ŽW, S . was introduced by Kazhdan and Lusztig in w6x. They define left, right, and two-sided cells, which play an important role in the study of the representations of the corresponding Hecke algebra. In w10x, Lusztig extends the concept of cells to the case in which there are integers c s G 1 associated to the simple reflections s g S satisfying c s s c t if s, t g S are conjugate. Integers subject to these conditions will be referred to as parameters and the cells as generalized cells. The original cells are obtained if all c s are equal. Generalized left, right, resp. two-sided, cells are certain equivalence classes in W which give rise to left, right, resp. two-sided, modules of the Hecke algebra H corresponding to ŽW, S . and parameters c s , s g S. If W is a finite Žaffine. Weyl group the representation theory of corresponding Hecke algebras is very relevant to the representation theory of reductive groups over finite Ž p-adic. fields. In this article, we are primarily concerned with affine Weyl groups. Let Wa be an affine Weyl group with a set S a of simple reflections and *Current address: Deutsche Bank, Credit Risk Management, 60262 Frankfurt, Germany. 149 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

150

KIRSTEN BREMKE

parameters c s , s g S a . The finite Weyl group corresponding to Wa will be denoted by W0 . For equal parameters, cells in Wa have been intensively studied Žsee, e.g., w12]15x.. They have been explicitly described for type A˜r , r g N Žsee ˜4 Žsee w21, 22, 4x.. w11, 19x., ranks 2, 3 Žsee w12, 1, 5x., and types B˜4 , C˜4 , D Much less is known for unequal parameters. However, several problems that have been solved for equal parameters arise for arbitrary parameters. It is the purpose of this paper to give solutions to some of these problems for unequal parameters. Partially, our results give answers to questions raised in w23x. The main results in Sections 5 and 6 are as follows. We describe the lowest generalized two-sided cell WT in Wa . Generalizing w23, Theorem 3.22x, we find reduced expressions for the elements in WT . This cell contains nearly all elements of Wa . We show that WT consists of at most < W0 < generalized left cells and provide a geometric interpretation of WT . We also give a description in terms of a numerical function a on Wa . For equal parameters, the function a is an important tool in the study of representations of Hecke algebras in w12x. We then study the case in which the parameters c s , s g S a , come from a graph automorphism. In this situation, the coefficients of the Kazhdan]Lusztig polynomials and of the structure constants of the canonical basis of H can be interpreted in terms of intersection cohomology sheaves. We derive a characterization of all generalized left cells, which also holds for finite Weyl groups. This characterization implies that there are only finitely many generalized left cells. We show that for parameters coming from a graph automorphism, WT consists of exactly < W0 < generalized left cells.

2. PRELIMINARIES ON AFFINE WEYL GROUPS AND A GEOMETRIC REALIZATION We start by collecting some basic material about affine Weyl groups which will be needed later on. The exposition follows w8, 23x, and we refer to these publications for more details and proofs. In particular, we recall a geometric realization of the affine Weyl group given in w8x and adapt some of the notions to the unequal parameter case. Let V be a Euclidean space of finite dimension r G 1. Let F ; V be an ˇ ; V U the dual root system. We irreducible root system of rank r and F denote the coroot corresponding to a g F by a ˇ , and we write ² x, y : for the value of y g V U at x g V. Let Q be the root lattice. The Weyl group W0 of F acts on Q Žon the ˇ left., and we write Wa s W0 h Q for the affine Weyl group of type F.

GENERALIZED CELLS

151

When l g Q is regarded as an element of Wa , we will write pl instead of l. The reflection in W0 along the hyperplane orthogonal to a g F will be denoted by sa . Geometrically, Wa can be described as follows. ŽWe will not distinguish between V and the underlying affine space.. Fix a set of positive roots Fq; F, and let P : Fq be the set of simple roots. For a g Fq and n g Z, we define a hyperplane Ha , n s  x g V N ² x, a ˇ : s n4 and write sa , n s sHa , n for the reflection along Ha , n . Let a 0 g Fq be ˇ and s0 s pa 0 sa 0 . Mapping sa , such that a ˇ0 is the highest coroot in F a g P, to sa , 0 and s0 to sa 0 , 1 establishes an isomorphism from Wa to the group V which is generated by sa , 0 , a g P, and sa 0 , 1. Denote the set of simple reflections in W0 by S0 . The group Wa is a Coxeter group with generating set S a s S0 j  s0 4 . We also need the following realization of Wa Žcf. w8x.. Let F s  Ha , n N a g Fq, n g Z 4 , and let X be the set of connected components of V y D H g F H. The elements of X are called alco¨ es. The group V acts on the set of faces of alcoves, and we denote the set of V-orbits by SXa . If f is a face contained in the orbit t g SXa , we say f is of type t. For A g X and t g SXa , there is a unique alcove tA g X, tA / A, such that tA shares with A its face of type t. The involutions st : A ¬ tA on X for t g SXa generate a group WaX . There is an isomorphism from Wa to WaX , which can be described as follows. Let Aqs  x g V N ² x, a ˇ : ) 0 for all a g P , ² x, aˇ0 : - 1 4 . For s s sa , a g P Žresp. s s s0 ., the hyperplane Ha , 0 Žresp. Ha 0 , 1 . contains a unique face of Aq, whose orbit in SXa we denote by t s . The isomorphism sends s g S a to the involution st s. Identifying Wa with V yields an action of Wa on V and thereby on X, which we consider as a right action. We also identify Wa with WaX from now on and write the action of Wa on X resulting from this identification on the left. The two actions of Wa on X can be seen to commute and to be simply transitive. We fix parameters c s , s g S a . LEMMA 2.1. Let H be a hyperplane in F , and suppose H supports faces of types s, t g S a . Then s and t are conjugate in Wa .

152

KIRSTEN BREMKE

Proof. The assumptions imply that there are alcoves A, AX g X such that sA s A sH and tAX s AXsH . Because of the transitivity of the left action of Wa on X, we can find an element w g Wa such that AX s wA. We have twA s tAX s AXsH s wA sH s wsA, and hence tw s ws, i.e., s and t are conjugate via w. As a consequence of this lemma, we can associate an integer c H g N to H g F , where c H s c s if H supports a face of type s. For a 0-dimensional facet l of an alcove, we define m Ž l. s

Ý

H , l gHg F

cH ,

and we call l a special point if mŽ l. is maximal. Note that, in general, the set of 0-dimensional facets of alcoves contains the weight lattice P as a proper subset. Let T ; V be the set of all special points. If all parameters are equal, the notion of special points coincides with the notion in w8x, so T s P and mŽ l. s < Fq< for l g T Žif c s s 1 for all s g S a .. The next lemma will enable us to determine T in all cases. Let G be the Coxeter graph of ŽWa , S a ., and identify the set of vertices of G with S a . If G is of type A˜1 or C˜r , r G 2, there is a unique non-trivial automorphism ˜ on G. LEMMA 2.2. Let H, H X be parallel hyperplanes in F and let s, sX g S a . If H supports a face of type s and H X supports a face of type sX , we ha¨ e either Ži. G is of type A˜1 or C˜r , r G 2, and  s, sX 4 s  s0 , ˜ s0 4 or X Žii. s and s are conjugate in Wa . ŽNote that if G is of type A˜1 or C˜r , r G 2, the parallel hyperplanes Ha 0 , 0 and Ha 0 , 1 , for example, indeed support faces of type s0 and ˜ s0 , respectively.. Proof. Suppose Ži. does not hold Žand s / sX .. Without loss of generality we can assume that there exists an element t g S a such that st has order 3. Let A be an alcove having its face of type s on H, and let H Y be the hyperplane containing the face of A of type t. Then H Y intersects H and hence H X at an angle "pr3, which implies H YsH X s H Y s H X . Therefore, H X supports a face of type t, and sX ; t by Lemma 2.1. Since Ž sts . sŽ sts . s t, we have s ; t. Thus sX ; s.

GENERALIZED CELLS

153

Throughout this paper, we refer to the situation in which the Coxeter graph G is of type A˜1 or C˜r , r G 2, and c s 0 / c ˜s 0 as Case 1 and all other situations as Case 2. In Case 1, let l r be the fundamental weight such that P is generated by Q and l r . Claim 2.3. We have T s Q or T s l r q Q in Case 1 and T s P in Case 2. We first notice that if l is a special point and m g Q, the point l q m s l pm is a special point as well. Next, since according to the definition of the weight lattice, P consists of all points l g V that lie in the intersection of < Fq< hyperplanes in F , we have T : P. Now suppose we are in Case 1 and G is of type C˜r . ŽFor the following data about roots and weights see, e.g., w3, Chap. VIx.. Take an orthonormal basis  e1 , . . . , e r 4 of V and write a i s e i y e iq1 , 1 F i F r y 1, and a r s e r for the simple roots in P. Then l r s 12 Ž a 1 q 2 a 2 q ??? qr a r . s 1 Ž . and a 0 s a 1 q ??? qa r s e1. Hence ² l r , a ˇ0 : s 1, i.e., 2 e1 q ??? qe r l r g Ha 0 , 1. More generally, we have l r g Ha , 1 for all short roots a g Fq. Since each Ha , 1 , a g Fq, a short, supports a face of type s0 , we conclude that T s Q if c s 0 - c ˜s 0 and T s l r q Q if c s 0 ) c ˜s 0 . ŽThe case G of type A˜1 follows by a simpler computation.. In Case 2, parallel hyperplanes have the same parameter, so the special points are the same as those in w8x. For the remainder of this paper, we assume that in Case 1 we have c s 0 - c ˜s 0 , so T s Q. ŽWe can always make this true by labeling the simple reflections accordingly.. A hyperplane H s Ha , n g F divides V y H into the two parts VHqs  x g V N ² x, a ˇ : ) n4 and VHys  x g V N ² x, a ˇ : - n4 . For l g T, a quarter with ¨ ertex l is a connected component of Vy

D

H.

H , l gHg F

Hyperplanes which are adjacent to a quarter C are called walls of C . The quarter

F

H , l gHg F

VHq

will be denoted by Clq, and Alq is the unique alcove in Clq such that l lies in the closure Alq .

154

KIRSTEN BREMKE

Let Wl, l g T, be the stabilizer of the set of alcoves containing l in their closure with respect to the left action of Wa . It can be shown that this group is a maximal parabolic subgroup of Wa . Let Sl s S a l Wl be the set of simple reflections and wl the longest element of Wl . We set Alys wl Alq, and write Cly for the quarter with vertex l containing Aly. ŽThe definitions of W0 and S0 are consistent with the definitions given before.. Let F U be the set of hyperplanes H g F such that H is a wall of Clq for some l g T. The connected components of V y D H g F U H will be called boxes. For l g T, we denote by Pl the box containing Alq. If l s 0, we also write Cq, Aq Žwhich is again consistent ., etc. An integer dŽ A, B . for A, B g X is defined as follows. Consider the set of hyperplanes H g F separating A from B. For each such hyperplane, we set « H s 1 if A ; VHy, B ; VHq and « H s y1 if A ; VHq, B ; VHy. Then dŽ A, B . is the sum of all « H . Finally, we have the following partial order on X. For A, B g X, we say A F B if and only if there exists a sequence A s A 0 , A1 , . . . , A n s B of alcoves such that dŽ A iy1 , A i . s 1 and A i s A iy1 sH i for some Hi g F , 1 F i F n.

3. HECKE ALGEBRAS, GENERALIZED CELLS, AND THE a-FUNCTION In this section, ŽW, S . is an arbitrary Coxeter system. Let c s , s g S, be parameters. For equal parameters, the following definitions and facts can be found in w12x. The generalization to the unequal parameter case is contained in w10x or w23x or is straightforward. Let A s Zw ¨ , ¨ y1 x be the ring of Laurent polynomials in an indeterminate ¨ . The generic Hecke algebra H corresponding to ŽW, S . and parameters c s , s g S, is a free A-module with basis T˜w N w g W 4 and multiplication given by

Ž T˜s y ¨ c .Ž T˜s q ¨ yc . s 0 s

T˜w T˜w X s T˜w w X

for s g S,

s

for w, wX g W , l Ž w . q l Ž wX . s l Ž wwX . ,

where l denotes the usual length function on W. If w g W has a reduced expression w s s1 s2 ??? sn , si g S for 1 F i F n, we set n

mŽ w . s

Ý cs , i

is1

GENERALIZED CELLS

155

which is known to be independent of the chosen reduced expression. Denote the Bruhat order on W by F , and set Aqs Zw ¨ x. It is shown in w10x that for w g W there exists a unique element Ž . Ž . Ý Ž y1. l w yl y ¨ mŽ w .ymŽ y .Py , w Ž ¨ y1 . T˜y ,

Cw s

yFw

where Py, w , y F w, is a polynomial in Aq and deg Py, w - mŽ w . y mŽ y . if y - w and Pw, w s 1. In particular, Cw g T˜w q ¨

Ý

Aq T˜y ,

y-w

T˜w g Cw q ¨

Ý

Aq C y ,

y-w

and  Cw N w g W 4 is an A-basis for H . For y, w g W, y F w, define polynomials Q y, w g Aq by Ž . Ž . Ý Ž y1. l z yl y Py , z Q z , w s d y , w ,

yFzFw

and let Dy s

Ý ¨ mŽ w .ymŽ y . Q y , w Ž ¨ y1 . T˜w , yFw

which is an element in the set H U of formal A-linear combinations of the elements T˜w , w g W. We have an A-linear map t : H U ª A, given by

t

žÝ

wgW

a w T˜w s a e

Ž aw g A for w g W . ,

/

where e is the identity element in W. Let x, y g W. It is easy to check that

t T˜x T˜y s d x , yy1

ž

/

and

t Ž C x Dy . s t Ž Dy C x . s d x , yy1 . Let FL be the preorder on W which is generated by x FL y for x, y g W, if there exists some s g S such that t Ž C s C y Dxy1 . / 0. The associated equivalence relation is denoted by ;L , and the equivalence classes with respect to ;L are called generalized left cells. Similarly, we define FR , ;R , and generalized right cells. We say x FLR y for

156

KIRSTEN BREMKE

x, y g W if and only if there exists a sequence x s x 0 , x1 , . . . , x n s y such that for all 1 F i F n we have x iy1 FL x i or x iy1 FR x i . We write ;LR for the associated equivalence relation, and the equivalence classes are called generalized two-sided cells. ŽWe use the attribute generalized whenever unequal parameters are involved.. For w g W, define R Ž w . s  s g S N ws - w 4 . Remark 3.1. Let x, y g W. Ži. Žw23, Corollary 1.20x. If x FL y, then RŽ x . = RŽ y .. Therefore, x ;L y implies RŽ x . s RŽ y .. Žii. If C x Dy / 0, then yy1 FL x. ŽUse w23, 1.15Ža.x.. Remark 3.2. Let M be an abelian group acting on W in a way such that mŽ S . s S for all m g M. The definitions in this section naturally extend to the extended Coxeter group W X s M h W Žcompare w15x.. Generalized cells in W X are then of the form Ž m, w . N m g M, w g G4 , where G is a generalized cell in W. A function a: W ª N 0 j  ` 4 is defined as follows Žcf. w12x for equal parameters and w17x for unequal parameters.. Let w g W. For x, y g W, express T˜x T˜y with respect to the basis  Cw N w g W 4 , and consider the coefficient of Cw y1 . If the order of the pole at 0 of these coefficients is bounded as x and y vary, we set aŽ w . equal to the largest such order. Otherwise, aŽ w . s `.

4. MULTIPLICATION OF STANDARD BASIS ELEMENTS OF H Let W s Wa be as in Section 2. We set n s l Ž w 0 ., n˜ s mŽ w 0 ., and j s s ¨ c s y ¨ yc s for s g S a . For x, y g Wa , we write T˜x T˜y s

Ý

m x , y , z T˜zy1 .

zgWa

Note that T˜s2 s j s T˜s q 1 for s g S a , so any m x, y, z is a polynomial in j s , s g S a.

GENERALIZED CELLS

157

The next theorem is an important step toward describing the lowest generalized two-sided cell of Wa in Section 5. PROPOSITION 4.1.

Let x, y, z g Wa .

Ži. As a polynomial in j s , s g S a , the degree of m x, y, z is at most n , and the coefficients are non-negati¨ e integers. Žii. The degree of m x, y, z in ¨ is at most n˜. The proof of part Ži. is analogous to the proof of Theorem 7.2 in w12x. The proof of part Žii. requires some preparation. Let F be the set of directions of hyperplanes in F. We denote the direction of a hyperplane H g F by iŽ H .. For i g F , we set ci s

max

Hg F , i Ž H .si

cH ,

and if J : F , we write mŽ J . s

Ý ci . ig J

For a quarter C , the set I Ž C . is said to contain all directions i g F such that C : VHy for some H g F with iŽ H . s i. In w8x, Lusztig proves that if C is a quarter and H g F has direction iŽ H . g I Ž C ., then < I Ž CsH .< < I Ž C . HC(X(sw))-b....

It can be proved that the maps a in (1.6.3) are zero; this fact will not be used here. Note that (1.6.1) and (1.6.3) are GF-equivariant and that GF acts trivially on Q,(-1). It followsthat for any g e GF: tr(g*, H*(X2)) = 0, tr(g*,Hc*(Xj)) = tr(g*, HC*(X(w)). If one uses the exact sequence ** *

-

, HfC-1(X1)

-

HI(X2)

-

HC(X(w'))

, HC(X1)

-*

it followsthat tr(g*, HC*(X(w')))= tr(g*,H*(X1)) + tr(g*, HC*(X2)) - tr(g*, HC*(X(w))) and 1.6 is proved in this case. The general case. It sufficesto treat the case where w' = swF(s) for a s. By permuting,if necessary,w and w' (w = sw'F(s)), fundamentalreflection we may even assume that l(w') ?> (w). If l(w') > I(w) we are in case 2. If l(w') = I(w), the followinglemma shows that eitherwe are in case 1 (withw and w' possiblyinterchanged)or that w = w'. LEMMA 1.6.4. Let s, t betwofundamentalreflectionsin W and letw G W

be such that l(w) = l(swt). Then eitherw = swt, or l(sw) = l(w)-1,

or l(wt) = I(w)-1.

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P. DELIGNE AND G. LUSZTIG

Let w = s1s2 s, be a reduced expression for w (0.4). Assume that l(wt) = 1(w) + 1; then wt = s1s2* * * sktis also a reduced expression. We have l(swt) = l(wt) - 1. It follows (cf. Bourbaki, [1, Ch. IV, ? 1, Lemme 31) that either there exists j, 1 < j < k with ss, . .. sj = s, ... sj-lsj or we have In the first case, we have w = ss, . . s_1sj+l . . . Sk and SS1 . . . Sk = Sis.t. l(sw) = 1(w) - 1; in the second case we have w = swt and the lemma is proved. ..

1.7. Let us choose a maximal torus T* in G and a Borel subgroup B* c G containing T*, with unipotent radical U*. The quotient E= G!U* is a T*-torsor (= right principal homogeneous space of T*) over X = GB *. For x e X, the fibre E(x) of the projection E X is E(x) = {geGIge*

=x)U*

where e* is the point of X corresponding to B*. Let tbe N(T*) define the element w in the Weyl group W via the isomorphism a(T*, B*): W > N(T*)/T*. If x, ye X are in relative position w, the g's in G such that ge* = x and giwe*= y form a torsor A(x, y) under B* nt B*tbl = T* (U* nrTU*ilrl). For g e A(x, y), the class of gtbin E(y) depends only on the class of g in E(x). This definesa map E(x) E(y), which we denote as right multiplication by tb. We have the formulas (1.7.1)

(ut)b = (uti) ad tb-r(t),

(1.7.2)

u(tbt)-

(utb)t.

We will express (1.7.1) by saying that *T is a w-map of T*-torsors. It is induced by a w-map of T*-torsors over O(w) *w: pr* EAssume

that w = w1w2, w = tbl2

pr* E.

and that

x, y, z c X with (x, y) c O(w1) and (y, z)

G O(w2),

1(W)

=

I(W1)

+

1(W2);

then for

we have (x, z) c O(w) and

uw = (u*1)i2.

(1.7.3)

1.8. We now assume that T* and B* are F-stable. The identification q(T*, B*) of T* and N(T*)!T* with the torus T and the Weyl group W is then compatible with F. For w in the Weyl group, we denote by T(w) the torus T, provided with the rational structure for which the Frobenius is

ad (w)F. We have T(W)F

{t c T* I ad (w)F(t) = t}

For x C X, the Frobenius map induces a map F: E(x) F(u)F(t). For x e X(w), we put

E(F(x)),

with F(ut)-

REDUCTIVE

GROUPS OVER FINITE

FIELDS

111

E(x, Th)= {u C E(x) I F(u) = uth}.

This is a T(w)F-torsor.The E(x, qb)are the fibresof a map wr:X2(wb)

>

X(w) , with X(wb) c E I X(w) a T(w)F-torsor over X(w) .

The actionof G on E restrictsto an action of GF on X(wl). Up to isomorphism,the GF-equivariant T(w)F-torsorX(wb)over X(w) is independentof the liftingqb of w in N(T*): for W' = wbt,there exists t1 with ad w-'(t,)F(t,)-' = t and the map u X(tw)

ut, induces an isomorphism

X(w').

The groups GF and T(w)F act on HC*(X(tb),Q) by transportof structure. For any 0 c Hom (T(w)F, Q*) we denote by H*(X(1k), Q)o the subspace of Ql) on whichT(w)F acts by 0. HC*(X(Wb) DEFINITION 1.9. RO(w) is the virtual representation

(tb) Q 1) of GE(an elementoftheGrothendieck groupofrepresentationsof GEoverQ1). E 1_])'H,(X

The character0 can be used to transformthe T(w)F-torsorX(b) into a local systemof Q1-vectorspaces of rank one To over X(w), provided with 0: X(il) -> ToY, (xt) = 0(x)0(t). The morphism w: X(b) X(w) is finiteand wr*Q1=

To

The sheaf Y0is the subsheafof r*Q,on which TF acts by 0, hence H,*(X(l), Qi)o

=

H,*(X(w), S0o)

In particular,for 0 = 1, R'(w) = E(-l)'H,(X(w), so that Definition1.9 is compatiblewith(1.5). Example 1.10. For w

=

qb = e, r: X(wh)

Q1) X(w) becomes the projection

wr:GF!U*F of GF on the space of GF!B*F, and RO(w) is the representation G functionson GF satisfying f (gtu) = d(t) lf(g)

GF acting by (g *f)(x) = f (g'x) (inducedrepresentation).

1.11. The Borel subgroup adgB* is in X(w) if and only if adgB* and ad FgB * are in relative positionw, i.e., ifand onlyif g'-Fg C B *hB * (where il e N(T*) representsw): (1.11.1)

X(w) = {g e G I g-'Fg C B *wB *JIB

If a Borel subgroupB is in X(w), one can findg e G such that adgB* is

P. DELIGNE AND G. LUSZTIG

112

B and adgadtwB*

=

FB: X(w) = {g e G I g-lFgeiB*}JB*

(1.11.2)

n adwB*

T* .( U*nadwkU*). Changing g to gt,we can normalize g so that we have also g-'Fg e twU*: where B* n adtwB*

(1.11.3)

X(w) = {ge G I glFg e w U*}!T(w).(*(U nadwlU*).

A pointin X(wl) is definedby a Borel subgroup B, plus g e G such that adgB* = B, adgadwhB* = FB and gw = Fg mod U*: (1.11.4)

X(Wb)= {g e G I g-'Fg

?wU*}!U* n adwhU*.

COROLLARY1.12. Thefollowingassertions are equivalent:

(i) X(w) is affine; (ii) X(wh)is affine; (iii) Let p be theaction of U* n adwhU*on U* definedby p(u)v = adil-'(u)vF(u-');

thenU*!p(U* n adwtU*) is affine. U*}. The mapf: S ) U*: g P->i-l1g-Fg induces Put S = {g e G I g-1Fge wb an isomorphism GF\S ) U* and is such that for ue U* nadwtU*, f(gu) = p(u)-lf(g). Hence, GF\X(h)

U*!(U* n adwhU*).

As X(b)!T(w)F = X(w), it only remainsto use the fact that a space and a quotientof it by a finitegroup are simultaneouslyaffineor not. We will have to use another descriptionof r: X(wb) X(w). First, an easy lemma: LEMMA 1.13. Let J be the set of pairs (T, B), T an F-stable maximal

torus and B a Borel subgroup containing T. The map h which to (T, B) associates therelative position of B and FB induces a bijection GF\j-,

W

The proofwill be given in (1.15). If we use (T, B) c J to identifyW with N(T)/T, we have (1.13.1)

h(T, adwl'B) = w-'h(T, B)F(w),

wherewlc N(T) representsw, hence COROLLARY1.14. The map h induces a bijection

{GF-conjugacyclasses of F-stable maximal tori}-* Here is anotherdescriptionof h: for (T, B) c J, if a: T

-+

Wo. T and o: W

W

REDUCTIVE

GROUPS OVER FINITE

FIELDS

113

(W = N(T)/T) are definedby (T, B), then > T. aFa-1 = ad h(T, B) oF: T To give h(T, B) is the same as to give adh(T, B) oFc W oFc End(T), and to give the F-conjugacy class of h(T, B) is the same as to give ad h(T, B) oF up to W-conjugacy. 1.15. The space of maximal tori of G can be identifiedwith the homogeneous space G/N(T*), and the space of maximal torimarkedby a containing Borel subgroup can be identifiedwith G/T*. The group T* being the connectedcomponentof N(T*), (1.13) is a special case of the general result describedbelow. Let G. be a connectedalgebraic group over Fq and let x: -0 E0 be a spaces. We denoteby G, E, E the correspondmorphismof G.-homogeneous ing objects over k, and we assume that the stabilizer S(e) of e c E is the connectedcomponentof the stabilizer S(w(ae)) of r(e) e E. Since any Gohomogeneousspace has a rational point, the existence of Eo imposes no conditionon E0. The groups S(e)/S(e)? forma local systemon E, whichbecomesconstant on E; we denote by W its constant value on E. For ~ e -E, we have an S(w(a ))/S(w(e)) and, for f = ge, we have ax(f) = isomorphismax(e): W W act on E on the right, by w = a(e)(w)e; in We the group let adgax(e). this way, E becomesa W-torsor(= principalhomogeneousspace) over E. The group W is acted on by F, with F(e w) = F(e )F(w). The set WSof F-conjugacy classes in W is the set of orbitsof the action of W on itself by w w-wF(w-)-1.

PROPOSITION1.16. For e e EF and

~ e E above it, defineh(e, e) e W by

F(e) = e.h(ee

).

(i) The map h induces a bijectionfrom the set of GF-orbits in e} to W. (ii) We have h(e,ew) = w-1h(e,e)F(w). Hence the map h induces a bijectionfrom GF\EF to the set of F-conjugacy classes in W. {(e, e) IeEF,w(E

)

We will only prove (i). Let Y be the set of ~e E such that w(e) ? EF. If e%e EF, the map g + ge0 identifiesX with {g G g-'Fg ? S(u(Q))}S0e0). The Lang isogeny g-'Fg is an isomorphismG'\G G, hence the map geo+ g-'Fg inducesa bijection GF\X

>

acting by s'xF(s)) . (Sw(CO))/(S(CO)

The orbitsof this actionof S(eO)are just the usual cosets, and the resulting

114

P. DELIGNE AND G. LUSZTIG

bijection GF\X -

W is the h above.

DEFINITION1.17. Let T be an F-stable maximal torus and let B be a Borel subgroupcontaining T, withunipotentradical U; let w betherelative position of B and FB. (i) XTCB is X(w). The map go adgB induces isomorphisms XTCB

{g ? G I g-'Fg ? BF(B)}/B = {ge GI g-'Fge FB}/B n FB ={g C G I g-'Fg C FU}/ TF*(unFU). =

(ii) XTCB is {g ? G I g-'Fg e FU}/un EU. is a We have a projectionmap w: XTCB XTCB, for which XTCBGF-equivariant TF-torsor over XTCB; GF acts by left multiplicationand TF by right multiplication(it normalizesU n EU). 1.18. Let ThC N(T*) be a representativeof w. If x' C G is such that ad x'(T *, B *) = (T, B) thenB and FB = ad Fx'B * are in relativepositionw, and FB contains T, henceFB = adx'ad hB* and x'-'F(x') C ubB*n N(T*) = wTT*. Replacing x' by x

=

x't (te T*) one can achieve x-'F(x) = h. The x

such that adx(T*, B*) = (T, B) and x-'F(x) = T form a T*F-torsor. For such an x, adx induces an isomorphismT(w) T (hence T(w)F TF); this isomorphismis independentof x. PROPOSITION1.19. Let T, B, U, w be as in (1.17) and x, Thas in (1.18). The map g gx- induces an isomorphismfrom the GF-equivariant T(w)F_ torsor X(wil)over X(w) (or rather its model (1.11)) to the GF-equivariant TF torsorXTCB over XTCB (or rather its model(1.17)).

This is a straightforwardcomputation. 1.20. The cohomologyof XTCB is acted on by GF and character8: TF Q* we put as in (1.8): RTCB =

E

TF.

For any

( 1) Hc(XTcB Qi)o

(an elementin the Grothendieckgroup of representationsof GF over Q). By (1.19), for x as in (1.18), we have Rf

=

Ro~adx(W)

We shall see in Chapter 4 that RTCBis independentof B. For g e GF such that adg carries T, B, and 8 to T', B', and 8', we have clearlyRTCB= R IcB'a hence RfCB will eventuallydependonlyon the GF-conjugacyclass of T and on the orbitof 8 under(N(T)/T)F. The end of this chapter will be used in the proofof 7.10 only.

115

REDUCTIVE GROUPS OVER FINITE FIELDS

1.21. Isogenies. Let T be an F-stable maximal torus of G, and let Z be thecentreof G. We denote G x ZT thequotientof G x T by thesubgroup {(z, z-) zG Z}. Let B be a Borel subgroupcontainingT. The actionx v-*gxtof GF x TF on XTCB is inducedby an action of (G x Z T)F, given by the same formula. COROLLARY1.22. On H*(XTCB, Qi)o, ZF acts bythecharacter8 | ZF. In particular, on any irreduciblerepresentationoccurringin R'CB, ZF acts by 8|

ZF.

- G be the simplyconnectedcoveringof the derived group of Let w: GO G. T = r-'(T), B = ;r1(B) and let Z be the centreof G.

PROPOSITION 1.23. One has TF/w(TF)

> GF/w(GF).

Injectivity is clear; we have to check the surjectivity of the map GF induced by q': Tx GC; G:t, g-twr(D). Via A', Tx G is a (Tx G)F T-torsorover G (with (t, g) * T = (ti, t`g)), and one applies Lang's theorem to the connectedgroup T. Let h

1.24.

A

B be a homomorphism of finitegroups and let X be a space on whichA acts. The induced space IndB (X) (unique up to unique =

-+

isomorphism)is any B-space I, providedwith an A-equivariantmap A: X I, such that for any B-space Y. HomB (I, Y) HomA (X, Y). One has IndA(X) = llbeB'A bp(X), and 9p(X) - Ker (h)\X. PROPOSITION 1.25. The (Gx

Z

XTCBis induced by the (G x

T)F-space

ZT)F

space XTCj.B LEMMA1.26. In the diagram 0

-

o->

Ker

TF

1~

(G/Z)F

I't' 1)

l(t, 1)

l(l)

Ker->

TF

(T X ZG)F

=_==

1.

-

(T x Z G)F

-

coker

-

0

1(2) coker->

0

(G/Z)F

the maps (1) and (2) are isomorphisms. Indeed, T x Z G (resp. T x Z G) is a T (resp. T)-torsorover G/Z = G/Z, hence, since T and T are connected, (T x

Z

G)F is a TFtorsor over (G!Z)F,

and (T x Z G)' is the induced TFtorsor. Proof of 1.25. By 1.26, we are reduced to prove that XT.B, as a TF_ are The spaces XTTB and X space, is induced by the TFSpace X

P. DELIGNE AND G. LUSZTIG

116

indeed respectively TF and TF-torsor over XTCB = XiCi the TF-torsor induced by the TF torsor XTCB.

and XTCBis hence

COROLLARY 1.27. Let 8 be a characterof GF/w(GF).We denoteagain by 8 its restriction to TF. One has RapB = $RCB It followsfrom1.25 that H*(XTC

B. Q I)

=

I nd ( Gx

ZT )F

(X

TcB

Q I)

The character 0(gt) of (G x Z T)F is trivial on the image of G x 2 T. The inducedrepresentationwe consideris hence isomorphicto its tensorproduct with 0(gt), and 1.27 is a formalconsequenceof this. 2. Examples in the classical groups 2.1. Let V be an n-dimenssional vectorspace over k and put G= GL( V). If b = (b1, *. .

,

b,) is a basis of V, we may take for T* the group of diagonal

matrices and for B * the group of upper triangular matrices. The Weyl group lifts into the subgroup of N(T*) consistingof the il's inducing a permutationof basis vectors. In this case, T, W (1.1), X (1.2), E, -lb (1.7), have the followingalternativedescription. are the transpositions (a) T = GA,W = e5, the fundamentalreflections (i, i + 1) and the actionof W on T is by permutation. (b) X is the space of completeflags D1 c ** c D,1 in V: a flag D is an of V with dimDi = i for 1 < i < n - 1. increasingfiltration (c) E is the space of complete flags marked by non-zero vectors ei C DiJDi-1= GrD(V) (1