Generic Blocks of Finite Reductive Groups

Generic Blocks of Finite Reductive Groups Michel BROUE Gunter MALLE Jean MICHEL Laboratoire de Mathematiques, UA 762 du CNRS Departement de Mathematiques et d'Informatique Ecole Normale Superieure LMENS - 92 - 8 March 1992

GENERIC BLOCKS OF FINITE REDUCTIVE GROUPS Michel Broue, Gunter Malle and Jean Michel

Ecole Normale Superieure and Universitat Heidelberg 30 March 1992 To Charlie Curtis

Contents

0. Introduction 1. Notation, prerequisites and complements A. Generic nite reductive groups Usual invariants of a generic nite reductive group Class functions on d{split Levi subgroups B. Generic characters Some consequences of Lusztig's results Generic unipotent characters, uniform functions The generic Deligne{Lusztig induction and restriction 2. d{cuspidality and the uniform theory A. d{cuspidality B. Regular unipotent characters C. Regular characters Introduction: the regular character of E (GF ; 1) The generic formalism D. First application to actual nite reductive groups 3. Generalized Harish{Chandra theory and applications A. d{Harish{Chandra theory for unipotent characters The fundamental theorem d {defect groups B. Regular characters of generic d{blocks C. First application to actual nite reductive groups 4. Application to `{blocks of nite reductive groups A. {blocks of nite reductive groups: applications of generic characters d{adapted sets of prime numbers {blocks when is d{adapted G

1991 Mathematics Subject Classi cation. 20, 20G. We thank the M.S.R.I. in Berkeley, and the second author thanks the Ecole Normale Superieure, for their hospitality during the elaboration of part of this work Typeset by AMS-TEX 1

2

Michel Broue, Gunter Malle and Jean Michel | 30 March 1992

B. Generalization to {blocks with abelian defect group, and Brauer morphisms General results about {idempotents Brauer morphisms C. Isotypies D. `{Blocks Appendix : d{Harish{Chandra theory for exceptional groups 0. Introduction

This work has been motivated by two problems, which interacted with one another: attempts to solve one helped to solve the other one. The rst one originated in [Br1], where is presented a conjecture about the structure of any block with abelian defect of any nite group (see [Br1], 6.1), which states that such a block should have the same type as the corresponding block of the normalizer of its defect group. One of our goals was to check this conjecture in the case of the nite reductive groups GF (we denote by G a connected reductive algebraic group over an algebraic closure of a nite eld q , by F : G ! G a Frobenius endomorphism de ning a rational structure on this nite eld, and by GF the group of rational points). This is achieved in the present work (see x4.D, 4.35), at least for \large" prime numbers (i.e., in the split case, prime numbers which do not divide the order of the Weyl group of G). By general properties of isotypies (see [Br1]), our results imply in particular that, for large prime numbers, conjectures as Alperin's weight conjecture or the Alperin{ McKay height conjecture are veri ed for unipotent blocks (and so for \almost" all blocks) of all nite reductive groups. The second question which motivated this work was to understand better the \generic" aspects of block theory of nite reductive groups. Throughout the intensive work which has been done recently by many authors about blocks of nite reductive groups (among whom Fong{Srinivasan, Schewe, Cabanes{Enguehard, Hi, Geck, and the authors), it had become gradually clear that, for large prime numbers `, properties of `{blocks of GF do not really depend on the prime number `, but rather depend on the cyclotomic factor of the \polynomial order" of GF which is divisible by ` (see [BrMa] and x1.A below). This remark had to be put in the perspective opened by the work of C.W. Curtis, who de ned in particular the \generic degrees" of characters of GF , and subsequently of his student R. Boyce (see [Boy]) who de ned characters \with d {defect zero". A few experiments convinced us that it was indeed very often possible to replace ` by the corresponding d in many of the congruence relations involved in modular character theory. One of the aims of the present work is to present further evidence for the existence of a \generic representation theory of generic groups". Using extensively [BrMa] (where are de ned what we call here \generic nite reductive groups", as well as their d {subgroups and their centralizers), we introduce here a formalism which allows us in particular to group the unipotent characters into d {blocks and to de ne a suitable notion of \d {defect group" of a unipotent character (see de nition 3.20 F

Generic blocks of nite reductive groups | 30 March 1992

3

below). From many points of view, this d {defect group (a rational torus in G) behaves like an ordinary defect group. Moreover, if ` is a large prime which divides d (q), then the actual `{defect group is just the Sylow `{subgroup of the group of rational points of the d {defect group. One of the by{products of this approach is indeed the fact that we can treat the case of {blocks ( a set of large prime numbers which divide the same cyclotomic factor) the same way we would treat `{blocks, and this is what we present here. With these two questions in mind, we came upon the main theorem of this work (see 3.2). This theorem is certainly the main tool to study our blocks, but it is also interesting in itself. It shows in particular that the usual so{called Harish{Chandra theory of characters, as well as many aspects of the Howlett{Lehrer{Lusztig theory about induction of cuspidal characters, must be viewed as a particular case (the case associated with the rst cyclotomic polynomial 1 ) of a general set of results which hold for every cyclotomic polynomial, provided one replaces the family of \Harish{ Chandra" Levi subgroups by the family of centralizers of d {subgroups (called here d{split subgroups), and the cuspidal characters by the \d{cuspidal characters". In particular, theorem 3.2 shows that, provided each irreducible unipotent character is equipped with an appropriate sign, Deligne{Lusztig inclusion from d{split subgroups is nothing but the ordinary induction in Weyl{type groups which are independent of q. 1. Notation, prerequisites and complements

A. Generic nite reductive groups.

We call \generic nite reductive group" what is called \donnee radicielle complete" in [BrMa]. Let us recall brie y the de nition and set our notation. A root datum of rank r is a quadruple (X; R; Y; R_ ) such that (rd.1) X and Y are free {modules of rank r, endowed with a duality X Y ! denoted by (x; y) 7!. (rd.2) R and R_ are nite subsets of X and Y respectively, endowed with a bijection R ! R_ denoted by 7! _ . (rd.3) For 2 R we have <; _>= 2. Let s be the involutive automorphism of X de ned by s (x) = x? , and let s_ be its adjoint, an automorphism of Y (one has s_ (y) = y? <; y> _ ). Then s (R) = R and s_ (R_ ) = R_ . The Weyl group of the root datum (X; R; Y; R_) is the subgroup of the group of automorphisms of Y generated by the s_ for 2 R. The following de nition is valid in the general case; for the particular twisted cases 2B2 , 2F4, 2G2, the reader may refer to [BrMa]. Z

Z

De nition. An automorphism of the root datum (X; R; Y; R_ ) is an automorphism of Y which stabilizes R_ and such that _ stabilizes R. A generic nite reductive group is a pair = (?G; WG), where ?G is a root G

datum, WG is its Weyl group, and is an automorphism of nite order of ?G. Let p be a prime number and let p be a chosen algebraic closure of p . To a root datum ?G is then associated a pair (G; T) where G is a connected reductive algebraic group over p , F

F

F

4


T is a maximal torus of G, and such a pair is uniquely determined up to inner automorphisms of G induced by the elements of T. It results from the isogeny theorem (cf. for example [Sp2], 11.4.9) that, (1.1) to a linear map of Y of the form q, where q is a power of p and an automorphism of ?G, is associated an isogeny Fq : G ! G of the algebraic group G, uniquely determined up to inner automorphisms of G induced by elements of T. If q = 1, the isogeny F is in fact an algebraic automorphism of G; otherwise, the isogeny Fq is the Frobenius endomorphism associated to some q -structure on G. Note that, since T is commutative, if and 0 commute then Fq commutes with Fq0 0 for any other power q0 of p. Thus, given a generic group , the choice of a power q > 1 of p and of an element in the coset W determines a triple (G; T; F), where F = Fq is a Frobenius endomorphism (such a construction is palso possible p in the twisted cases, where q has to be replaced by an odd power of 2 or 3, and F is a special isogeny of the corresponding group of type B2, F4 or G2 cf. [BrMa], x2). Let us call such a triple a \(q; ){triple associated to ". As in [BrMa], x2 we will write (q) for GF . Usual invariants of a generic nite reductive group (cf. [BrMa]). Preliminary remark. What follows is written in the \general case". In order to apply it to the twisted cases ( 2B2, 2F4 , 2G2 ), one has to performpa few modi cations, such p ?1 as to replace the ring and the eld by, respectively, [ 2 ] and ( 2) (for 2B2 p p ? 1 and 2F4 ), or by [ 3 ] and ( 3) (for 2G2 ). We leave this work to the reader. Let = ((X; R; Y; R_); WG) be a generic nite reductive group. We set V = Y and W = WG. We denote by SV the symmetric algebra of V , by (SV )W the graded subalgebra of W{invariant elements of SV , and by AG the ideal of (SV )W consisting of elements without degree zero terms. The vector space AG=A2G has dimension r, and is endowed with an action of the image of modulo W. We set "G := (?1)r det : F

G

G

G

Z

Q

Z

Q

Q

Z

G

Q

We set R := SV=(SV AG) . Then R is a nite dimensional graded algebra which, as a W{module, is isomorphic to the regular representation of W. We denote by Rn the subspace of elements of degree n of R , and by 2N( ) the number of elements of the root system R. Then we have NX (G) R = Rn : G

G

Q

G

G

G

n=0

G

G

The polynomial order of (denoted by OG(x) in [BrMa]) is the polynomial denoted here by j j and de ned by the formula N (G) : (1.2) j j := 1 P "Gx 1 jW j w2W detV (1 ? xw) G

G

G


5

If (q; ) (whence (G; T; F)) is chosen, one has jGF j = j j(q) : We recall now how various de nitions are translated in the setting of generic groups. G

Tori and Levi subgroups.

A generic torus is a generic group such that R = R_ = ; (so W reduces to a

single element). Generic toric subgroups of (called \sous{donnees toriques" in [BrMa], and often abbreviated herein \toric subgroups" or simply \tori") are generic tori of the form (X 0 ; Y 0; wjY 0 ) where w 2 W and Y 0 is a w-stable direct factor of Y and X 0 its dual (a quotient of X). Generic Levi subgroups of (called in [BrMa] \sous{donnees de Levi", and often abbreviated herein \Levi subgroups") are generic groups of the form ((X; R0 ; Y; R0_ ); WR0 w), where w 2 W, where R0_ is a parabolic subsystem of R_ which is w-stable and where WR0 is the Weyl group of R0. G

G

Lifting scalars.

Let ? be a root datum, and let = (?; W) be an associated generic nite group. Let a 2 . (1.3) We de ne the generic nite group (a) = (?(a) ; (a) ) by the following rules. (ls.1) ?(a) := ? ? (a times), (ls.2) (a) is the product of (acting diagonally on ? ?) by the a{cycle which permutes the various factors ? of ?(a) . We have j (a) j(x) = j j(xa ) : Let (G; T; F) be a (q; ){triple associated with , let (G; T; Fa) be a (qa ; ){triple associated with , and let (G(a) ; T(a) ; F (a)) be a (q; (a) ){triple associated with (a) . Then (1.4) G(a)F a ' GFa : In other words, we have (a) (q) ' (qa ) : Changing q into ?q. Let = (?G; WG) be a generic nite group. We recall (see [BrMa]) that we de ne the generic nite group ? by ? := (?G; WG(?)) : (1.5) Note that if ?Id 2 WG, then ? = . We have j ? j(x) = (?1)r j j(?x). G

N

G

G

G

G

G

G

( )

G

G

G

G

G

G

G

G

G

The radical, the semi{simpli ed group, the adjoint group, the dual group.

If = ((X; R; Y; R_); W) is a generic nite reductive group, we denote by Q(R) the {submodule of X generated by R. The radical of is by de nition the toric group de ned by Rad( ) := ((X=Q(R)?? ; Q(R)?); jQ R ? ) : G

Z

G

G

( )

For (G; T; F) a triple associated with , the algebraic group associated with Rad( ) is Z o (G) (the connected component of 1 in the center Z(G) of G). G

G

6


The semi{simpli ed generic nite reductive group of is \the quotient of by its radical Rad( )", namely G

G

G

:= ((Q(R)?? ; R; Y=Q(R)?; R_ ); W) (here \" stands for the automorphism of Y=Q(R)? induced by , and R_ stands for the image of R_ in Y=Q(R)? ). For (G; T; F) a triple associated with , the algebraic group associated with ss is G=Z o(G). We have j ss j = j j=jRad( )j : The dual group is de ned by G

ss

G

G

G

G

G

G

:= ((Y; R_ ; X; R); W_?1 ) ;

where _ is the adjoint of , an automorphism of X, and where W is identi ed with its contragredient action on X, i.e., W is the group generated by the s for 2 R (note that is covariant in ). We denote by P(R) the dual of Q(R_ ) in Q(R), i.e., the set of all v 2 Q(R) such that 2 for all _ 2 R_ . The adjoint generic nite group of is de ned by _ _ ad := ((Q(R); R; P(R ); R ); W) : For (G; T; ) a triple associated with , the algebraic group associated with ad is Gad. We de ne the generic nite reductive group D( ) by (with obvious abuse of notation) D( ) := ((X=Q(R_ )? ; R; Q(R_)?? ; R_); W) : Thus we have a generic nite torus =D( ) = ((Q(R_ )? ; Y=Q(R_)?? ); jY=Q R_ ?? ) : G

G

Q

Q

Z

G

G

G

G

G

G

G

G

(

)

We have (1.6) Rad( ) = ( =D( )) : Let (G; T; F) be a (q; ){triple associated with . Let [G; G] be the derived group of G. Then the algebraic group associated with D( ) (resp. =D( )) is [G; G] (resp. G=[G; G]). G

G

G

G

G

G

G

Good and excellent primes.

The following de nition can be found in [SpSt], 4.1. It depends only on the root system of . (1.7) We say that a prime ` is good for if there is no closed root subsystem R1 of R such that ` divides the order of the torsion subgroup of Q(R)=Q(R1). It results from [SpSt], 4.4, that if ` is good for , then there is no closed root subsystem R1 of R such that ` divides the order of the torsion subgroup of Q(R_ )=Q(R_1 ). If ` is good for it is also good for the dual generic group . The following de nition does depend on and not only on its root system. G

G

G

G

G

G


7

(1.8) We say that a prime ` is excellent for if, whenever R1 is a closed root subsystem G

of R, then neither the order of the torsion subgroup of X=Q(R1) nor the order of the torsion subgroup of Y=Q(R_1 ) is divisible by `. Hence if ` is excellent, it is good. We shall use the following property of excellent primes. (1.9) Let (G; T; F) be a (q; ){triple associated with , and let (G ; T; F ) be a (q; ){triple associated with . Let H be a connected reductive subgroup of G containing T, and let H0 be a connected reductive subgroup of G containing T . Then the excellent primes do not divide jZ(H)=Z o (H)j nor jZ(H0 )=Z o (H0 )j. (Indeed, the group of characters of Z(H)=Z o (H) is isomorphic to the p0{part of the torsion subgroup of X=Q(RH), where RH denotes the root system of H.) The connection index fR of R is by de nition (see [Bou], ch. VI, x1, 9) the order of the nite group P(R)=Q(R). Since the group P(R_ )=Q(R_ ) is naturally in duality with P(R)=Q(R), we see that we have fR = fR_ . By [Bou], ch. VI, x2, Prop. 7, we see that fR divides jW j. Since the torsion subgroup of X=Q(R), namely the group Q(R)?? =Q(R), is a subgroup of P(R)=Q(R), we thus see in particular that: (1.10) If ` is a prime number which does not divide jWGj, then ` is excellent for . G

G

G

Intersections of Levi subgroups. Let L = ((X; RL; Y; R_L); WLv) and

subgroups of . (1.11) We say that \ G

L

M

M

= ((X; RM; Y; R_M); WMw) be two Levi

is de ned if WLv \ WMw 6= ; :

In that case, choosing an element u 2 WLv \ WMw, we de ne L

\ := ((X; RL\ RM; Y; R_L\ R_M); (WL\ WM)u) ; M

Note that WLv = WLu and WMw = WMu, whence WLv \ WMw = (WL\ WM)u. It is easy to see that \ is a well{de ned (i.e., independent of the choice of u) Levi subgroup of both and . By the preceding de nition, it is clear that (1.12) \ is de ned if and only if there exists a maximal generic torus of such that is contained in both and . In this case, is contained in \ . Remark. If (G; T; F) is a (q; ){triple associated to , we recall (cf. [BrMa], 2.1) that there is a well{de ned bijection between WG{classes of generic Levi subgroups of and GF {conjugacy classes of F{stable Levi subgroups of G. Then \ is de ned if and only if there exists L and M in the corresponding classes of Levi subgroups in G such that L \ M contains a maximal torus; in that case, L \ M is a Levi subgroup of G, whose rational conjugacy class corresponds to the WG{conjugacy class of \ . L

L

L

T

M

M

M

T

L

M

T

L

M

L

M

G

G

G

L

M

8


Class functions on . Let CF ( ) be the space of all W{invariant functions on the coset W with values G

G

in [x], called class functions on . We shall see later on that this space can be \specialized" onto the space of unipotent uniform functions for a chosen (q; ){triple associated to . P If and 0 2 CF ( ), we set <; 0>G:= jW1 j w2W (w)0(w) : Q

G

G

G

Induction and restriction. Let L = ((X; RL; Y; R_L); WLw) be a Levi subgroup of G , and let 2 CF (G ) and

2 CF ( ). We denote by ResGL the restriction of to the coset WLw, by IndGL the class function on de ned by X ~ ?1 (1.13) IndGL (u) = jW1 j (vuv ) for u 2 WG; L v2W L

G

~ where (x) = (x) if x 2 WLw; and ~(x) = 0 if x 2= WLw : In other words, we have X ( v (u)) : (1.14) (IndGL )(u) = v2WG=WL; v(u)2WLw

We have the Frobenius reciprocity: (1.15) <; IndGL >G=L: The Mackey formula.

For and two Levi subgroups of , we denote by SWG( ; ) the set of all w 2 WG such that w \ is de ned. It is clear that WLacts on SWG( ; ) from the right, while WMacts on SWG( ; ) from the left. If we let w run over a chosen double coset WMvWL for some v 2 SWG( ; ), we see that w \ is de ned up to W wL{conjugation as a subgroup of w and up to WM{conjugation as a subgroup of , which proves that the operations w IndM M\ wLand ResML \ wLdepend only on the double coset of w. This gives sense to the following formula L

M

L M

G

L

M

L M

L M

L M

L

M

L

(1.16)

M

ResGMIndGL=

X

w IndM M\ wL ResML \ wLad(w) ;

w2WMnSWG(L;M)=WL

whose proof goes like the proof of the Mackey formula for ordinary induction and restriction (note that SWG( ; ) may be empty). L M


9

Some important class functions. We denote by trRG the class function on G de ned by the character of the \regular graded representation" RG (see above). Thus the value in w of the function trRG is P N ( G ) trRG(w) := tr(w; RnG )xn : n=0

We shall consider the following polynomial (which will be identi ed later on with the \fake degree" Deg(G )) :

(1.17)

<; trRG>G=

NX (G)

X

n=0

w2W

( jW1 j

(w)tr(w; Rn ))xn : G

We have (cf. [BrMa], prop. 1.6') 1 X (w) 1 1 X =<; tr > (1.18) R G G jW j w2W detV (1 ? xw) jW j w2W detV (1 ? xw) ; or, in other words

<; trSV >G=<; trRG>GG : For a Levi subgroup of , we set G :=G (1.19) IL G will be identi ed later on with the generic degree Deg(RGL(1))). (the polynomial IL By the Frobenius reciprocity (1.15), we have NX (G) G == tr(w; (Rn )WL)xn : R G L L

G

G

n=0

It follows from 1.2, 1.15 and 1.18 (see also [BrMa], prop. 1.7) that G: (1.20) j j=j j = "G"LxN (G)?N (L)IL G

L

Every element w 2 W de nes a maximal toric subgroup (or, equivalently, a minimal Levi subgroup) w := ((X; Y ); w) : We have (see 1.19 above) G : trRG(w) := IT T

w

So we have (1.21)

j j=j G

N (G)trRG(w) ; wj = "G"T w x

T

and we see by 1.19 that trRG(w) =G, where w is jCW (w)j times the characteristic function of the W{conjugacy class of w. It follows from 1.20 and 1.21 that G trRL: (1.22) ResGLtrRG = IL

10


G < ; trRL>L: (1.23) For 2 CF ( ), we have G= IL L

G < ; trRL>L : Indeed, G=< ; ResGLtrRG>L= IL Class functions on G ? . The map G: 7! ?, where ? (w(?)) := (w), is an isometry from CF (G ) onto CF (G ? ). The map L 7! L? is a WG{equivariant bijection from the set of all generic Levi subgroups of G onto the set of all generic Levi subgroups of G ? . Then it is clear that ? G IndGL= IndGL? L ; L ResGL= ResGL?? G ; (1.24) G??(x) = IL G(?x) : IL Class functions on G (a) .

Let a 2 . We have (see 1.3) WG a = (WG)a , and it is clear that the map (w1; w2; : : : ; wa)(a) 7! w1w2 wa de nes a bijection between the set of WG a { conjugacy classes on WG a (a) and the set of WG{conjugacy classes on WG. Thus it induces an isometry G(a) : CF ( (a) ) ?!CF ( ): 1.25. Proposition. We have N

( )

( )

( )

G

G

a (a) G(a) IndGL= IndGLa L (a) ResG a : ResGLG(a) = L La ( )

( )

( )

( )

d{split Levi subgroups. Let = ((X; Y ); ) be a generic torus. We recall that in this case the polynomial order j j is just the characteristic polynomial of on Y . A generic toric d {group (cf. [BrMa], x3; we will often abbreviate here to \d {group" or even to \d-group") is a generic torus whose polynomial order is a power of d . It is proved in [BrMa], 3.3 that a d {group is completely speci ed up to isomorphism by its polynomial order. Remark. If (T; F) is an associated algebraic torus, the fact that is a d {group translates to the fact that T splits \exactly" over qd (\exactly" in the sense that no subtorus splits over a smaller eld). We call d{split Levi subgroups of the centralizers in of the generic toric d { subgroups of (\centralizer", as in [BrMa], means that their root system consists of the roots orthogonal to the considered subtorus). In particular, the centralizers of the Sylow d {subgroups are the minimal d{split Levi subgroups. We shall use the following technical remark. By [BrMa], proposition 3.8, we know that whenever is a minimal d{split Levi subgroup of , we have "G"LxN (G)?N (L) 1 mod d : It then follows that the preceding congruence holds for every d{split Levi subgroup of , from which we deduce by 1.20 T

T

T

F

G

G

G

L

G

G


11

(1.26) For a d{split Levi subgroup of , we have G mod d : j j=j j IL L

G

G

L

The Mackey formula becomes particularly simple when we restrict ourselves to pairs of d{split Levi subgroups. 1.27. Proposition. Let 1 and 2 be two d{split Levi subgroups of . (1) 1 \ 2 is de ned if and only if 1 and 2 contain a common Sylow d { subgroup of . In particular SWG( 1 ; 2 ) 6= ;. (2) If 1 \ 2 is de ned, then it is a d{split Levi subgroup of . (3) Let 1 and 2 be two d{split Levi subgroups containing the minimal d{ split Levi subgroup . Then we have SWG( 1 ; 2 ) = WM :NWG( ):WM : In particular, we have X wM IndM ResGM IndGM = M \ wM ResM \ wM ad(w) : w2WM (L)nWG(L)=WM (L) M

M

M

G

M

M

M

G

M

M

M

M

G

M

M

M

L

2

M

2 2

1

L

2

1

1

1

2

1

1

2

Proof of 1.27. For M a generic Levi subgroup of G , let us denote by Zd (M ) the Sylow d {subgroup of its center. Then M is d{split if and only if M = CG(Zd (M )).

(1) Suppose rst that 1 \ 2 is de ned. By 1.12, there exists a generic maximal torus of contained in both 1 and 2 . Then the generic groups Zd ( 1 ) and Zd ( 2 ) are both contained in the Sylow d {subgroup d of . Now if is a Sylow d {subgroup of containing d, we see that 1 and 2 both contain CG( ) hence . Conversely, if 1 and 2 both contain a minimal d{split Levi subgroup, a fortiori they both contain a maximal torus , thus 1 \ 2 is de ned by 1.12. The last assertion in (1) follows from the Sylow theorems in [BrMa]. (2) Let be a common maximal torus for both 1 and 2 . Then the generic subgroup Zd ( 1 ):Zd ( 2 ) of is de ned in an obvious way (cf. [BrMa], x3.E), and it is then clear that 1 \ 2 = CG(Zd ( 1 ):Zd ( 2 )). (3) Set = Zd ( ) (thus is a Sylow d {subgroup of ). Assume that w 2 SWG( 1 ; 2 ). Then by what precedes, there exists a Sylow d {subgroup 0 of such that 0 is contained in both w 1 and 2 . By Sylow theorems (cf. [BrMa], th. 3.4), there exist w1 2 WM and w2 2 WM such that w? 0 = w and 0 = w , from which it follows that w2?1ww1 2 NWG( ). The assertion now follows from the fact that NWG( ) = NWG( ) (cf. [BrMa], th. 3.4). M

T

M

G

M

M

M

M

T

G

S

T

M

M

M

M

S

M

T

M

M

M

M

M

S

M

T

S

T

M

T

M

L

M

M

S

G

M

S

S

M

1

M

2

1

S

1

S

S

2

G

S

S

S

L

B. Generic characters. Some consequences of Lusztig's results. For chosen q and , whence a triple (G; T; F) (cf. x1.A), one has a well{de ned bijection between the W{conjugacy classes of the coset W and the GF {conjugacy classes of F{stable maximal tori of G (cf. [BrMa], 2.1). Thus in particular, for F G G w 2 W, the virtual Deligne-Lusztig character RTw (1), denoted here Rw , is well

de ned. The following result is a particular case of a more general theorem which will be proved later (see 1.35).

12


1.28. Theorem. (Lusztig) There exists a nite set, denoted by Uch( ), and a map Uch( ) ! CF ( ), denoted by 7! m , with the following property: whenever (G; T; F) is a (q; ){triple associated to , there is a bijection 7! G F from Uch( ) onto the set E (GF ; 1) of unipotent characters of GF such that, for every w 2 W, we have m (w) = (G F ; RGwF )GF : Thus in particular (1) for all 2 Uch( ), m takes only integral values, (2) by the known orthogonality relations on the virtual characters RGwF , we have G

G

G

G

G

G

X

2Uch(G)

m (w)m (w0 ) =

(

jCW (w)j if w and w0 are W{conjugate ; 0

if not :

Sketch of proof of theorem 1.28. Let Fam(W) be the set of families of ordinary irreducible representations of W. One knows (see [Lu1], chap. 4, and also [DiMi1], x4) how to associate to each element F 2 Fam(W) a well{de ned nite group G (F ). 1. Assume rst that is chosen in W. Let Fam(W) be the set of all {invariant families. The choice of de nes, for each F 2 Fam(W) , an automorphism F of G (F ) (see [DiMi1], 4.1). We denote by Uch( ; ) the union (for all F 2 Fam(W) ) of the sets of G (F ){conjugacy classes of pairs (gF ; ), where g 2 G (F ) and 2 Irr(CG (F )(gF )). It is one of Lusztig's main theorems in [Lu1] (cf. theorem 4.23) that there is a map Uch( ; ) ! CF ( ), say 7! m , such that for each of a suitable q, there is Gchoice F GF F a bijection Uch( ; ) ! E (G ; 1) with m (w) = ; Rw GF : 2. If 0 is another element of W (de ning a triple (G; T; F 0)) there is a well{ de ned bijection ;0 from Uch( ; ) onto Uch( ; 0 ), such that G

G

G

G

G

GF

G

0

0

F GF ; RGwF GF = G; 0 ( ) ; Rw GF 0 : The system (Uch( ; ); ;0 );0 2W is projective. We de ne Uch( ) as the limit of this system. Remark. It results from [Lu1] chap. 4, that the set Uch( ), as well as the \multiplicities" m , depend only on the action of on WG and on the order of on XG. In particular, to compute the multiplicities we may reduce to the case where XG = Q(RG) and RG is irreducible. Also, (1.29) any automorphism 0 of the generic group which commutes with de nes a permutation of the set Uch( ), which depends only on the action of 0 on WG and on the order of 0. G

G

G

G

G

F Remark. We refer to [DiMi1], 6.4, for what follows. The map 7! G

is uniquely F F determined by the condition (G ; RGw )GF = m (w) for all rational unipotent characters G F . This includes all unipotent characters in the case of classical groups.


13

To ensure unicity in the case of exceptional groups for non-rational characters, the following supplementary conditions have to be added: To 2 Uch( ) there is associated a root of unity (cf. [DiMi1], before 6.4 for the de nition | we just mention that when F acts trivially on G (F ) it is de ned for the element (gF ; ) by = (g)=(1)). We ask that, if F is the smallest split power of F, the eigenvalues of F associated to G F in any Deligne-Lusztig variety of G be equal to up to some power of q=2 . To deal with the principal series unipotent characters corresponding to characters q of degree 512 of the Hecke algebra of a group of type E7 or of degree 4096 of the Hecke algebra of a group of type E8, we must add another condition: Lusztig ([Lu1], chapter 4) identi es F to a subset of the couples (gF ; ); the we are looking at is an element of F via this identi cation, which gives us a character of W. We ask that be the specialization for q = 1 of the character q associated to G F . In what follows we will assume that these conditions are satis ed. This has the following consequence: 1.30. Proposition. Let 0 be an automorphism of ?G which commutes with . Let us denote by U(0 ) the corresponding permutation of Uch( ) (cf. 1.29). Then, for any power q0 of p, and for 2 Uch( ) we have Fq0 0 (G F ) = GU (F0 )( ) . In particular we get that the action of Fq0 0 on E (GF ; 1) does not depend on q0 . G

G

G

Generic unipotent functions, uniform functions. Generic generalized unipotent functions.

We denote by [x]Uch( ) the free [x]{module on Uch( ), endowed with the quadratic form for which the canonical basis f g 2Uch(G) is orthonormal. The scalar product of two elements and 0 of [x] Uch( ) is denoted by ( ; 0)G. The elements of [x] Uch( ) are called the generic generalized unipotent functions. Q

G

Q

G

Q

Q

G

G

Generic uniform functions. P For w 2 W, we set RGw := 2Uch(G) m (w) : Thus RGw depends only on the WG{conjugacy class of w, and the system fRGw gw, where w runs over a set of representatives for the WG{conjugacy classes of WG, is an orthogonal system in Q[x]Uch(G ) (by the remark (2) following theorem 1.28). For all 2 Uch(G ), we have m (w) = ( ; RGw)G and more generally, for 2 G Q[x]Uch(G ), we set m (w) = ( ; Rw )G : We denote by UF (G ) the free Q[x]{submodule of Q[x] Uch(G ) consisting of the G Q[x]{linear combinations of the R w . The elements of UF (G ) are called the generic

uniform functions on . For 2 CF ( ), we set G

G

(1.31)

G := jW1 j

X w2W

(w)RGw ;

so that = mG . The linear map G : CF ( ) ! [x] Uch( ) such that G : 7! G is an isometric embedding whose image is UF ( ). G

Q

G

G

14


The orthogonal projection from [x]Uch( ) onto UF ( ) is denoted by uG. Thus, for all 2 [x] Uch( ), we have Q

Q

uG( ) = jW1 j

(1.32)

G

G

G

X

w2W

( ; RGw)G RGw ;

and uG( ) = Gm . Generic degrees. For all w 2 WG, we set (see x1.A) Deg(RGw) := trRG(w) : We extend linearly the map Deg to all of UF (G ), by setting Deg(G ) :=<; trRG>G ; and then to all of G Q[x]Uch(G ) by composition with u (this corresponds to the fact that, for the actual

nite reductive groups, the degree of a character equals the degree of its uniform projection). Thus we have (1.33)

Deg( ) := DeguG( ) = jW1 j

X

w2W

m (w)Deg(RGw) = Deg(Gm ) ;

and we see in particular that for 2 Uch( ), we have Deg( ) 2 jW1 j [x]. Z

G

Remark.

Whenever (q; ) is chosen, we have Deg( )(q) = G F (1). Indeed, it follows from 1.33 that Deg( )(q) is equal to the scalar product of G F P with the character jW1 j w2W Deg(RGwF (1))RGwF (1) ; which is the unipotent projection of the regular character of GF . The polynomial Deg( ) is called the generic degree of . The following statement is a consequence of a result of Boyce ([Boy], cf. [Ge] for a short | but not \generic" | proof of Boyce's result). 1.34. Theorem. Whenever 2 Uch( ) and w 2 WG are such that m (w) 6= 0, then the polynomial Deg( ) divides xN (G)Deg(RGw ) (in [x]). G

Q

The generic Deligne-Lusztig induction and restriction.

If = ((X; RL; Y; R_L); WLw) is a Levi subgroup of , a choice of 0 2 WG is said {adapted if 0 2 WLw. If this is so, any choice of q determines a quadruple (G; L; T; F) where G is a connected reductive algebraic group over p , F : G ! G is a surjective endomorphism, L is an F{stable Levi complement of some parabolic subgroup of G and T is an F{stable maximal torus of L. The following theorem translates into our language the known fact that the DeligneLusztig induction is \generic". 1.35. Theorem. For each Levi subgroup of there exists a linear map RGL: [x] Uch( ) ! [x] Uch( ) L

G

L

F

L

Q

L

G

Q

G


15

with the following properties: P (1) whenever 2 Uch( ) and RGL() = 2Uch(G) n ; then n 2 , and for P any {adapted choice of , RGL (LF ) = 2Uch(G) n G F : (2) RGLsends UF ( ) into UF ( ) and it induces IndGLon the level of class functions, G i.e., we have RG L uL= uG RG Land RGL L= G IndL. (3) If is a Levi subgroup of and if is a Levi subgroup of , then RGL= RGM RM L: Z

L

L

L

G

M

G

L

M

Sketch of proof of 1.35.

Notice that once (1) is proved, (2) and (3) are just translations of known properties of the map RGL . We prove (1). Let (G; F) be associated with as above. ~ where G ~ has a connected center and (G) There exists an embedding : G ! G ~ ~ contains the derived group of G. Let L = (L), and let L be the restriction of to L. Since for any 2 Uch( ) we have G F = G ~ F (cf. [DiMi2], 13.20), and RGL~~ (L ~ F ) = RGL (L ~ F L ) (cf. [DiMi2], 13.22), it is enough to prove the theorem for G~ , i.e., for a group with connected center. Applying the same arguments to the ~ ad is the adjoint group) which has a connected quotient map G~ ! G~ ad (where G kernel (so [DiMi2], 13.22, still applies), we may assume G adjoint. When G is adjoint, (G; F) is a direct product of groups of type (Gi(ai ) ; F (ai) ) (restriction of scalars from qai to q of (Gi; F)) where Gi is simple. Since all our constructions behave nicely with respect to products, we may assume that the group is of the form (G(a); F (a)) with G simple. (a)F a ' LFa (see 1.4) \commute" L Now the isomorphisms G(a)F a ' GFa and with RGL , and map isomorphically E (G(a)F a ; 1) to E (GFa ; 1). So we can reduce our problem to the case of (G; Fa), i.e., to the case where G is simple. We will use results of Shoji [Sho1], [Sho2] in this case. Let m be suciently divisible so that (G; F m) is split, i.e., the generic group from 1.30 that there exists a 0 associated to it is such that = 1.m It follows Fm F G G map U() on Uch( 0 ) such that F( ) = U()( ) : Moreover, the set of xed points Uch( 0 )U() has the same cardinality as Uch( ). In this context, for any

2 Uch( 0 )U() , Shoji ([Sho2], 3.1.2 and 3.1.3) de nes an element of [x]Uch( ) (that he calls Rx or Rx^ depending on the case), which we shall denote here by Sh0 (here Sh stands for \Shintani" | see [DiMi1] for the notation used here, such as ShF m =F ), such that: 7! Sh0 is an isometry, and the set fSh0 ; ( 2 Uch( 0 )U() )g spans [x]Uch( ), For mm divisible enough, there is a well-de ned GF m {class function ~ Fonm GF :F (whose values are those of a product of a suitable extension of G to the group GF m by a root of unity) such that ShF m =F (~ ) = GShF (where this last function is de ned by extending by linearity the notation G F ). We shall give a formula for RGL in the basis GShF which will make clear it is generic. G

G

F

F

( )

( )

( )

G

G

G

G

G

Q

G

G

Q

G

0

0

16


Let m be divisible enough so that (L; F m ) is split and let 0 be the corresponding generic group. Let 2 Uch( 0)U() . By Harish-Chandra theory, there exists a Levi subgroup M of L, such that (M; F m ) is split, and an element m2 Uch( 0 ) (where m0 F is cuspidal and LF is the generic group corresponding to (M; F m )) such thatm M m F F is a constituent of RLM (M ). Furthermore, since L is F-stable, we may choose m F (M; M ) to be F-stable. Let be a generic group corresponding to (M; F) and let WMv be the corresponding coset. Set WG ( 0 ; ) := NWG ( 0 ; )=WM . Then the F-stability of implies a natural action of v on WG ( 0 ; ) and on WL( 0 ; ), and the extension ~ corresponds naturally to a function on WL( 0 ; )v (an extenm sion of the v-invariant irreducible character of WL( 0 ; ) corresponding to LF , by Howlett{Lehrer{Lusztig theory). Conversely any WL( 0 ; )-invariant function on WL( 0 ; )v corresponds to a linear combination of extensions ~ . It is proved in [DiMi3], 9.7, that (assuming Shoji's results) we have L

L

M

M

M

0 M

0

M

0

0 M

0 M

0 M

0 M

0 M

0 M

F

F

RGL (LSh ) = GSh 0 0

0

where 0 is the linear combination of extensions ~ corresponding to the function G (M ;)v IndW WL (M ;)v . 0

0

0

0

Note that Uch( w) consists of only one element denoted by 1, and RGTw (1) = RGw ; G hence (by 1.35, (3)), for all w0 2 WLw, we have RGL(RL w 0 ) = Rw 0 ; G (as promised after 1.19). Deg(RGL(1)) = IL 1.36. Proposition. For all 2 [x]Uch( ), we have DegRGL( ) = Deg(RGL(1))Deg( ) : T

Q

L

G Proof. It is an immediate consequence of 1.23 and of the formula RG ) = IndGL ; L(L

for all 2 CF ( ). L

Taking the adjoint map of RGL, we get a map RGL: [x]Uch( ) ! [x]Uch( ) which is in particular such that for all 2 CF ( ), we have RGL(G ) = L ResGL : G for all w 2 WG, we have RTw ( ) = m (w)1 : The following theorem, which extends the Mackey formula (see 1.16) to the preceding context, is a consequence of an unpublished result of Deligne (see [DiMi2], 11.6) and of 1.35, (1) (and of 1.27, (3)). Q

G

G

1.37. Theorem. (1) Let and L

M

be two Levi subgroups of . Then

RG R G = M L

X

G

w RM M\wL RML \ wL ad(w) :

w2WMnSWG(L;M)=WL

Q

L


17

(2) Suppose that 1 and 2 are two d{split Levi subgroups containing the minimal d{split Levi subgroup . Then X wM RG R G = RM M M M \ wM RM \ wM ad(w) : w2WM (L)nWG(L)=WM (L) M

M

L

2 2

1

2

1

1

2

1

1

2

(Indeed, Deligne proved the Mackey formula for actual nite reductive groups under the assumption that q is large. Thus 1.37 results from the genericity of RGL.) The arguments at the beginning of the proof of 1.35, together with 1.30, prove the following two propositions that we will use later to reduce some questions to the case of simple groups. Going to the adjoint group.

_ G = ((X; R; Y; R ); W) be a generic group, and let 0 0_ 0 = ((X; R ; Y; R ); WR w) be a generic Levi subgroup of G . The sets Uch(G )

1.38. Proposition. Let L

and Uch( ad ) may be identi ed (cf. remark before 1.29), and similarly the sets Uch( ) and Uch( ), where is the generic Levi subgroup of ad de ned by = ((Q(R); R0; P(R_); R0_ ); WR0 w). Then the following diagram is commutative G

L

L

L

G

L

Uch( ) ????! Uch(

x RGL? ?

Z

G

Z

G

ad

Uch( ) ????!

Z

x? Gad ) ?RL :

L

Uch( )

Z

L

Remark. If G is the algebraic group associated to G , and : G ! Gad is the natural

isogeny, and if L is associated to , then (L) is associated to . L

L

Lifting scalars.

1.39. Proposition. Let be a generic nite reductive group and let a 2 . For all there is a natural identi cation G(a) : 7! (a) between Uch( ) and Uch( (a) ) G

N

G

G

such that the following diagram is commutative:

G

Ga Uch( ) ????! Uch( (a) ) ( )

x RGL? ?

Z

G

Z

x? G a ?RL a : G

( ) ( )

La Uch( ) ????! Uch( (a)) ( )

Z

L

Z

L

2. d{cuspidality and the uniform theory

A. d{cuspidality.

Let G F be a unipotent character of GF . We say that G F is d{cuspidal if, whenever T is a maximal F{stable torus of G such that RGT (G F ) 6= 0, then the Sylow d {subgroup of T is contained in Z(G). The translation in terms of generic groups (independent of q) is achieved by the following de nition.

18


2.1. De nition.

(1) A class function on is d{cuspidal if whenever w 2 W is such that (w) 6= 0, then ker d (w) is orthogonal to all the roots of . (2) Let 2 [x]Uch( ); we say that is d{cuspidal if the corresponding class function m is d{cuspidal, i.e., whenever is a maximal torus of such that RG( ) 6= 0, then the Sylow d {subgroup of is contained in Rad( ). T G

G

Q

G

T

G

T

G

Remarks.

1. Note that for d = 1 we get the usual notion of cuspidality for the elements of Uch( ). Indeed, condition (2) above can be reformulated as follows: \ RGT( ) = 0 whenever contains some non-central d -subtorus ". Introducing the centralizer of in , condition (2) becomes: \whenever is a proper d-split Levi subgroup of , the uniform projection of the function RGL( ) is zero" (in particular, being d-cuspidal is a property of u ( )). Now, for d = 1 and 2 Uch( ) the \positivity" of ordinary Harish-Chandra restriction shows that if RGL( ) 6= 0, then its degree is non zero, and so its uniform projection cannot vanish; thus we see that is 1{cuspidal if and only if RGL( ) = 0 for any proper 1-split Levi subgroup, which is the usual notion of cuspidality. 2. In the case where = GLn , then W = Sn , = 1, and a d{cuspidal class function on Sn is a function which vanishes on all elements whose cycle decomposition contains a cycle of length a multiple of d. Moreover, Uch( ) may be viewed as the set of partitions of n. Then a partition of n is d{cuspidal if and only if it has no d{hook (i.e., the d{cuspidal partitions are the d{cores of length n). Thus this paragraph may be considered as the generalization to all Weyl groups of the \yoga" of hooks for the symmetric groups. 2.2. De nition. Let be a d{split Levi subgroup of . We denote by CF d ( ; ) the set of all class functions on with the following property: if w 2 WG is such 0 0 w that (w) 6= 0, then there exists w 2 WG such that ker d (w) is orthogonal to all roots of . G

T

S

G

L

L

S

G

G

G

G

L

G

G L

G

L

Remarks.

Thus 2 [x] Uch( ) is such that m 2 CF d ( ; ) if and only if whenever is a maximal torus of such that RGT ( ) 6= 0, then the maximal d {subgroup of is contained in a WG{conjugate of Rad( ). CF d ( ; ) is the set of all d{cuspidal functions, if is the centralizer of a Sylow d {subgroup of , then CF d ( ; ) = CF ( ). 2.3. Proposition. Let be a d{split Levi subgroup of . (1) Let 2 CF d ( ; ). Then ResGL is d{cuspidal. (2) Let be a d{cuspidal class function on . Then IndGL 2 CF d ( ; ), and X w : ResGLIndGL = w2WG(L) Q

G

G L

T

G

T

L

G G

L

G

L

G L

G

G

G L

L

G L

Proof of 2.3. The rst assertion is obvious. The second assertion is an application of

formula 1.14.


19

Generic degrees and d{cuspidality.

For a polynomial P 2 [x], we denote by Pd the largest power of the cyclotomic polynomial d which divides P (i.e., the d {part of P). 2.4. Proposition. If 2 [x]Uch( ) is d{cuspidal, then j ss jd divides Deg( ). Proof of 2.4. Set := m . We have Deg( ) = Deg(G ), where 2 CF ( ) is d{cuspidal. Since by 1.31 Q

Q

G

G

G

Deg(G ) = jW1 j

X

w2W

(w)Deg(RGw) ;

it follows from de nition 2.1 that already

X Deg(G ) = jW1 j (w)Deg(RGw ) ;

where the sum is taken over those w's such that the Sylow d {subgroup of w is contained in Rad( ), which implies that j ss jd divides j j=j wj. Since (cf. 1.21 and the de nition of RGw) j j=j wj = xN (G)Deg(RGw), we deduce that j ss jd divides Deg(G ). We introduce a new de nition which will be technically useful. For 2 [x]Uch( ), we set T

G

G

G

Q

G

T

T

G

G

X Degd ( ) := jW1 j m (w)Deg(RGw ) ;

(2.5)

where the sum is taken over those w's such that the Sylow d {subgroup of w is contained in Rad( ). It is clear from the proof of 2.4 above that (2.6) j ss jd divides Degd ( ), and if is d{cuspidal, then Degd ( ) = Deg( ). T

G

G

The following result is speci c to the elements of Uch( ). 2.7. Proposition. If 2 Uch( ) and if j ss jd divides Deg( ), then is d{cuspidal. In particular, for 2 Uch( ), the following statements are equivalent: (i) is d{cuspidal (ii) Deg( )d = j ss jd ; (iii) Deg( ) = Degd ( ) . G

G

G

G

G

Proof of 2.7. Assume that 2 Uch( ) is such that j ss jd divides Deg( ). Whenever w 2 W is such that m (w) 6= 0, it follows from theorem 1.34 that (Deg( ))d divides Deg(RGw ). Thus we see that whenever m (w) 6= 0, j ss jd divides j j=j wj, which proves that m , whence , is d{cuspidal. G

G

G

G

T

20


B. Regular unipotent characters.

The unipotent regular character of is the element of [x]Uch( ) de ned by the formula X Deg(RGw):RGw : (2.8) URegG := jW1 j G

w2W

Q

G

Since, by 1.33, Deg( ) = ( ; URegG)G, we see that X Deg( ): : URegG =

2Uch(G) We change the notation in formula 2.8 : instead of summing over w 2 W, we sum over the corresponding maximal tori Tw, taken up to WG{conjugation. Following the notation of [BrMa], x1.B, for L a Levi subgroup of G , we put WG(L) = NWG(L)=WL. We recall that WG(Tw) = CW (w) and we get : X Deg(RGT(1)) G (2.9) URegG = jWG(T)j RT(1) ; [T ]WG where the notation \[T]WG" means that T has to be taken up to WG{conjugation. Similarly, we de ne X Cd URegG = jW1 j Deg(RGw ):RGw ; where the sum is taken over those w's such that the Sylow d {subgroup of Tw is contained in Rad(G ). We have X Degd ( ): : Cd URegG =

2Uch(G) We perform a similar change of notation as in 2.8. Let us call \d{anisotropic maximal tori" of G the maximal tori T such that the Sylow d {subgroup of T is contained in Rad(G ). We denote by Td (G ) the set of all d{anisotropic maximal tori. Then X Deg(RGT(1)) G (2.10) Cd URegG = RT(1) : [T 2Td(G)]WG jWG(T)j 2.11. Proposition. We have X Deg(RGM(1)) G RM(Cd URegM) : URegG = j W ( M )j G [Md{regular]WG Proof of 2.11. The formula 2.9 may be written as follows X Ud;MRegG ; URegG = [Md{regular]WG where X Deg(RGT(1)) RG(1) : Ud;MRegG := jWG(T)j T [T ;T d=WGRad(M)d]WG GM(1)) G M Thus it suces to prove that Ud;MRegG = Deg(R jWG(M )j RM(Cd UReg ) : We use the following lemma, which will also be useful later on.


21

2.12. Lemma. Let ' be a WG{stable function on the set of all maximal tori of .

Let

M

be a d{split Levi subgroup of . Then X X '( ) = 1 '( ) : jWG( )j jWG( )j [T2Td(M)]W jWM( )j [T ;T d=WGRad(M)d]WG M

G

G

T

T

T

M

T

Proof of 2.12. Since M = CG(Rad(M )d ), we see that Td = Rad(M )d if and only if T

is a d{anisotropic maximal torus of . Moreover, since NWG(Rad( )d ) = NWG( ), two such tori are WG{conjugate if and only if they are NWG( ){conjugate. It then follows that X X '( ) '( ) = jWG( )j [T; Td=Rad(M)d]N M jWG( )j [T ;T d=WGRad(M)d]WG WG X j W M: NWM( )j '( ) = jNWG( ) : NWG( )j jWG( )j [T ;T d=Rad(M)d]WM X '( ) : = jW 1( )j j W G [T 2Td(M)]WM M( )j M

M

M

M

T

T

T

(

T

)

T

T

T

M

T

T

M

T

We apply the preceding lemma with '( ) = Deg(RGT(1))RGT(1) : It suces now to apply the transitivity (RGT= RGM RM T) of the Deligne{Lusztig induction and the multiplicativity of the degrees (Deg(RGT(1)) = Deg(RGM(1))Deg(RM T(1))) to end the proof of proposition 2.11. We may notice the following lemma, which is immediate by 1.32. 2.13. Lemma. An element 2 [x]Uch( ) is uniform if and only if, whenever is a maximal torus in , we have X 1 G G = jW ( )j RT( RT( )) : [T ] WG G T

Q

G

T

G

T

2.14. Proposition. Let be a generic Levi subgroup of . Then RG(URegG) = Deg(RG(1))URegL: L L L

Proof of 2.14.

G

For 2 Uch( ), we have G RL(URegG); L= URegG; RGL( ) G = Deg(RGL( )) = Deg(RGL(1)):Deg( ) = Deg(RGL(1)) URegL; L : L

The following statement generalizes proposition 2.4.

22


2.15. Proposition. Let 2 [x] Uch( ) be such that m 2 CF d ( ; ). Then (1) j ssjd divides both Deg( ) and Deg( RGL( )), Q

G

G L

L

(2) We have

Deg( ) Deg(RGL(1)) Deg( RGL( )) mod : d j ssjd jWG( )j j ssjd L

L

L

Proof of 2.15. We know (cf. 2.3) that RG L( ) is d{cuspidal, and it follows from 2.4 that jLssjd divides Deg( RGL( )).

Moreover, we have

Deg( ) = ( ; URegG)G =

X Deg(RGM(1)) G M jWG( )j ( ; RM(Cd UReg ))G :

[M]WG

M

Since (by 2.15) ( ; RGM(Cd URegM))G =

X

Deg(RM T(1)) ( ; RGT (1))G ; j W ( M )j [T ;T 2Td(M)]WM T

we see that this scalar product is zero unless d WG Rad( ), whence that d divides ( ; RGM(Cd URegM))G unless = Rad( ). L

T

j ssjd L

d WG

T

L

G d URegM))G 0 mod d unless This proves that ( ; RM(C j ssjd 2.15 is immediate. L

M

=WG , from which L

C. Regular characters. Introduction: the regular character of E (GF ; 1).

In what follows we shall give a \generic version" of the series E (GF ; 1) introduced in [BrMi], and we start by computing the regular character of this series in a particular case, in order to justify the notation which will be introduced in the generic case. Notation and prerequisites.

Let (G; T; F) be a (q; ){triple associated with . We recall that there is a natural indexation of the set of minimal closed non trivial unipotent subgroups of G normalized by T by the root system R of . For 2 R, we denote by U the corresponding unipotent \root subgroup". Notice that, since [G; G] is connected, we have G

G

(G=[G; G])F = GF =[G; G]F :


23

2.16. De nition. Let be a characteristic zero eld, containing enough roots of unity for GF . (1) We denote by AbIrr(GF ) the group of {characters of the abelian group GF G PF =[G; G]F , considered as characters of GF . We denote by AbReg = K

K

2AbIrr(GF ) the corresponding regular character. (2) For a set of prime numbers, we denote by Ab Irr(GF ) the subgroup of AbIrr(GF )F consisting of characters whose order is a {number. We denote by P G Ab Reg = 2Ab Irr(GF ) the corresponding regular character.

Notice that the notation \AbIrr(GF )" is slightly abusive, since GF =[G; G]F depends on G and F, and not only on GF . Whenever M is a Levi subgroup of G, we have M = Z o (M)[M; M]. In particular for L a Levi subgroup of M we have M = L[M; M], and M=[M; M] = L=L \ [M; M]. Thus MF =[M; M]F is a quotient of LF =[L; L]F , and the restriction of characters induces inclusions AbIrr(GF ) AbIrr(MF ) AbIrr(LF ) : From now on, we assume that (2.17) is a set of good prime numbers for . G

Centralizers of {elements.

Since T = p Y , the group p X is identi ed with Hom(T; p ). For s 2 T and 2 X, we denote by (s) the corresponding element of p . Let s 2 T. We set F

F

F

F

(2.18)

Rs := f ; ( 2 R)((s) = 1)g ;

and we denote by G(s) the connected centralizer CGo (s) of s in G. 2.19. Proposition. Let s be a {element of T. (1) Rs is a parabolic subsystem of R. (2) G(s) is the Levi subgroup of G generated by T and by the U for 2 Rs . Proof of 2.19. (1) cf. [GeHi], 2.1.

(2) See for example [BoTi], 3.4, and also [DiMi2], 2.3. Centralizers of {characters.

Let be a character of TF , whose order is divisible only by primes in : we then say that is a {character of TF . ( = ) 0 : Then it follows from Suppose chosen once for all an isomorphism p ?! p [DeLu], (5.2.3), that we have an exact sequence F

(2.20)

Q Z

?1 Y ?! TF ?! f1g ; f0g ?! Y F?!

24


which allows us to view TF as a quotient of Y , and thus to de ne the image (y) of an element y 2 Y through a character of TF . For such a character, we then de ne the closed subsystem (2.21)

R := f ; ( 2 R)((_ ) = 1)g ;

and the subgroup G(T; ) of G by

G(T; ) := T: < U >2R : 2.22. Proposition. Let be a {character of TF .

(1) The subsystem R is parabolic in R. (2) The group G(T; ) is the largest of the F{stable Levi subgroups M of G such that T M, belongs to the image of AbIrr(MF ) in AbIrr(TF ).

Proof of 2.22.

(1) We may assume that T is part of a (q; ){triple associated to and let (G ; T ; F ) be a (q; _?1 ){triple associated with . We choose an isomorphism G

G

(*)

Irr(TF ) : TF ?!

The reciprocal image of through the isomorphism (*) is a {element s 2 T F . The system R is then the dual of R_s (see 2.18), hence is parabolic by 2.19, (1). (2) The second assertion of 2.22 results from [DeLu], proposition 5.11, (i), and from the fact that a closed connected subgroup H of G containing T is generated by T and by fU ; ( 2 R)(U H)g (see [BoTi], 3.4). We set GF (T; ) := G(T; )F : We recall that, following Lusztig's notation, we denote by E (GF ; 1) the set of unipotent characters of GF . For any 2 E (GF (T; ); 1) we then have (see [Lu2]) a well de ned irreducible character of GF , de ned by the formula G(GF(T;);;) := "G "G(T;) RGG(T;) () (this is a particular case of the \Jordan decomposition of characters"). Two characters G(GF(T;);;) and G(GF(T0 ;0 );0 ;0 ) are equal if and only if the triples (G(T; ); ; ) and (G(T0 ; 0 ); 0 ; ) are GF {conjugate (see [Lu2], and also [DeLu], 5.20). A particular case.

We assume now that (2.23) Every {element of GF is GF {conjugate to an element of TF . We shall see in x4 that the preceding hypothesis is satis ed if = f`g for ` a \large" prime number (\d-adapted" in the sense of x4) and T contains a Sylow d -subgroup of G.


25

It is then easy to see that the regular character of the series E (GF ; 1) (see [BrMi]) is given by the following formula: (2.24)

X

F

RegG :=

[(G(T;);;)]GF

G(GF(T;);;) (1):G(GF(T;);;) :

Let us call {split Levi subgroups of (G; F) (or, by abuse of notation, \of G") the F{stable Levi subgroups M of G for which there exists an element 2 Ab Irr(TF ) such that M is GF {conjugate to G(T; ). For M an F{stable Levi subgroup of G containing T, we de ne (2.25)

F M G; :=

X

f ; (2Ab Irr(MF ))(M=G(T;))g

:

2.26. Lemma. Under the preceding hypothesis, we have F

RegG = =

X Deg(RGM(1)) G MF F R ( : URegM )

[M]GF

jWGF (M)j M G;

X Deg(RGM()) G MF R ( :) ;

[(M;)]GF

jWGF (M; )j M G;

where M runs over the set {split Levi subgroups of G, where runs over the set E (MF ; 1) of irreducible unipotent characters of MF , where we set

X

F

URegM :=

2E (MF ;1)

(1)

:

and WGF (M) := NGF (M)=MF ; WGF (M; ) := NGF (M; )=MF : Proof of 2.26. Left to the reader (it is immediate once the notation is understood).

The generic formalism.

Lemma 2.26 justi es the introduction ofF the following formalism, meant to de ne a \generic" version of the character RegG : The group WG acts naturally on the set of pairs ( ; ) (cf. 1.29), where is a d{split generic Levi subgroup of and where 2 Uch( ); we set WG( ; ) = NWG( ; )=WM. We denote by Rd ( ) the free [x]{module on the set of WGconjugacy classes of such pairs, with basis denoted by fG(M;)g. We de ne (cf. 2.26) X Deg(RGM()) G (2.27) RegGd := jWG( ; )j :(M;) : [(M;)]WG M

G

M

M

M

G

Q

M

M

26


For any generic Levi subgroup of , we denote by Ad;G( ) the free [x]{module on the set of pairs ( ; ) such that is a d{split Levi subgroup of containing and 2 Uch( ), with basis denoted by fN G;M g : N N We set G;M := G;M 1 : For a d{split generic Levi subgroup of , we set M G := M G;M. F NF MF Remark. N G;M should be viewed as the formal version of ResNF (M G; ) : We de ne RGN: Ad;G( ) ! Rd ( ) by the formulae ( RG(M ) = G for a d{split Levi subgroup of , M G (M;) (2.28) G N RN(G;M ) = RGM(M G RM N( )) for , d{split. N

G

M

N

Q

M

G

N

N

M

N

G

G

M

G

N

We set Abd RegN:=

(2.29)

X fM; (N4M)(Md{split)g

M

M

N G;M:

F [Notice that Abd RegNshould be viewed as the generic version of Ab RegN := P F .]

2Ab Irr(N )

With these notation, it follows from de nition 2.27 that X Deg(RGM(1)) G M RegGd = jW ( ; )j RM(G Deg():) [(M;)]WG G (2.30) X Deg(RGM(1)) G M RM(G URegM) : = j W ( ) j G [M]WG M

M

The following theorem is the analogue of 2.11. 2.31. Theorem. We have X Deg(RGT(1)) G RT(Abd RegT ) ; where runs (1) RegGd = j W ( ) j G [T ] WG T

T

over the set of maximal tori of . G

(2) RegGd = = where

M

X Deg(RGM(1)) G RM(Abd RegM Degd ():) j W ( ; ) j G [(M;)]WG X Deg(RGM(1)) G M M M

[M]WG

jWG( )j RM(Abd Reg Cd UReg ) ; M

runs over the set of d{split Levi subgroups of .

Proof of 2.31.

G


27

(1) By 2.30, we see that X RegGd = jW1 j jWMjDeg(RGM(1))RGM(M G URegM) : G M Also, by 2.9, we see that

X Deg(RM URegM= jW1 j T(1))RM T(1) : MT M

Thus we get, using 2.28, X X RGT(T

1 ) (1)) RegGd = jW1 j Deg(RGT G;M GT G fM; T Mg and the result follows from 2.29. (2) Let us de ne

X

Deg(RGT(1)) RG(Ab RegT ): T d j W ( ) j G [T 2Td(G)]WG By lemma 2.12 applied with '( ) = Deg(RGT(1))RGT(Abd RegT ), we see that Cd RegGd :=

T

T

RegGd =

X

Deg(RGM(1)) RG(C RegM) : jWG( )j M d d [M; Md{split]WG M

Thus it suces to prove the following lemma. 2.32. Lemma. We have Cd RegGd = Abd RegG CdURegG : Proof of 2.32. We have X Deg(RGT(1)) Abd RegG RGT (1) ; Abd RegG Cd URegG = j W ( ) j G [T 2Td(G)]WG T

and it suces to check that, for 2 Td ( ), we have Abd RegG RGT(1) = RGT(Abd RegT ); T

G

which is immediate.

In order to de ne a generic version of the adjoint RGNof RGNwhich may be \specialized" to the ordinary RGN , we must introduce more notation. Let Rd;G( ) be the free [x]{module on the set of WN{conjugacy classes of pairs ( ; ), where is a d{split Levi subgroup of such that \ is de ned, 2 Uch( \ ) , N

Q

M

M

G

M

N

M

N

28


with basis denoted by fN (M;)g : N (M Remark. We must see N G;\MN ) . (M;) as RM\N We de ne RGN: Rd;G( ) ! Rd ( ) by the formula N

G

G M RGN(N (M;)) := RM(G ) ;

(2.33)

and then we de ne RGN: Rd ( ) ! Rd;G( ) in such a way as to t with the Mackey formula, i.e., for ( ; ) any d{split pair of , we set G

N

M

(2.34)

G

RG(N ) := N (M;)

w \N wM w RN N\ wM GM ;wM R wM\N( ) :

X

WNnSWG(M;N)=WM

2.35. Theorem. Let be a generic Levi subgroup of . Then N

G

RG(RegG) = Deg(RG(1))RegN:

N

N

d

d

Sketch of proof of 2.35.

Since, by 2.31,

RegGd =

X Deg(RGT(1)) G T jWG( )j RT(Abd Reg ) ;

[T ] WG

T

and since Abd RegT= RegT d , it follows that RG(RegG) = Deg(RG(1))RegT

T d

and also that

RG(RegG) =

N

d

X

T

d

RN ( RN ( RGN(RegGd ))) : T T

[fT ;T Ng]WN

Thus the result follows from the transitivity of both RGNand RGN.

D. First applications to actual nite reductive groups. Let (G; T; F) be a (q; ){triple associated with . Using notation introduced at the beginning of x2.C, we denote by (G=[G; G])d0 the product of all the Sylow e {subgroups (e = 6 d) of the torus G=[G; G]. We then denote by Abd Irr(GF ) the group of all characters of GF =[G; G]F (viewed as characters of GF ) whose kernel F Pcontains the group of rational points of (G=[G; G])d0. We denote G

by Abd RegG := 2Abd Irr(GF ) the corresponding regular character.


29

2.36. De nition{Proposition. (1) Let N be an F{stable Levi subgroup of G. The class function NGF on NF is

de ned inductively by the formula

X

F

fM ; (M d{split)(NM)g

F MF N ResM NF (G ) = Abd Reg :

(2) (Equivalent de nition) For N an F{stable Levi subgroup of G, we de ne Irr(NGF ) := Abd Irr(NF ) ?

P

[

fM ; (M d{split)(NM)g

Abd Irr(MF ) :

Then we have NGF = 2Irr(NGF ) : For a unipotent character of MF , we set WGF (M; ) := NGF (M; )=MF and (with obvious notation) F

RegGd :=

X

[(M;)]GF

DegRGM () RG (MF :) : jWGF (M; )j M G

The following result is a kind of \generic version" of [BrMi], thm. 2.2. 2.37. Theorem. Let Fda(d) be the largest power of d which divides j j. Then all the values of jWG j RegGd are divisible by d (q)a(d) . Proof of 2.37. It depends on the following general property of the Deligne{Lusztig induction. 2.38. Proposition. Let L be an F{stable Levi subgroup of G. If is a class function on LF which takes integral values, then the function jGF jp RGL () takes integral values. Proof of 2.38. Let P be a parabolic subgroup of G such that L is a Levi complement of P, and let U be the unipotent radical of P. We denote by VU the associated Deligne{Lusztig variety, on which the nite group GF acts by left multiplication, while the nite group LF acts by right multiplication. We denote by U the class function on GF LF de ned by the formula G

U (g; l) :=

X i

tr((g; l?1 ) ; H i (VU ; `)) : Q

It is known (cf. [DeLu], prop. 3.3) that the function U takes integral values, which are independent of ` (` 6= p). Moreover, the map RGL is de ned by U as follows (cf. for example [Br1], x1). For a class function on LF , the value at g 2 GF of RGL () is given by the following formula: RGL ()(g) = (U (g; ); )LF where we denote by (g; ) the class function on LF whose value at l 2 LF is (g; l).

30


By [Br1], 4.1, it suces then to prove that U is \`{perfect" for every ` 6= p. This follows from [Br1], prop. 2.1 (see also [DeLu], prop. 3.5). We can now prove Theorem 2.37. F By 2.31 and 1.30 (which shows that we may identify WG( ; ) and WG (M; M )), we see (note that our notation has been chosen so that everything specializes nicely) that M

F

RegGd =

(2.39)

X

[(M;)]GF

DegRGM (1) RG (Ab RegMF :Deg ()(q):) : d jWGF (M; )j M d

It suces to prove the following lemma. 2.40. Lemma. Whenever M is Ga d{split Levi subgroup of G and is a unipoM (1) RG (Abd RegMF:Degd ()(q):) takes values tent character of MF , jWM j jWDegR GF (M; )j M a ( d which are divisible by d (q) ) . Proof of 2.40.

On one hand, since WGF (M; ) = NGF (M; )=MF , we have "G "M jGF : N F (M; )j 0 : DegRGM(1) = p G jWGF (M; )j jWGF (M; )jp

On the other hand, byF 2.5, we see that jWM jDegd ()(q) is divisible by j and thus jWM jAbd RegM :Degd ()(q): takes values divisible by j jd (q). Since j jd (q) = j jd (q), we see that 2.37 results from 2.38.

M

ss jd (q),

M

M

G

3. Generalized Harish{Chandra theory and applications

A. d{Harish{Chandra theory for unipotent characters. The fundamental theorem.

We call d{split pairs of the pairs ( ; ) where is the centralizer in of a d {torus of , and where 2 Uch( ). We say that such a pair is d{cuspidal if is d{cuspidal. 3.1. De nition. Let ( 1 ; 1) and ( 2 ; 2 ) be two d{split pairs. We write G

M

G

M

M

M

M

( 1 ; 1 ) 4 ( 2 ; 2 ) M

if

(1) 1 is a subgroup of 2 , (2) (RM M (1 ); 2)G 6= 0. M

M

2 1

M

G


31

The Weyl group WG of acts on the set of d{split pairs of and it stabilizes the relation 4 . If ( 1 ; 1) and ( 2 ; 2) are two d{split pairs, we set ( 1 ; 1) 4WG ( 2 ; 2) if and only if there exists w 2 WG such that ( 1 ; 1)w 4 ( 2 ; 2). For ( ; ) a d{cuspidal pair of , we denote by Uch( ; ( ; )) the set of 2 Uch( ) such that ( ; ) 4 ( ; ). Note that Uch( ; ( ; )) depends only on the WG{conjugacy class of ( ; ). The following statement is fundamental. It expresses that, provided each unipotent character is equipped with the appropriate sign, and if we work only with d{split Levi subgroups of , then the Deligne{Lusztig induction can be viewed as ordinary induction through generalized Weyl groups. G

G

M

M

M

M

M

L

M

G

G

L

G

G

G

L

L

L

G

3.2. Fundamental Theorem.

(1) For each d, the sets Uch( ; ( ; )) (where ( ; ) runs over a complete set of representatives of the WG{conjugacy classes of d{cuspidal pairs of ) constitute a partition of Uch( ). (2) There exists a collection of isometries G

L

L

G

G

I(M L;) : Irr(WM( ; )) ! Uch( ; ( ; )) Z

L

Z

M

L

where

runs over the set of all d{split generic subgroups of , ( ; ) runs over the set of d{cuspidal pairs of , M

G

L

such that (a) for all

M

M

and all ( ; ), we have L

WG(L;) RGM I(M L;) = I(GL;) IndWM(L;) :

(b) The collection I(M L;) M;(L;) is stable under the conjugation action by WG. (c) I(LL;) maps the trivial character of the trivial group WL( ; ) to . L

We believe that 3.2 will have a general and nice proof, involving in particular a generalization of the notion of Hecke algebras (cf. [Br1] last paragraph, and [Br2]) and a suitable generalization of Howlett{Lehrer theory (which must be understood as adapted to the case d = 1 | see below) to a \d{Howlett{Lehrer theory". Unfortunately such a theory seems out of reach for us at the moment. Indeed, the case d = 1 is known: the rst assertion is the so{called Harish{Chandra theory, and the second assertion can be deduced from theorem 5.9 in [HoLe]. We provide a case by case proof of the fundamental theorem. The case of type A will be treated rst; for the classical groups, we give a rough sketch (we refer to [En2] for a more detailed proof), and a proof for the exceptional groups is contained in the appendix. It will be convenient to prove 3.2 together with the following result, stating a general \Ennola"{duality via G for Deligne{Lusztig induction:

32


3.3. Theorem1 (\Ennola"{duality for RGL). There exists a bijective isometry G : Uch( ) ! Uch( ? ) such that the following diagram is commutative : G

G

G Uch( ) ????! Uch( ? )

x RGL? ?

Z

G

Z

L

x? G? ?RL? : G

Uch( ) ????! Uch( ?) We de ne a sign " and ? by G( ) = " ? where 2 Uch( ) and ? 2 Uch( ? ). Note that the G de ned above extends the one of 1.24 through the identi cation of CF ( ) with UF ( ). Proof of 3.2 and 3.3. First note that it suces to prove both statements in the case where is simple. Namely, by 1.38, we may reduce to the case of adjoint generic groups, which are direct products of simple generic groups (a) . Since RGLis compatible with products, we may hence assume that we have just one factor (a) , and nally, by 1.39, that a = 1. Z

L

Z

L

G

G

G

G

G

G

G

The case of the type A.

Since the property we want to prove concerns only the unipotent characters, we may as well assume that we are dealing with the linear or the unitary group. Let us denote by GLn the generic nite group corresponding to the general linear groups of rank n. Thus GLn = (( n; R; n; R_ ); Sn ) with R = R_ = f(ei ? ej ) ; (1 i 6= j n)g, where fe1; : : : ; eng is the canonical basis of n. Then the generic nite group corresponding to the unitary groups of rank n is GUn := GL?n , where GL?n = (( n; R; n; R_); Sn :(?Id)) (cf. [BrMa], x2). If we have shown 3.3, then it suces to prove 3.2 for GLn. But 3.3 is well known in the case of type A. Namely, by [LuSr] (see for example [DiMi2], 15.4) if we take as reference torus the \diagonal" maximal torus of GLn( q ) (which is stable by both the standard and the \twisted" Frobenius endomorphism), the irreducible unipotent characters of GLn (q) are the functions 1 X (w)RGLn (q) for 2 Irr(S ) ; (q) GL n w n := n! w2Sn Z

Z

Z

Z

Z

F

while the irreducible unipotent characters of GUn(q) are the " GUn (q) where " = 1 and X (q) for 2 Irr(S ) : GUn (q) := n!1 (w)RGU n w n w2Sn ? Identifying Uch(G ) and Uch(G ) with Irr(Sn ), we then see that the map G we are

looking for is the map 7! " , which shows 3.3. The natural bijection 7! ? between the set of generic Levi subgroups of GLn and the set of generic Levi subgroups of GUn is such that the d (x){subgroups of M

M

1 At present, when G contains a factor of exceptional type, we have only veri ed this result when L is d{split for some d.


33

GLn correspond to the d (?x){subgroups of GUn (recall that, for d > 2, d (?x) = 2d (x); d=2 (x) ; or d (x) if d is respectively odd, congruent to 2 modulo 4, or divisible by 4). So to prove 3.2 for type A, we may as well assume that = GLn, which we do from now on. We identify CF ( ) and UF ( ) through the map 7! G (see 1.31). Thus the map RGLis identi ed to the induction IndGLon class functions (see 1.35, (2)) and RGL to ResGL. Let be an irreducible character of Sn and let w 2 Sn . Then RGT w ( ) = (w)1 and we see then by 2.1 that is d{cuspidal if and only if (w) = 0 whenever the cycle decomposition of w contains a cycle of order divisible by d. Thus it results from the Murnaghan{Nakayama formula (see for example [JaKe]) that is d{cuspidal if and only if the partition of n corresponding to is a d{core. Moreover, a d{split Levi subgroup of is of type GL(ad) GL(ad) GL(adm) GLs (cf. 1.3). It is then easy to see that a d{cuspidal pair of is a pair ( ; ) such that is the direct product GLr where is a torus such that j j = (xd ? 1)a , (with the appropriate identi cations) is a character of Sr corresponding to a d{core of cardinal r. Thus n = ad + r, and = (?L; WLv), where ?Lis a suitable root datum, WL' Sr , v is an element of cycle type da of Sad (here we identify Sad with the obvious subgroup of the Young subgroup Sad Sr in Sn ). We have 2 Uch( ; ( ; )) if and only if is a d{core of . This proves in particular the rst assertion of 3.2. A d{split Levi subgroup containing is of type GL(ad) GL(adm) GLs ; where M s r. We can compute RGM by computing successively RM M , then RM , etc., where (d) ( d ) ( d ) ( d ) 1 is of type GLa GLam? GLdam +s ; 2 is of type GLa GLam? GLdam? +dam +s ; and so we see that it is enough to assume (since RGLbehaves nicely with respect to products) G

G

G

G

2

1

G

L

L

T

T

T

L

G

L

M

M

L

1

M

1

1

2 1

1

1

2

1

= GLn and = GL(bd) GLs with ; which we do from now on. Thus n = ad + r = bd + s where r s, = (?G; WG) where WG ' Sn , and = (?M; WMw), with WM' (Sb )d Ss , a Young subgroup of Sn w is an element of cycle type db in Sbd (note that WMw = WMv). The class functions on are the usual class functions on Sn , while the set CF ( ) may be identi ed with the set of usual class functions on Sb Ss . Indeed, every 2 CF ( ) (i.e., is a WM{stable function on WMw) de nes a class function b on Sb as follows: for (1; 2; : : : ; d) 2 (Sb )d , we set b (12 d ) := ((1 ; 2; : : : ; d):w) : We have WG( ; ) = CSad r (v)=Sr ' (( =d ) o Sa ) ; G

L

M

M

G

M

G

M

M

L

+

Z

Z

34


and

WM( ; ) ' Sb (( =d ) o Sa?b ) : With the preceding identi cations, the description of I(GL;) and I(M L;) amounts to the description of corresponding maps Id;m (m = n; s), where Id;n : Irr(( =d ) o Sa ) ! Irr(Sn ; (d; )) ; Id;s : Irr(( =d ) o Sa?b ) ! Irr(Ss ; (d; )) ; where Irr(Sm ; (d; )) denotes the set of irreducible characters of Sm (partitions of m) with d{core . We want these maps to be such that the following diagram is commutative : Id;n Irr(( =d ) o Sa )) ?! Irr(Sn ) (3.4) # # s IdSb Id; Irr(Sb ( =d ) o Sa?b ) ?! Irr(Sb Ss ) where the vertical arrows are the restriction maps. With appropriate choices (see [Os]), an irreducible character of ( =d )oP Sa (resp. of ( =d )Po Sa?b) is de ned by a family of partitions (i ` ai )i=1;:::;d where di=1 ai = a (resp. di=1 ai = a ? b). Such a collection may be interpreted as the \d{quotient" of a partition of n (resp. of s) with d{core , and de nes a sign " (cf. for example [En1], x3) in a way which is \compatible" with the Murnaghan{Nakayama formula, and this de nes the maps Id;m . Assume rst that a = b. In this case 3.4 is equivalent to the following property : For all 2 Irr(( =d ) o Sa ), for all (1 ; 2; : : :; d ) 2 (Sa )d and for all 2 Sr , we have Id;n ()((1 ; 2; : : :; d ):w:) = (1 2 d )() : This equality results from the Murnaghan{Nakayama formula for wreath products (see [Os] or also [Rou]) and from the fact that the cycle type of ((1; 2; : : :; d ):w) 2 Sad is d times the cycle type of 1 2 d 2 Sa . The general case a > b is analogous, and is translated as follows. We let w be as above. Then for 2 Irr(( =d ) o Sa ) the value of Id;n () on ((Sb )d (Sa?b )d Sr ):w is given as follows: For (1 ; 2; : : : ; d ; 10 ; 20 ; : : : ; d0 ) 2 (Sb )d (Sa?b )d and 2 Sr we have Id;n ()((1 ; 2; : : : ; d; 10 ; 20 ; : : : ; d0 ):w:) = ((1 2 d )(10 20 d0 ))() : Let now w0 be an element of cycle type (d)a?b in S(a?b)d , and let b denote the restriction of to Sb ( =d ) o Sa?b . Then the value of (IdSb Id;s )(b ) on (Sb (Sa?b )d Sr ):w0 is given as follows: For (1 ; 2; : : : ; d ; 10 ; 20 ; : : : ; d0 ) 2 (Sb )d (Sa?b )d and 2 Sr we have (IdSb Id;s )(b )((1 2 d )(10 ; 20 ; : : : ; d0 ):w0 :) = ((1 2 d )(10 20 d0 ))() : It is clear that the preceding formulae prove the commutativity of 3.4. Note also that statement (b) of the theorem holds trivially since all automorphisms of GLn x all unipotent characters. L

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z


35

The case of classical groups.

We use Asai's results on the decomposition of the Lusztig functor RGLas cited in the paper [FoSr] of Fong and Srinivasan. Let be a generic group of type Bl , Cl , Dl or 2Dl . Lusztig has shown that the unipotent characters Uch( ) of these groups may be parametrized by means of so called symbols. A symbol is an unordered set fS; T g of two strictly increasing sequences S = x1 < < xa , T = y1 < < yb of nonnegative integers, usually denoted as = xy 1 :: :: :: xya : G

G

b

1

Two symbols are considered to be equivalent if they can be transformed into each other by a sequence of steps

x ::: x 0 x + 1 ::: x + 1 1 a 1 a y1 : : : yb 0 y1 + 1 : : : yb + 1 :

The rank of a symbol is rank() :=

X

xi +

X

yi ?

"

a+b?1 2

2#

;

and ja ? bj is called the defect of . The unipotent characters of groups of type Bl and Cl are in bijection with the equivalence classes of symbols of rank l and odd defect. The unipotent characters of groups of type Dl are parametrized by (classes of) symbols of rank l and defect divisible by 4, except that if the two rows of are equal, two unipotent characters correspond to the same symbol; such symbols are called degenerate. Finally the unipotent characters of groups of type 2Dl are parametrized by symbols of rank l and defect congruent 2 (mod 4) (cf. [Ca], 13.8). Let us now rst assume that the integer d of the theorem is odd. We will need the decomposition of RGLin the particular case of a Levi subgroup of of type , where is a generic classical group of the same type as (or possibly ? in the case of even orthogonal groups) and is a torus such that j j = xf ? 1 (we will always have djf). In this particular situation the decomposition of RGL( ), where = S;T denotes the unipotent character of , hence , indexed by the symbol = (S; T), can explicitly be described by a simple formula of Asai, which is given in (3.1) of [FoSr]. Namely we have L

H

G

T

H

(3.5) RGL( ) =

X

x2S x+f 2= S

G

T

H

G

T

L

(?1)MS;x S ?fxg[fx+f g;T +

X

y2T y+f 2=T

(?1)MT;y S;T ?fyg[fy+f g ;

where MS;x := jfs 2 S j x < s < x + f gj. Also by convention if a degenerate symbol occurs on the right hand side of the formula, then it is understood that both unipotent characters corresponding to it appear in the sum. The formula above has a convenient interpretation in terms of f-hooks: The symbols occurring as indices of unipotent characters on the right hand side are precisely those which are obtained

36


from by adding an f-hook. (The concept of hooks, cohooks, cores and cocores of symbols is explained in detail in [Ol] and [FoSr], for example.) The proof will be in several steps. We rst describe the d-cuspidal pairs. Each symbol has a uniquely determined d-core. We now claim that the d-cuspidal unipotent characters of are precisely those where itself is a d-core. Indeed it is easily seen that the generic d-split subgroups of all have type G

L

G

GL(ad) GL(adr) ; H

1

where has the same type as (or possibly ? ). Now by 1.39 the d-cuspidal unipotent characters of GL(ad) are in natural bijection with the 1-cuspidal characters of GLa , and as we already saw above such only exist if a = 1. Hence d-split subgroups can have d-cuspidal unipotent characters only if is of type H

G

G

L

L

where j j = (xd ? 1)a ;

;

(3.6)

T

H

T

where has the type of , and in turn if a > 0 by induction these d-cuspidal characters have the form = 1 1 , where is a d-core. Now RGLbehaves well with respect to direct products. So the constituents of RGL() for ( ; ) as above may be computed by repeated application of Asai's formula 3.5 in the particular case f = d, using transitivity. Clearly all constituents of RGL() have the same d-core , so we obtain a partition of the unipotent characters of into series, such that two characters lie in the same series if and only if their associated symbols have the same d-core. This completes the description of the d-cuspidal pairs and of the d-series and proves 3.2(1). We next show that the series are indeed disjoint even with respect to induction from intermediate d-split Levi subgroups. Obviously by induction it suces to do this for maximal subgroups (see the reasoning for type A). Up to -conjugacy any maximal d-split generic subgroup of containing has the form H

G

L

G

M

G

G

(3.7)

M

L

= GL(bd) 0 H

where b a and 0 has the same type as . Although the formula of Asai doesn't explicitly yield the decomposition of RGM() for a unipotent character of , we can still deduce it. Namely, the unipotent characters of GL(bd) are indexed by the irreducible characters of Sb = W(GLb). The unipotent character of GLb corresponding to is 1 X (w)RGL(1); (3.8)

= T w jSbj w2Sb H

G

M

Q

Q

where j wj = i (xdbi ? 1) when w has cycle type i (bi). It is now clear by repeated application of Asai's formula for f = dbi that the constituents of RGM() are indexed by symbols obtained from by adding dbi-hooks, and since the addition of a dbi-hook may be mimicked by the addition of bi d-hooks, RGMpreserves the series de ned above, which proves disjointness. T


37

Thus it remains to establish the existence of the collection of maps I(M L;). Arguing as in the case of type A above it is again sucient to do this for maximal. Let be a xed d-split subgroup of as in 3.6 with of rank r, so that l = ad + r, and = a d-cuspidal character of . Then has the form 3.7. Let us assume for the moment that has type Bl or Cl . Then we have WG( ; ) = Z2d o Sa (where implicitly we have already made all the identi cations as in the proof for type A). The charactersPof this wreath product are in bijection with 2d-tuples of partitions (!i ` ai ) with ai = a. As we have seen, the unipotent characters in Uch( ; ( ; )) are parametrized by symbols with given d-core and hence weight a. As a symbol = fS; T g is uniquely determined (up to equivalence) by the defect and by the two partitions represented by the -numbers S, T, we get a bijection from the set of symbols with a given d-core onto the set of pairs of partitions of a1; a2 with prescribed d-cores and a1 + a2 = a. These in turn are in bijection with the set of P pairs of dquotients (see [JaKe]), hence with 2d-tuples of partitions (!i ` ai ) of a = ai . This de nes a bijection between Uch( ; ( ; )) and Irr(WG; ( ; )). For as above we also have WM( ; ) = Sb (Z2d o Sa?b). The set Uch( ; ( ; )) is now parametrized by pairs (!; ?) of partitions ! ` b and symbols ? for 0 with d-core . We have an obvious bijection from these to Irr(WM( ; )). We now have to compare the decomposition of RGM(), for a constituent of RM L(), with the corresponding decomposition of induced characters in the Weyl group. We rst note that by (3B) of [FoSr] there is associated a well de ned sign to each unipotent character of so that no cancelling takes place in the induction. We make again use of the decomposition 3.8 for the unipotent characters of GLb . The dierent tori w appearing in the sum correspond on the side of Weyl groups to the one element subsets fwg. Induction to WG( ; ) just yields the characteristic function on the corresponding class, so that the multiplicity of any irreducible character of WG( ; ) is just equal to the value (w). This value may be computed by the generalized Murnaghan-Nakayama formula of Osima [Os] Q for wreath products, by successively removing hooks of length ni from w of type i ni. On the other hand, the multiplicity of a unipotent character of in RGT w(1) is seen by 3.5 to be obtained (up to sign) by the same procedure of adding hooks of length dni . Adding a hook of length dni is equivalent to adding a hook of length ni to one of the partitions of the d-quotient. Hence indeed the multiplicities on both sides agree and the assertion of 3.2 follows for types B and C. In types D and 2D, two complications arise. Firstly, degenerate symbols may occur in 3.5, and secondly, sometimes the normalizer WG( ; ) is not the full wreath product. It will become clear presently that both problems are intimately linked. But if both do not occur, the above proof carries through precisely as for types B and C. As the d-core of a degenerate symbol is easily seen to be also degenerate, the rst complication occurs when the cuspidal character is indexed by a degenerate symbol. But also, the normalizer WG( ; ) is not the full wreath product precisely when is degenerate. So it remains to check the existence of I(M L;) in this case. Then WG( ; ) is a subgroup of index two in Z2d o Sa , generated by the elements of Sa and the even elements (products of two elements) in Z2d a . It hence suces to see that precisely P those characters of Z2d o Sa indexed by 2d-tuples of partitions (!i ` ai ) of a = ai with ai = ai+d and !i = !i+d , 1 i d, split when restricted to the subgroup described above. This in turn is equivalent to saying that precisely those vanish on M

G

L

H

H

M

G

L

G

G

M

L

L

L

L

M

L

H

L

G

T

L

L

G

L

L

L

38


all odd elements of Z2d o Sa . This nal fact follows from the Murnaghan-Nakayama formula in [Os], and the proof for odd d is completed. It remains to treat the case when d is even. It resembles very much the previous case, so we do not go into any details. (Indeed, the particular case where d is twice an odd number will follow immediately from 3.3, the proof of which is given below.) Let be a generic Levi subdatum of of type , where is a generic classical group of the same type as (or possibly ? ) and is a torus such that j j = xf +1. Again, the decomposition of RGL( ), where denotes the unipotent character of indexed by the symbol , was explicitly given in [FoSr], (3.2). Namely, RGL( ) equals (3.9)X X (?1)NS;x +NT;x f S ?fxg;T [fx+f g + (?1)NS;y f +NT;y S [fy+f g;T ?fyg : L

G

G

T

G

H

H

T

T

L

+

+

x2S x+f 2= T

y2T y+f 2=S

Here NS;x := jfs 2 S j s < xgj, and the same convention concerning degenerate symbols as in 3.5 applies. This may again be expressed in terms of f-cohooks, so that the symbols occurring as indices of unipotent characters on the right hand side are precisely those which are obtained from by adding an f-cohook. Each symbol has a well de ned f-cocore, and arguments as in the previous part now show that indeed the d-cuspidal pairs of are the unipotent characters = 1 1 of G

= ; where j j = (xe + 1)a ; e := d=2; and where is an e-cocore. As above this is an easy consequence of 3.9. In types Bl and Cl we now have WG( ; ) = Z2e o Sa . The same holds for Dl and 2Dl , except that when is degenerate, we only obtain the subgroup of even elements as previously. It is well known that for symbols the mechanism of d{hooks when d is odd is completely parallel to the one of d=2{cohooks when d is even. Since this same parallelity also holds for Asai's formula 3.9, the previous arguments may be applied with only minor changes. A more detailed account is contained in [En2]. We next prove the general Ennola{duality for RGL in the classical groups. Let = (S; T) be a symbol. De ne ? = (S ? ; T ?) to be the symbol obtained from by moving all odd entries from S to T and vice versa. In fact, as (S; T) = (T; S) by de nition, we might as well exchange the even entries of S and T to obtain ? . We now claim that the map G : 7! ? ; L

T

H

T

L

where denotes the parity of the degree in x of Deg( ), is the right one. (Note that is the sign by which one has to adjust Deg( )(?q) to obtain a positive value.) To prove the commutativity of 3.3, observe that any Levi subgroup of a classical group contains at most one factor of classical type, i.e., not of type A. Since all unipotent characters of type-A groups are uniform, the formulae of Asai 3.5, 3.9, enable us to compute the decomposition of RGLfor any Levi subgroup of any classical group. For type A the desired commutativity has already been stated, and the de nition of the sign agrees with the one given there. So by induction it therefor suces to prove the commutativity of 3.3 in the case that has the particular form assumed in Asai's formula. L

L


39

As the operation of going from to ? clearly interchanges the concept of odd hooks and odd cohooks, while leaving invariant even hooks and cohooks, it follows that the symbols for constituents of RGL( ? ) are just the images under ? of the constituents of RGL( ). Hence we are reduced to checking the correctness of the signs. So assume that is of the type considered in 3.5, and is obtained by adding an f-hook at position x of . By shifting the symbol if necessary we may assume that x is even. Evaluating 3.5 and 3.9, the equality to be proved is (?1)MS;x = (?1)NS?;x +NT ?;x f : By the de nition of S ? , T ? , M and N, this may be written as = (?1)1+NS;x +NT;x +N ; where N1 := jfs 2 S j x s < x + f; s evengj + jft 2 T j x t < x + f; t evengj. But by the degree formula for given in [Ca], 13.8, the right hand side expresses precisely the change in parity of the degree of Deg( ) occurring when is replaced by , hence the desired result follows for odd length hooks. (Note that if is of type D, exactly three of the four groups , ?, and ? will have the same orthogonal type. This fact accounts for a factor -1 in the above equation.) If f is even, one obtains similarly = (?1)1+N ; where N2 := jfs 2 S j x s < x + f; s oddgj + jft 2 T j x t < x + f; t oddgj and again this identity is veri ed using the degree formulae. This completes the proof of 3.3 for classical groups. The following statements are a sharp generalization of the usual so{called Harish{ Chandra theory, which corresponds here to the case d = 1. They were recently conjectured or partially proved by many authors (see for example [Sch] or [FoSr]). They are immediate consequences of theorem 3.2. 3.10. Theorem. (transitivity) Let ( ; ) be a d{cuspidal pair, and let ( ; ) and ( ; ) be d{split pairs such that ( ; ) 4 ( ; ) and ( ; ) 4 ( ; ). Then we have ( ; ) 4 ( ; ). This theorem, together with the disjointness asserted in 3.2(1), shows the existence of a \d{Harish{Chandra theory". In what follows, we draw some straightforward consequences of the existence of such a theory. 3.11. Proposition. Let ( ; ) be a d{cuspidal pair, and let ( ; ) and ( ; ) be d{split pairs such that ( ; ) 4 ( ; ) and ( ; ) 4 ( ; ). Then we have ( ; ) 4WG ( ; ). Proof of 3.11. By 3.2, (1), there exists a d{cuspidal pair ( 0; 0) such that ( 0; 0) 4 ( ; ). Then by 3.10 we have ( 0; 0) 4 ( ; ). By 3.2, (1) again, we then see that ( 0; 0) is WG{conjugate to ( ; ). 3.12. Corollary. The d{cuspidal pairs are the minimal d{split pairs for the relation 4. Proof of 3.12. Apply the preceding proposition with ( ; ) = ( ; ). The following consequence of d-Harish-Chandra theory is a result of \control of fusion". L

+

1

L

L L

G

G

2

L

G

L

L

M

M

M

G

G

L

M

L

L

G

M

M

L

M L

G

G

L

L

G

L

L

G

40


3.13. Proposition. Let ( ; ) and ( ; ) be two d{split pairs. We assume that ( ; ) is d{cuspidal, and that ( ; ) 4 ( ; ). (1) Assume that w 2 WG is such that also ( ; ) 4 ( ; )w . Then there exists v 2 WM and n 2 NWG( ; ) such that w = vn. (2) The natural map NWG(( ; );( ; ))=NWM( ; ) ! NWG( ; )=WM is an isomorphism. In particular (with obvious notation) we have NWG(L;)( ; )=WM( ; ) = WG( ; ) : L

L

M

L

M

L

M

L

L

M

M

L

M

L

M

Proof of 3.13. If (L; ) is contained in both (M ; ) and (M ; )w , then (L; ) and w(L; )

are two d{cuspidal pairs both contained in ( ; ), whence (see 3.12) are conjugate under WM. This proves (1). We get (2) by applying (1) to the case where w 2 NWG( ; ). The next result is also an immediate consequence of 3.11. 3.14. Proposition. Suppose that ( ; ) is a d{cuspidal pair such that ( ; ) 4 ( ; ). Then X w RG( ) = ( ; RG())G ; L L w2WG(L)=WG(L;) where the symbol for the summation means that the sum is taken over a complete set of representatives for the classes of WG( ) modulo the stabilizer WG( ; ) of . In particular, we have Deg( RGL( )) = ( ; RGL())GjWG( ) : WG( ; )jDeg() ; and Deg( RGL( )) 6= 0. M

M

L

L

G

L

L

L

L

Again on regular unipotent characters.

Let ( ; ) be a d{cuspidal pair. We denote by prGL; the orthogonal projection from [x]Uch( ) onto the submodule with basis Uch( ; ( ; )). We de ne X Deg( ): ; (3.15) URegGL; := prGL;(URegG) = f ; (L;)4(G; )g and (see 2.10) X Degd ( ): : (3.16) Cd URegGL; := prGL;(Cd URegG) = f ; (L;)4(G; )g The following statement is a re nement of 2.11, from which it follows using 3.2(i). 3.17. Proposition. We have X Deg(RGM(1)) RG(Deg ():) : URegGL; = jW ( ; )j M d [(M;); (L;)4WG(M;)]WG G Finally, the preceding machinery allows us to prove a re ned version of the Mackey formula (see 1.37). The proof of the following statement follows from 1.37 and from 3.13 and is left to the reader. L

Q

G

G

M

L


41

3.18. Proposition. If ( 1 ; 1) and ( 2 ; 2) are two d{split pairs containing the M

M

d{cuspidal pair ( ; ), we have (RGM (1 ); RGM (2 ))G = L

X

2

1

w ( RMM\ wM ( w1 ); RM M \ wM (2 ))M \ wM : 1

2

w2WM2(L;)nWG(L;)=WM1(L;)

2 2

1

2

1

1

The preceding statement can be reformulated in terms entirely analogous to the previous \Mackey formulae", provided we consider maps \over ( ; )". For a d{split generic Levi subgroup containing , let us denote by RGM( ; ): Uch( ; ( ; )) ! Uch( ; ( ; )) the restriction of RGM, and by RG( ; ): Uch( ; ( ; )) ! Uch( ; ( ; )) M its adjoint, i.e., the restriction to Uch( ; ( ; )) of prM L; RGM . 3.19. Proposition. Let ( ; ) be a d{cuspidal pair, and let 1 and 2 be two d{split Levi subgroups of containing . Then RG ( ; ) RG ( ; ) = M M X wM RM M \ wM ( ; ) RM \w M ( ; ) ad(w) : w2WM (L;)nWG(L;)=WM (L;) d {defect groups. The d{Harish{Chandra theory allows us to de ne the notion of \d {defect groups" or \defect (generic) tori" of an element 2 Uch( ), as follows. 3.20. De nition. Let 2 Uch( ). (1) We denote by Sd ( ) the set of all d {tori of for which there exists a ( ) 6= 0. maximal torus containing such that RGT (2) We call defect tori of the maximal elements of Sd ( ). 3.21. Theorem. Let 2 Uch( ), and let ( ; ) be a d{cuspidal pair (unique up to WG{conjugation) such that 2 Uch( ; ( ; )). Then 2 Sd ( ) if and only if is WG{conjugate to the maximal d {subgroup of Rad( ). In particular the defect tori of are the WG{conjugates of Rad( )d. Remark. An element 2 Uch( ) is d{cuspidal if and only if the Sylow d {subtorus of Rad( ) is its defect torus. Proof of 3.21. We set := CG( ). Assume rst that 2 Sd ( ). Let be a maximal torus of containing and such ( ) 6= 0. Since , we have RGM( ) 6= 0 and there exists 2 Uch( ) that RGT such that ( ; ) 4 ( ; ). By 3.2(i) above, we deduce that there exists w 2 WG such that is a subgroup of w , from which it follows that w is contained in the maximal d {subtorus of Rad( ). Assume now that is contained in the maximal d {subgroup of Rad( ). Then . Since (cf. 3.14) Deg( RGL( )) 6= 0, we see that there exists a maximal torus of , whence of , such that RGT ( ) 6= 0, which shows that 2 Sd ( ). For the notation used below, the reader may refer to x2.A above. L

M

L

L

Z

L

M

L

Z

G

G

L

Z

M

Z

Z

G

L

L

L

L

M

G

2

L

M

L

L

1

2 2

1

1

L

2

1

L

1

2

G

G

S

T

G

S

G

L

G

L

S

S

L

L

G

G

M

S

S

T

T

M

G

S

M

M

G

L

M

S

L

S

L

M

T

L

M

L

S

42


3.22. Theorem. Let 2 Uch( ), with defect group . Let := CG( ). Then (1) = Rad( )d and m 2 CF d ( ; ), (2) Deg( )d = j jd =j j, G

S

S

L

L

S

G L

G

S

(3) whenever is a d{split Levi subgroup of such that Zd ( ) is not WG{ conjugate to a subgroup of , then RGM( ) = 0. M

G

M

S

Proof of 3.22.

(1) results from the de nition of a defect group. (2) Since j jd =j j = j ssjd, we see by 2.15 that Deg( ) Deg(RGL(1)) Deg( RGL( )) mod ; d j ssjd jWG( )j j ssjd G and it suces to prove that Deg(j RjL( )) is not divisible by d . But this results ss d from 3.14 and from 2.7. (3) Zd ( ) is not WG{conjugate to a subgroup of if and only if is not conjugate to a subgroup of . Thus (3) is an easy consequence of 3.2(i). G

S

L

L

L

L

L

M

S

L

M

Remark. We shall see later on that this generic defect group does behave, in many aspects, as the right \generization" of an actual defect group. Moreover, for suitable choices of q and ` (see x4), the Sylow `{ subgroups of the rational points of a generic defect group are the defect groups of a corresponding actual `{block. Assertions (2) and (3) of 3.22 may be stated in the following way, which generalizes the properties of d{cuspidal elements of Uch( ). 3.23. Proposition. Let 2 Uch( ; ( ; )). Then (1) Deg( )d = j ssjd , (2) for a d{split generic Levi subgroup of , we have RGM( ) = 0 unless WG . G

G

L

L

M

L

G

M

B. Regular characters of generic d{blocks. Following the notation introduced in x2.C above, we have a decomposition Rd ( ) = G

M

[L;]WG

RL;( ) G

where ( ; ) runs over the set of d{cuspidal pairs of , and where we denote by L

G

RL;( ) the free [x]{submodule of Rd ( ) generated by fG(M;)g(L;)4WG(M;). We \extend" the de nition of the orthogonal projection prGL; (see x3.A above) by G

de ning

Q

G

( G if (L; ) 4 (M ; ) WG (M;) G G prL;((M;)) := 0 if not : The following property of prGL; is a translation of the consequences of 3.2 listed in x3.A above.


43

3.24. Lemma. Whenever and are two d{split generic Levi subgroups of such that , and 2 Uch( ), we have ( RG(N ) if ( ; ) 4 ( ; ) WG G G N prL;(RN(G;M )) = N G;M N

N

M

M

G

N

N

L

0

if not:

Proof of 3.24. By 2.28, we know that RGN(N G;M ) = RGM(M G RM N( )) ;

or, in other words, RGN(N G;M ) =

X

(RM N( ); )MRGM(M G RM N( )) :

f ; (N; )4(M;)g

The result then follows from the following property: Assume ( ; ) 4 ( ; ). Then N

M

( ; ) 4WG ( ; )if and only if ( ; ) 4WG ( ; ) ; L

M

N

L

which is immediate by 3.12. We de ne (cf. 2.26) RegG(L;) := prGL;(RegGd ) : Thus by de nition, X Deg(RGM()) :G RegG(L;) = jWG( ; )j (M;) [(M;); (L;)4G(M;)]WG X Deg(RGM(1)) RG(M Deg():) ; (3.25) = jWG( ; )j M G [(M;); (L;)4G(M;)]WG and also X RegG(L;) ; (3.26) RegGd = [(L;)]WG M

M

where ( ; ) runs over the set of d{cuspidal pairs of . 3.27. Theorem. We have X Deg(RGM(1)) RG(Ab RegM Deg ():) : RegG(L;) = d jWG( ; )j M d [(M;); (L;)4G(M;)]WG L

G

M

Proof of 3.27. It suces to apply prG L; to both sides of the corresponding formula G for Reg (see 2.31), and to use lemma 3.24 above. d

44


C. First application to actual nite reductive groups. The reader may refer to x2.D above for the notation used here. Let (G; T; F) be a (q; ){triple associated with , chosen in such a way that T contains a Sylow d {subgroup of G. Let (L; ) be a d{cuspidal pair of G. G

We set

X

F

RegGL; :=

3.28. Theorem.

[(M;) ; (L;)4GF (M;)]GF F

DegRGM () RG (MF :) ; jWGF (M; )j M G

P

(1) We have RegGd = [(L;)]GF RegGL;F : (2) Let d a(d) be Fthe largest power of d which divides j j. Then all the values of jWG j RegGL; are divisible by d (q)a(d) . Proof of 3.28. The rst assertion is immediate. As for the proof of 2.37, the second assertion is a consequence of 2.40. G

4. Applications to `{blocks of finite reductive groups

The preceding \generic" approach shows that the group GF (at least as far as unipotent characters are concerned) makes hardly any dierence between two prime numbers which both divide the same factor d (q). Thus, many usual objects and properties of the classical `{modular representation theory can be replaced, without any harm, by corresponding objects or properties when the single prime ` is replaced by a set of primes, as long as these primes divide the same factor d (q). This is what we do next, in x4.A, 4.B, 4.C. Since many notions (Brauer morphisms, generalized decomposition maps, isotypies, etc.) are often known only for the case of a single prime, we give (or recall) when needed their de nitions for a set of primes. A. {blocks of nite reductive groups: applications of generic characters. d{adapted sets of prime numbers. Let be a generic nite reductive group, let p be a prime number, and let (G; T; F) be a (q; ){triple associated to such that q is a power of p. Let ` be a prime number such that ` 6= p, ` divides jGF j. If ` jWGj, there exists a unique integer d such that d divides j j and ` divides d (q). In other words, we have jGF j` = j jd (q)` : 4.1. De nition. Let be a set of prime numbers such that p 2= . We say that is d{adapted for G if (1) no element of divides jWGj, (2) we have jGF j = j jd (q) : G

G

-

G

G

G

Remarks.

(1) If is d{adapted for G and if L is an F{stable Levi subgroup of G, then is d{adapted for L.


45

(2) If is d{adapted, it consists only of prime numbers which are excellent for G (cf. 1.10). When is d{adapted, the centralizers of {elements are d{split Levi subgroups of G, as is shown by the following statement. 4.2. Proposition. Let S be an F{stable torus of G, and let d be an integer. Let be a set of prime numbers. We assume that each element of is excellent for G, and that is d{adapted for S. Then, for every {subgroup S of SF , the group CGo (S) is a d{split Levi subgroup of G. Proof of 4.2. Since every element of is excellent for G, we have S Z o (CGo (S)) and CGo (S) is a Levi subgroup of G which contains CG (S). Thus Z o (CGo (S)) S, and is d{adapted for Z o (CGo (S)). It follows that S is contained in the Sylow d {subgroup Z o (CGo (S))d of Z o (CGo (S)), since the quotient of the corresponding group of rational points is a 0 {group. Then we have CGo (S) = CGo (Z o (CGo (S))d ), which shows that CGo (S) is a d{split Levi subgroup of G. {blocks when is d{adapted. Let (L; ) be a d{cuspidal pair of G. We want to prove that, if is d{adapted F G for G, the projection of RegL; (see x2.C above) onto the subspace generated by characters in [ E (GF ; (s)) ; E (GF ; 1) := s2(G F )

(cf. [BrMi]) is the regular character of a central \{idempotent" of GF (which will turn out to be a block if = f`g). From now on, we assume that (G; T; F) is chosen so that T contains a Sylow d { subgroup of G. So T is contained up to GF {conjugation in all d{split Levi subgroups of G. 4.3. Theorem. Let be a set of primes which is d{adapted for G. Then F F (1) The projection of RegGd onto E (GF ; 1) is the character RegG . (2) Let (LF ; ) be a d{cuspidal pair of G such that T L. The projection F G G Reg;(L;) of RegL; onto E (GF ; 1) is given by the following formula: F

RegG;(L;) =

X

[(M;;)]GF

F

F

G(M;;) (1):G(M;;)

where (M; ) is a d{split pair such that (L; ) 4GF (M; ), runs over Ab Irr(LF ), M = G(T; ). All its values are divisible by jGF j . In other words, this character is the F G character associated to a central {idempotent e;(L;) . (3) We have X F F RegG = RegG;(L;) ; [(L;)]GF

46


P (in other words, eG F = [(L;)]GF eG;F(L;) ), where (L; ) runs over the set of all d{cuspidal pairs of G.

Proof of 4.3.

F (1) For M a d{splitF Levi subgroup ofF G, let us denote by Irr (M G ) the set of {elements of Irr(M denotes the projection onto the subspace G ). Then, if prG generated by characters in E (GF ; 1), we have X F F prG (RGM (M RGM(:) : G :)) = F 2Irr (M G )

The following lemma follows from proposition 4.2 (applied to a dual group L of L). The reader may refer to 2.36 for the notation. 4.4. Lemma. Let L be an F{stable Levi subgroup of G containing T. Let be a set of prime numbers. We assume that each element of is excellent for G and that is d{adapted for Z o (L). Let M be a d{split Levi subgroup of G containing L. Let F M be a {element of Irr(G ). Then M = G(T; ). It then results from 4.4 that X F F prG (RGM (M RGM (:) : G :)) = f ; ( {element)(G(T;)=M)g

The rst assertion, as well as the rst part of the second assertion of 4.3 are now immediate. F (2) Let us now prove that the values of RegG;(L;) are divisible by jGF j . Let eG F denote the central idempotent corresponding Fto E (GF ; 1). We know (see [BrMi], F G th. 2.2) that e is a {idempotent, i.e., eG 2 [1=r] r20 . By 3.28, and since F G is d{adapted for G, we know that all values of RegL; are divisible by jGF j . For F F all g 2 GF , we have RegG;(L;) (g) = RegGL; (g:eG F ) and this proves now the second assertion of 4.3. (3) Finally the third assertion of 4.3 is an immediate consequence of 3.26. Z

B. Generalization to {blocks with abelian defect groups, and Brauer morphisms. General results about {idempotents. Generalities.

Let be a set of prime numbers such that p 2= . We denote by 0 the complement of into the set of all prime numbers (so p 2 0). 4.5. De nition. We denote by G0F the class function on GF de ned by ( F jG j if g is a 0 {element F G 0 (g) = 0 if not It is clear that G0 F (g) depends only on the semi{simple part of g. It then follows from [BrMi], 2.5, that


47

(4.6)

(1) whenever L is an F{stable Levi subgroup of G and is a class function on LF , then RG ( G0 F ) = Deg(RG (1)) L0F RG ( ) L L L and F G G LF G0 RG L ( ) = Deg(RL (1)) RL (0 ) ;

(2) G0 F is uniform, and G0

F

=

X Deg(RGT(1)) G TF R ( 0 ) :

[T]GF

T

jWGF (T)j

Watch out that with our conventions the -part of a number is always positive, so in particular Deg(RGL (1)) Deg(RGL (1))0 = "G "L Deg(RGL (1)). Let us denote by uG0 F the projection of G0 F onto the space generated by the unipotent characters of GFF . Since T0F = Ab RegT , we also have (4.7)

uG0F =

X Deg(RGT (1)) G R (1) :

[T]GF

jWGF (T)j

T

Let us now come back to the regular character of the series E (GF ; 1) de ned in [BrMi]. Recall that we denote by prG F the projection from the set of all class functions on F G onto the subspace generated by characters in

E (GF ; 1) := and that we set

[

s2(G F ) F

F

F RegG := prG RegG = F

E (GF ; (s)) ;

X 2E (GF ;1) F

(1) :

Finally we denote by URegG the projection of RegG onto the space generated by the unipotent characters of GF . The second assertion of the following proposition has already been proved in [BrMi]. 4.8. Proposition. We have G F P T (1) RG (RegT F ) ; (1) RegG = [T]GF DegR jWGFF (T)j T F F G (2) Reg = G0 D(u(G0 ?fpg)0 ) ; where D denotes the Curtis-Alvis-Kawanaka duality.

48


Proof of 4.8.

(1) is got by application of prG F to the formula X DegRGT(1) G TF F RegG = jWGF (T)j RT (Reg ) ; [T] F G

since

F

F

F

F

F

prG (RGT (RegT )) = RGT (prT (RegT )) = RGT (RegT ) : (2) By 4.7, we see that X Deg(RGT (1))0 G (4.9) u(GF0 ?fpg)0 = jWGF (T)j RT (1) : [T] F G

Multiplying both sides of the preceding equality by G0 F and using 4.6,(1), and that D(RGT (1)) = "G "T RGT (1) gives the result. Case when is good.

Suppose now that every element of is good for G. Then whenever S is an abelian {subgroup of GF , the group CGo (S) is an F{stable Levi subgroup of G. We recall that we call {split Levi subgroups of G the connected centralizers of abelian {subgroups of G. A group M is a {split Levi subgroup of G if and only if there exists an F{ stable maximal torus T of G such that M = CGo (TF ), where TF denotes the Sylow {subgroup of TF .

4.10. De nition.

(1) We say that an F{stable maximal torus T of GF is {anisotropic if the Sylow {subgroup TF of TF is contained in Z o (G)F . We denote by T (G) the set of all {anisotropic maximal tori of G. (2) For a uniform class function on GF we denote by c the \{anisotropic projection" of , namely X 1 RG ( RG ( )) : c := T T j W F G (T)j [T2T (G)] F G

The following formulae are immediate consequences of the de nitions.

4.11. Lemma.

(1) We have c G0 F = jGF : Z o (G)F j

(2) and

X [T2T (G)]GF

c uG0 F = jGF : Z o (G)F j = jGF

1 TF G jWGF (T)j RT (Ab Reg ) ;

X [T2T (G)]GF

: Z o (G)F j c 1GF

:

1 G TF jWGF (T)j RT (1 )


49

In a way which is quite analogous to 2.5, we de ne the \{anisotropic degree" Deg ( ) of a unipotent character of GF by the formula

F

Deg ( ) := ; c RegG GF :

(4.12)

Like in 2.6, we have: (4.13) For 2 E (GF ; 1), Deg ( ) is divisible by jGF : Z o (G)F j . The following proposition is the analogue of 2.31.

4.14. Proposition. (1) We have F

G0

=

and

X

[fM ; (M {split)g]GF

uG0F = (2) We have

X [fM ; (M {split)g]GF

X

F

RegG = and

DegRGM (1) " " RG (Ab RegMF c u MF ) ; 0 jWGF (M)j G M M


DegRGM (1) RG (Ab RegMF c URegMF ) ; jWGF (M)j M

X

F

URegG =

DegRGM (1) " " RG (c u MF ) : jWGF (M)j G M M 0


DegRGM (1) RG (c URegMF ) : jWGF (M)j M

Proof of 4.14. The proof is analogous to the proof of 2.11: it uses lemma 2.12, as well

as the following analogue of 2.32 (which is immediate).

4.15. Lemma. F

F

(1) c G0 = Ab RegG cF uG0 F : (2) c RegG F = Ab RegG c URegG F :

The case when is d{adapted.

De nition 4.1 can be relativized to a given d{cuspidal pair (L; ) of G. 4.16. De nition. Let (L; ) be a d{cuspidal pair of G. Let be a set of prime numbers such that p 2= . We say that is (d; (L; )){adapted for G if (1) every element of is excellent for G, (2) is d{adapted to L, (3) no element of divides jWGF (L; )j. With this de nition, theorem 4.3 can be partly generalized, as follows.

50


4.17. Theorem. Let (L; ) be a d{cuspidal pair of G. Let be a set of prime numbers which is (d; (L; )){adapted for G. Then F F (1) The projection RegG;(L;) of RegGL; onto E (GF ; 1) is given by the following formulae:

X

F

RegG;(L;) = =

[(M;;)]GF

X

[(M;)]GF

G(MF;;) (1):G(MF;;)

DegRGM () RG (MF :) jWGF (M; )j M G;

where (M; ) is a d{split pair such that (L; ) 4GF (M; ), runs over Ab Irr(LF ), M = G(). (2) We have RegG;F(L;) =

X

[(M;)]GF

DegRGM () RG Ab RegMF:Deg () : jWGF (M; )j M

F

(3) The values of RegG;(L;) are all divisible by jGF j . In other words, this character is the regular character associated with a central {idempotent eG;F(L;) of GF . Q

Proof of 4.17.

The proof of (1) is identical to the proof of the corresponding assertion in 4.3 (see above). (2) is an easy consequence of 4.14. (3) We prove that for every d{split pair (M; ) such that (L; ) 4 (M; ), the DegRGM () RG Ab RegMF:Deg () are all divisible by jGF j . values of jW GF (M; )Fj Mo F Since jM =Z (M) j divides Deg (), we see by 2.38 that the function (cf. 4.13) F M G RM Ab Reg :Deg () takes values all divisible by jMF j0 . Thus the values of DegRGM () RG Ab RegMF:Deg () jWGF (M; )j M

are all divisible by jGF j jGF : MF j([fpg)0 , and it suces to check that jWGF (M; )j is a 0{group. This follows from 3.13, (2), since we see that WGF (M; ) is a quotient of a subgroup of WGF (L; ).

Brauer morphisms.

Let be a set of prime numbers, and let s be a {element of GF . Whenever e is a central {idempotent of GF , we denote by Brs (e) the image of e through the corresponding Brauer morphism (cf. [Rob], and also [BrMi], x3). Thus Brs (e) is a Q


51

central {idempotent of CGF (s). For the convenience of the reader, we recall its de nition. We denote by $sGF the class function on GF whose value is jCGF (s)j on the GF {conjugacy class of s and 0 elsewhere. For a class function on GF we denote by decs; GF ( ) the class function on CGF (s) such that Q

F decs; G (

( (st) if t is a 0{element )(t) = 0

if not.

For a class function on GF , we denote by e: the function de ned by e: (g) :=

(eg). Then the following formula de nes the idempotent Brs (e) : F

F

Brs (e): RegG = decG (e:$sGF ) :

(4.18)

The iteration of the preceding formula 4.18 allows to de ne (see for example [Rob]) an idempotent BrS (e) for all abelian {subgroup S of GF . In the case where = f`g, this de nition coincides with the original de nition of Brauer. 4.19. Theorem. Let (L; ) be a d{cuspidal pair of G. Let be a set of prime numbers which is (d; (L; )){adapted for G. Let S be a subgroup of Z o (L)F ; we set M := CGo (S). Then we have BrS (eG;F(L;) ) =

X

(L0 ;0 )

F eM ;(L0 ;0 )

where (L0; 0) runs over a complete set of representatives for the MF {conjugacy classes of d{cuspidal pairs of M which are GF {conjugate to (L; ). Proof of 4.19. We may assume that S is cyclic. Let s be a generator of S. By \Curtis{type formula" (see [Br3], 4.3), we know that F

F

decs; G = decs; M RGM : So by 4.18 we must prove that s;MF

(dec

0 1 X F MF : eM RGM )(e;(L;) :$s ) = @ ;(L0 ;0 ) A : Reg GF

GF

(L0 ;0 )

Let us denote by EF (GF ; (L; )) the subspace of the space of all class functions on F G that eG;(L;) : = . Since RGM and RGM interchange E (GF ; (L; )) and S 0 such F 0 0 (L ;0 ) E (M ; (L ; )), we see that

1 0 X F G GF RG (e eM M ;(L;) :$s ) = @ ;(L0 ;0 ) A : RM ($s ) ; GF

GF

(L0 ;0 )

52


and so

0 1 X F F F F s;MF RG )($GF ) eM (dec s; M RGM )(eG;(L;) :$sG ) = @ M s ;(L0 ;0 ) A :(dec (L0 ;0 ) 0 1 X F s;GF GF eM =@ ;(L0 ;0 ) A :dec ($s ) 0 0 0(L ; ) 1 X F MF : eM =@ ;(L0 ;0 ) A : Reg (L0 ;0 )

C. Isotypies.

Preliminaries.

The following notation and hypothesis are in force throughout this section. Let (L; ) be a d{cuspidal pair of G. Let be a set of prime numbers which is (d; (L; )){adapted for G. Let T be an F{stable maximal torus of G which contains Z o (L).

4.20. Proposition.

(1) The restriction map from LF down to Z o (L)F induces an isomorphism Irr(Z o (L)F ) : Ab Irr(LF )?!

(2) Let be a character of Z o (L)F , and let still denote the linear character of TF which corresponds to it by (1). Then the group WGF (T;)(L; ) is the inertial subgroup of in WGF (L; ). Proof of 4.20. We know that L = [L; L]Z o(L). Moreover, Z o (L) \ [L; L] is a subgroup of the nite group Z([L; L]). Since consists of prime numbers which are excellent for G, we see that Z o (L)\[L; L] is a 0{group. Thus the projection Z o (L)F ! LF =[L; L]F is injective. Since jZ o (L)F j = jLF =[L; L]F j , it follows that the image of Z o (L)F into LF =[L; L]F is actually a supplement of (LF =[L; L]F )0 , which proves 4.20. In order to simplify the notation, we set

G() := G(T; ) : We denote by ~ the character of the group Z o (L)F WGF () (L; ) which is de ned by ~(zw) := (z) : Then, whenever is an irreducible character of WGF () (L; ), o

o F GF (L;) ~ the character IndZZ o((LL))F ooW WGF (L;) ( :) is an irreducible character of the group Z o (L)F WGF (L; ). We have (with the above notation) o F GF (L;) ~ Irr(Z o (L)F WGF (L; )) = fIndZZ o((LL))F ooW WGF (L;) ( :)g[(; )]WGF L; : ( )

o

o

( )

(

)


53

We denote by eG;F(L;) the central {idempotent of GF associated with and (L; ) (cf. 4.3). For x 2 GF we set G(x) := CGo (x) and GF (x) := CGo F (x) : Notice that the centralizer in Z o (L)F WGF (L; ) of an element x 2 Z o (L)F is Z o (L)F WGF (x) (L; ). The next proposition is a result about control of fusion. F 4.21. Proposition. The map x 7! (x; eG;(L(;x)) ) induces a bijection between the conjugacy classes of {elementsF of the group Z o (L)F WGF (L; ) and the GF { conjugacy classes of pairs (x; eG;(L(;x)) ) (for x a {element of Z o (L)F ). o

o

o

Proof of 4.21. F F Since eG;(L(;x)) = eG;(L(0x;)0 ) if and only if (L; ) and (L0; 0) are GF (x){conjugate,

we see that we must check the following property: two elements x and x0 of Z o (L)F are conjugate under Z o (L)F WGF (L; ) if and only if there exists g 2 GF such that (a) x0 = xg , (b) (L; )g is GF (x0 ){conjugate to (L; ). This is obvious since WGF (L; ) = NGF (L; )=LF . o

{perfect isometries. We shall use the notion of \perfect isometry", as de ned in [Br1]. Nevertheless, since this was only de ned for `{idempotents (and not for {idempotents, a set of primes), we give here brie y a whole account of the necessary de nitions and elementary properties of {perfect isometries. Let H be a nite group and let be a set of prime numbers. Let K be a eld with characteristic zero which is a splitting eld for the group H. Let O denote the integral closure in K of the ring [f1=`g`2]. Let e be an idempotent of the center Z OH of the group algebra OH of H over O (such an idempotent is sometimes called a \central {idempotent" for H). We denote by CF(H; e; K) the K{vector space consisting of class functions on H with values in K and such that (eh) = (h) for all h 2 H. Let CF0 (H; e; K) be the subspace of functions vanishing outside of the set H0 of 0 {elements of H. One de nes in an analogous way the O{modules CF(H; e; O) and CF0 (H; e; O). We denote by Irr(H; e) the set of irreducible characters of H on K which belong to CF(H; e; K) (and so to CF(H; e; O)). 4.22. De nition. (see [Br1], 4.1) Let H and H 0 be two nite groups, and let e and e0 be two central {idempotents for H and H 0 respectively. Let Irr(H 0 ; e0) I : Irr(H; e)?! be an isometry. We say that I is {perfect if the following two conditions are ful lled: (P1) I extends linearly to an isometric isomorphism CF(H 0; e0 ; O) : CF(H; e; O)?! Z

Z

Z

54


(P2) By restriction to CF0 (H; e; O), it then induces an isometric isomorphism CF 0 (H 0; e0 ; O) : CF0 (H; e; O)?! The proof of the following property of {perfect isometries is the same as the proof of the corresponding result given in [Br1], 1.5, for the case = f`g. 4.23. Proposition. We use the notation of 4.22. Assume that Irr(H 0 ; e0) I : Irr(H; e)?! is a {perfect isometry. Then the algebra isomorphism from ZKHe onto ZKH 0 e0 which is de ned by I induces an algebra isomorphism from Z OHe onto Z OH 0 e0 . In particular e is primitive if and only if e0 is primitive. Z

Z

Generalized decomposition maps.

For the convenience of the reader, we also recall the de nition of generalized decomposition maps in the particular settings we are working in (cf. for example [Br1], 4A).

4.24. De nition.

F

(1) Whenever is a class function on GF , we denote by decx;;G(L;) ( ) the class function on GF (x) de ned as follows (cf. for example [Br1] 4.3) decx;;G(LF;) ( ) vanishes outside of the set of 0{elements of GF (x), for x0 2 GF (x)0 , we have F

decx;;G(L;) ( )(x0 ) := (xx0 eG;(L(;x)) ) : F

(2) Similarly, whenever ' is a class function on Z o (L)F WGF (L; ), we denote x;Z o(L)F oWGF (L;) by dec (') the class function on the group Z o (L)F WGF (x) (L; ) de ned as follows o F decx;Z (L) oWGF (L;) (') vanishes outside of the set of 0 {elements of the group Z o (L)F WGF (x) (L; ), for x0 a 0 {element of Z o (L)F WGF (x) (L; ), we have o(L)F oW F (L;) G decx;Z (')(x0 ) := '(xx0) : o

o

o

o

F

F

Let tdecx;;G(L;) denote the adjoint of decx;;G(L;) . Thus for ' a class function on GF (x) which vanishes outside of GF (x)0 , the function tdecx;;G(LF;) (') is the class function which vanishes outside of the {section of GF de ned by x, and which is such that tdecx;GF (')(xx0 ) = '(x0 ) ;(L;) 0 F for all x 2 G (x)0 . o(L)F oW F (L;) o F G Similarly, we denote by tdecx;Z the adjoint of decx;Z (L) oWGF (L;) . The following result is a classical consequence of Brauer's second main theorem (see for example [Br1], 4A).


55

(4.25) (1) Whenever is a linear combination of elements of E (GF ; (L; )), we have X F F = ( tdecx;;G(L;) decx;;G(L;) )( ) : [x2Z o (L)F ]WGF (L;)

(2) Whenever is a class function on Z o (L)F WGF (L; ), we have X o(L)F oW F (L;) o(L)F oW F (L;) G G ( tdecx;Z decx;Z )() : = o

[x2Z o (L)F ]WGF (L;)

The main theorem.

4.26. Theorem. (1) The collection of maps (where N runs over the set of d{split Levi subgroups of G containing L) IN;F(L;) : Irr ?Z o(L)F WNF (L; ) ?! Irr(NF ; eN;F(L;) ) Z

such that

o

Z

IN;F(L;) : IndZZo((LL))F ooWWNNFF (L;(L);) (~:) 7! RNN()(:I(NL(;)) ()) o

F

( )

\commutes" with the generalized decomposition maps, in the following sense: whenever x 2 Z o (L)F , we have o(L)F oW F (L;) F F G decx;;G(L;) IG;F(L;) = IG;(L(;x)) decx;Z : (2) Each of these maps IN;F(L;) is a {perfect isometry between (NF ; eN;F(L;) ) and (Z o (L)F WNF (L; ); 1). Remark. Had we given the general de nition of a \-isotypie" (see [Br1], 4.6, for the de nitionF of an `{isotypie), assertions (1) and (2) above would certainly mean that (NF ; eN;(L;) ) and (Z o (L)F WNF (L; ); 1) are {isotypic. o

o

Proof of 4.26, (1).

Let us denote the last formula in the preceding statement by (a). For x 2 Z o (L)F , we have the \Curtis type formula" (cf. [Br3], 4.3) F F (4.27) decx;;G(L;) = dec;x;G(L;(x) ) RGG(x)(L; ) and similarly it is easy to see that o(L)F oW F (L;) o(L)F oW F (L;) o F G x G decx;Z = decx;Z ResZZ o((LL))F ooWWGGFF (xL;(L);) : Thus we see that (a) is equivalent to F dec;x;G(L;(x) ) RGG(x)(L; ) IG;F(L;) = o(L)F oW F (L;) o F F G x ResZZ o((LL))F ooWWGGFF (xL;(L);) : (b) IG;(L(;x)) decx;Z The proof of 4.26 is in several steps. Step 1. The rst step allows us to reduce the proof of (b) to the case where x 2 Z o (G)F . ( )

( )

( )

( )

56


4.28. Proposition. Whenever N is a d{split Levi subgroup of G, we have RG (L; ) IGF = INF ResZ oo (L)FF oWGF (L;) : N ;(L;) ;(L;) Z (L) oWNF (L;)

Proof of 4.28. By adjunction, it suces to prove that o F GF (L;) RGN (L; ) IN;F(L;) = IG;F(L;) IndZZ o((LL))F ooW WNF (L;) :

Thus we must check that, for all d{split Levi subgroups M of N such that L M, for 2 Ab Irr(GF ()) such that N() = M, and for all irreducible characters of WM (L; ), we have o F NF (L;) ~ F RGN (L; ) IN;(L;) IndZZ o((LL))F ooW WMF (L;) ( ) = o F o F IG;F(L;) IndZZo ((LL))F ooWWGNFF ((LL;;)) IndZZo ((LL))F ooWWMNFF ((LL;;)) (~) :

This last equality is equivalent to

G() (I M ()) = RG I G() IndWGF (L;) () : RGG() RM (L;) G() (L;) WMF (L;) This follows now from the fundamental theorem 3.2 applied in G(). By 4.28, we see that (b) is equivalent to o F F F GF (L;) dec;x;G(L;(x) ) IG;(L(;x)) ResZZ o((LL))F ooW WGF x (L;) = o(L)F oW F (L;) o F F G x ResZZ o((LL))F ooWWGGFF (xL;(L);) : IG;(L(;x)) decx;Z ( )

( )

( )

( )

Thus we see that it is sucient to prove that, for x 2 Z o (G)F , o(L)F oW F (L;) F G : (c) decx;G IGF = IGF decx;Z ;(L;)

;(L;)

;(L;)

Step 2. The usual decomposition maps are de ned by the formulae o F o F F F decG;(L;) := dec1;;G(L;) and decZ (L) oWGF (L;) := dec1;Z (L) oWGF (L;) : F , we denote by tx;GF ( ) the For x 2 Z o (G)F , and for a class function on G class function on GF de ned by the formula tx;GF ( )(g) := (xg) : Similarly, foro x F2 Z o (G)F , and for ' a class function on Z o (L)F o WGF (L; ), we denote by tx;Z (L) oF WGF (L;) (') the class function on Z o (L)F o WGF (L; ) de ned by the formula tx;G (')(h) := '(xh) :

Thus for x 2 Z o (G)F we have

8 x;GF < dec;(L;) = decG;F(L;) tx;GF o(L)F oW F (L;) o F : decx;Z G = decZ (L) oWGF (L;) tx;Z o(L)F oWGF (L;) :


57

Equation (c) becomes then F

F F decG;(L;) tx;G IG;(L;) = o F IG(LF;) decZ (L) oWGF (L;) tx;Z o(L)F oWGF (L;) :

We see that it suces to prove F F F o F tx;G IG;(L;) = IG;(L;) tx;Z (L) oWGF (L;)

(d) and

decG;(L;) IG;F(L;) = IG;F(L;) decZ (L) oWGF (L;) : o

F

(e)

F

Equation (d) is an immediate consequence of the values of Deligne{Lusztig and ordinary induction. Indeed, we see that

x;GF t

IG;F(L;) (IndZZ o((LL))F ooWWGGFF (L;(L);) (~:)) = GF o F I;(L;) tx;Z o(L)F oWGF (L;) (IndZZo((LL))F ooWWGGFF (L;(L);) (~:)) = o

F

( )

( )

(x)RGG() (:I(GL(;)) ()) :

Step 3 : proof of (e).

The proof is by induction on rkss(G) ? rkss (L), where we denote by rkss(G) the semi{simple rank of G. 1. The case G = L. This is the case where rkss (G) = rkss(L). The irreducible characters of LF to be considered are the : where 2 Ab Irr(LF ). The equality (e) is then proved by the following lemma.

4.29. Lemma.

(1) For a d{cuspidal irreducible character of LF and 2 Ab Irr(LF ), we have F F decL;(L;) (:) = jZ o (1L)F j Ab RegL = jZ o (1L)F j

X

2Irr(Z o(L)F )

: :

(2) For 2 Irr(Z o (L)F ), we have

X o F o F : decZ (L) () = jZ o (1L)F j RegZ (L) = jZ o (1L)F j 2Irr(Z o(L)F )

Proof of 4.29. The second assertion is obvious. To prove the rst one, we notice the

following property of d{cuspidal characters.

58


4.30. Let be a d{cuspidal irreducible unipotent character of GF . Let be a set of primes which is d{adapted for G. Let g 2 GF . Then (g) = 0 unless the {component g of g belongs to Z o (G)F . Proof of 4.30. It results from the Curtis type formula (see 4.27) that, for M := CGo (g ), we have

(g) = RG M ( )(g) : Since M is a d{split Levi subgroup of G (by 4.2), and since is d{cuspidal, we see by 3.23, (2), that RGM ( ) = 0 unless M = G, which proves 4.30.

Remark. The preceding result is a particular case of the following property, which

also follows from 3.23, (2). ? Let be a unipotent character of GF such that ; RGL () GF 6= 0 ; where L is a d{split Levi subgroup of G, and where is a d{cuspidal unipotent character of LF . Let be a set of primes which is (d; (L; )){adpated for G. Let g 2 GF . Then

(g) = 0 unless the {component g of g belongs to Z o (L)F . This property can also be viewed as a consequence of Brauer's second main theorem and theorem 4.19 (see for example formula 4.25, (1)). 2. Applying the induction hypothesis. By the induction hypothesis, we assume now that (e) holds whenever G is replaced by any of its proper d{split Levi subgroups. It follows that (c) (whence (b) and (a) as well) holds for all x 2 Z o (L)F ? Z o (G)F . The formulae 4.25 applied with F G = I;(L;) () then show that

0 1 X o F o F z;Z ( L ) o W ( L ; ) z;Z ( L ) o W ( L ; ) tdec GF I;(L;) @ dec GF A = o F 0z2Z (G) 1 X F F tdecx;G decz;G A IGF @ ;(L;) : ;(L;) ;(L;) GF

(4.31)

z2Z o (G)F

To prove that (e) holds, again by the induction hypothesis and by 4.28, it suces to prove that, for 2 Irr (Z o (G)F ) and 2 Irr(WGF (L; )), we have o F F (decG IGF )(:) = (IGF decZ (L) oWGF (L;) )(:) ; ;(L;)

i.e.,

;(L;)

;(L;)

F (decG;(L;) (:I(GL;) ()) = (IG;(L;) decZ (L) oWGF (L;) )(:) : It is easy to check that X X t z;Zo(L)F oWGF (L;) z;Zo(L)F oWGF (L;) dec dec () = F

2Irr (Z o (G)F ) z2Z o(G)F

o

F

jZ o (G)F jdecZ o(L)F () ;


and also that

X

X

2Irr (Z o(G)F ) z2Z o(G)F

Since and

59

tdecz;GF decz;GF (I GF ()) = ;(L;) ;(L;) (L;)

jZ o (G)F jdecG;F(L;) (I(GLF;) ()) :

o F o F decZ (L) oWGF (L;) (:) = decZ (L) oWGF (L;) () F

F

F decG;(L;) (:I(GL;) ()) = decG;(L;) (I(GL;) ()) ; we see that (e) follows then from what precedes. Proof of 4.26, (2). . As in the proof of step 3 above, the proof is by induction on rkss(G) ? rkss (L), where we denote by rkss(G) the semi{simple rank of G. 1. The case where G = L. We use notation previously introduced (see 4.22). We shall need the following property of d{cuspidal characters. 4.32. Lemma. Let be a d{cuspidal unipotent character of GF . Let be a set of prime numbers which is d{adapted for G. Then for all g 2 GF , jCGF (g) : Z o (G)F j divides (g) (in other words, jC F (g) (g) : Z o (G)F j 2 O). F

G

Proof of 4.32.

Since is d{adapted, we see by 2.7 that for each ` 2 , we have (1)` = jGF : and so is a character with `{defect zero of the group GF =Z o (G)F . Henceforth, we know that, for all g 2 GF , jCGF (g) : Z o (G)F j` divides (g), from which it follows that jCGF (g) : Z o (G)F j divides (g). We must check that the map Z o (G)F j` ,

Irr(LF ; eLF ) IL(FL;) : Irr(Z o (L)F )?! ;(L;) Z

Z

such that 7! (here we identify Ab Irr(LF ) and Irr(Z o (L)F ) after proposition 4.20, (1)), is a {perfect isometry. Theo OF{module CF(Z o (L)F ; O) is generated by the set of characteristic functions xZ (L) of the elements x of Z o (L)F , owhile the submodule CF0 (Z o (L)F ; O) is Z (L)F generated by the characteristic function 1 . We have X o F (x?1 ) : (a) xZ (L) = jZ o (1L)F j 2Irr(Z o(L)F )

The O{module CF(LF ; eL;F(L;) ; O) is generated by the set of \projections" of the characteristic functions of the conjugacy classes of LF \onto the idempotent eL;F(L;) ".

By 4.30, we see that all these characteristic functions project onto zero, except the

60


characteristic functions (LxxF 0 ) of elements xx0 2 LF whose {component x belongs to Z o (L)F . The projection of such a function is then 0?1 X F F (x?1 ) : (b) eL;(L;) (Lxx0 ) := jC(xF (x0))j L 2Ab Irr(LF ) Since 0?1 eL;F(L;) (LxxF 0 ) := jC(xF (x0))j eL;F(L;) (LxF) ; L

we see by 4.32 that the module CF(LF ; eL;F(L;) ; O) is actually generated by the functions eL;F(L;) (LxF) for x 2 Z o (L)F . The submodule CF0 (LF ; eL;F(L;) ; O) is generated by the functions eL;F(L;) (LxF0 ) . LF . For the same reasons as above, it is actually generated by eL;F(L;) (1) By (a) and (b), we see that F F Z o(L)F ) ; LF eL;(L;) (Lx) = jLF :Z(1) o (L)F j I;(L;) (x and this proves indeed that IL;F(L;) induces a bijection between CF(LF ; eL;F(L;) ; O) and CF(Z o (L)F ; O) as well as between CF0 (LF ; eL;F(L;) ; O) and CF0 (Z o (L)F ; O). 2. Applying the induction hypothesis The following proof is the analogue of the proof of lemma 4.5 in [Br1]. We rst need to prove an easy general result about nite groups. Let H be a nite group, and let e be a central {idempotent for H. Let us denote by decH : CF(H; e; O) ! CF0 (H; e; O) the \{decomposition map", de ned as usual (for f 2 CF(H; e; O), and for h 2 H0 , we have decH (f)(h) = f(h)). 4.33. The O{module CF0 (H; e; O) is generated by decH (Irr(H; e)). Proof of 4.33. Since decH commutes with the \projection onto e", we may assume H be the characteristic function of the e = 1. Let h be a 0 {element of H, and let ;h 0 H {section of h (namely, we have ;h (g) = 1 if the 0 {component of g is conjugate to H (g) = 0 otherwise). It suces to prove that H is a linear combination h, and ;h ;h with coecients in O of elements of Irr(H). This is an easy consequence of Brauer's characterization of characters. We now assume that for every proper d{split Levi subgroup N of G containing L, the map IN;F(L;) is a {perfect isometry. Then by the de nition of a perfect isometryF (4.22) and by assertion (1) of theorem 4.26, we see that it suces to prove that IG;(L;) induces an isomorphism between CF0 (GF ; eG;F(L;) ; O) and CF0 (Z o (L)F WGF (L; ); O). By 4.33, this follows from the fact (4.26, (1)) that o F F F F decG(L;) IG;(L;) = IG;(L;) dec;Z ((LL;)) oWGF (L;) : o


61

4.34. Corollary. The idempotent eG;F(L;) is a {block of GF (i.e., it is a primitive central {idempotent). Proof of 4.34. By 4.26, (2), and by 4.23, it suces to prove that 1 is a {block of Z o (L)F WGF (L; ). By [Rob], thm. 9 (a non trivial generalization to any of a result which is well known in the case where consists of a single prime), it suces to check that WGF (L; ) acts faithfully on Z o (L)F . This follows from the fact that L = CG(Z o (L)F ). o

`{Blocks. The following hypothesis are in force in what follows. (H1) Let G be a connected reductive algebraic group over an algebraic closure of q , endowed with a Frobenius endomorphism F : G ! G de ning a rational structure over q . (H2) Let ` be a prime number which does not divide q, which divides jGF j, and which is excellent for (G; F). We denote by a generic nite reductive group corresponding to (G; F), and we set = (?; W). Following [BrMi], we call unipotent `{blocks of GF the primitive central `{idempoF G tents which are constituents of the \unipotent idempotent" e` . We denote by O the extension of the ring of `{adic integers byFa root of unity whose order is the l.c.m. of the orders of the elements of GF . Thus eG` 2 Z OGFF , and the unipotent `{blocks are the primitive idempotents of the algebra Z OGF eG` . The following omnibus theorem is an immediate consequence of the preceding sections (specially of theorem 4.26), as well as a consequence of general results about isotypies (see [Br1]). We make free use of the notation which has been introduced previously. F

F

G

G

4.35. Theorem. (I). Let (L; ) be a d{cuspidal pair of G. We assume that ` is (d; (L; )){adapted.

(1) The set of all irreducible constituents of the virtual characters RGL (), where 2 Ab` Irr(LF ), is the Fset of all irreducible characters of an `{block of GF , which we denote by eG`;(L;) . (2) (Z o (L)F` ; eL`;F(L;) ) is a maximal eG`;(FL;) {subpair of GF . In particular, Z o (L)F` is a defect group of eG`;(FL;) . (3) Whenever S is a subgroup of Z o (L)F` , its centralizer M in G is a d{split Levi subgroup, and the Brauer correspondent BrS (eG`;(FL;) ) of eG`;(FL;) is given by the following formula: BrS (eG;F(L;) ) =

X

(L0 ;0 )

F eM ;(L0 ;0 )

where (L0; 0) runs over a complete set of representatives for the MF {conjugacy classes of d{cuspidal pairs of M which are GF {conjugate to (L; ).

62


(4) (GF ; eG`;(FL;) ) and (Z o (L)F` WGF (L; ); 1) are isotypic, and so in particular (see [Br1], thm. 1.5F and x4.D) (a) the block eG`;(L;) of GF has respectively the same numbers of ordinary and modular characters as the group Z o (L)F` WGF (L; ), (b) all irreducible characters in eG`;(FL;) have height zero, (c) the Cartan matrix of eG`;(FL;) de nes the same integral quadratic form as the Cartan matrix of the group Z o (L)F` WGF (L; ), (d) the generalized decomposition matrix of (GF ; eG`;(FL;) ) de nes the same quadratic form (over [e2i=`a ], where a is the valuation in ` of d (q)) as the generalized decomposition matrix of (Z o (L)F` WGF (L; ); 1), (e) the algebras Z OGF eG`;(FL;) and Z O[Z o (L)F` WGF (L; )] are isomorphic. (II). Assume that ` does not divide jWj. Let d be the unique integer such that d (x) divides j j and ` divides d (q). Then o

o

o

Z

o

o

G

eG` F =

X

[(L;)]GF

eG`;(FL;) ;

where (L; ) runs over the setF of all d{cuspidal pairs of GF (module GF {conjugation). In other words, (L; ) 7! eG`;(L;) is a bijection between the set of all unipotent `-blocks of GF and the GF {conjugacy classes of d{cuspidal pairs Notice that, still assuming ` is large (i.e., ` jWj), the preceding theorem implies that several of the current conjectures in modular representation theory are true for groups of type GF and unipotent blocks : Alperin conjecture, Alperin{McKay conjecture, all implied in these cases (blocks with abelian defect groups) by conjecture 6.1 in [Br1]. Notice also that the preceding remark can be strengthened by extending it to non{ unipotent blocks of nite reductive groups, using the fact (see [Br1], theorems 2.3 and 5.6) that in many cases the \Jordan decomposition of blocks" is an isotypie. -

5. Appendix : d{Harish{Chandra theory for exceptional groups

In this section we prove the fundamental theorem 3.2 for exceptional groups of Lie type, as well as the "Ennola"{duality 3.3 in the case of d{split Levi subgroups. We assume that has a connected Dynkin diagram of exceptional type, so that the corresponding groups of Lie type are simple of exceptional type. In this case we do not have such a nice result on the decomposition of RGL() as Asai's formula for the classical groups (cf. 3.5, 3.9). But since we are dealing with groups of bounded rank, it can be replaced by the Mackey formula for twisted induction 1.37 (which had been already proved by Deligne if q is large enough, and which follows in general from Shoji's results, cf. x1.B). Using the Mackey formula the decomposition may uniquely be determined except for some ambiguities stemming from algebraic conjugate eigenvalues of Frobenius. These we can not resolve, but the attainable information is sucient to prove the fundamental theorem 3.2. G


63

Our approach will consist in a case by case analysis, but some frequently used arguments are collected at the beginning. Clearly if we know the decomposition of RGL() for each d{split generic Levi subgroup of and each unipotent character 2 Uch( ), then the assertion of the theorem may be checked. Namely the disjointness and the positivity will follow from the lists, and the existence of the collection of maps I(M L;) just by comparing these nitely many decompositions with tables showing the decompositions of induced characters in the corresponding Weyl groups. This trivial step of veri cation will not be made explicit in the proof. We shall just sketch how to nd the decomposition of RGL, and list the relevant centralizers WG( ; ) in the Weyl group. If our d{split Levi subgroup happens to be a maximal torus , then RGLis just the usual Deligne{Lusztig induction RGT. The decomposition of RGTis known in all cases by the work of Lusztig. This already accounts for about half the cases in the exceptional groups. When is contained in a Harish{Chandra generic subgroup , i.e., centralizes a 1{subgroup (see [BoTi], Sect. 4), then RM Lis known by induction and RGM coincides with ordinary Harish{Chandra induction, so the decomposition follows from the result of Howlett and Lehrer [HoLe] (the case d = 1). Finally, if lies in the space of uniform functions of then it may be written as a linear combination of Deligne{Lusztig characters RL (1). By transitivity of Lusztig{induction we can thus T also express RGL() as a linear combination of RGT (1)'s. Hence the determination of the decomposition is reduced to computations with Weyl group characters. So far we don't even have to assume the Mackey formula to obtain the decompositions. Unfortunately, not all cases ( ; d; ; ) are of one of the above types. We now rst collect a list of all quadruples ( ; d; ; ), where is simple exceptional, d an integer with d (x) dividing j j, a d{split generic Levi subgroup and 2 Uch( ) is d{cuspidal (this last information is either already known if is classical, or by induction for exceptional ). This list appears as Table 1. The notation for the unipotent characters of the exceptional groups follows [Ca], 13.9. For groups of type Al , the unipotent characters are indexed by partitions, for types Dl and 2Dl by symbols (see the proof of 3.2). However we have omitted the cases with d = 1, as these are known by the classical Howlett{Lehrer theory, and the cases where = , (hence) = 1, (then d is regular for in the sense of Springer [Sp1]), because here the decomposition is also clear due to = . The cases are numbered, and a indicates that in this case the decomposition of RGL() follows from the above considerations, i.e., without having to apply to the results of Deligne and Shoji. Note that the groups WG( ; ) occurring in case d = 2 are all well known Coxeter groups, as they should be by the "Ennola"{duality, since we have G(2; 1; 2) = W(B2 ), G(1; 1; 3) = W(F4 ) and G(6; 6; 2) = W(G2). = W(A2 ), G(2; 1; 3) = W(B3 ), G28 The table already shows that for the types 2B2 , 2G2 , G2, 3D4 and 2F4 the decomposition of RGL() for all cuspidal pairs ( ; ) can be determined from the knowledge of RGT . But this is not the case for the decomposition of all RGM(), where < is maximal d{split but not cuspidal. In particular we will have to return to 2F4 later on. 5.1 Theorem. The decomposition of the RGL() for the ( ; d; ; ) listed in Table 1 is given by Table 2. L

G

L

L

L

T

L

M

L

G

L

G

G

L

G

L

L

L

L

L

T

G

L

T

L

L

M

G

L

G

64


From this, the fundamental theorem 3.2 follows for exceptional groups. At the same time, the "Ennola"{duality 3.3 follows for all Levi subgroups which are d{split for some d. L

Remark.

In the cases marked by a in the last column, the decomposition can be proved without using the general Mackey formula. Remark.

The decompositions for not of type E8 , i.e., cases 1{38, were already obtained by Schewe [Sch] assuming a conjecture on Shintani descent, but note that some of his tables contain inaccuracies. G

Outline of the proof of 5.1. In all cases marked with a in the next to last column of Table 1, the d{cuspidal character of L lies in the space of uniform functions of GL can L. So by our above remarks on transitivity of Deligne{Lusztig induction, R

be computed purely mechanical from information inside the Weyl group. All these computations with characters of Weyl groups were done with the computer algebra system CAS [NPP]. This leaves 28 cases ( ; d; ; ) to be treated. These can not be solved with the methods discussed so far. We shall now assume in addition the validity of the Mackey formula for RGLin full generality, as allowed by 1.37 and 1.35. It enables us to calculate the norm of RGL() from the knowledge of WG( ). Now for 2 [x] Uch( ), denote by uL() its projection to the space of uniform class functions on . By de nition, this projection may be written as a linear combination of Deligne{Lusztig characters RGT (1), so as above we may calculate RGL(uL()) from the character tables of WLand WG. Since we have RGL(uL()) = uG(RGL()) by 1.35 (2), the uniform part of RGL() is known in all cases, as is the norm by the Mackey formula. These two informations suce to determine the decomposition except for some small ambiguities resulting from irrationalities. Namely, it will turn out that there exists an essentially unique element 2 [x]Uch( ) with uG( ) = uG(RGL()) of minimal norm, and that this norm coincides with the norm of RGL() calculated from the Mackey formula. By transitivity of Deligne{Lusztig induction we may recover the decomposition of RGLif for some maximal Levi subgroup < containing we know the decomposition of RGM (and the one of RM L by induction). We will therefore from now on assume that is already maximal in . If is a direct product of two groups = 1 2 , and all unipotent characters of 1 happen to be uniform, then RGM may be written as a sum of RGT M , where is maximal toric in 1 . If moreover 2 is contained in another maximal Levi subgroup 0 for which we know RGM0, then by induction we can also compute RGM. This is for example useful if RGM0 tends to contain only few constituents (and hence is easily determined), while RGM has large norm. As an example we cite the case d = 2 in of type E8, where has type D5 + A2 and 0 has type E7. We now consider the restriction of the uniform part uG(RGL()) to a family F of unipotent characters as de ned by Lusztig (see [Ca], chap. 13). The characters in a family, as well as their uniform projections, span pairwise orthogonal spaces of class functions. If a family consists of just one unipotent character, then this G

L

L

Q

L

L

Q

M

L

G

G

L

G

M

M

M

M

M

2

M

T

M

M

G

M

M

T


65

character coincides with its uniform projection and hence its multiplicity in RGLmay be recovered from the uniform projection. Next assume that the family F contains four unipotent characters 1 ; : : : ; 4 , i.e., belongs to the group S2 . Then except for one case in E7 and two cases in E8 the orthogonal complement of uniform functions has a one dimensional intersection with F and is spanned (after a suitable renumbering of the i ) by 1 ? 2 ? 3 + 4 . It turns out that in all cases of interest to us the uniform projections of the RGL(), restricted to a four element family, coincide with the uniform projection of exactly one i . It is now clear that the minimal norm is achieved if indeed RGL()jF = i . In case that adding up the minimal norms one just arrives at the norm of RGL(), the decomposition is uniquely determined. This accounts for the cases (17), (27), (62), (63), (64), (65). For cases (42) and (43) the decomposition can not be deduced unambiguously, since there two unipotent characters in the four{element family have the same uniform projection, but at least the sum of the two decompositions follows from the norm condition. Since the pairs of characters concerned do only occur with equal multiplicity in all other RGL, the ambiguity here does not harm the proof of the theorem. Whichever of the possibilities holds, the collection of maps I(M L;) exists. Next assume we consider the eight member family associated to the group S3 . Here the orthogonal complement to the space of uniform functions has dimension three, except for type G2, but there all unipotent characters of every proper Levi subgroup are uniform. In the other cases, the vector of shortest length in the sublattice spanned by generalized characters is 7 ? 8 of length two; all other vectors having length at least four. Checking the cases one sees that (5), (9), (11) and (14) follow from this observation. In the case of the 21{member family in F4 associated to S4 , the shortest vectors in the lattice of generalized characters orthogonal to all uniform functions are F4[] ? F4[2 ] and F4 [i] ? F4[?i] of norm two, and all other vectors have length at least four. This determines the cases (1), (2) and (3). Finally in the 39{member family F in E8 associated to S5 one again nds that all vectors of length two come from pairs of unipotent characters with algebraically conjugate eigenvalues of Frobenius. Moreover, all other vectors have length at least four. This information is not yet sucient do obtain all missing decompositions. So for example in case (47) we have (q) = E8(q), (q) = 3E6 (q), and it turns out that the norm of RGM() restricted to the family F may be as big as seven. But at least we know how to decompose RGM( ) for those lying in the principal series of E6 (q). Also, by the Mackey formula, the scalar products of these with the still unknown RGM() are known. This, combined with the information on nonuniform characters in F , allows to compute the decomposition completely. The same type of reasoning takes care of the 13{member family of characters in groups of type 2F4 for 08 {split subgroups with structure (q) = 08 2B2 (q). Although some ambiguities with respect to characters with algebraically conjugate eigenvalues of Frobenius remain, these are not relevant to the proof of the theorem, since it holds whichever of the dierent possible cases occurs. After the determination of all RGM it remains to compare the decompositions for maximal with those of the corresponding characters induced from WM( ; ) to WG( ; ). The groups WG( ; ) turn out to be irreducible complex re ection groups in all cases. We have indicated these groups, using the standard notation given for G

M

M

M

M

L

L

L

66


example in [Be] or [Co], in the last column of Table 1 for the cases listed there, and collected them in all other cases where d 6= 1; 2 and WG( ; ) is noncyclic in Table 3. (For d = 1; 2 they are the usual Weyl groups of or ? .) In the two cases (22) and (23), WG( ) is the re ection group G8 but the stabilizer of the d{cuspidal character of is a Sylow 2{subgroup of G8 of index three, isomorphic to the imprimitive re ection group G(4; 1; 2). With the help of the re ection representations for the relevant re ection groups given in [Be] it is now possible to nd the fusion of WM( ; ) into WG( ; ) and then compute the decomposition of all induced characters, using the computer algebra system CAS [NPP]. Then a straightforward veri cation proves the existence of the maps I(M L;). The details are omitted. This completes the proof of 5.1. L

G

G

L

L

L

L

It might be noted that the bijections I(M L;) are not at all unique if WG( ; ) is cyclic; indeed on this level any bijective map is admissible. But if we have once chosen the bijections on the level of cyclic groups, then it is "almost" determined for all bigger groups WG( ). To be more precise, the characters in the ( ; ){series are determined by their multiplicities in the RGM( ) for ( ; ) 4 ( ; ) (provided that some numbering is chosen on the 1{dimensional level), up to a small number of cases which basically result from algebraically conjugate eigenvalues of Frobenius, making characters indistinguishable by induction. This is again a remarkable analogy to the case d = 1, where Benson and Curtis have proved a similar result (here the "cyclic" case is just the trivial group), also up to one ambiguity in E7 and two in E8. L

L

L

L

M


d F4 2 4 E6 2 3 4 5 2E6 2 4 6 10 E7 2 2 3 3 4 G

5 6 6 8 10 12 E8 2 2 2 3 3 4 5 6 6 7 8 9 10 12 14 18

Table 1. Some d-series for exceptional groups.

(q) (q + 1)2 :B2 (q) (q2 + 1):B2 (q) (q + 1):A5(q) (q2 + q + 1): 3D4 (q) (q2 + 1)(q ? 1): 2A3 (q) (q5 ? 1):A1(q) (q + 1)2 :D4(q) 2 (q + 1)(q + 1):A3(q) (q2 ? q + 1): 3D4 (q) (q5 + 1):A1(q) (q + 1)3 :D4(q) (q + 1): 2E6(q) (q3 ? 1): 3D4 (q) (q2 + q + 1):A5(q) (q2 + 1)2:A1(q)3

case # 1 2; 3 321 4 3D4 [?1] 5 22 6 2 ; 11 7 ; 8 13;02 9 22 10 2;1 11 2 ; 11 12; 13 13;02 14 2E6[]; 2E6[2 ] 15; 16 3D4 [?1] 17 42; 2211 18; 19 3 3 2 ; 11 20; 21 2 2 211; 211 22; 23 (q5 ? 1):A2(q) 3; 21; 111 24; 25 ; 26 3 3 (q + 1): D4 (q) 2;1 27 (q2 ? q + 1): 2A5 (q) 42; 2211 28; 29 4 2 2 2 (q + 1):A1(q ):A1 (q) 2; 211; 112; 11 30; 31 ; 32; 33 (q5 + 1): 2A2 (q) 3; 21; 111 34; 35 ; 36 4 2 3 (q ? q + 1):A1(q ) 2 ; 11 37; 38 4 (q + 1) :D4(q) 13;02 39 2E6[]; 2E6[2 ] (q + 1)2: 2E6(q) 40; 41 (q + 1):E7(q) 512;11; 512;12 42; 43 2 2 3 3 (q + q + 1) : D4 (q) D4 [?1] 44 (q2 + q + 1):E6(q) 81;6; 81;10; 90;8 45; 46 ; 47 2 2 (q + 1) :D4(q) 3;1; 123;013; 23;01; 12;03 48; 49 ; 50; 51 4 3 2 (q + q + q + q + 1):A4(q) 32; 221 52; 53 (q2 ? q + 1)2 : 3D4 (q) 2;1 54 (q2 ? q + 1): 2E6(q) 09;6; 009;6; 006;6 55; 56 ; 57 7 (q ? 1):A1(q) 2 ; 11 58; 59 4 2 (q + 1): D4 (q) 13;?; 0123;13 60; 61 023;1; 123;0; 013;2; 012;3 62; 63; 64; 65 (q6 + q3 + 1):A2(q) 3; 21; 111 66; 67 ; 68 4 3 2 2 (q ? q + q ? q + 1): A4(q) 32; 221 69; 70 4 2 3 0 00 3 (q ? q + 1): D4 (q) 1;3; 1;3; 2;2; D4 [1] 71; 72 ; 73; 74 (q7 + 1):A1(q) 2 ; 11 75; 76 6 3 2 (q ? q + 1): A2(q) 3; 21; 111 77; 78 ; 79 L

1;1 11;?; ?;2

67

WG( ; ) G(2; 1; 2) Z4 Z2 Z3 Z4 Z5 G(1; 1; 3) Z4 Z3 Z5 G(2; 1; 3) Z2 Z6 Z6 G8 G(4; 1; 2) Z10 Z6 Z6 Z8 Z10 Z12 G28 G(6; 6; 2) Z2 G5 Z6 G8 Z10 G5 Z6 Z14 Z8 Z8 Z18 Z10 Z12 Z14 Z18 L

68


Table 2. Decomposition of some RGL().

RGL() 4;1 + 4;13 ? 004;7 ? 04;7 ? 2B2 ; r 02;4 + 02;16 ? 04;7 + B2 ; 0 002;4 + 002;16 ? 004;7 + B2 ; 00 64;4 ? 64;13 D4 ; 1 + D 4 ; ? D 4 ; r 20;2 ? 60;5 + 60;11 ? 20;20 1;0 ? 24;6 + 81;10 ? 64;13 + 6;25 6;1 ? 64;4 + 81;6 ? 24;12 + 1;36 08;3 + 008;9 + 216;5 4;1 + 04;7 ? 004;7 ? 4;13 08;3 + 008;9 ? 16;5 1;0 ? 008;3 + 009;6 ? 2A5 ; ? 002;16 02;4 ? 2A5 ; 1 ? 09;6 + 08;9 ? 1;24 56;3 ? 120;4 ? 3216;9 ? 2336;11 + 2336;14 + 3216;16 + 120;25 ? 56;30 +3280;8 ? 3280;17 15 + 16 E6 []; 1 ? E6 []; + E6 [2 ]; 1 ? E6 [2 ]; 17 D4 ; 1 + D4 ; 1 ? D4 ; 2 ? D4 ; 20 + D4 ; 2 + D4 ; 18 27;2 + 189;5 ? 378;9 + 189;20 ? 216;16 + 189;17 19 189;7 ? 378;14 + 189;22 + 27;37 ? 216;9 + 189;10 20 1;0 ? 2210;6 ? 3105;6 ? 3189;22 + 21;36 + 256;3 ? 2D4 ; 1 + 3405;8 + 3189;10 ?4336;11 ? 4D4 ; 2 + 2120;25 ? 2D4 ; 2 + 315;16 + 270;18 + 35;22 21 21;3 ? 3189;5 ? 3105;21 ? 2210;21 + 1;63 + 2120;4 ? 2D4 ; 1 ? 4336;14 ?4D4 ; 20 + 3405;15 + 3189;17 + 256;30 ? 2D4 ; + 315;7 + 270;9 + 35;13 22 7;1 ? 189;7 + 2378;9 ? 105;15 + 2210;13 + 27;37 ? 15;7 + 105;5 ? 2216;16 ?2D4 ; r2 ? 35;31 + 21;33 ? 2280;8 ? 2D4 ; r 23 27;2 + 2210;10 ? 105;12 + 2378;14 ? 189;20 + 7;46 ? 35;4 + 21;6 ? 2216;9 ?2D4 ; r1 ? 15;28 + 105;26 ? 2280;17 ? 2D4 ; r 24 1;0 + 21;3 ? 189;7 ? 189;22 ? 84;12 + 336;11 + 216;16 ? 189;17 + 56;30 + 21;33 25 7;1 + 27;2 ? 168;6 ? 378;9 ? 378;14 ? 168;21 + 27;37 + 7;46 + 512;11 + 512;12 26 ?189;5 ? 189;20 + 21;36 + 1;63 + 56;3 + 21;6 + 216;9 ? 189;10 ? 84;15 + 336;14 27 56;3 ? 120;4 + 336;11 ? 336;14 + 120;25 ? 56;30 28 27;2 ? 189;5 + 378;9 + 189;20 ? 405;15 + D4 ; r2 29 189;7 + 378;14 ? 189;22 + 27;37 ? 405;8 + D4 ; r1 30 1;0 ? 105;6 + 189;22 ? 21;36 + 216;9 + D4 ; r1 ? 280;17 ? D4 ; r 31 7;1 + 189;7 ? 105;15 ? 27;37 ? 120;4 + D4 ; 1 + 56;30 ? D4 ; case 1 2 3 4 5 6 7 8 9 10 11 12 13 14


69

27;2 + 105;12 ? 189;20 ? 7;46 ? 56;3 + D4 ; 1 + 120;25 ? D4 ; 2 21;3 ? 189;5 + 105;21 ? 1;63 ? 216;16 ? D4 ; r2 + 280;8 + D4 ; r 1;0 ? 21;3 + 189;7 ? 189;22 ? 420;10 + D4 ; 2 + 405;15 ? D4 ; r2 + 35;31 + D4 ; ?7;1 + 27;2 ? 168;6 + 378;9 ? 378;14 + 168;21 ? 27;37 + 7;46 + E7 [] + E7 [?] ?189;5 + 189;20 ? 21;36 + 1;63 + 35;4 + D4 ; 1 + 405;8 ? D4 ; r1 ? 420;13 + D4 ; 20 1;0 + 210;6 + 21;36 ? 56;3 + D4 ; 1 ? 336;11 ? D4 ; 2 ? 120;25 + D4 ; 2 + 280;18 +E6 []; + E6 [2 ]; 38 21;3 + 210;21 + 1;63 ? 120;4 + D4 ; 1 ? 336;14 ? D4 ; 20 ? 56;30 + D4 ; + 280;9 +E6 []; 1 + E6 [2 ]; 1 39 112;3 ? 2160;7 ? 400;7 ? 91296;13 ? 42240;10 ? 23360;13 + 92800;13 + 83200;16 +83200;22 ? 23360;25 + 92800;25 ? 42240;28 ? 91296;33 ? 400;43 ? 2160;55 + 112;63 +41344;8 ? 8D4 ; 08;3 ? 8D4 ; 008;9 + 41344;38 ? 41344;19 ? 62016;19 ? 65600;19 ?12448;25 ? 16D4 ; 16;5 40 + 41 E6 []; 1;0 ? E6 []; 01;3 ? E6 []; 001;3 + E6 []; 1;6 ? 2E8 [] ? 2E8 [?]+ E6 [2 ]; 1;0 ? E6 [2 ]; 01;3 ? E6 [2 ]; 001;3 + E6 [2 ]; 1;6 ? 2E8[2 ] ? 2E8 [?2 ] 42 + 43 4096;11 ? 4096;26 + 4096;12 ? 4096;27 44 D4 ; 1;0 ? D4 ; 02;4 + D4 ; 01;12 ? 2D4 ; 04;7 ? D4 ; 002;4 + 2D4 ; 008;3 + 2D4 ; 08;9 ? D4 ; 02;16 ?2D4 ; 004;7 + D4 ; 001;12 ? D4 ; 002;16 + D4 ; 1;24 ? 2D4 ; 4;1 + 2D4 ; 08;3 + 2D4 ; 008;9 ?2D4 ; 4;13 + D4 ; 4;8 ? 2D4 ; 16;5 ? 3E8 [?] ? 3E8 [?2 ] + 3E8 [?1] 45 567;6 + 3240;9 ? 4536;13 ? 2835;22 + 2268;30 + 1296;33 46 ?2835;14 ? 4536;23 + 3240;31 + 567;46 + 2268;10 + 1296;13 47 1008;9 + 1575;10 + 1575;35 + 1008;39 ? 3150;18 ? 2016;19 48 8;1 + 2560;5 + 34536;13 + 4200;21 ? 33240;31 ? 31400;11 + 3840;13 ? 43200;16 ?4D4 ; 008;3 + 22240;28 ? 2D4 ; 004;7 ? 21344;8 ? 2D4 ; 4;1 + 1400;37 + 21008;39 + 56;49 49 ?33240;9 + 4200;15 + 34536;23 + 2560;47 + 8;91 + 22240;10 ? 2D4 ; 04;7 ? 43200;22 ?4D4 ; 08;9 ? 31400;29 + 3840;31 + 1400;7 + 21008;9 + 56;19 ? 21344;38 ? 2D4 ; 4;13 50 28;8 + 2160;7 + 300;8 ? 3972;12 + 2840;14 ? 3700;16 + 2840;26 ? 3700;28 ? 3972;32 +300;44 + 2160;55 + 28;68 + 41344;19 ? 2D4 ; 06;6 ? 2D4 ; 006;6 ? 4D4 ; 12;4 51 84;4 ? 2D4 ; 02;4 ? 700;6 + 32268;10 ? 24200;12 ? 32100;16 ? 24200;24 ? 32100;28 +32268;30 ? 700;42 ? 2D4 ; 002;16 + 84;64 + 22016;19 + 25600;19 + 4448;25 ? 4D4 ; 4;8 52 35;2 + 560;5 ? 3240;9 + 2835;22 ? 840;14 + 3360;13 ? 2240;28 ? 840;31 + 210;52 + 160;55 53 2835;14 ? 3240;31 + 560;47 + 35;74 + 210;4 + 160;7 ? 2240;10 ? 840;13 ? 840;26 + 3360;25 54 112;3 + 160;7 ? 400;7 + 22240;10 + 3360;13 + 23200;16 + 23200;22 + 3360;25 +22240;28 ? 400;43 + 160;55 + 112;63 ? 21344;8 ? 2D4 ; 08;3 ? 2D4 ; 008;9 ? 21344;38 ?37168;17 ? 1344;19 + 2D4 ; 16;5 + 3E6 []; 2;2 + 3E6 [2 ]; 2;2 55 567;6 ? 3240;9 + 4536;13 ? 2835;22 + 972;32 + D4 ; 9;10 56 ?2835;14 + 4536;23 ? 3240;31 + 567;46 + 972;12 + D4 ; 9;2 32 33 34 35 36 37

70


57 1008;9 ? 1575;10 ? 1575;34 + 1008;39 + 1134;20 + D4 ; 06;6 58 1;0 ? 3240;9 ? 6075;14 + 8;91 + 400;7 ? 300;8 + 4096;11 + 4096;12 + 3200;22 ?2400;23 ? 972;32 + 1296;33 + 50;56 ? 160;55 59 8;1 ? 6075;22 ? 3240;31 + 1;120 + 50;8 ? 160;7 ? 972;12 + 1296;13 + 3200;16 ?2400;17 + 4096;26 + 4096;27 + 400;43 ? 300;44 60 8;1 + 4536;13 ? 4200;21 + 3240;31 ? 2240;10 + D4 ; 04;7 ? 1344;38 ? D4 ; 4;13 61 ?3240;9 + 4200;15 ? 4536;23 ? 8;91 + 2240;28 ? D4 ; 004;7 + 1344;8 + D4 ; 4;1 62 84;4 ? 400;7 + D4 ; 9;2 + 700;16 ? D4 ; 009;6 ? 972;32 + 700;42 ? 112;63 63 D4 ; 1;0 + 300;8 ? 2268;10 + 2800;13 ? 2100;28 + 1296;33 ? D4 ; 001;12 ? 28;68 64 112;3 ? 700;6 + 972;12 + D4 ; 09;6 ? 700;28 ? D4 ; 9;10 + 400;43 ? 84;64 65 28;8 + D4 ; 01;12 ? 1296;13 + 2100;16 ? 2800;25 + 2268;30 ? 300;44 ? D4 ; 1;24 66 1;0 ? 560;47 ? 50;8 + 160;7 + 2800;13 + 700;16 ? 5600;21 ? 3200;22 + 4096;26 + 4096;27 +112;63 + 28;68 ? 1008;9 ? E6[]; 1;0 ? E6 [2 ]; 1;0 ? 1575;34 ? E6 []; 001;3 ? E6 [2 ]; 001;3 67 8;1 + 35;2 + 35;74 + 8;91 ? 700;6 ? 400;7 + 2240;10 + 1400;11 + 2240;28 + 1400;29 ?700;42 ? 400;43 ? 3150;18 ? 2016;19 ? E6 []; 2;1 ? E6 [2 ]; 2;1 ? E6 []; 2;2 ? E6 [2 ]; 2;2 68 ?560;5 + 1;120 + 112;3 + 28;8 + 4096;11 + 4096;12 ? 5600;15 ? 3200;16 + 2800;25 + 700;28 ?50;56 + 160;55 ? 1575;10 ? E6 []; 01;3 ? E6 [2 ]; 01;3 ? 1008;39 ? E6 []; 1;6 ? E6 [2 ]; 1;6 69 35;2 ? 560;5 + 3240;9 + 2835;22 ? 4200;12 + D4 ; 002;4 ? 1400;29 ? D4 ; 004;7 + 50;56 + D4 ; 002;16 70 2835;14 + 3240;31 ? 560;47 + 35;74 + 50;8 + D4 ; 02;4 ? 1400;11 ? D4 ; 04;7 ? 4200;24 + D4 ; 02;16 71 8;1 ? 560;5 + 4200;21 ? 3200;16 ? D4 ; 008;3 ? 2240;28 + D4 ; 004;7 + 1344;8 + D4 ; 4;1 +448;39 + E6 []; 1;6 + E6 [2 ]; 1;6 72 4200;15 ? 560;47 + 8;91 ? 2240;10 + D4 ; 04;7 ? 3200;22 ? D4 ; 08;9 + 1344;38 + D4 ; 4;13 +448;9 + E6[]; 1;0 + E6 [2 ]; 1;0 73 84;4 + D4 ; 02;4 ? 700;6 + 4200;12 + 4200;24 ? 700;42 + D4 ; 002;16 + 84;64 ? 7168;17 ?D4 ; 4;8 + E6 []; 2;2 + E6 [2 ]; 2;2 74 28;8 ? 160;7 + 300;8 ? 840;14 ? 840;26 + 300;44 ? 160;55 + 28;68 + 1344;19 ?E8 [?] ? E8 [?2 ] + E8 [?1] 75 1;0 + 3240;9 ? 6075;14 ? 8;91 ? 700;6 + D4 ; 01;12 + E7 []; 1 + E7 [{]; 1 + 5600;21 ?D4 ; 08;9 ? 2268;30 + D4 ; 9;10 + 210;52 ? D4 ; 002;16 76 8;1 + 6075;22 ? 3240;31 ? 1;120 ? 210;4 + D4 ; 02;4 + 2268;10 ? D4 ; 9;2 ? 5600;15 +D4 ; 008;3 + E7 []; + E7 [{]; + 700;42 ? D4 ; 001;12 77 1;0 + 560;47 ? 210;4 + D4 ; 02;4 ? 2100;16 ? D4 ; 09;6 + 2400;23 + D4 ; 08;9 ? E7 []; ? E7 [{]; ?84;64 ? D4 ; 1;24 + 1008;9 + E6 []; 1;0 + E6 [2 ]; 1;0 ? 1575;34 ? E6 []; 001;3 ? E6 [2 ]; 001;3 78 ?8;1 + 35;2 + 35;74 ? 8;91 ? 300;8 ? D4 ; 01;12 + 840;13 + D4 ; 04;7 + 840;31 + D4 ; 004;7 ?300;44 ? D4 ; 001;12 ? 1134;20 ? D4 ; 06;6 ? E8 [{] ? E8 [{2 ] ? E8 [] ? E8[2 ] 79 560;5 + 1;120 ? 84;4 ? D4 ; 1;0 + E7 []; 1 + E7 [{]; 1 + 2400;17 + D4 ; 008;3 ? 2100;28 ? D4 ; 009;6 ?210;52 + D4 ; 002;16 ? 1575;10 ? E6 []; 01;3 ? E6 [2 ]; 01;3 + 1008;39 + E6[]; 1;6 + E6 [2 ]; 1;6


71

Table 3. WG( ) for d regular and noncyclic. T

G

3D4 2F4

F4 E6

j j

d 3 6 4 80 800 3 4 6 3 4 6

T

23 26 24 082 008 2 23 24 26 33 2421 263

T

WG( ) G4 G4 G12 G8 G8 G5 G8 G5 G25 G8 G5 T

G

2E6

E7 E8

d 3 4 6 3 6 3 4 5 6 8 10 12

j j T

236 2422 36 331 362 43 44 25 46 28 210 212

WG( ) G5 G8 G25 G26 G26 G32 G31 G16 G32 G9 G16 G10 T

References [Be] [BoTi] [Bou] [Boy] [Br1] [Br2] [Br3] [BrMa] [BrMi] [Ca] [Co] [DeLu] [DiMi1] [DiMi2] [DiMi3] [DiMi4]

M. Benard, Schur indices and splitting elds of the unitary re ection groups, J. Algebra 38 (1976), 318{342. A. Borel and J. Tits, Groupes reductifs, Publ. Math. I. H. E. S. 27 (1965), 55{151. N. Bourbaki, Groupes et algebres de Lie, chap. 4, 5 et 6, Hermann, Paris, 1968. R. Boyce, Cyclotomic polynomials and irreducible representations of nite groups of Lie type, unpublished manuscript. M. Broue, Isometries parfaites, types de blocs, categories derivees, Asterisque 181{182 (1990), 61{92. M. Broue, Conjectures, Preprint (1992). M. Broue, Les {blocs des groupes GL( ) et U( 2 ) et leurs structures locales, Asterisque 133{134 (1986), 159{188. M. Broue and G. Malle, Theoremes de Sylow generiques pour les groupes reductifs sur les corps nis, Math. Ann. (1992) (to appear). M. Broue and J. Michel, Blocs et series de Lusztig dans un groupe reductif ni, J. reine angew. Math. 395 (1989), 56{67. R. W. Carter, Finite groups of Lie type: Conjugacy classes and complex characters, John Wiley and Sons, Chichester, 1985. A. M. Cohen, Finite complex re ection groups, Ann. scient. E c. Norm. Sup. 9 (1976), 379{ 436. P. Deligne and G. Lusztig, Representations of reductive groups over nite elds, Annals of Math. 103 (1976), 103{161. F. Digne and J. Michel, On Lusztig's parametrization of characters of nite groups of Lie type, Asterisque 181{182 (1990), 113{156. F. Digne and J. Michel, Representations of nite groups of Lie type, Cambridge University Press, Cambridge, 1990. F. Digne and J. Michel, Lusztig functor and Shintani descent, Rapport de Recherche du LMENS 2 (1989), Publications du DMI, ENS Paris. F. Digne and J. Michel, Foncteurs de Lusztig et caracteres des groupes lineaires et unitaires sur un corps ni, J. Algebra 107 (1987), 217{255. `

n; q

n; q

72


[En1] [En2] [FoSr] [Ge] [GeHi] [HoLe] [JaKe] [Lu1] [Lu2] [LuSr] [NPP] [Ol] [Os] [Rob] [Rou] [Sch] [Sho1] [Sho2] [Sp1] [Sp2] [SpSt]

M. Enguehard, Isometries parfaites entre blocs de groupes symetriques, Asterisque 181{182 (1990), 157{171. M. Enguehard, Positivite de l'application de Lusztig, Preprint expected (1992). P. Fong and B. Srinivasan, Generalized Harish{Chandra theory for unipotent characters of nite classical groups, J. Algebra 104 (1986), 301{309. M. Geck, On the classi cation of {blocks of nite groups of Lie type, J. Algebra (to appear). M. Geck and G. Hi, Basic sets of Brauer characters of nite groups of Lie type, J. reine angew. Math. 418 (1991), 173{178. R. B. Howlett and G. I. Lehrer, Representations of generic algebras and nite groups of Lie type, Trans. Am. Math. Soc. 280 (1983), 753{779. G. James and A. Kerber, The representation theory of the symmetric group, Addison Wesley, London, 1981. G. Lusztig, Characters of reductive groups over a nite eld, Annals of Mathematical Studies, no 107, Princeton University Press, Princeton, New Jersey, 1984. G. Lusztig, On the niteness of the number of unipotent classes, Invent. math. 34 (1976), 201{213. G. Lusztig and B. Srinivasan, The characters of the nite unitary groups, J. Algebra 49 (1977), 167{171. J. Neubuser, H. Pahlings and W. Plesken, CAS; Design and use of a system for the handling of characters of nite groups, in: Computational group theory, Academic Press, 1984, pp. 195{247. J. B. Olsson, Remarks on symbols, hooks and degrees of unipotent characters, J. Combinatorics A 42 (1986), 223{238. M. Osima, On the representations of the generalized symmetric group, Math. J. Okayama Univ. 4 (1954), 39{56. G. Robinson, Group algebras over semi{local rings, J. Algebra 117 (1988), 409{418. R. Rouquier, Sur les blocs a groupe de defaut abelien dans les groupes symetriques (1991) (to appear). K. {D. Schewe, Blocke exzeptioneller Chevalley-Gruppen, Dissertation, Bonner Mathematische Schriften, nr 165, Bonn, 1985. T. Shoji, Some generalization of Asai's result for classical groups, in: Advanced Studies in Pure Math. { Algebraic groups and related topics, vol. 6, 1985, pp. 207{229. T. Shoji, Shintani descent for exceptional groups over a nite eld, J. Fac. Sci. Univ. Tokyo 34 (1987), 599{653. T. A. Springer, Regular elements of nite re ection groups, Invent. math. 25 (1974), 159{ 198. T. A. Springer, Linear algebraic groups, Birkhauser, Boston{Basel{Stuttgart, 1981. T. A. Springer and R. Steinberg, Conjugacy classes, in: Seminar on algebraic groups and related nite groups, Lecture Notes in Mathematics, vol. 131, Springer{Verlag, Berlin Heidelberg, 1970. `

M. B. and J. M. : D.M.I., 45 rue d'Ulm, F{75005 Paris, France

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G. M. : I.W.R., Im Neuenheimer Feld 368, D{6900 Heidelberg, Germany

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Generic Blocks of Finite Reductive Groups

Generic Blocks of Finite Reductive Groups

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