On a Theorem of Shintani

Article ID jabr.1998.7797, available online at http:rrwww.idealibrary.com on

On a Theorem of Shintani

Cedric Bonnafe*´ ´

Department of Mathematics, The Uni¨ersity of Chicago, Chicago, Illinois 60637 Communicated by Michel Broue´

Received June 13, 1998

Ž d.

Let ␹ be an irreducible character of G_dsGL_n⺖_q invariant under the automorphism␾ofG_dinduced by the field automorphism⺖_q^dª⺖_q^d,x¬x^q, and letebe a divisor ofd. By a theorem of Shintani, there exists an extension ␹˜e of␹

² ^e:

to G_di ␾ whose Shintani descent to G_e is, up to a sign ␧, an irreducible character ofGe. It is shown in this paper that ␹˜e may always be chosen such that

␧s1. With this particular choice,␹˜e is the restriction of␹˜1. Our methods rely on the work of Digne and Michel on Deligne᎐Lusztig theory for nonconnected reductive groups. 䊚1999 Academic Press

Ž .^d

Let G⬚sGL_n⺖ , where ⺖ is an algebraic closure of a finite field and where nand dare natural numbers. The symmetric group ᑭd acts on G⬚ by permutations of the components of G⬚. We denote by G the semidirect product GsG⬚iᑭ_d. It is a nonconnected reductive group, with neutral component G⬚. We denote by F₀: GªG the natural split Frobenius

Ž .

endomorphism on G acting trivially on ᑭ_d , and we choose an element

␴gᑭ_d. Let F: GªG denote the Frobenius endomorphism defined by

Ž . ^␴ Ž .

F g s F g₀ .

F Ž

In this paper we discuss the irreducible characters of G the unipotent

F w x.

characters of G were described in B . We first prove that there exists a

Ž .

Jordan decomposition of characters this result is well-known for G⬚; moreover, this decomposition commutes with Lusztig generalized induc-

Ž Ž .. ^F

tion cf. 3.2.1 . We also prove that all the irreducible characters of G are linear combinations of generalized Deligne᎐Lusztig characters this gener-Ž

* Current address: Universite de Franche-Comte, Departement de Mathematiques, 16´ ´ ´ ´ Route de Gray, 25 030 Besanc¸on, France.

229

0021-8693r99 $30.00

Copyright䊚1999 by Academic Press All rights of reproduction in any form reserved.

w

alizes the well-known result of G. Lusztig and B. Srinivasan LS, Theorem .x

3.2 about irreducible characters of the general linear group over a finite field ..

As an application of these results, we obtain new results about Shintani w x

descent in the case of the general linear group. In S , Shintani proved that Ž d.

any irreducible characters of the finite groupG_dsGL_n ⺖_q stable under the automorphism␾ induced by the field automorphism⺖_q^dª⺖_q^d, x¬x^q

² :

can be extended to G_d ␾ in such a way that its Shintani descent is, up to Ž .

sign, an irreducible character ofG₁sGL_n⺖_q . In Theorem 4.3.1 we prove that this sign can always be chosen to be equal to 1 and get precise formulas for the corresponding extension. As a consequence, we obtain that the Shintani descent of this particular extension to G_e is an irre-

Ž .

ducible character of G_e where e divides d .

0. NOTATION 0.1. General Notation

Let ⺖ be an algebraic closure of a finite field. We denote by p its characteristic. We also fix a power q of p, and we denote by ⺖_q the subfield of ⺖ with q elements. All algebraic varieties and all algebraic

Ž .

groups will be considered over ⺖. If H is an algebraic group over ⺖, we will denote by H⬚its connected component containing 1. If H is endowed with an⺖_q-structure, we also define

Ž .

⺖q-rank H⬚

␧_H⬚s yŽ 1. .

Let ll be a prime number different from p. We denote by ⺡_ll an algebraic closure of the ll-adic field ⺡_ll. If G is a finite group, all representations and all characters of G will be considered over ⺡_ll. For instance, a G-module is a ⺡_llG-module of finite dimension. We will denote by IrrG the set of irreducible characters of G.

If n is a positive integer, we denote by GL_n the group of invertible matrices with coefficients in⺖, and if ggGL , we will denote by_n g^Žq. the matrix obtained from g by raising all coefficients to the qth power. We will denote by T the split maximal torus of GL_{n n} consisting of diagonal matrices and by B the rational Borel subgroup of GL consisting of upper_{n n} triangular matrices.

0.2. The Problem

Let r be a positive integer and let d₁, . . . ,d_r and n₁, . . . ,n_r also be positive integers. Throughout this paper G⬚ will denote the following

connected reductive group:

r

G⬚s

Ł Ž

GLⁿi=⭈⭈⭈=GLⁿi

.

.

is1

^ ` _

d_itimes

We endow G⬚with the split Frobenius endomorphism

F0: G⬚ ª G⬚

.

Žq. Žq.

g , . . . ,g ¬ g , . . . ,g

Ž ^{i1 i d}i.1FiFr

Ž

^{i1 i d}i

.

1FiFr

We will denote by T^T₀ and B^T₀ the maximal torus and the Borel subgroup of G^T, defined, respectively, by

T r

T⁰ s

Ł Ž

Tⁿi=⭈⭈⭈=Tⁿi

.

is1

^ ` _

d_itimes

and

T r

B0s

Ł Ž

Bn_i=⭈⭈⭈=Bn_i

.

.

is1

^ ` _

d_itimes

The group ᑭsᑭ_d =⭈⭈⭈=ᑭ_d acts on G⬚ in the natural way. More

1 r

Ž . Ž .

explicitly, if ␴s ␴₁, . . . ,␴_r gᑭ and if g_i1, . . . ,g_{i d 1FiFr}gG⬚, we put

i

␴Žgi1, . . . ,gi di.1FiFrsŽgi␴y1i Ž1., . . . ,gi␴y1i Ždi..1FiFr.

The elements of ᑭ induce automorphisms of G^T, which stabilize T^T₀ and

T Ž w Ž .x.

B , so they are quasi-semisimple cf. DM2, Definition 1.1 i . In fact, they₀

Ž w x w

are all quasi-central cf. DM2, Definition-Theorem 1.15 and B, Lemma x.

7.1.1 .

We extend the Frobenius endomorphism F₀ to G^Tiᑭ by letting F₀ act trivially on ᑭ. We fix once and for all an element ␴gᑭ, and we denote by F the Frobenius endomorphism on G^Tiᑭ given by

F gŽ .s␴F0Žg.␴^y1s^␴F0Žg. for all ggG^Tiᑭ.

We will denote by G an F-stable subgroup of G^Tiᑭ containing G^T. Hence G is a reductive group with neutral component G^T. Moreover, there

Ž .

exists an F-stable that is, a ␴-stable subgroup A of ᑭ such that GsG^TiA.

Thus we have G^FsG^TFiA^FsG^TFiA^␴.

Problem. Parametrize the irreducible characters of G^F.

For this purpose we can make the following hypothesis without loss of generality:

HYPOTHESIS. The Frobenius endomorphism F acts tri_¨ially on GrG^T, that is, A is contained in the centralizer of ␴ in ᑭ.Consequently,

G^FsG^TFiA.

Remark 0. Let Nsd n_{1 1}q⭈⭈⭈qd n_{r r}. Then G^T is isomorphic to a

T Ž

rational Levi subgroup H of a parabolic subgroup of GL_N endowed with

Žq.

the split Frobenius endomorphism g¬g ., and G is isomorphic to a rational subgroup H of the normalizer of H^T in GL , containing H_N ^T and such that all elements of HrH^T are rational. Conversely, if H is such a rational subgroup of GL , then there exist positive integers_N r, d₁, . . . ,d_r,n₁, . . . ,n_r; an element ␴ of ᑭ_d =⭈⭈⭈=ᑭ_d; and a subgroup A

1 r

of ᑭ^␴ such that H is isomorphic to the group G constructed as above. In particular, if L is an F-stable Levi subgroup of a parabolic subgroup of G Žcf. B, Definitions 6.1.1 and 6.1.2 for the definitions of parabolic sub-w x groups and Levi subgroups of nonconnected reductive groups , then all of. the results proved for G hold in L.

1. JORDAN DECOMPOSITION OF CHARACTERS OF G^F 1.1. Dual of G

Ž ^{TU TU} . Ž ^{T T} .

Let G , T0 ,F* be a dual triple of G , T ,0 F . The elements ␣ of ᑭ induce automorphisms ␣* of G^TU. The group ᑭ* of automorphisms of G^TU induced by ᑭ is isomorphic to the opposite group of ᑭ. We extend the action of F* to G^TUiᑭ^U so that it acts on ᑭ* by conjugation by

␴*^y1. We denote by G* the semidirect product G^TUiA*, where A* is the image of A under the preceding anti-isomorphism. In particular, G^UTs G^TU!

1.2. Lusztig Series of G^F

UTF*

Ž . Ž Ž . UF*

Let s be a semisimple element of G . We denote by s or s _G

. ^F* Ž .^T Ž

if confusion is possible the G* -conjugacy class of s and by s or

Ž .^T . ^UTF

U UTF*

s_G the G -conjugacy class of s.

Ž ^F Ž .. ^F

DEFINITION1.2.1. The Lusztig series EE G , s of G associated with s Žor Ž ..s is the set of irreducible characters of G^F occurring in some

GT^FF T T Ž ^TF Ž ^X.^T.

Ind_G ␥ , where␥ is an element of a usual Lusztig series EE G , s with s⬘gŽ .s.

Ž ^F .

The characters of the Lusztig series EE G , 1 are called unipotent; this

w x w

definition agrees with definitions given in DM2, Section 5 or B, Defini- xŽ w x.

tion 6.4.1 cf. B, Lemma 6.4.2 .

The following lemma follows immediately from the definitions:

LEMMA1.2.2. Let s be a semisimple element of G^UTF*,␥^T be an element

Ž ^TF Ž .^T. ^{␣ T} Ž ^TF Ž^␣Uy1 .^T.

of EE G , s , and ␣gA.Then ␥ gEE G , s . COROLLARY1.2.3.

Irr G^Fs

D

EE

Ž

G^F,Ž .s

.

,

Ž .s

Ž . ^UF

U UTF^U

where s runs o_¨er the set of G -classes of semisimple elements of G . Moreo_¨er, this union is disjoint.

Proof. The equality follows easily from the corresponding fact for G^TF. Let us prove now that the union is disjoint. Let sand t be two semisimple elements of G^UTFU and let␥ be an irreducible character of G^F belonging

Ž ^F Ž .. Ž ^F Ž ..

to both EE G , s and EE G , t . Then by definition there exist irre- ducible characters ␥^T₁ and␥^T₂ of G^TF occurring in the restriction of ␥ to

TF T Ž ^TF Ž .^T. ^T Ž ^TF Ž .^T. ^X Ž . G such that ␥₁gEE G , s⬘ and ␥₂gEE G , t⬘ , where s g s

X Ž . and t g t .

But by Clifford theory there exists ␣gA such that␥^T₂ s^␣␥^T₁. It follows from Lemma 1.2.2 and from the fact that Corollary 1.2.3 holds in G_Uy ^T that

␣ 1 T

Ž . Ž .

t⬘g s⬘ , so tg s .

COROLLARY 1.2.4. Let s be a semisimple element in G^UTF*, and let Ž ^F Ž ..

␥gEE G , s .

Ž .a Let␥^T be an irreducible component of the restriction of␥ to G^TF,

UTF* T Ž ^TF Ž .^T. and let t be a semisimple element of G such that ␥ gEE G , t . Then tgŽ .s.

Ž .b There exists an irreducible component of the restriction of␥ to G^TF Ž ^TF Ž .^T.

belonging to EE G , s .

Ž . Ž .

Proof. a is a reformulation of Corollary 1.2.3, and b is an easy consequence of a and of Lemma 1.2.2.Ž .

1.3. Nice Elements

Let sbe a semisimple element of G^UTF*. The centralizer of sin G^UT is connected and is a Levi subgroup of a parabolic subgroup of G^UT. The image of C_G*Ž .s by the morphism

C_G^UŽ .s ªG^UªA^U

UŽ . ^UT

is denoted by A s. Then the G -conjugacy class of s is stable under

UŽ . ^UTA

UŽs. UŽ . ^UT

A s. If we denote by G the group of fixed points of A s on G , then the G^UT-conjugacy class of s in G^UT meets G^UTAUŽs. in a single

UTA^UŽs. UŽ .

G -conjugacy class because A s acts by permutations on the com- ponents of G^UT. This conjugacy class is F^U-stable and G^UTAUŽs. is con- nected, so there exists an F^U-stable element t in the G^UT-conjugacy class

Ž . UTŽ . Ž .^T

of s centralized by A* s. Moreover,C_G s is connected, so tg s . It

UŽ . ^UŽ .

also implies that A t contains A s . Because they are conjugate under A^U, they are equal.

Ž ^U .

DEFINITION 1.3.1. The element s is said to be nice or G -nice if

UŽ .

A s centralizes s.

The preceding discussion shows that there exists a nice element in every semisimple G^UTFU-conjugacy class. If s is a nice element of G^UTFU and if

U U ␣^UŽ .^T Ž .^{T U U}Ž .

␣ gA is such that s s s , then ␣ gA s. 1.4. The GroupGŽ .s

Until the end of this section, we fix a nice semisimple element s in

UTF^U

Ž . ^UŽ .

G . Let A s be the subgroup of Acorresponding to A s. The group

Ž . Ž .^{T U}

UT U

C_G s sC_G s if an F -stable Levi subgroup of a parabolic subgroup

UT TŽ .

of G . Let G s be an F-stable Levi subgroup of a parabolic subgroup of

T UTŽ . Ž . ^TŽ .

G dual toC_G s; we can assume that A s normalizes G s. We define GŽ .s to be the semidirect product

GŽ .s sG^TŽ .s iA sŽ .. Ž1.4.1.

Ž . ^T

Because A s acts on G by permutations of the components, there exists

T TŽ .

a parabolic subgroup of G that has G s as a Levi subgroup and is stable

Ž . Ž .

under A s . Hence, G s is a Levi subgroup of a parabolic subgroup of G.

Ž .^{T T}Ž . Moreover, G s sG s.

With the semisimple element s is associated a linear character ˆs^T of

TŽ .^F Ž w x. ^UŽ .

G s cf. DM1, Proposition 13.30 . Since s is centralized by A s, the

T Ž . Ž .^F

character ˆs is invariant by A s , so it extends to a character ˆs of G s ,

Ž . Ž .

where sˆ␣ s1 for ␣gA s . 1.5. A Lemma

TŽ . ^TŽ .^F w

Let␥ s be a unipotent character of G s . By B, Theorem 7.3.2 and

x Ž . ^TŽ .

Definition 7.3.3 , there exists a canonical extension ␥˜ s of ␥ s to

TŽ .^F Ž ^TŽ .. Ž ^TŽ .. ^TŽ . Ž .

G s iA s,␥ s , where A s,␥ s is the stabilizer of␥ s in A s .

G^TiAŽs,␥^TŽs..

Ž Ž . .

T T T T

LEMMA1.5.1. ␧G Žs.␧G RG Žs.iAŽs,␥ Žs.. ˜␥ s mˆs is an irreducible char-

TF

Ž ^TŽ .. ^TF

acter of the groupG iA s,␥ s . Its restriction to G is the irreducible

G^T

Ž ^TŽ . ^T. Ž ^TF Ž .^T.

T T T

character␧G Žs.␧G RG Žs. ␥ s mˆs which belongs to EE G , s .

w x Ž . ^TŽ .^F Remark. By B, Theorem 7.3.2 , the unipotent character␥˜ s of G s

Ž ^TŽ ..

iA s,␥ s is a uniform function, that is, a linear combination of generalized Deligne-Lusztig characters. Hence the class function

G^TiAŽs,␥^TŽs.. Ž Ž . .

T T T T

␧G Žs.␧G RG Žs.iAŽs,␥ Žs.. ␥˜ s msˆ is independent of the choice of a

TŽ . Ž ^TŽ ..

parabolic subgroup of G having G s iA s,␥ s as Levi subgroup.

That is why the Lusztig functor is denoted without reference to the parabolic subgroup the notion of a Lusztig functor for disconnectedŽ

w x

reductive groups has been defined in DM2 , and slightly generalized for w x.

the purpose of this article in B .

Proof of Lemma1.5.1. To simplify notation, we can assume that As Ž ^TŽ ..

A s,␥ s . Let

␥˜s␧G^TŽs.␧G^TRGŽs.^G

Ž

␥˜Ž .s mˆs

.

and

␥^Ts␧G^TŽs.␧G^TRGG^TTŽs.

Ž

␥^TŽ .s mˆs^T

.

.

w x ^TF

It follows from DM2, Corollary 2.4 that the restriction of␥˜ to G is

T w x ^T

equal to␥ . Moreover, by LS, Theorem 3.2 ,␥ is irreducible and lies in

Ž ^TF Ž .^T. ^F

E

E G , s . So we need only prove that ␥˜ is a character of G .

Ž . Ž . Ž .

Let P s be a parabolic subgroup of G s that has G s as Levi subgroup, and let U be its unipotent radical. We define

< ^y1

YUs

ggG g F gŽ .gU

4

and

T T< ^y1

YUs

ggG g F gŽ .gU .

4

iŽ .

Let H_c Y_U be the ith cohomology group with compact support with

Ž . F

coefficients in the constant sheaf ⺡_ll where ig⺞. The group G Žrespectively, GŽ .s ^F. acts on Y_U by left respectively, right translation.Ž .

iŽ . ^F Ž .^F

Hence H_c Y_U inherits the structure of a G -module-G s . Let V be an

Ž .^F Ž .

irreducible G s -module affording ␥˜ s as character. Then the virtual G^F-module

i i

y1 H Y m F V

Ž . Ž .

Ý

^c U ⺡_llGŽs.

ig⺞

Ž . ^TF

affords ␥˜ as virtual character. We have similar results for G . We denote by V^T the restriction of V to G^TF.

w Ž .x

By DM1, Theorem 13.25 i there exists j in⺞such that

i T _T T

HcŽYU.m⺡_llG Žs.F V s0

if i/jand such that

j T _T T

HcŽYU.m⺡_llG Žs.F V

Ž w x

is irreducible in DM1 , the statement and the proof of Theorem 13.25 are not entirely correct; a precise value for j is given, and it is not clear that this value is correct. However, the existence of j satisfying the above conditions has been established in a revised version of their book . More-.

Ž .^{j T T} w x

over, y1 s␧_{G Žs.}␧_G . But by DM2, Proof of Proposition 2.3 we have

T

i F i

TF

HcŽYU.s⺡llG m⺡_llG HcŽYU. as a G^F-module. Hence we have

i _{T F} T

HcŽYU.m⺡_llG Žs. V s0 for all i/j. But

T F T

i i

T F F T F

H^cŽY^U.m⺡_llG Žs. V s

ž

HcŽYU.m⺡_llGŽs. ⺡llGŽ .s

/

m_⺡llG Žs. V

F T

i _T

F F

sH^cŽY^U.m^⺡llGŽs.

ž

⺡^llGŽ .s m^{⺡llG Žs.} V

/

i GŽs._T ^F T

F F

sHcŽYU.m⺡_llGŽs. IndG Žs. V . Ž .^{F GŽ}T^s.FF ^T

SinceV is a direct summand of the G s -module Ind_{G Žs.} V , it follows that

i F

HcŽYU.m⺡_llGŽs. Vs0 if i/jand that ␥˜ is the character of the module

j F

H_cŽY_U.m_{⺡ GŽs.} V.

ll

1.6. Clifford Theory

T Ž ^TF Ž .^T. w x

Let ␥ gEE G , s . By LS, Theorem 3.2 there exists a unique

TŽ . ^TŽ .^F

unipotent character␥ s of G s such that

␥^Ts␧G^TŽs.␧G^TR^GG^TTŽs.

Ž

␥^TŽ .s mˆs^T

.

. Ž1.6.1.

Ž ^T. ^{T U}Ž ^T.

Let A ␥ be the stabilizer of ␥ in A. Its dual A ␥ stabilizes the

UTF*

G -conjugacy class of s and hence is contained in A*Ž .s. By duality

Ž ^T. Ž . ^TŽ . Ž ^T.

A␥ is contained in A s . The uniqueness of␥ s implies that A␥ is

Ž ^TŽ .. ^TŽ . Ž .

the stabilizer A s,␥ s of ␥ s in A s .

Ž . ^TŽ . ^TŽ .^F Ž ^T. We denote by␥˜ s the canonical extension of␥ s to G s iA ␥ Žas defined in B, Definition 7.3.3 . We putw x.

␥˜s␧G^TŽs.␧G^TR^GG^TTŽⁱs.i^AŽ␥AŽ␥^T.T.

Ž

␥˜Ž .s mˆs

.

. Ž1.6.2.

TF

Ž ^T. Then, by Lemma 1.5.1, ␥˜ is an irreducible character of G iA ␥ extending ␥^T.

TF

Ž ^T. DEFINITION 1.6.3. The irreducible character ␥˜ of G iA ␥ will be called the canonicalextension of ␥^T.

Ž ^T.

If ␰ is an irreducible character of A ␥ , then by Clifford theory␥˜m␰

TF

Ž ^T. ^GTF^F T Ž .

is an irreducible character of G iA␥ , and IndG iAŽ␥ . ␥˜m␰ is

F Ž ^TF Ž ^T.

an irreducible character of G where ␰ is lifted to G iA␥ in the natural way . Moreover,.

Ind^GG^TFF ␥^Ts

Ý

␰Ž .1 Ind^GG^TFFiAŽ␥^T.Ž˜␥m␰.. Ž1.6.4.

Ž T.

␰gIrrA␥

1.7. Jordan Decomposition

Ž ^F Ž ..

Let␥ be an irreducible character in EE G , s . By Corollary 1.2.4 there

T Ž ^TF Ž .^T.

exists an irreducible character␥ gEE G , s occurring in the restric-

TF T TF

Ž ^T. tion of␥ to G . Let␥˜ be the canonical extension of␥ to G iA ␥ defined in Definition 1.6.3. Then by Clifford theory there exists a unique

Ž ^T.

irreducible character ␰ of A ␥ such that

␥sIndG^GTFF iAŽ␥^T.Ž␥˜m␰..

TŽ . ^TŽ .^F Ž .

Let␥ s be the unipotent character of G s satisfying 1.6.1 , and let

Ž . ^TŽ .^F Ž ^T. Ž Ž ^T.

␥ s be its canonical extension to G s iA ␥ recall that A ␥ is the

˜ TŽ . Ž ..

stabilizer of␥ s in A s . Then

␥Ž .s sInd^GŽG^Ts.Žs.^FFiAŽ␥^T.

Ž

␥˜Ž .s m␰

.

Ž .^F

is an irreducible character of G s and is unipotent by definition. It

w x

follows from B, Propositions 6.3.2 and 6.3.3 that

␥s␧G^TŽs.␧G^TR^GGŽs.

Ž

␥Ž .s mˆs

.

. Ž1.7.1. Remark. The remark following Lemma 1.5.1 shows that the Lusztig functor R^G_GŽs. does not depend on the choice of a parabolic subgroup of G that has GŽ .s as Levi subgroup.

Ž .

THEOREM 1.7.2 Jordan Decomposition . With the abo_¨e notation the map

F F

ⵜG ,s: EE

Ž

G ,Ž .s

.

ªEE

Ž

GŽ .s , 1

.

␥¬␥Ž .s

Ž .

is well-defined and bijecti_¨e.The in_¨erse map is gi_¨en by Formula 1.7.1 . Proof. First we have to prove that ⵜ_G,s is well defined. There is one ambiguity in the construction of ␥Ž .s: in the first step, we chose an

T Ž ^TF Ž .^T.

irreducible character␥ gEE G , s occurring in the restriction of␥ to

TF T Ž ^TF Ž .^T.

G . If ␦ is another element of the Lusztig series EE G , s occurring in the restriction of␥ to G^TF, then there exists ␣gA such that ␦^Ts^␣␥^T.

Ž ^TF Ž .^T. Ž . ^TŽ .

But both lie in EE G , s , so we have ␣gA s . If we construct ␦ s,

˜Ž . Ž . ^TŽ . Ž . Ž .

␦ s, and ␦ s in the same way as␥ s ,␥˜ s , and␥ s, respectively, then

TŽ . ␣ TŽ . Ž . ˜Ž . ^␣ Ž . Ž . ^␣ Ž . Ž .

␦ s s␥ s by uniqueness , so ␦ s s˜␥ s, and so␦ s s␦ s s␦ s because ␣gA sŽ .. Thusⵜ_G,s is well defined.

Ž .

ⵜ_G,s is injective by Formula 1.7.1 and surjective by Lemma 1.5.1, which

Ž . Ž F Ž ..

proves that Formula 1.7.1 always defines an element of EE G , s .

2. UNIFORM FUNCTIONS

w x ^F

In B, Formula 7.3.1 and Theorem 7.3.2 the unipotent characters of G are described as linear combinations of generalized Deligne᎐Lusztig char-

Ž .

acters. It is possible using Formula 1.7.1 to describe all of the irreducible characters of G^F as linear combinations of generalized Deligne᎐Lusztig characters.

2.1. Notation

Let s be a nice semisimple element of G^UTFU.

Ž . ^TŽ . ^TŽ .

We fix an F-stable and A s -stable Borel subgroup B₁ s of G s and

Ž . ^TŽ . ^TŽ .

an F-stable and A s -stable maximal torus T₁ s of B₁ s. We denote by

Ž . Ž ^TŽ .. Ž . Ž ^TŽ ..

W s respectively, W s the Weyl group of G s respectively, G s

TŽ . to T₁ s .

Ž . ^TŽ .

For each ␣gA s , we define T₁ s,␣ to be the semidirect product

TŽ . ² : ^TŽ .^␣ Ž ^TŽ .

T₁ s i ␣ . For each wgW s that is, the subgroup of W s

. Ž .

consisting of elements centralized by ␣ , we denote by T_w s,␣ the

TŽ . ² : w

quasi-maximal torusof G s i ␣ associated withw as in DM2, Propo-

x Ž w

sition 1.40 for the definition of a quasi-maximal torus, cf. B, Definition

x. Ž . Ž Ž .^␣.^T

6.1.3 . T_w s,␣ is defined by the following property: T_w s,␣ is an

TŽ .^{␣ T}Ž .^␣

F-stable maximal torus of G s of type w with respect to T₁ s .

TŽ . Ž .

The groupW s is a product of symmetric groups, and A s and F act

TŽ . Ž ^TŽ . .

onW s by permutations of the components F acts onW s as ␴ . By

w x

the argument used in B, Sect. 7.3 we can associate canonically with each

T TŽ .^F Ž ^T.

irreducible character ␹ of W s and each ␣ in the stabilizer A s,␹

T Ž . ^TŽ .^␣ ² :

of ␹ in A s an irreducible character ␹˜_␣ of W s i ␴ . Ž ^TF Ž .^T.

2.2. Irreducible Characters in EE G , s as Uniform Functions

T TŽ .^F

Let ␹ be an irreducible character of W s . We define

␧ ^T ␧ ^T

T G Žs. G T

T_T G_T G T

R␹ Ž .s sR␹ Ž .s s WTŽ .s wgW

Ý

TŽ .s ␹˜1Žw␴.RTwŽs, 1.Žˆs .. Ž2.2.1.

Ž w x. Ž .

PROPOSITION2.2.2 Lusztig᎐Srinivasan LS, Theorem 3.2 . a For all

T TŽ .^{F T}TŽ . ^TF Ž ^TF Ž .^T.

␹ gIrrW s ,R_␹ s is an irreducible character of G in EE G , s . Ž .b The map

T

T F TF

IrrW Ž .s ªEE

Ž

G ,Ž .s

.

␹^T¬R^T_␹^TŽ .s is bijecti_¨e.

Ž . ^{T T}Ž .^F Ž . ^{␣ T}TŽ .

COROLLARY2.2.3. a If ␹ gIrrW s and␣gA s ,then R_␹ s

T␣ TŽ . sR_␹ s.

Ž .b If ␹^TgIrrW^TŽ .s ^F, then A RŽ ^T_␹TŽ ..s sA s,Ž ␹^T.. 2.3. Canonical Extensions as Uniform Functions

T TŽ .^F

Let ␹ be an irreducible character of W s . We define a function

˜TŽ . ^TF Ž ^T. R_␹ s on G iA s,␹ by

G^TFTFiAŽs,␹^T.˜T

ResG ⭈␣ R␹ Ž .s

␧G^TŽs.␧G^T GT_TFFi²␣: GTi²␣:

s WTŽ .s ␣ wgW

Ý

_TŽ .s_␣˜␹␣Žw␴. ResG ⭈␣ RT^wŽs,␣. Ž .ˆs Ž2.3.1.

Ž ^T. for all ␣gA s,␹ .

˜TŽ . ^TF Ž ^T.

PROPOSITION2.3.2. R_␹ s is an irreducible character of G iA s,␹

TTŽ . Ž .

and is in fact the canonical extension of R_␹ s cf. Definition1.6.3 .

Ž . w

Proof. This follows immediately from Formula 1.6.2 , from B, Theo-

x w x

rem 7.3.2 , and from DM2, Proposition 2.3 .

Ž ^F Ž ..

2.4. Parameterization of EE G , s

Ž . Ž ^T . ^T

We denote by II s the set of pairs ␹ ,␰ where ␹ is an irreducible

TŽ .^F Ž ^T.

character of W s and ␰ is an irreducible character of A s,␹ . The

Ž . Ž . Ž .

group A s acts by conjugation on II s , and we denote by II s the set of

Ž . Ž . Ž ^T . Ž . ^T

orbits of A s in II s. Moreover, if ␹ ,␰ gII s , we denote by ␹ )␰ its orbit under A sŽ ..

T Ž .

For all ␹ )␰gII s, we define

GT T GT^FF T ˜T

R^{␹ )␰}Ž .s sR^{␹ )␰}Ž .s sInd^{G iAŽs,␹ .}

ž

R^␹ Ž .s m␰

/

. Ž2.4.1.

Ž . T Ž .

It follows from Corollary 2.2.2 a that R_{␹ )␰} s only depends on the orbit

Ž ^T . Ž .

of ␹ ,␰ under A s . Moreover, it follows from Clifford theory and from Corollary 2.2.2 b that we haveŽ .

LEMMA2.4.2. The map I F

IŽ .s ªEE

Ž

G ,Ž .s

.

␹^T)␰¬R_␹^T_)␰Ž .s is bijecti_¨e.

w x ^T

By B, Proposition 2.3.1 , ␹ has a canonical extension ␹˜ to the

TŽ . Ž ^T.

semidirect productW s iA s,␹ . By Clifford theory again we have LEMMA2.4.3. The map

I F

IŽ .s ªIrrW sŽ .

␹^T)␰¬Ind^WŽs.W^TŽs.^FFiAŽs,␹^T.Ž␹˜m␰. is bijecti_¨e.

Lemmas 2.4.2 and 2.4.3 imply the following:

THEOREM2.4.4. There is a well-defined bijection

F F

IrrW sŽ . ªEE

Ž

G ,Ž .s

.

␹¬R_␹Ž .s .

GŽ .

Remark. If necessary, we will write R_␹ s for the irreducible character

Ž . ^F Ž .

R_␹ s of G . By applying Theorem 2.4.4 in the case where GsG s and ss1, we obtain a bijection,

F F

IrrW sŽ . ªEE

Ž

GŽ .s , 1

.

␹¬R^GŽ_␹ ^s.Ž .1 ,

and it is easy to check that the following diagram is commutative:

IrrW sŽ .F

; 6

;

6 F

F 6

EE G s , 1 . EE

Ž

G ,Ž .s

.

_ⵜG,s

Ž

Ž .

.

2.4.5

Ž .

2.5. Induction from a Particular Subgroup of G

Let G⬘be a subgroup of G containing G^T. It is F-stable because F acts trivially on A. There exists a subgroup A⬘of A such that

G⬘sG^TiA⬘.

If we construct G^XU in the way we construct G, then G^XU may be identified

U XŽ .

with a subgroup of G . We can also construct G s so that it is contained

Ž . Ž . Ž . ^TŽ .

in G s, and we denote by W⬘ s the Weyl group of G⬘ s relative to T₁ s

Ž . Ž .

so that W⬘ s is a subgroup of W s .

Ž .^F PROPOSITION 2.5.1. Let ␹ ⬘ be an irreducible character of W⬘ s . Sup- pose

Ind^WŽs.W⬘Žs.^FF␹ ⬘s

Ý

n␹␹.

Ž .F

␹gIrrW s

Then

Ind^GGF⬘^FR^G␹ ⬘^⬘Ž .s s

Ý

n␹R^G␹Ž .s .

Ž .F

␹gIrrW s

3. LUSZTIG FUNCTORS

HYPOTHESIS. Throughout this section, and only in this section, A will be assumed abelian.

3.1. Notation

Let L be an F-stable Levi subgroup of a parabolic subgroup P of G. Let A_L be the image of L through the composite morphism

LªGªGrG^TªA

ŽA_L is a subgroup of A.. Because A is abelian, we can use the same

w x ^U

argument as in B, 7.6 to assume that L contains A_L. Let A_L be the image of A_L under the anti-isomorphism AªA*.

Let L^TU be an F^U-stable Levi subgroup of a parabolic subgroup of G^UT that is a dual of L^T. We can choose L^TU to be A^U_L-stable, and we define

L^UsL^TUiA^U_L.

then L^U is an F^U-stable Levi subgroup of a parabolic subgroup of G^U and L^UTsL^TU.

3.2. Jordan Decomposition and Lusztig Functors

Let s be a semisimple element in L^UTF*. We may assume that sis nice

U Ž .

in G . Then the subgroup L s of L following the construction of Section

Ž . Ž .^F

1.4 can be chosen as a subgroup of G s. The linear character of L s associated with s as defined in Section 1.4 is then the restriction of ˆs to

Ž .^F

L s . It results from this remark and from the transitivity of Lusztig

Ž w x.

induction functors cf. B, Proposition 6.3.3 that the following diagram is commutative:

ⵜL,s F

F _U 6

E F*

E

Ž

L ,Ž .s ^L

.

EE

Ž

LŽ .s , 1

.

6

G ␧T ␧ T RGŽs.

T T LŽs. GŽs. LŽs.

␧ ␧_{L G} R_L

6 F

F 6

F* EE G s , 1

EE

Ž

G ,Ž .s ^G*

.

_ⵜG,s

Ž

Ž .

.

3.2.1

Ž .

GŽs. w x

The description of the functor R_LŽs. in B, Theorem 7.6.1 thus provides

G Ž .

a description of the functor R_L via the commutative diagram 3.2.1 .

4. SHINTANI DESCENT IN THE GENERAL LINEAR GROUP

In this section, we explain the link between the theory of irreducible characters of G^Fand the theory of Shintani descent for the general linear group. For this purpose, we need to consider a particular case:

HYPOTHESIS AND NOTATIONS. Throughout this section, we assume that rs1.We will denote dsd and n₁ sn for simplicity.₁ We also assume that

Ž .

␴s 1, . . . ,d and that A is generated by␴. 4.1. The GroupG^F

We denote by G the general linear group GL , and we endow it with_{1 n} the split Frobenius endomorphism:

F₀: G₁ªG₁ g¬g^Žq..

We denote by␾₀ the automorphism of G^F₁^0d induced by F₀. Then the map

␪: G^F₁^0dªG^TF

g¬

Ž

g,F0Žg., . . . ,F0^dy1Žg.

.

is an isomorphism of groups and the following diagram is commutative:

␪

d TF

F₀ 6

G₁ G

6 ^y1

␾0 ␴

6 _d

TF

F₀ 6

G . G₁

␪

This implies that ␪ can be extended to an isomorphism denoted by

˜ ^F0d ² : ^F

␪: G₁ i ␾₀ ªG g␾0^k¬␪Žg.␴^yk

for all ggG^F₁^0d and kg⺪. 4.2. Shintani Descent

Let ggG1^F0. By Lang’s theorem, there exists xgG such that1 gs

y1 dŽ . Ž . ^{y1 F0d}

x F₀ x . Then g⬘sF x x₀ belongs to G , and the map that sends₁ the conjugacy class of g in G₁^F0 to the ␾₀-conjugacy class of g⬘in G^F₁^0d is well-defined and is bijective. We denote it by

N^F₀rF₀^d: Cl G

Ž

1^F0

.

ªH¹

Ž

␾0, G1^F0d

.

,

1Ž ^F0d. ^F0d

where H ␾₀, G₁ denotes the set of ␾₀-conjugacy classes of G₁ and

Ž ^F0. ^F0

Cl G₁ denotes the set of conjugacy classes of G . If we denote by₁

Ž ^F0d . Ž Ž ^F0.. ^F0d

C

C G1 ⭈␾0 respectively,CC G1 the space of class functions on G1 ⭈␾0 F₀^d² : Ž obtained by restrictions from class functions on the group G₁ ␾₀ re-

F₀

. d

spectively, G₁ , then N_{F rF} induces an isomorphism

0 0

Sh^F₀^drF₀:CC

Ž

G^F1^0d⭈␾0

.

ªCC

Ž

G1^F0

.

, called the Shintani descent from F₀^d to F₀.

We recall the following theorem:

Ž . ^F0d

THEOREM 4.2.1 Shintani . Let ␥₁ be an irreducible character of G₁

F₀^d ² : stable under␾0.Then there exists an extension␥˜1 of␥1 toG1 i ␾0 such that ShF₀^drF₀˜␥1 is, up to a sign, an irreducible character of G1^F0.