Shintani Descent for Special Linear Groups

199, 175]228 Ž1998. JA977174

JOURNAL OF ALGEBRA ARTICLE NO.

Shintani Descent for Special Linear Groups Toshiaki Shoji Department of Mathematics, Science Uni¨ ersity of Tokyo, Noda, Chiba, 278, Japan Communicated by Michel Broue´ Received December 10, 1996

In this paper, the Shintani descent of irreducible characters of finite special linear groups is discussed. We parametrize irreducible characters of SL nŽFq . by making use of Žmodified. generalized Gelfand]Graev characters. Shintani descent of irreducible characters of SL nŽFq m . is determined by investigating the Shintani descent of generalized Gelfand]Graev characters. Our results hold for arbitrary p and q. Q 1998 Academic Press

INTRODUCTION In this paper, we consider the Shintani descent of irreducible characters of special linear groups G F s SL nŽFq . over finite fields Fq with chŽFq . s p. In wAx, under the assumption that q ' 1 mod n, T. Asai parametrized the irreducible characters of G F appearing in the decomposition of regular characters of GLnŽFq . in terms of ‘‘modified’’ Gelfand]Graev characters. Using this parametrization, he studied the twisting operators on G F. But his arguments contained some errors. Recently in his thesis wBx, C. Bonnafe ´ described the decomposition of the twisted induction R GP for G F. He followed a similar strategy as Asai, but filled the gaps by making use of the results due to F. Digne, G. Lehrer, and J. Michel wDLMx concerning the Gelfand]Graev characters of G F Žwhich in turn depends on the result of G. Lusztig wL4x on generalized Gelfand]Graev characters ., and of a certain Mackey formula for R GP . In order to apply these two subjects, he had to assume that p and q are large enough. Our approach in this paper is somewhat different from theirs. It relies on the extensive use of modified generalized Gelfand]Graev characters of G F. We parametrize a wider class of irreducible characters of G F than those obtained from regular characters as above in terms of generalized 175 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

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Gelfand]Graev characters. On the other hand, we describe the Shintani m descent of generalized Gelfand]Graev characters of G F for sufficiently large m. Combining this with some results from Shintani descent identities, we are able to describe the Shintani descent of irreducible characters m of G F . Our results are valid for arbitrary p and q. In wK3x, N. Kawanaka proposed a strategy to use modified generalized Gelfand]Graev characters to parametrize irreducible characters of reductive groups in general. Our result is in some sense a realization of his idea in the case of SL n . He also pointed out the importance of generalized Gelfand]Graev characters in the theory of Shintani descent. The paper is organized as follows. In Section 1, we give a general result concerning the Shintani descent for certain induced characters, which will be applied in later sections to the case of generalized Gelfand]Graev characters. Since our result in Section 1 is of independent interest in the theory of Shintani descent, we give it in a general form for arbitrary reductive groups. Lusztig’s non-abelian Fourier transform appears quite naturally in our context. In Section 2, we give a parametrization of a certain class of irreducible characters of G F in terms of generalized Gelfand]Graev characters, which is a generalization of Asai’s results mentioned above. After getting some results from Shintani descent identities in Section 3, we parametrize irreducible characters of G F, and then determine the Shintani descent for them in Section 4. Through Shintani descent identities, our results give some description of the decomposition of the twisted induction R GP . A part of this work was done during the author’s stay in the Department of Mathematics, University of Sydney, in the Spring of 1996. He is very grateful for its hospitality. Also he is indebted to Bonnafe ´ for valuable discussions concerning the parametrization of irreducible characters of SL n . He thanks the referee who suggested the improvement of the results in Section 1 in the first version of this paper. Some Notations. Let A be a finite group and F be an automorphism on A. We denote by Ar;F the set of F-twisted classes in A, where for x, y g A, x is F-twisted conjugate to y Ždenoted as x ;F y . if there exists z g A such that y s zy1 xF Ž z .. If F is identity we denote it simply as Ar; , which is the set of conjugacy classes in A. For a finite group A, we denote by A n the set of irreducible characters of A on Q l Žor by abuse of the notation, the set of irreducible representations of A up to isomorphisms.. For a finite set X, we denote by C Ž X . the Q l-space of all the functions on X.

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1. SHINTANI DESCENT FOR CERTAIN GROUPS 1.1. Let H be a connected algebraic group over a finite field Fq with Frobenius map F. We prepare some notations from the theory of Shintani descent. Let F9 and F0 be two Frobenius maps on H such that F9F0 s F0 F9. Then F9 acts on the F0-fixed point subgroup G F 0 of G, and similarly for F0. One can define a bijection NF 9r F 0 : H F 9r;F 0 ª H F 0 r;F 9y1 as follows. Take x g H F 9 and write x s ay1 F0 Ž a . with a g H. Then ˆ x s F9Ž a . ay1 lies in H F 0 and the correspondence x ¬ ˆ x induces a well-defined bijection NF 9r F 0 . The map NF 9r F 0 is called a norm map from H F 9 to H F 0 . Now taking the transpose of NF 9r F 0 , one gets a linear isomorphism NFU9r F 0 : C Ž H F 0 r;F 9y1 . ª C Ž H F 9r;F 0 .. The typical case is that F0 s F, F9 s F m for some positive integer m. In this case m NF m r F is a map from H F r;F to H Fr; . We note that in this case ˆ x as above is given by

ˆx s F m Ž a . ay1 s a xF Ž x . F 2 Ž x . ??? F my 1 Ž x . ay1 .

Ž 1.1.1.

We define a linear map Sh F m r F by F Sh F m r F s NFUmy1 r;F . ª C Ž H Fr; . rF : CŽ H m

m

and call it the Shintani descent map from H F to H F. m Let s be the restriction of the map F on H F . We consider the m m semidirect product H˜ F s H F ² s :, where ² s : is the cyclic group of m m order m generated by s , and s acts on H F via F. The coset H F s of m m m H F in H˜ F is stable under the conjugation of H F , and we have a m m bijection H F r;F , H F sr; via the map x ¬ x s . Now each F-stable m m character r of H F can be extended to a character of H˜ F , in m distinct m way. Let r˜ be one of them. Then, the restriction r˜ < H F ms of r˜ to H F s m m gives rise to an element of C Ž H F sr;. , C Ž H F r;F .. Note that r˜ < H F ms does not depend, up to an mth root of unity multiple, on the choice of the extension r˜ of r . 1.2. We consider the case where H is an abelian group. Let Ž H F 9 .F 0 be the quotient of H F 9 by the subgroup generated by all the elements xy1 F0 Ž x . for x g H F 9, which coincides with the largest quotient of H F 9 on which F0 acts trivially. Then the set H F 9r;F 0 may be identified with Ž H F 9 .F 0 . The correspondence x s ay1 F0 Ž a . ¬ ˆ x s F9Ž a . ay1 induces a F9 F 0 surjective homomorphism from H to Ž H .F 9 which we denote by n F 0 r F 9. The map n F 0 r F 9 induces an isomorphism NF 0 r F 9: Ž H F 9 .F 0 , Ž H F 0 .F 9. In the case where F9 s F m , F0 s F, in view of Ž1.1.1. the map m n F m r F : H F ª Ž H F .F m s H F is nothing but the usual norm map x ¬ ˆ xs my xF Ž x . ??? F 1 Ž x ..

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Let m be a positive integer large enough so that F m s Ž F9. r , F m s Ž F0 . s m for some positive integers r, s. Let n F m r F 9: H F ª H F 9, n F 0 r F 9: H F 0 ª m Ž H F 9 .F 0 be the maps defined as above. We define n F m r F 0 : H F ª Ž H F 0 .F 9 F0 as the composite of n F m r F 0 and the natural surjection H ª Ž H F 0 .F 9. Then the following equality is easily verified Žcf. wS4, Ž2.3.1.x.: n F m r F 0 s n F 9r F 0 ( n F m r F 9 .

Ž 1.2.1.

1.3. In the remainder of this section, we assume that H is a semidirect product of F-stable connected subgroups L and U, where U is a unipotent normal subgroup of H. We shall consider the Shintani descent of certain m characters of H F . The simplest non-trivial example of such a group is that H is a two dimensional solvable group with L , G m and U , G a , which was already treated in wS4, 5x. The result in this section is in some sense a generalization of it. 1.4. Let k be an algebraic closure of Fq . We denote by u the Lie algebra of U. u has a natural Fq-structure whose Frobenius map is again denoted by F. We pose the following assumption. Ž1.4.1. There exists a bijective Fq-morphism log: U ª u, compatible with the action of L, satisfying the property logŽ u¨ . g logŽ u. q logŽ ¨ . q u9 for any u, ¨ g U, where u9 is an L-stable subalgebra of u. Let l: u ª k be an F-stable linear map which vanishes on u9. Then by Ž1.4.1., the map l (log: U ª k is a homomorphism of algebraic groups. Let us fix a non-trivial additive character c : Fq ª QUl , once and for all. Then the function L s c ( l (log: U F ª QUl turns out to be a linear character of U F . Now l may be regarded as an element in Ž uru9.*, the dual space of uru9 over k. The adjoint action of L on u induces an action of L on Ž uru9.*. We denote by ZLŽ l. the stabilizer of l in L, which is an F-stable closed subgroup of L. Put A s ZLŽ l.rZL0 Ž l.. Hence A is a finite group and F acts naturally on it. Let Ol be the L-orbit of l in Ž uru9.*. Then Ol is F-stable, and the set of LF -orbits contained in OlF is in one to one correspondence with the set Ar;F of F-twisted conjugacy classes in A. The correspondence is explicitly given as follows. For each c g A choose a representative ˙ c g ZLŽ l.. We find a c g L such that Ž . ay1 F a s c, and define l : u ª k by l c s l (Ad ay1 ˙ c c c c , where Ad denotes the adjoint action of L on u. Then l c g OlF, and the correspondence c ¬ l c induces the well-defined bijection. For each c g A, we define a function L c : U F ª QUl by L c s c ( l c (log. Then up to LF-conjugacy, L c depends only on c, and not on ˙ c nor a c . Also it is easily verified that Ž1.4.2. L c and L c9 are LF -conjugate exactly when c and c9 are F-twisted conjugate in A.

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Note that for each c g A, ZLŽ l c . F is isomorphic to ZLŽ l. c˙F under the y1 map x ¬ ay1 induces an isomorphism c x a c . It follows that ad a c ZLŽ l c . rZL0 Ž l c . , ZLŽ l . ˙ rZL0 Ž l . ˙ , Ac F . F

F

cF

cF

Ž 1.4.3.

Let j be an irreducible character of Ac F . Under the above isomorphisms, j induces a character of ZLŽ l c . F which is trivial on ZL0 Ž l c . F . We denote this character of ZLŽ l c . F by j h. We now consider a subgroup ZLŽ l c . F U F of H F . The character j h gives rise to a character of ZLŽ l c . F U F under the natural surjection ZLŽ l c . F U F ª ZLŽ l c . F , which we denote also by j h. On the other hand, it is easy to see that ZLŽ l c . F coincides with the stabilizer ZLF Ž L c . of L c in LF Žunder the conjugate action of LF on the characters of U F .. Hence the linear character L c may be extended to a linear ˜ c of ZLŽ l c . F U F by L ˜ c Ž xu. s L c Ž u. Ž x g ZLŽ l c . F, u g U F ., character L which we call the trivial extension of L c . For each c g Ar;F , j g Aˆc F , we define a character rŽ c, j . of H F by

˜c . rŽ c , j . s Ind ZHL Ž l c . F U F j h m L F

ž

/

We have the following lemma, which is an easy consequence of Clifford theory and so the proof is omitted. LEMMA 1.5. Let M s Ž c, j .< c g Ar;F , j g Aˆc F 4 . Then for each Ž c, j . g M , rŽ c, j . is an irreducible character of H F . Moreo¨ er, all the rŽ c, j . are distinct. 1.6. For a positive integer m, let cm : Fq m ª QUl be the additive character of Fq m given by cm s c (Tr Fq m rFq. We shall construct similar objects as m rŽ c, j . for H F which are F-stable. From now on, we pose the following assumption on m. Ž1.6.1. m is large enough so that Ž aF . m acts trivially on A for any a g A. Let us take c g A such that the class of c g A is F-stable. Then there exists a g A such that c is aF-stable. We choose a representative a ˙g m ZLŽ l. F . Then we choose a representative ˙ c g ZLŽ l. a˙F , and find bc g L mŽ such that by1 bc . s ˙ c. We define lŽcm. : u ª k by lŽcm. s l (Ad by1 c F c , m and define a linear character LŽcm. : U F ª QUl by LŽcm. s cm ( lŽcm. (log. m Note that LŽ1m. coincides with LŽ m. up to LF -conjugate Žhere c s 1 Ž m. denotes the unit element in A, and L s cm ( l (log.. We also note that

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Ž1.6.2. Let ˆ . g LF . Then LŽcm. is ˆ c s bc aF cF-stable. ˙ Ž by1 c In fact, we have m

lŽcm. ( Ž aF ˙ .

y1

s l (Ad by1 ˙ . c ( Ž aF

y1

s Fy1 ( l (Ad Ž aF ˙ Ž by1 ˙y1 . c .a s Fy1 ( lŽcm. (AdŽ ˆ˙ cay1 . , and so LŽcm. ( Ž aF ˙ .

y1

s cm ( Ž aF ˙ .

y1

( lŽcm. (Ad Ž ˆ˙ cay1 . (log s LŽcm. (ad Ž ˆ˙ cay1 . ,

since AdŽ ˆ˙ cay1 .(log s log(adŽ ˆ˙ cay1 . and cm ( Fy1 s cm . Statement Ž1.6.2. follows from this. m We consider the structure of ZLŽ lŽcm. . F . The map ad by1 gives an c isomorphism from ZLŽ lŽcm. . to ZLŽ l.. By this isomorphism, F m Žresp. ˆ cF . is transferred to ˙ cF m Žresp. aF ˙ .. Since ˙c g ZLŽ l. a˙F, ˙cF m and aF ˙ stabilize ZLŽ l. and commute each other. It follows that ˆ cF acts on ZLŽ lŽcm. ., m commuting with F . So, by Ž1.6.1. we have an isomorphism ZL Ž lŽcm. .

Fm

rZL0 Ž lŽcm. .

Fm

cF m

cF m

, ZLŽ l . ˙ rZL0 Ž l . ˙

, Ac ,

Ž 1.6.3.

m

and the action of ˆ cF on ZLŽ lŽcm. . F induces an action of aF on Ac. Let j be an aF-stable irreducible character of Ac. It follows from m Ž1.6.3. that j may be regarded as a ˆ cF-stable character of ZLŽ lŽcm. . F m which is trivial on ZL0 Ž lŽcm. . F . We denote this character again by j h. We m define, as in 1.4, an irreducible character rŽŽc,m.j . of H F by

h ˜Ž m. , H m m j m L rŽŽcm. , j . s Ind Z L Ž lŽcm . . F U F c Fm

ž

/

m

m

˜Žcm. is the extension of LŽcm. to ZLŽ lŽcm. . F U F as in Subsection 1.4. where L ˜Žcm. are both ˆcF-stable, that rŽŽc,m.j . is F-stable. Note, since j h and L Ž m. m. Ž . Žg g . Moreover, we have rŽ c, j . s rŽŽc9, j 9. if c9, j 9 s c, j for some g g A. F Let us denote by Ž Ar;. the set of F-stable conjugacy classes in A. For each c g Ž Ar;. F we pick up its representative in A, and denote it also by c. Then c g A a F for some a g A, and we define Aˆcex as the set of aF-stable irreducible characters of Ac. Note that Aˆcex depends only on c. Let A² F : be the semidirect product of A with the cyclic group ² F : of order m, where we use the same symbol F to denote the induced automorphism on A by the Frobenius map. Let Ac ² aF : be the subgroup of A² F :. Then Aˆcex is nothing but the set of irreducible characters of Ac which are extendable to Ac ² aF :. We put M s  Ž c, j . < c g Ž Ar;. , j g Aˆcex 4 . F

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By Lemma 1.5, we see that all the rŽŽc,m.j . are distinct for Ž c, j . g M . Summarizing the above discussion, we have the following. LEMMA 1.7. Let M be as abo¨ e. Then for each Ž c, j . g M , rŽŽc,m.j . is an m F-stable irreducible character of H F . Moreo¨ er, all the rŽŽc,m.j . are distinct. m

m

1.8. We shall construct an extension r˜ŽŽc,m.j . of rŽŽc,m.j . to H˜ F s H F ² s :. m m For each c g Aa F , take ˆ c g LF as in Ž1.6.2., and set ˆ c0 s Ž ˆ c s . m g LF . m m We also put Mc s ZLŽ lŽcm. . F and Mc0 s ZL0 Ž lŽcm. . F . We note that ˆ c0 g Mc . In fact, since LŽcm. is ˆ cF-stable by Ž1.6.2., LŽcm. is stable by Ž ˆ cF . m s ˆc0 F m , and so stable by ˆc0 . Since Mc s ZLF m Ž LŽcm. ., we get ˆc0 g Mc . Since Ž . m and F m act trivially on Ac, ˆcF is transferred to aF ˙ by ad by1 c , and aF we see that ˆ c 0 commutes with any element in Mc modulo Mc0 . m m We now consider the subgroup Mc U F ² ˆ c s : of H F ² s : generated by m Mc U F and ˆ c s . Since j h g Mcn is ˆ c s-stable, and since Ž ˆ cs . m s ˆ c 0 g Mc , h h ˜ j may be extended to an irreducible character j of Mc² ˆ c s : in m distinct way. Since the image of ˆ c 0 in McrMc0 lies in its center, ˆ c 0 acts as a scalar multiplication on the representation space of j h. The scalar is given by j hŽ ˆ c 0 .rj Ž1.. We choose an mth root of j hŽ ˆ c 0 .rj Ž1. and denote it by c ˆ mŽ c, j . . Now j g A ex may be extended to an irreducible character j˜ on Ac ² aF : in an m distinct way. We can define a unique bijection between the set of j˜h on Mc² ˆ c s : and the set of j˜ on Ac ² aF : in such a way that h ˜ ˜ Ž . Ž . j ˆ c s s mŽ c, j . j aF . We fix an extension j˜ of j to Ac ² aF :, and let j˜h be the corresponding m extension of j h to Mc² ˆ c s :. Since Mc U F ² ˆ c s : is the semidirect product m m F h ˜ ² : of Mc ˆ c s and U , j may be regarded as a character of Mc U F ² ˆ c s :. Ž m. Ž m. ˜ ˜ On the other hand, let L c be as in Subsection 1.6. Since L c is m ˆc s-stable, it is further extended to a linear character of Mc U F ² ˆc s : by ˜Žcm. Ž ˆc s . s 1. We denote this extension by the same symbol. We consider L ˜Žcm. of Mc U F m² ˆc s : which is an extension an irreducible character j˜h m L m ˜Žcm. on Mc U F , and let of j h m L

˜h ˜Ž m. . m r˜ŽŽcm. , j . s Ind M c U F ² cˆs : j m L c m

H F ²s :

Then rŽŽc,m.j . gives rise to an extension Ž m. function my1 Ž c, j . rŽ c, j . depends only on

˜

˜

ž

rŽŽc,m.j .

/

m

of to H F ² s :. Note that the the choice of Ž c, j˜..

1.9. For each pair Ž c 0 , j 0 . g M , choose a0 g A such that c0 g A a 0 F . We fix an extension j˜0 to Ac 0 ² a0 F :. We shall define a pairing  , 4 : M = M ª Q l as follows. For Ž c 0 , j 0 . g M , Ž c1 , j 1 . g M , put

 Ž c 0 , j 0 . , Ž c1 , j 1 . 4 s < ZA Ž c 0 .
Ý ggA gc 0 gy1 gZ AŽ c 1 F .

j˜0 Ž gy1 c1 Fg . j 1 Ž gc0 gy1 . .

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TOSHIAKI SHOJI

Note that gy1 c1 Fg g Ac 0 ² a0 F : since Ac 0 ² a0 F : s Ž A² F :. c 0 . Also note that this pairing is not the same as the one defined in Lusztig wL2x. Let  , 4 # be the pairing given in wL2x. Then the relationship between  , 4 and  , 4 # is as follows. For x s Ž c 0 , j 0 . g M put x* s Ž cy1 . 0 , j 0 g M . Then ) gives an involution on M , and we have  x, y4 # s  x*, y4 for x g M , y g M . For a pair Ž c1 , j 1 . g M and for an integer m satisfying Ž1.6.1., we define Ž .ym is a central element in a root of unity lŽŽ m. c 1 , j 1 . as follows. Since c1 F c1 F A , it acts on the representation space of j 1 by a scalar multiplication by ŽŽ .ym .rj 1Ž1.. We also consider j 1ŽŽ c1 F .ym .rj 1Ž1.. We put lŽŽ m. c 1 , j 1 . s j 1 c1 F the number mŽ c 0 , j 0 . given in Subsection 1.8. The following theorem describes the Shintani descent of r˜ŽŽcm. . 0, j 0. THEOREM 1.10. Assume that m satisfies Ž1.6.1.. Then for each x s Ž c 0 , j 0 . g M , we ha¨ e Sh F m r F Ž my1 ˜xŽ m. < H F ms . s x r

Ý  x, y 4 lŽym.r y . yg M

1.11. The remaining part of this section is devoted to the proof of Theorem 1.10. First we prepare some lemmas. Let F9 be a Frobenius map ˜F 9 according to the on H stabilizing L. We choose either d g LF 9, or d g L m case where F9 s F or F9 s F . Let K be an F9-stable subgroup of H normalized by d. We consider the sets K F 9dr; and H F 9dr; Žthe symbol ; means the conjugation under K F 9 or H F 9, respectively.. We define an induction F9

d F9 F9 Ind s Ind H K F 9 d : C Ž K dr; . ª C Ž H dr; .

by

Ž Ind f . Ž xd . s < K F 9
f Ž gy1 xdg . .

Ý F9

ggH gy1 xdggK F 9d

Note that this definition of the induction is similar to the one given in wS4, 1.13x, but not the same. In wS4, 1.13x some additional scalar factor is multiplied for the above one. m .. We put Now take dˆg LF , and let b d g L be such that dˆs b d F Ž by1 d F m Fm dˆF ˆ . Ž . d s F m Ž by1 b g L . Then d s s d g L l L , where d0 s d d 0 y1 ˆ Ž dˆ. ??? F my 1 Ž dˆ. s bd F m Ž by1 . dF , and d s b d b . One sees easily from d d 0 d ˆ this that ad b d induces isomorphisms H Fdr;, H d Fd 0r; , U Fdr;, ˆ U d Fd 0r; . Then by simple computations, we obtain the following commutative diagrams Ždiagram Ž1.11.2. is a special case of wS4, Lemma 1.4x;

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SHINTANI DESCENT

Ž1.11.3. was used also in the proof of Lemma 5.12 in wS4x.: Žad b d .*

C Ž H Fdr;. 6

6

ˆ

C Ž H d Fd 0r;. 6 Ind

Ž1.11.1.

Ind Žad b d .*

C ŽU Fdr;.,

6

ˆ

C ŽU d Fd 0r;. m

m

C Ž H F dˆsr;. 6

C Ž H F sr;. 6 Ž1.11.2.

NFUm r F

NFUm r dˆF Žad b d .*

NFUm r dˆF

ˆ

m

C Ž H F dˆsr;. 6

6

C Ž H d Fd 0r;. 6

C Ž H Fdr;.,

6

ˆ C Ž H d Fd 0r;.

Ind

Ž1.11.3.

Ind NFUm r dˆF

m C ŽU F dˆsr;..

6

ˆ C ŽU d Fd 0r;.

m

m

1.12. Take c g A a F and let ˆ c g LF , Mc s ZLŽ lŽcm. . F , etc., be as before. For an element z g Mc , we put dˆs zc. d, d 0 be as in Subsection ˆ Let m ˆ 1.11. We consider the norm map NF m r dˆF : U F r;dˆF ª U d F r;F y m . Since m ˆ F the former Žresp. the latter . is isomorphic to U dˆsr; Žresp. U d Fd 0r;., we have an isomorphism m ˆ NFUm r dˆF : C Ž U d Fd 0r; . ª C Ž U F dˆsr; . .

Combining with the isomorphism Žad b d .* in Ž1.11.1., we get 1 F mˆ F m Ž ad b d . *( NFUy r dˆF : C Ž U d sr; . ª C Ž U dr; . .

We have the following lemma. LEMMA 1.13.

˜Žcm. g ŽU F ² dˆs :. n be as before. Then Let L m

1 ˜Ž m. ˜ m Ž ad b d . *( NFUy r dˆF Ž L c . s L c 1 ,

for some c1 g A such that c1 F commutes with c in A² F :. Here c1 is determined as the image of by1 ˙ g ZLŽ l. to A under the isomorphism in c z bc a ˜ c1 is the tri¨ ial extension of L c1 g UˆF to U F² d :. Ž1.6.3.. L Proof. The proof of Ž1.6.2. shows that lŽcm. ( ˆ cF s F ( lŽcm.. Since z g Ž m. . F m Ž m. Ž m. ˆ Ž Mc s Z L l c , we see that l c ( dF s F ( l c . This implies that lŽcm. ˆ induces a map from u d F to Fq . We now define a linear character

184

TOSHIAKI SHOJI

ˆ ˆ ˆ ˆ . m s d 0 F m , we have LŽcd. : U d F ª QUl by LŽcd. s c ( lŽcm. (log. Since Ž dF Ž m. m m Ž m. Ž m. m l c ( d 0 F s F ( l c s l c ( F . It follows that d 0 g Mc , and so LŽcd.ˆ is d 0-stable. First we show that 1 ˜Ž m. ˜Ž d. m NFUy r dˆF Ž L c . s L c ,

ˆ

˜Ž d.ˆ

ˆ Ž d.

Ž 1.13.1.

dˆF

where L c is the trivial extension of L c to U ² d 0 :. ŽNote, since d 0 g L, ˆ ˆ that U d F² d 0 : is the semidirect product of U d F with ² d 0 :.. Let U be the quotient of U by Uder , where Uder is the derived subgroup m of U. Then one can consider the norm map NF m r dˆF : U F dˆsr;ª ˆ U d Fd 0r; , which is compatible with the original map NF m r dˆF with respect m m ˆ to the natural surjections U F dˆsr;ª U F dˆsr; , U d Fd 0r; dˆF Ž m. ª U d 0r; . Recall that w s l c (log is a homomorphism from U to k. Hence it factors through a homomorphism w : U ª k. It follows that LŽcm. m Žresp. LŽcd.ˆ . factors through a linear character U F ª QUl Žresp. U dˆF ª QUl .. We use the same notation to denote these maps also. In order to prove Ž1.13.1., it is enough to show the corresponding statement for U. Let ˆ i.e., F m s s Ž dF ˆ . m s. s be a positive integer such that F m s is a power of dF, ms Since U is abelian, one can define homomorphisms such as n F m s r F m : U F m ms ª U F , given in 1.2. Let LŽcm s. be a homomorphism from U F ª QUl defined by LŽcm s. s cm s ( w . Then we have nUF m s r dˆF Ž LŽcd. . s LŽcm s. ,

nUF m s r F m Ž LŽcm. . s LŽcm s. .

ˆ

We show the first equality. For x g U nUF m s r dˆF

Ž

ˆ LŽcd.

.Ž x. s

ˆ LŽcd.

Ž nF

ms

r dˆF

F ms

,

Ž x. .

ˆ . Ž x . ??? Ž dF ˆ . s c ( w x Ž dF

ž

Ž 1.13.2.

m sy1

Ž x. /

ˆ Ž x . q ??? qw ( Ž dF ˆ . s c w Ž x . q w ( dF

ž

m sy1

Ž x. / .

ˆ s F ( w , the last sum is equal to But since w ( dF c Ž w Ž x . q F Ž w Ž x . . q ??? qF m sy1 Ž w Ž x . . . s cm s ( w Ž x . s LŽcm s. Ž x . . The second equality is shown similarly. ˜Žcd.ˆ . s LŽcm.. Here we Now Ž1.13.2. implies, by Ž1.2.1., that nUF m r dˆF Ž L ˜Žcd.ˆ as a function on ŽU dˆF .d 0 , the quotient of U dˆF by the subgroup regard L ˆ generated by xy1 d 0 xdy1 for x g U d F . Since nUF m r dˆF induces the isomor0 U phism NF m r dˆF , we get Ž1.13.1.. ˜Žcd.ˆ . s L ˜ c1. Since d 0 is sent to d by Next we shall show that Žad b d .*Ž L ˆ y1 Ž d. Ž . Ž ad b d , it is enough to show that ad b d * L c . s L c1. We have ˆ

LŽcd. (ad b d s c ( lŽcm. (log(ad b d s c ( l (Ad Ž by1 c b d . (log.

185

SHINTANI DESCENT

If we put g s by1 d b c , we have y1 gy1 F Ž g . s by1 c b d F Ž b d . F Ž bc .

ˆ s by1 c dF Ž b c . s by1 ˙ c z b c a. Ž . c˙F and since aF commutes with c, gy1 F Žg . deterSince by1 c z bc g ZL l mines an element c1 g A as in the lemma by the natural surjection . Ž . Ž Ž d.ˆ . s ZLŽ l. ª A. Hence l (AdŽ by1 c b d s l c 1 , and we see that ad b d * L c L c1. Combining with Ž1.13.1., the lemma follows. m

As a corollary for Lemma 1.13, Fm

ˆ

H ds ˜Ž m. . s Ind UH F dd L ˜ c1. Here we m Ž m LEMMA 1.14. We ha¨ e NFUy1 r F Ind U F dˆs L c m F ˆ F . Fm Ž regard a function on H d s resp. H d as the one on H s Ž resp. H F ., respecti¨ ely. F

Proof. H F m dˆs

m

H F dˆs

˜Ž m. s Ž ad bd . *( NFUy1 ˜Ž m. m m m m NFUy1 r F Ind U F dˆs L c r dˆF Ind U F dˆs L c

ž

ž

/

/

by Ž 1.11.2.

ˆ

H dF d

˜Ž m. m s Ž ad b d . *(Ind U dˆF d 00 NFUy1 r dˆF L c

ž

˜Ž m. m s Ind UH F dd Ž ad b d . *( NFUy1 r dˆF L c F

F

ž

˜ c1 s Ind UH F dd L

/

/

by Ž 1.11.3. by Ž 1.11.1.

by Lemma 1.13.

. 1.15. We note that d g ZLŽ l c1 . F . In fact, since l c1 s l (AdŽ by1 c b d , we have m l c1 (Ad dy1 s l (Ad Ž by1 c F Ž bd . .

s l (Ad Ž ˙ c . (Ad Ž F m Ž by1 c bd . . s l (Ad Ž F m Ž by1 c bd . . since ˙ c g ZLŽ l.. But gy1 F Žg . s by1 ˙ g ZLŽ l. by the proof of Lemma c z bc a m Ž y1 . 1.13, where g s by1 b . It follows that F bc b d s F m Žgy1 . g ZLŽ l.gy1 , d c and so, the last formula is equal to l (Ad gy1 s l c1. Hence d g ZLŽ l c1 . F . As in Ž1.4.3., ad gy1 induces an isomorphism ZLŽ l c1 . F rZL0 Ž l c1 . F , Ac1 F . Then we have the following. Ž1.15.1. The image of d g ZLŽ l c . F under the natural surjection 1 ZLŽ l c1 . F ª Ac1 F coincides with c1 F Ž c1 . ??? F my 1 Ž c1 . cy1 .

186

TOSHIAKI SHOJI

Ž Ž m. . , ZLŽ l c ., d is the In fact, under the isomorphism ad by1 d : ZL l c 1 Ž m. y1 .Ž d 0 .. But we have image of d 0 g ZLŽ l c .. Thus Žad g .Ž d . s Žad by1 c . and so d 0 s b d F m Ž by1 d y1 m by1 F Ž g . ? F m Ž by1 c d 0 bc s g c . bc

s gy1 F Ž g . ? F Ž gy1 F Ž g . . ??? F my 1 Ž gy1 F Ž g . . ˙ cy1 . This implies Ž1.15.1.. Let Mc and Mc0 be as in Subsection 1.8. We also put Nc1 s ZLŽ l c1 . F ˜ c1 be and Nc01 s ZL0 Ž l c1 . F . We now consider the group U F Nc01 ² d : and let L the trivial extension of L c1 to U F Nc01 ² d :. By making use of Lemma 1.14, one can show Fm

1 H m ds ˜Ž m. . s Ind UH F Ndc0 d L ˜ c1. Ž m NFUy r F Ind U F M c0 dˆs L c

PROPOSITION 1.16.

ˆ

F

1

Proof. For a given y g Mc0 , put dˆU s ydˆs yzc. ˆ Replacing z by yz, we ˆ as d, d 0 for d. ˆ First we note that get similar elements d*, dU0 for d* Ž1.16.1. d* may be chosen so that d* s dy* for some y* g Nc0 . 1 m

ˆ In fact, since y g Mc0 s ZL0 Ž lŽcm. . F and ZL0 Ž lŽcm. . is dF-stable, one can 0 Ž Ž m. . y1 y1 ˆ ˆ ˆ ˆ find d g ZL l c such that y s d dF Ž d . d , i.e., d dF Ž dy1 . s ydˆs d*. y1 y1 y1 ˆ ˆ Since d s b d F Ž b d ., we have d* s db d F Ž b d d .. Hence, we can choose d* g LF as y1 d* s F m Ž by1 . dbd d d m y1 s F m Ž by1 . d . bd d .ŽF Ž d m y1 s d by1 . d . bd . d ŽF Žd

Ž F m Ž dy1 . d . b d . We have F m Ž dy1 . d g ZL0 Ž lŽcm. . dˆF . Since Put y* s by1 d Ž m. 0 Ž Ž m. . l c (Ad b d s l c1, we see that by1 b d s ZL0 Ž l c1 .. As ad by1 sends d ZL l c d 0 ˆ to F, we conclude that y* g Nc1 . This shows Ž1.16.1.. dF We also note the property that c1 g A commutes with cF remains ˆ since c1 only depends on z modulo unchanged when dˆ is replaced by d* Mc0 . Let Xz be the left hand side of Proposition 1.16. Then it follows from the definition of the induction given in Subsection 1.11, that Xz s < Mc0
Ý ygM c0

s

Fm

H ms ˜Ž m. m NFUy1 r F Ind U F y dˆs L c

ž

/ m

Ý ygM c0r;dF ˆ

H Fm s ˜Ž m. . < ZM 0 Ž ydˆs .
ž

/

187

SHINTANI DESCENT

Now by Lemma 1.14 together with Ž1.16.1., we see that Fm

1 H ms ˜Ž m. s Ind UH F ddy*y* L ˜ c1 m NFUy r F Ind U F y dˆs L c

ž

F

/

with y* g Nc01 . Moreover, the correspondence y ¬ y* gives rise to a bijection between the set Mc0r;dˆF and the set Nc01r;d Ž;d means the twisted conjugation by the action of d .. Since < ZM c0 Ž ydˆs .< s < ZNc0 Ž y*d .<, we 1 have Xz s

F ˜ c1 < ZN 0 Ž y*d .
Ý

1

y*gNc01r;d

s < Nc01
F

˜ c1 Ind UH F ddy*y* L

Ý y*gNc01

F

˜ c1 . s Ind UH F Ndc0 d L 1

This proves the proposition. 1.17. Let r˜xŽ m. for x s Ž c, j . g M be as in the theorem, and we shall consider its Shintani descent. First we note that Fm

my1 ˜ŽŽcm., j . s < Ac
˜Žcm. . Ž 1.17.1. j˜Ž c1 F . Ind UH F m Mz cˆc0sz cˆs L

Ý zgM crM c0

m

In fact, for x g H F , we have

r˜ŽŽcm. , j .Ž xs . s < U F Mc
ž j h m L˜ / Ž y

m

Ž m. c

Ý

Fm

y1

xs y .

ygH m yy1 x s ygU F M c cˆs

s < Mc
Ý zgM c



< U F
Ý

Fm

y

y1

ygH m x s ygU F zcˆs

˜Žcm. Ž yy1 x s y . . j˜hŽ zcˆs . L

0

Under the isomorphism ad by1 in Ž1.6.3., zcˆs is transferred to c1 F g c A² F :. Hence under the identification j h l j given in 1.8, j˜hŽ zcˆs . coincides with mŽ c, j . j˜Ž c1 F .. Note that this value depends only on z

188

TOSHIAKI SHOJI

modulo Mc0 . Hence the last sum is equal to < Ac
mŽ c , j . j˜Ž c1 F . < Mc0 U F
Ý



zgM crM c0

˜Žcm. Ž yy1 x s y . L

m

Ý

Fm

y

y1

ygH m x s ygU F M c0 zcˆs

0

m

s < Ac
F ˜Žcm. Ž x s . . mŽ c , j . j˜Ž c1 F . Ind UH F m Mz cˆc0sz cˆs L

Ý

ž

zgM crM c0

/

This implies Ž1.17.1.. Note that the correspondence z ¬ c1 given in Lemma 1.14, for a fixed Mc0 and the set of c1 g A c g Aa F , induces a bijection between the set McrM such that c1 F commutes with c. Now it follows from Ž1.17.1., by making use of Proposition 1.16, that we have Sh F m r F my1 ˜ŽŽcm., j . < H F ms s < Ac
ž

/

F F

Ý j˜Ž c1 F . Ind UH N d L˜ c , Ž 1.17.2. c1

0 c1

1

where c1 runs over all the elements in A such that c1 F commutes with c, and d g Nc1 is determined by c1 up to Nc01 Žcf. Ž1.15.1... We fix a pair Ž c 0 , j 0 . with c 0 , a representative on an F-stable class in A, and j 0 g Aˆcex0 , and consider the pair Ž c9, j 9. s Žg c 0 , gj 0 . for each g g A. Then clearly we have

r˜ŽŽcm. s < A
Ý r˜Ž c , j g

0

g

0.

,

ggA

since Žg c 0 , gj 0 . gives the same character as Ž c 0 , j 0 .. In the following, we use F ˜ c1 for each c1 and c, since it a notation I Ž c1 , c . to denote Ind UH F Ndc0 d L 1 depends only on c1 and c. Let X s < A < < ZA Ž c 0 . < Sh F m r F my1 ˜ŽŽcm.0 , j 0 . < H F ms . Ž c0 , j 0. r

ž

/

Ž 1.17.3.

Then by Ž1.17.2., together with the above remark, we have Xs

Ý Ý gj˜0 Ž c1 F . I Ž c1 , g c0 . , ggA c1

where c1 runs over all the elements in A such that c1 F commutes with gc0 gy1 . Hence we have Xs

s

Ý

Ý

c1gA

ggA gc 0 gy1 gZ AŽ c 1 F .

Ý

Ý

c1gA

ggZ AŽ c 1 F ._A gc 0 gy1 gZ AŽ c 1 F .

j˜0 Ž gy1 c1 Fg . I Ž c1 , g c0 . j˜0 Ž gy1 c1 Fg .

Ý hgZ AŽ c 1 F .

I Ž c1 , h g c 0 . . Ž 1.17.4.

189

SHINTANI DESCENT

1.18. Let J be the third sum in the last equality in Ž1.17.4.. In order to determine X , we need to describe J in terms of characters in H F. For this we introduce a function Jq as follows. For each c1 , c as in 1.17, we put U F Nc d

˜ c1 . Iq Ž c1 , c . s Ind U F Nc01 d L 1

y1

Fixing c1 g A and g g A such that gc0 g tion Jq on U F Nc1 by Jqs

g ZAŽ c1 F ., we define a func-

Iq Ž c1 , h g c 0 . .

Ý hgZ AŽ c 1 F .

For c s hgc0 gy1 hy1, we fix d s dŽ h. g Nc1, and define an element D in the group algebra Q l w Nc1 x by Ds

dŽ h.

Ý

y1

.

hgZ AŽ c 1 F .

Then we have U F Nc

˜ c1 , Jqs D* Ind U F Nc01 L

ž

/

1

Ž 1.18.1.

where D*f Ž x . s f Ž xD . for a function f on U F Nc1 and for x g U F Nc1. We show Ž1.18.1.. For each x g U F Nc1, we have Jq Ž x . s < U F Nc01
Ý

˜ c1Ž yy1 xy . . L

Ý F

hgZ AŽ c 1 F .

ygU Nc 1

y y1 xygU F Nc01 d Ž h .

But since l c1 is stable by dŽ h., L c1 is stable for the twisted dŽ h.-action of U F. Hence Jq Ž x . s < Nc01
˜ c1Ž yy1 xyd Ž h . L

Ý

Ý

hgZ AŽ c 1 F .

ygNc1 y y1 xyd Ž h .y1 gU F Nc0

y1

.

1

s

< Nc0
LUc1

Ý ˜ Žy

y1

xyD . ,

ygNc1

˜Uc1 denotes the function on Q ˜ l wU F Nc1 x which coincides with L ˜ c1 on where L F 0 Q l wU Nc1 x and is zero outside. Let D be the image of D under the natural surjection Nc1 ª ZAŽ c1 F .. It follows from Ž1.15.1. that Ds

ž

Ý hgZ AŽ c 1 F .

hgc0 gy1 hy1 Ž c1 F .

/

ym

.

190

TOSHIAKI SHOJI

ŽHere Ž c1 F .ym g A since F m s 1 on A.. This implies that D is a central element in Q l w ZAŽ c1 F .x, and so, D commutes with any y g Nc1 up to a ˜ c1 is trivial on Nc01 , we see that factor in Nc01 . Since L Jq Ž x . s < Nc01
Ý L˜Uc Ž yy1 xD y . s 1

ygNc1

ž Ind

U F Nc1 U F N c01

˜ c1 Ž xD . . L

/

This proves Ž1.18.1.. U F Nc

˜ c1 is decomposed as Now Ind U F N 01 L c1

U F Nc

1 ˜ c1 s Ind U F N 0 L c1

Ý j 1gZ AŽ c 1 F

.n

˜ c1 , j 1 Ž 1 . j 1h m L

where j 1h is the character of U F Nc1 corresponding to j 1 under the map ˜ c1 is the trivial extension of L c1 to U F Nc1. Since D is a U F Nc1 ª Ac1 F , and L central element in Q l w Ac1 F x, D acts by a scalar multiplication on the representation space Vj 1 for each j 1 g Ž Ac1 F . n. This scalar is given as

j 1 Ž D . rj 1 Ž 1 . s j 1 Ž 1 .

y1

j 1 Ž hgc0 gy1 hy1 ? Ž c1 F .

Ý

ym

.

hgZ AŽ c 1 F .

s < Z A Ž c1 F . < j 1 Ž 1 .

y1

j 1 Ž gc0 gy1 . lŽŽ m. c1 , j 1 .

since Ž c1 F .ym is also a central element, and acts on Vj 1 as a scalar ŽŽ .ym .rj 1Ž1.. It follows from the above multiplication by lŽŽ m. c 1 , j 1 . s j 1 c1 F discussion that we see easily Jqs < ZA Ž c1 F . <

Ý j 1gZ AŽ c 1 F

.n

y1 lŽŽ m. . j 1h m L˜ c1 . Ž 1.18.2. c 1 , j 1 . j 1 Ž gc 0 g

Now we can describe the function J. Since Js

I Ž c1 , hgc0 gy1 hy1 . s Ind UH F Nc Jq , F

Ý

1

hgZ AŽ c 1 F .

it follows from Ž1.18.2. that we have J s < Z A Ž c1 F . <

Ý j 1gZ AŽ c 1 F

.n

j 1 Ž gc0 gy1 . lŽŽ m. c 1 , j 1 . rŽ c 1 , j 1 . .

Ž 1.18.3.

191

SHINTANI DESCENT

Hence, by substituting J in Ž1.18.3. into Ž1.17.4., we have

Xs

Ý

Ý

c1gA j 1gZ AŽ c 1 F . n

ž

Ý

j˜0 Ž gy1 c1 Fg . j 1 Ž gc0 gy1 .

ggA gc 0 gy1 gZ AŽ c 1 F .

/

= lŽŽ m. c 1 , j 1 . rŽ c 1 , j 1 . s < A < < ZA Ž c0 . <

Ý

 Ž c0 , j 0 . , Ž c1 , j 1 . 4 lŽŽ m. c , j . rŽ c , j . , 1

1

1

1

Ž c 1 , j 1 .g M

by the definition of the pairing  , 4 given in 1.9. In view of Ž1.17.3., this proves the theorem.

2. GENERALIZED GELFAND]GRAEV CHARACTERS 2.1. For a connected reductive group G defined over Fq , we denote by g its Lie algebra. The Frobenius maps on G and g are denoted by F. For a nilpotent element N in g F, Kawanaka defined a generalized Gelfand]Graev character GN on G F in wK1, K2x, and also their modifications in wK3x. In this section, we shall parametrize certain irreducible characters of SL nŽFq . by making use of generalized Gelfand]Graev characters. For this purpose, we need to consider the modified ones, i.e., the refinement of GN , given in wK3x. 2.2. We assume that G is a group of type A ny 1 of split type such that the derived subgroup Gder is isomorphic to SL n . We pose no restriction on p nor q. We shall construct GN by modifying in part the argument in wK1x. First note that there exists a bijective Fq-morphism from the unipotent variety Guni of G to the nilpotent variety g nil of g. We choose such a map explicitly as a map x ª x y 1, and denote it by log. Note that this map is not the same as the usual log map. Let S be a root system of g with P a set of simple roots compatible with the Fq-structure. By Dynkin]Kostant theory Žcf. wSS, IIIx. one can associate to each orbit ON ; g containing a nilpotent element N g g F, a Z-linear map h: ZS ª Z which has the property that hŽ a . g  0, 1, 24 for a g P. The Dynkin diagram with hŽ a . attached to each vertex corresponding to a g P is called a weighted Dynkin diagram. Now the function h gives a grading of g, g s [i g Z g i over Fq , where g i is the sum of root spaces g a such that hŽ a . s i. Let u i s [jG i g j for i G 1. Then u i is a nilpotent subalgebra of g, and there exists a connected unipotent subgroup Ui of G defined over Fq such that logŽUi . s u i . Also one can find an F-stable parabolic subgroup P s PN

192

TOSHIAKI SHOJI

such that P s LU1 , where L is an F-stable Levi subgroup of P such that LieŽ L. s g 0 , and that U1 is the unipotent radical of P. Moreover, by replacing N by its conjugate if necessary, we may assume that N lies in g 2 . Let ² , : be a fixed G-invariant non-degenerate bilinear form g = g ª k such that its restriction on the Lie algebra of SL n coincides with the Killing form. Let N* g gy2 be the image of N under the opposition Fq-automorphism of g Žcf. wK1, 1.2x.. We define a linear map l: u 1 ª k by lŽ x . s ² N*, x :. Then the map Ž x, y . ¬ lŽw x, y x. gives rise to a nondegenerate symplectic form on g 1. 2.3. In order to construct GN , we need to choose a Lagrangian subspace of g 1 with respect to the above symplectic form. In what follows, we shall find a Lagrangian subspace s which enjoys some good properties. First we give an explicit description of the weighted Dynkin diagram in the case of type A ny 1 following wSS, IVx. Let Sq be the set of positive roots with respect to P, which is written as Sqs  « i y « j <1 F i - j F n4 for certain basis vectors « 1 , . . . , « n . Then we have P s  a 1 , . . . , a ny14 with a i s « i y « iq1. Now assume that the orbit ON corresponds to a partition m s Ž m 1 G m 2 G ??? G m r . of n via the Jordan’s normal form of N. For each m i , we consider a set Yi s  m i y 1, m i y 3, . . . , ym i q 3, ym i q 1 4 consisting of m i integers. Then Y s @i Yi is a set of n integers Žwith multiplicities., and we arrange its elements in a decreasing order, Y s  n 1 G n 2 G ??? G nn 4 . We put, for each i Ž1 F i F n y 1., hŽ a i . s n i y n iq1. Then h: P ª Z gives the weighted Dynkin diagram. It follows from this construction that one sees easily that Ž2.3.1. The weighted Dynkin diagram is invariant under the graph automorphism s : P ª P. Furthermore in the case where n is even, we have hŽ a . / 1 for a unique a such that s Ž a . s a . Let P 1 Žresp. P 0 . be the set a g P such that hŽ a . s 1 Žresp. hŽ a . s 0.. We also put S 1 s  a g S < hŽ a . s 14 . Then we have g 1 s [a g S 1 g a . We shall determine S 1 explicitly. For a given a i g P 1 , let j be the smallest integer such that j ) i and that hŽ a j . / 0, and let k be the largest integer such that k - i and that hŽ a k . / 0. We define a subset Ci of Sq by Ci s  « s y « t < k q 1 F s F i , i q 1 F t F j 4 .

SHINTANI DESCENT

193

ŽIf j Žresp. k . does not exist, we put j s n Žresp. k s 0... Then it is easy to see that the Ci are mutually disjoint for a i g P 1 , and that S1 s

@

a igP 1

Ci .

For a i , a j g P 1 , i - j, we say that Ci and Cj are adjacent if hŽ a k . s 0 for any a k g P such that i - k - j. Then the following facts are easily verified. Ž2.3.2. For any a , b g Ci , we have a q b f S. Furthermore, if Ci and Cj are not adjacent, then a q b f S for any a g Ci , b g Cj . Ž2.3.3. Let a g Ci , b g "P 0 . If a q b g S, then a q b g Ci . Now in view of Ž2.3.1., s permutes the set Ci , and no Ci are stabilized by s . Hence we can find a subset C of S 1 such that S 1 s C @ s Ž C . and that C is a union of the Ci which are not adjacent to each other. We define a subspace s of g 1 as s s [a g C g a . Then it follows from Ž2.3.2. and Ž2.3.3. that Ž2.3.4. s is an abelian subalgebra of u 1 stable by the action of L on u 1. In particular, s is an F-stable Lagrangian subspace in g 1. We note that there exists exactly two such subspaces s in g 1 Žif s /  04.. In fact, this follows from the following property of hŽ P .; the set hŽ P . is partitioned into three parts, the left, right, and central parts. In the central part of hŽ P ., only the weights 1 or 0 appear, while in the left or right part, only the weights 2 or 0 appear, i.e., the weights 2 and 1 are never mixed. We now show that s also occurs in connection with a Levi subgroup of some parabolic subgroup of G. Let X i s  n j g Y < n j s 2 i or n j s 2 i y 1 4 for each integer i. Then those X i give a partition of Y, and the sequence of the cardinalities < X i < for X i / B gives, by arranging the order if necessary, the partition of n, which coincides with the dual partition m* of m. Now put P Ž i. s  a j g P < n j , n jq1 g X i 4 for each i such that < X i < G 2. Then the P Ž i. are mutually disjoint, and one can define a subset P M s @i P Ž i. of P. We define a parabolic subgroup Q s MUQ of G so that M is defined by the subroot system S M of S generated by P M , and UQ is defined by the set Sqy S M . Let SŽ i. be the subroot system of S generated by P Ž i.. Then the following properties are easily verified. Each P Ž i. contains at most one root a j such that hŽ a j . s 1, and hŽ a . s 0 for any a g P Ž i. y  a j 4 . Moreover, if a j g P Ž i. is such that

194

TOSHIAKI SHOJI

hŽ a j . s 1, then SŽ i. l S1 coincides with Cj defined as above. The adjacent Cj do not occur as the form of SŽ i. l S 1. It follows from these properties that S M l S1 coincides with a set C. We note, since s Ž C . s s Ž S M . l S 1 , that s Ž Q . s s Ž M .Us ŽQ. determines the set s Ž C . by the above procedure, where s Ž Q . is the parabolic subgroup defined by the set s Ž P M . ; P. Summing up the above argument, we have LEMMA 2.4. Let Q s MUQ be as in Subsection 2.3, and put m s Lie M. Then s s m l u 1 coincides with the abelian subalgebra of u 1 with respect to C gi¨ en in Ž2.3.4.. Moreo¨ er, the other abelian subalgebra s9 with respect to s Ž C . is obtained in a similar way by using the parabolic subgroup s Ž Q . s s Ž M .Us ŽQ. . 2.5. Following Kawanaka, we put u 1.5 s s q u 2 . Then u 1.5 is a subalgebra of u 1 , and we have an F-stable closed subgroup U1.5 of U1 such that logŽU1.5 . s u 1.5 . Note, in view of Ž2.3.4., that L normalizes U1.5 and that U1.5 is a normal subgroup of U1. Using the definition of the map log and the explicit description of s, one can verify the following property. For any u, ¨ g U1.5 , we have log Ž u¨ . g log Ž u . q log Ž ¨ . q u 3 .

Ž 2.5.1.

We now consider the map l (log: U1.5 ª k. Since l vanishes on u 3 , it follows from Ž2.5.1. that l (log is an F-stable homomorphism from U1.5 to k. Hence, by letting L N s c ( l (log Ž c is as in 1.4., we obtain a linear F character on U1.5 . The generalized Gelfand]Graev character GN is F F L . Following wK3x, we shall construct modified defined as GN s Ind G U1.5 N generalized Gelfand]Graev characters. Let Al s ZLŽ l.rZL0 Ž l.. ŽHere we consider l: u 1.5 ª k.. We consider the setting in 1.4 with U s U1.5 and L. Note that the assumption Ž1.4.1. is satisfied by Ž2.5.1.. Let M be the set given in Lemma 1.5 for A s Al. Then it follows from the discussion in 1.4, F ˜ c on ZLŽ l c . F U1.5 that for each Ž c, j . g M , we obtain a character j h m L . We define a modified generalized Gelfand]Graev character GŽ c, j . by

˜c . GŽ c , j . s Ind GZ L Ž l c . F U1F.5 j h m L F

ž

/

F

Note that GŽ c, j . s Ind GH F rŽ c, j . under the notation of 1.4. Also note that GN and GŽ c, j . do not depend on the choice of s since the ˜ c to ZLŽ l c . F U1F do not induction of L c to U1F and the induction of L depend on it Žcf. wK1, Lemma 1.3.6x.. Remark 2.6. In wK3x, Kawanaka uses an arbitrary F-stable Lagrangian subspace to define U1.5 . In this case L does not necessarily normalize U1.5 .

195

SHINTANI DESCENT

He constructs GŽ c, j . by constructing a representation directly on ZLŽ l c . F U1F , where he had to appeal the argument similar to the case of constructing Weil representations. In our construction, we can avoid using the representation on ZLŽ l c . F U1F which makes accessible the discussion in Section 1. However, our construction is not available to the case where F is of non-split type. 2.7. We consider the structure of the group Al. Let A G Ž N . be the centralizer of N in G modulo the connected centralizer of N. Then Žby choosing a good representative N in ONF . we have A G Ž N . , ZLŽ N* . rZL0 Ž N* . , Al .

Ž 2.7.1.

˜ of type A ny 1 with In fact, G can be embedded in a connected group G ˜ 1 be the parabolic subgroup of G˜ associated connected center. Let P˜ s LU ˜ l G. It is known by wSS, IVx, that to N. Then P s P˜ l G and L s L ZG˜Ž N . is a semidirect product of ZL˜Ž N . and ZU1Ž N ., where ZU1Ž N . is the unipotent radical of ZG˜Ž N ., and ZL˜Ž N . is connected reductive. Since ZU1Ž N . is contained in ZG Ž N ., we have ZG Ž N . s ZLŽ N . ZU1Ž N ., a semidirect product, and Z G0 Ž N . s Z L0 Ž N . Z U 1Ž N .. Hence A G Ž N . , ZLŽ N .rZL0 Ž N .. On the other hand, by choosing a suitable representative N with respect to the opposition automorphism, we may assume that ZLŽ N . s ZLŽ N*.. But by the definition of l, we see that ZLŽ N*. s ZLŽ l.. This implies Ž2.7.1.. Let Z be the center of G, and put G s GrZ. Then ZG Ž N .rZ , ZG Ž N ., and we have the following exact sequence, 1 ª ZrZ l ZG0 Ž N . ª A G Ž N . ª A G Ž N . ª 1. In our case we have A G Ž N . s  14 . It follows that Ž2.7.2. A G Ž N . , ZrZ l ZG0 Ž N .. In particular, A G Ž N . is abelian. 2.8. We assume throughout the rest of this section, except 2.22 and 2.23, ˜ s GLn with split Fq-structures. We fix a pair of that G s SL n , and that G F-stable Borel subgroup B˜ and an F-stable maximal torus ˜ and let W s NG˜ŽT˜.rT. ˜ Put B s B˜ l G, and T s T˜ l G. T˜ contained in B, Then W may be identified with NG ŽT .rT. We fix a root system S and the ˜ T˜.. First of all we shall describe set of positive roots Sq associated to Ž B, ˜ F. Let G* ˜ , GLn be the dual group of G˜ the irreducible characters of G ˜ F . n of irreducible characters of G˜ F is partitioned over Fq . Then the set Ž G F ˜ ,  s4., parametrized by F-stable semisimple classes  s4 in into the sets E Ž G ˜ We shall describe the set E Ž G˜ F,  s4. for each semisimple class  s4. We G. ˜ Then the Weyl group NG* ˜*.rT˜* fix a dual torus T˜* of T˜ over Fq in G*. ˜ ŽT may be identified with W. For any semisimple element s g T˜* such that

196

TOSHIAKI SHOJI

˜ is F-stable, let the conjugacy class  s4 of s in G* Ws s  w g W < w Ž s . s s 4 , Z s s  w g W < Fw Ž s . s s 4 . Corresponding to s, we have a subroot system S s , and the set of positive q roots Sq s ; S . Now Z s can be written as Z s s w 1Ws for some w 1 g W. q We choose w 1 so that w 1 maps Sq s into S . Let us take a positive integer m F m large enough so that s g ŽT˜*. . By the duality of the torus, s m determines an irreducible character u 0 of ŽT˜*. F , which is stable by Fw for any w g Z s . Then by making use of the norm map n F m r F w given in Subsection 1.2, we get an irreducible character uw on T˜ F w , T˜wF . ŽHere T˜w ˜ obtained by twisting T˜ by denotes an F-stable maximal torus of G w g W.. ˜ G We consider the Deligne]Lusztig virtual character R T˜wŽ uw . for uw g T˜wF . Let g be an automorphism on Ws obtained by g Ž w . s w 1wwy1 for 1 w g Ws , and consider the semidirect product group Ws²g : of Ws with g . n We denote by ŽWs .ex the set of irreducible characters of Ws which is n extendable to Ws²g :. For each E g ŽWs .ex let E˜ be an extension of E to Ws²g :. Following Lusztig wL2x, we define a function R E˜ by R E˜ s < Ws
Ý

˜ G

Tr Ž g w, E˜. R T˜w 1 w Ž uw 1 w . .

Ž 2.8.1.

wgWs

˜ F up to a Then it is known that R E˜ gives an irreducible character of G ˜F scalar multiple. We denote the corresponding irreducible character of G by r s, E . Then we have n

˜ F ,  s 4 . s  r s, E < E g Ž Ws . ex 4 . E ŽG 2.9. In wL2, 13.4x, Lusztig defined, for a connected reductive group H with connected center, a map from the set of irreducible characters of H F to the set of F-stable unipotent classes in H. We now describe this map ˜ s GLn . Let rs, E g E Ž G˜ F,  s4.. Put E9 s E m « explicitly in the case of G for the sign character « on Ws . Then IndW W s E9 contains a unique irreducible character Eˆ of W such that bE9 s bEˆ. ŽHere for a Coxeter group W1 , let V be the space of the reflection representation of W1. Then for each E1 g W1n , bE1 is defined as the smallest integer i G 0 such that E1 occurs in the ith symmetric power of V.. Now by the Springer correspon˜ more precisely, if Eˆ is dence, Eˆ corresponds to a unipotent class O in G; an irreducible character of W s Sn corresponding to a partition m of n, then O is a unipotent class of GLn of type m. The correspondence

SHINTANI DESCENT

197

r s, E ¬ O gives the required map. We denote by Or the unipotent class in ˜ Žor the nilpotent orbit in g . corresponding to an irreducible character r G ˜ F. on G In wK3x, Kawanaka announced a result, for GLn or adjoint groups of exceptional type under the condition that p is good, that a map from the set of irreducible characters to the set of unipotent classes can be defined by making use of generalized Gelfand]Graev representations Ža ‘‘wave front set’’ of irreducible characters in his terminology., and that it actually coincides with the map defined by Lusztig as above. This result was generalized by Lusztig wL4, Theorem 11.2x to the case of arbitrary reductive groups, but subject to the condition that p and q are large enough. In the case of GLn , the following fact holds without any assumption on p nor q.

˜N be the generalTHEOREM 2.10 ŽKawanaka wK3, Theorem 2.4.1x.. Let G ˜ F corresponding to N. Then for r g Ž G˜ F . n, ized Gelfand]Grae¨ character of G we ha¨ e ˜N , r :G˜ F s ²G

½

1

if Or s ON ,

0

unless ON : Or ,

where Or denotes the closure of the nilpotent orbit Or in g. Proof. Since Kawanaka did not publish the proof of this fact, we give the proof here for the sake of completeness. ŽHe proved the related fact for GLn in wK1x, but some assumption is attached there.. Let Q w be the F F Ž ˜ i.e., the restriction of R TG˜˜wŽ1. to G˜uni Green function of G, s Guni the set F. of unipotent elements in G . Then there exists a polynomial Q w, N Ž t . g F F Zw t x such that Q w Žexp N . s Q w, N Ž q ., where exp: g nil ª Guni is the inNŽ . verse of the map log. We now define a polynomial X w t g Zw t x by X wN Ž t . s t d N Q w , N Ž ty1 . , where d N s Ždim ZG˜Ž N . y n.r2 is the degree of Q w, N . The main ingredient for the proof is the following formula due to Kawanaka ŽwK1, 3.2.14x, ˜N into irreducible see also wK3, 2.3.2x., which enables us to decompose G ˜ F. characters of G

˜N s < W
Ý Ž y1. sŽ G˜.ysŽT˜ . < T˜wF < X wN Ž q . Qw , w

Ž 2.10.1.

wgW

where sŽ?. denotes the split rank of the indicated group. ŽNote, since the F ˜N as a function character GN has non-zero value only on Guni , we regard G F . on Guni .

198

TOSHIAKI SHOJI

˜

˜

Now in our case, Žy1. sŽG.ysŽT w . s « Ž w . for the sign character « of W. ˜ F,  s4., we shall compute the inner product ² G ˜N , r :G˜ F . For r s r s, E g E Ž G ˜ G F First note that the restriction of R T˜w wŽ uw 1 w . to the set Guni coincides with 1 Q w 1 w . Then by Ž2.8.1., Ž2.10.1. together with the orthogonality relations for Green functions,

˜ F
Qw Ž g . Qw 9 Ž g . s

Ý F ggGuni

½

< T˜wF
if w ; w9, if w ¤ w9,

we see easily that

˜N , r :G˜ F s < Ws
Ý « Ž w1 y . X wN y Ž q . Tr Ž g y, E˜. . 1

ygWs

Let X be the right hand side of this equality. Now Ws²g : is a quotient of the subgroup Ws² w 1 : of W generated by Ws and w 1 , and E˜ is regarded as a character of Ws² w 1 :. Then X can be written as the form of inner product on Z s s Ws w 1 , i.e.,

˜ :Z s , X s ² X ?N Ž q . , E9 ˜ is where X ?N Ž q . is a class function on W defined by w ¬ X wN Ž q ., and E9 an extension of E9 s E m « to Ws² w 1 :. Now using the Frobenius reciprocity, we have ˜ :Z s s ² X ?N Ž q . , IndWZ s E9 ˜ :W . ² X ?N Ž q . , E9 ˜ is defined by the similar formula as in Subsection 1.11. But Here IndWZ s E9 ˜ is a character of ² : Ws w 1 is again a semidirect product, and since E9 ² : Ws w 1 , we see easily that ˜ s IndWZ ss² w 1 : E9

Ý

˜ w Ž wy1 1 . Ž w m E9 . . n

wg ² w 1:

It follows that

˜ s IndWZ s E9

Ý

w g ² w 1 :n

W ˜ w Ž wy1 1 . Ind W s ² w 1 : Ž w m E9 . .

In particular, the irreducible characters occurring in the decomposition of ˜ are the ones occurring in the decomposition of IndWW s E9. MoreIndWZ s E9 over, if E1 g W n appears in IndW W s E9 with multiplicity one, then the inner ˜ : is a root of unity. product ² E1 , IndWZ s E9

199

SHINTANI DESCENT

For each E1 g W n, let OE1 be the orbit in g nil corresponding to E1 under the Springer correspondence. Then it is known that if E1 occurs in IndW W s E9, then OE1 ; OEˆ s Or . On the other hand, it follows from the description of Green functions Q w in terms of Springer representations of W, we see that if E1 appears in X ?N Ž q ., then ON ; OE1, and that if ON s OE1, we have ² X ?N Ž q ., E1 : s 1 Žsee, e.g., wS1, Corollary 3.3x.. Now NŽ . assume that E1 occurs in both of IndW W s E9 and X ? q . Then we have ON ; Or . This shows the second statement of the theorem. Next assume that ON s OEˆ. Then Eˆ is the unique constituent contained in both of NŽ . IndW W s E9 and X ? q , and the multiplicities in them are both one. It ˜ ² follows that GN , r :G˜ F is equal to a root of unity. Since it is a positive ˜N , r :G˜ F s 1. This proves the theorem. integer, we have ² G 2.11. In order to obtain similar results for the case of SL n with respect ˜N . We denote by to GŽ c, j . , we need to consider a certain refinement of G ˜ ˜ ˜ P s LUP the parabolic subgroup of G associated to N, where UP coincides ˜N is defined as G ˜N s Ind GU˜1.5FF L. Let w be an with U1 in 2.2. Then G irreducible character of ZL˜Ž l. F , which we regard as a character of F ZL˜Ž l. F U1.5 . We now consider an induced character ˜F

˜N , w s Ind GZ L˜ Ž l. F U1F.5Ž w m L ˜ .. G ˜ s MU ˜ Q be the parabolic subgroup of G˜ as given in Lemma 2.4. Let Q ˜ s M˜ l w 0 Mw ˜ y1 Then from the construction in 2.3, we see that L 0 . Here y1 ˜ 0 s s Ž M˜ .. We have w 0 is the longest element in W and we have w 0 Mw the following lemma. ˜ F such that u < Z L˜ Ž l. F s w . LEMMA 2.12. Let u be a linear character of M F ˜ We regard u as a character of Q in a natural way. Then ˜F

˜N , w , Ind GQ˜ F u :G˜ F / 0. ²G Proof. Let ˜F

˜ .. IN , w s Ind ZP L˜ Ž l . F U1F.5Ž w m L Then by the Mackey formula, we have ˜F

˜N , w , Ind GQ˜ F u :G˜ F s ²G

y1

Ý

wgWM_WrWL

² IN , w , w u :w y1 Q˜ F l P˜F .

200

TOSHIAKI SHOJI

Hence it is enough to show that ² IN , w , w 0u :w 0 Q˜ F l P˜F s 1.

Ž 2.12.1.

˜ l U1 s 0 M˜ l U1 , and w 0 Q˜ l L ˜ s L. ˜ If we put S sw 0 M˜ l U1 , Now 0 Q then by Lemma 2.4, Lie S s s9 s s Ž m . l u 1 is another Lagrangian subspace of g 1 constructed in 2.3. Hence we can write w

w

w0

˜ l P˜ s Ž w 0 Q˜ l L ˜ . Ž w 0 Q˜ l U1 . s LS. ˜ Q

F . ˜FS F ? Ž ZL˜Ž l. F U1.5 ˜F S F l Then it is easy to see that P˜F s L and that L F F F ZL˜Ž l. U1.5 s ZL˜Ž l. . Again by using the Mackey formula, we have

IN , w < L˜F S F s Ind ZL L˜ ŽSl . F w . ˜F

F

It follows that ² IN , w , w 0u :L˜F S F s ² w , w 0u :Z L˜ Ž l. F . The last term is equal to 1 if the restriction of w 0u to ZL˜Ž l. F coincides with w . Now ZL˜Ž l. F is a product of various general linear groups, and so a linear character of ZL˜Ž l. F is determined by its restriction on the center of ZL˜Ž l.. Since w 0 acts trivially on the center, we see that w 0u s u on ZL˜Ž l. F . This shows Ž2.12.1. and the lemma follows.

˜ F. Now, Ws is a 2.13. Let r s r s, E be an irreducible character of G ˜ determines a parabolic subgroup of W, and the conjugacy class of s in G* Ž . type b s b1 , . . . , bt , i.e., Ws , Sb 1 = ??? = Sb t , where S j is the symmetric group of degree j. Then E g Ws n may be expressed as E s E1 G ??? G Et , where Ei g Sbni corresponds to a partition bi of bi . Using the explicit description of the map r ¬ Or in the case of A ny1 , we see that the sequence of the partitions Ž b 1 , . . . , bt ., regarded as a partition of n by arranging the order if necessary, gives the dual partition of the partition corresponding to Or . We now assume that Or s ON . Let m be the partition of n corresponding to N as before. Then Ž b 1 , . . . , bt . gives rise to the partition m*. In particular, the partition m* turns out to be a refinement of the partition b Žby neglecting the order.. This implies, by replacing s by its conjugate if necessary, that we may regard WM as a subgroup of Ws . In ˜ . of the dual group M* ˜ of other words, s is contained in the center ZŽ M* ˜ Let us take an integer m large enough so that s g ZŽ M* ˜ . F m . Then s M. ˜ F m . We denote by wˆ the restriction of determines a linear character uˆ of M m F uˆ to ZL˜Ž l. . The following proposition describes a finer decomposition of generalized Gelfand]Graev characters.

201

SHINTANI DESCENT

˜ F . n, and assume that Or s ON . PROPOSITION 2.14. Let r s r s, E g Ž G ˜N, w , r :G˜ F Then there exists a unique linear character w on ZL˜Ž l. F such that ² G ˜ s 1, and that ² GN, w 9 , r :G˜ F s 0 for any irreducible character w 9 different from w . Here w is characterized by the property that NFUm r F Ž w . s w ˆ. ˜N , r :G˜ F s 1. Since G ˜N is decomProof. By Theorem 2.10, we have ² G posed as ˜N s G

˜ ., w 1 Ž 1 . Ind GZ˜L˜ Ž l. F U1F.5Ž w 1 m L F

Ý

w 1g Ž Z L˜Ž l . F . n

the first assertion is clear. We show the second assertion. Let w be as in the statement in the proposition. First we consider the case where s g ˜ . F. Then one obtains a linear character u of M˜ F corresponding to s. ZŽ M* ˜F The endomorphism algebra of Ind u s Ind GQ˜ F u is isomorphic to the endoWs morphism algebra of Ind W M 1. It follows that r appears in the decomposition of Ind u with multiplicity one, and any irreducible character r 9 appearing in Ind u satisfies the property that Or 9 ; Or . Thus, by Theorem ˜N, w , Ind u :G˜ F s 1. Note, since ² G ˜N , Ind u :G˜ F s 1, that 2.10 we have ² G w g ZL˜Ž l. F is characterized by the property that the inner product as above is non-zero. Hence by Lemma 2.12, we see that w s u < Z L˜ Ž l. F . ˜ . is not necessarily Next we consider the general case, i.e., s g ZŽ M* ˜ . F m . The irreducible F-stable. We choose an integer m such that s g ZŽ M* ˜ F m are parametrized as in 2.8. In particular, since s g characters of G m F ˜ . , the set E Ž G˜ F m ,  s4. is in one to one correspondence with the set Ž G* m Ž ˜ F ,  s4. Ws n . We denote by r s,Ž m. E the irreducible character belonging to E G n Ž m. corresponding to E g Ws . Note that r s, E is F-stable if and only if E is g-stable. Now it is known by the theory of Shintani descent that NFUm r F Ž r s, E . coincides with r s,Ž m. E up to a scalar multiple. On the other ˜NŽ m. on hand, one can construct the generalized Gelfand]Graev character G m F ˜ as G ˜F m

˜NŽ m. s Ind GU1F.5m LŽ m. , G where LŽ m. s cm ( l (log Žsee 1.6.. Then we have

˜N . s G ˜NŽ m. , NFUm r F Ž G

˜N , w s G ˜NŽ m. NFUm r F G ,w ˆ

ž

/

˜N,Ž m.wˆ is defined in a similar way as G ˜N, w , by up to scalar factors, where G U Fm Ž . Ž . m using a linear character w on Z l such that N w s w . ˆ ˆ ŽThis is ˜ L F rF ˜NŽ m. has an an abbreviation of the notation. By the discussion in 1.6, G ˜ F m² s : which we fix and denote by the same symbol. Then extension to G U ˜N . coincides with G ˜NŽ m. < G F ms up to a scalar multiple. A similar fact NF m r F Ž G

202

TOSHIAKI SHOJI m

˜N, w . We also choose an extension of r s,Ž m.E to G˜ F ² s : holds also for G which will be denoted again by the same symbol. The choice of such extensions does not affect the discussion below.. Since the Shintani descent preserves the inner product, we have Ž m. : F m ˜NŽ m. ²G ,w ˆ , r s, E G˜ s / 0.

˜NŽ m., we see that On the one hand by applying Theorem 2.10 to G Ž m. Ž m. m ˜N , r s, E :G˜ F s 1. This implies that ²G Ž m. : F m ˜NŽ m. ²G s 1. ,w ˆ , r s, E G˜

Then the argument in the former part of the proof can be applied to ˜ . F m , replacing F by F m . So, we have a linear character uˆ on s g ZŽ M* m ˜ F corresponding to s, and we must have uˆ< Z L˜ Ž l. F m s wˆ by the uniqueM ness of w . ŽNote that uˆ is not necessarily F-stable. But uˆ< Z L˜ Ž l. F m turns out to be F-stable by the above argument.. This proves the proposition. 2.15. From now on, we consider the characters concerning G F s SL nŽFq .. Let P s P˜ l G be the parabolic subgroup of G. We have ˜ l G. We consider the structure of the subgroups P s LU1 , where L s L 0Ž . Ž . ZL l and ZL l . Let m be the partition of n corresponding to N as before. We express m as m s Ž1m 1 , 2 m 2 , . . . . in the increasing order. Then by wSS, IVx, ZL˜Ž l. , ZL˜Ž N . is described as ZL˜Ž l . ,

Ł

m i)0

GLm i .

Under this isomorphism, ZLŽ l. is given as ZLŽ l . , Ž . . . , x i , . . . . g

½

Ł

m i)0

GLm i < Ł Ž det x i . s 1 , i

5

i

and if we denote by d the greatest common divisor of  i < m i ) 04 , ZL0 Ž l. may be expressed as ZL0 Ž l . , Ž . . . , x i , . . . . g

½

Ł

m i)0

GLm i < Ł Ž det x i . i

ird

s 1 . Ž 2.15.1.

5

In particular, we see that ZLŽ l.rZL0 Ž l. , Zrd p9 Z, where d p9 is the p9 part of d, and that ZLŽ l. F rZL0 Ž l. F , Zrd9Z, where d9 s gcdŽ d p9 , q y 1. s gcdŽ d, q y 1.. Note that all the isomorphisms are defined over Fq . We now consider the generalized Gelfand]Graev character GŽ c, j . of G F defined in 2.5. We shall compute the inner product of GŽ c, j .

203

SHINTANI DESCENT

˜ F. The following with the restriction r < G F for irreducible characters r of G is a generalization of a result of Asai wA, Proposition 3.1.1x, where he considered the special case where N is a regular nilpotent element, i.e., GŽ c, j . is a modified Gelfand]Graev character, and r is a regular character ˜ F. He used this result to parametrize irreducible characters of G F of G appearing in the restriction of r to G F. ˜ F . n. Then for each pair Ž c, j . g M , THEOREM 2.16. Let r s r s, E g Ž G we ha¨ e the following. Ži. ² GŽ c, j . , r < G F :G F s 0 unless ON : Or . Žii. Assume that Or s ON . Let w be the linear character of ZL˜Ž l. F gi¨ en in Proposition 2.14. Then ² GŽ c, j . , r < G F :G F s 0 if w < Z L0 Ž l. F is non-tri¨ ial. If w < Z L0 Ž l. F is tri¨ ial, then w < Z L Ž l. F determines a character j 0 g Aˆlc F , which satisfies the formula ² GŽ c , j . , r < G F :G F s

½

if j s j 0 if j / j 0 .

1 0

Ž Here Alc F s AlF since Al is abelian, and so we regard j 0 g Aˆlc F as the character of ZLŽ l. F ¨ ia the isomorphism AlF , ZLŽ l. F rZL0 Ž l. F .. F ˜Ž c, j . s Ind GG F GŽ c, j . . Since j h is a character of ZLŽ l c . F U1.5 Proof. Let G 0Ž F F which is constant on ZL l c . U1.5 , we see that

˜F

˜Ž c , j . s < AlF
Ý j Ž a. IndGU˜

F F 0 F ˙ 1 .5 Z L Ž l c . a

˜c, L

Ž 2.16.1.

agAlF

where a ˙ is a representative of a in ZLŽ l c . F under the isomorphism F ˜ c from Ž . ZL l c rZL0 Ž l c . F s Alc F s AlF . First we consider the induction of L F 0Ž F F F 0Ž . Ž . Ž . U1.5 ZL l c a ˙ to U1.5 ZL˜ l c . It follows from 2.15.1 that ZL l c . is a normal subgroup of ZL˜Ž l c . and A˜l s ZL˜Ž l c .rZL0 Ž l c . is abelian. This implies that U F Z ˜Ž l . F

˜c s Ind U11F.5.5 Z LL0 Ž lcc . F a˙ L

Ý

w 9 Ž a˙.

y1

AlF . n

Ž w 9 m L˜ c . .

Ž 2.16.2.

w 9g Ž ˜

Substituting Ž2.16.2. into Ž2.16.1., we have

˜Ž c , j . s G s

Ý

˜ ² j , w 9 < AlF :AlF Ind G U F Z ˜Ž l

Ý

˜ Ind G U1F.5 Z L˜ Ž l c . F Ž w 9 m L c . ,

F

w 9g Ž A˜lF . n

w 9g Ž A˜lF . n w 9 < AlF s j

1 .5

˜F

L

c.

F

Ž w 9 m L˜ c . Ž 2.16.3.

204

TOSHIAKI SHOJI

where AlF is naturally regarded as a subgroup of A˜lF . We now compute the inner product in the theorem. By the Frobenius reciprocity, we have

˜Ž c , j . , r :G˜ F s ² GŽ c , j . , r < G F :G F s ² G

Ý

w 9g Ž AlF . n w 9 < AlF s j

˜N , w 9 , r :G˜ F . ²G

˜

˜N, w 9 is a direct Now the first assertion follows from Theorem 2.10 since G ˜ summand of GN . Next assume that Or s ON . Then the second assertion follows from Proposition 2.14 in view of the above formula. This proves the theorem. 2.17. We shall now look for the characters r s, E such that the restriction ˜ ª G* s G*rZ* ˜ ˜ be the natural of w to ZL0 Ž l. F is trivial. Let p : G* ˜ is the center of G*, ˜ and G* is the dual group of G. projection, where Z* ˜*.rT˜* is Let T * s p ŽT˜*. be the maximal torus of G*. Then W s NG* ˜ ŽT naturally identified with NG* ŽT *.rT *. Let s s p Ž s . g G*. Then the stabilizer Ws of s in W turns out to be a semidirect product Ws i V s , where V s 0 Ž . is a cyclic group isomorphic to ZG* Ž s .rZG* s . We consider r s r s, E satisfying the following condition. Ž2.17.1. Ws is of type b s Ž b1 , . . . , bt . with b1 s ??? s bt s nrt. Ws s Ws V s with V s , ² w 0 :, where w 0 is an element of order t permuting the factors of Ws transitively. Furthermore, E s E1 G ??? G Et g Ws n with E1 s ??? s Et . We now assume that Or s ON , and let m be the partition corresponding to N. Then the dual partition m* is of the form m* s Ž b 1 , . . . , bt . where b 1 s b 2 s ??? s bt s a is the partition of nrt corresponding to Ei . We now put F9 s Fw 1. Then s is F9-stable Žsee 2.8.. Let t9 be the order of V Fs 9. We define a subgroup ZL1 Ž l. of ZL˜Ž l. by ZL1 Ž l . , Ž . . . , x i , . . . . g

½

Ł

m i)0

GLm i < Ł Ž det x i . i

irt 9

s 1 . Ž 2.17.2.

5

Then ZL1 Ž l. is a subgroup of ZLŽ l. containing ZL0 Ž l.. We have the following lemma. LEMMA 2.18. Let the embedding WM ; Ws be gi¨ en as in 2.13, and let w be the linear character of ZL˜Ž l. F corresponding to s as gi¨ en in Proposition 2.14. Then the restriction of w to ZL1 Ž l. F is tri¨ ial. In particular, the restriction of w to ZL0 Ž l. F is tri¨ ial. Proof. We express m* as a sequence of partitions, m* s Ž a 9, . . . , a 9., t9 copies of a 9, where a 9 is a partition of nrt9 consisting of trt9 copies of a . ˜ is decomposed as M˜ , M˜1 = ??? = M˜t 9. According to this decomposition, M

205

SHINTANI DESCENT

n ˜i are all Since E g ŽWs .ex , we can choose the decomposition such that M X F9-stable, and if we denote the partition a 9 as a 9 s Ž a 1 G ??? G a kX ., we have

k

˜i , M

Ł GLa . js1

Ž 2.18.1.

X j

˜ is expressed as Under this isomorphism an element x in M t9

xs

k

Ł Ł yi j

is1 js1

Ž yi j g GLa . . X j

Using Ž2.17.2. it is verified that if x g ZL1 Ž l., then yi j satisfies the condition that Ł kjs1 detŽ yi j . s 1. We now consider the F9-stable linear charac˜ F m as in 2.13. Then uˆ can be expressed as uˆs uˆ1 G ??? G uˆt 9 , ter uˆ on M ˜iF m , while M˜i is decomposed where uˆi is an F9-stable linear character of M ˜i , M˜0 = ??? M˜0 Ž trt9-times., and M˜0 is a subgroup of M˜i correas M sponding to the partition a of nrt. Note that F9 acts transitively on the ˜i . Thus uˆi g Ž M˜iF m . n is expressed as set of components of M

uˆi s f G f q G ??? G f q

ay 1

, m

˜0F . It is easily where a s trt9, and f is an F a-stable linear character of M F9 Fa ˜ ˜ verified that Mi is naturally isomorphic to M0 , and under this isomorphism, we have Sh F m r F 9 uˆi s Sh F m r F a f ,

Ž 2.18.2.

where the left hand side Žresp. the right hand side. denotes the Shintani ˜iF m Žresp. M˜0F m . to M˜iF 9 Žresp. M˜0F a ., respectively. Now, descent from M m since Ws is of type b , there exists a linear character f 9 on GLFn r t such that m F ˜0 . ŽHere we regard M˜0 as a subgroup of f is the restriction of f 9 on M m GLn r t under the isomorphism Ž2.18.1... Since f˜ is trivial on SLFn r t , we see X Fa that f 0 s Sh F m r F a f 9 is trivial on SL n r t . This implies that f 0 s Sh F m r F a f ˜0F a l SLFn ra t . On the other hand, let wˆ be the restriction of uˆ is trivial on M m F ˜ F 9 l ZL˜Ž l. s ZL˜Ž l. F since w1 acts trivially on to ZL˜Ž l. . We note that M ZL˜Ž l.. It follows that Sh F m r F 9 Ž uˆ. < Z L˜ Ž l . F s Sh F m r F Ž w ˆ. .

Ž 2.18.3.

˜ F 9 ª M˜iF 9. Then the image of ZL˜Ž l. F by We consider the projection M F ˜0 under the isomorphism M˜iF 9 , M˜0F a. Hence, if this map lies in M 1Ž . F F ˜i is regarded as an element of M˜0F, and x g ZL l ; ZL˜Ž l. , Ł j yi j g M

206

TOSHIAKI SHOJI

˜0F l SLFn r t . In view of Ž2.18.3., by the preceding remark it is contained in M this implies that w s Sh F m r F Ž w ˆ . is trivial on ZL1 Ž l. F. So the lemma follows. 2.19. Let r s r s, E be as in Ž2.17.1.. We denote by Tr the set of irreducible characters of G F appearing in the decomposition of r < G F . By making use of Theorem 2.16, we shall give a parametrization of Tr . First note that since Al is abelian, the set Alr;F may be naturally identified ˜ FrG F ª Ž Al .F as follows. We with Ž Al .F Žsee 1.2.. We define a map f : G F ˜ as g s g 1 z Ž g 1 g G, z g Z˜.. Then gy1 ˜ Ž . write g g G 1 F g 1 g Z l G s Z, y1 Ž Ž . Ž . . the center of G, and so gy1 F g g Z N . The image of g 1 1 G 1 F g 1 in 0 Al , ZG Ž N .rZG Ž N . depends only on g up to F-twisted conjugacy on Al , hence it determines a unique element f Ž g . in Ž Al .F . It is easy to see that f is a su rjective h om om orph ism from G˜ F r G F to Ž A l . F . ˜F as a representative of G˜ FrG F. Let Nc be a Now we can choose g g L twisted nilpotent element in g corresponding to c g Ž Al .F . Then g Nc corresponds to the class cc9 g Ž Al .F with c9 s f Ž g .. It follows from this that we see easily that

˜F, and f Ž g . s c9 g Ž Al.F . Then we have g GŽ c, j . s Ž2.19.1. Let g g L GŽ cc9, j . . ˜ FrG F acts transitively on the set Tr . We note that On the other hand, G Ker f acts trivially on it. In fact, by Theorem 2.16, there exists a unique irreducible character in Tr appearing in the decomposition of GŽ c, j 0 . , which we denote by r 0 . Then all other characters in Tr are obtained as gr 0 for ˜ FrG F. Hence, by Ž2.19.1. any character in Tr is characterized as the ggG unique common constituent with r < G F and GŽ c9, j 0 . . Hence Ž2.19.1. implies that Ker f acts trivially on Tr . It follows that Ž Al .F acts transitively on Tr . Hence a quotient group Ž Al .XF of Ž Al .F is in bijection with Tr . It is known by wL3, Proposition 5.1x that r < G F is multiplicity free and Tr consists of t9 elements, where t9 is the order of V Fs 9 as in 2.17. ŽNote that V Fs 9 stabilizes E by Ž2.17.1... Now Al , Zrd p9 Z is described in 2.15, and so Ž Al .XF may be identified with AlrAlt 9. It follows, under the notation of Ž2.17.2., that X

Ž Al . F , ZLŽ l . rZL1 Ž l . .

Ž 2.19.2.

Note that the group Ž Al .XF can be interpreted as follows. Let Al s ZLŽ l.rZL2 Ž l., where ZL2 Ž l. is the subgroup of ZLŽ l. defined in a similar way as ZL1 Ž l. Žsee. Ž2.17.2.., but replacing t9 by t. Then Ž Al .XF is naturally isomorphic to Ž Al .F , the largest quotient of Al on which F acts trivially. Now summing up the above discussion, we have the following parametrization of the set Tr .

SHINTANI DESCENT

207

Ž2.19.3. Let rŽ c, j . be the unique irreducible character in Tr such that 0 rŽ c, j 0 . appears in the decomposition of GŽ c, j 0 . . Then we have Tr s  rŽ c, j 0 . < c g Ž Al .F 4 . 2.20. We consider s s p Ž s . g G* F 9. Then G* F 9-conjugacy classes in G* F 9 which is contained in the geometric conjugacy class of s are parametrized by F9-twisted classes V sr;F 9 in V s . Since V s is abelian, this set is identified with Ž V s .F 9. Note that Ž V s .F 9 is a cyclic group of order t9. We regard V s as a subgroup of Ws , and for each x g Ž V s .F 9 , choose a ˜*. corresponding to x g Ws ; W. Then s g representative ˙ x g NG* ˜ ŽT ŽT *. ˙x F 9, and since ker p is connected, one can find s x g ŽT˜*. ˙x F 9 such that p Ž s x . s s. Clearly we have Ws s Ws x, and V s˙xxF 9 s V Fs 9 since V s is abelian. ˜ For each Moreover, we note that the s x are mutually non-conjugate in G. ˜ F . n. Let Ts, E pair Ž s x , E . satisfying Ž2.17.1., we consider r x s r s x , E g Ž G be the set of irreducible characters of G F appearing in the decomposition of r x < G F for various x g Ž V s .F 9. Hence Ts, E is the disjoint union of various Tr x. Using the group Al , we define a set Ms, N as Ms, N s Ž Al .F = Ž AlF . n. Then the set Ž AlF . n is regarded as a subset of Ž AlF . n. Also, we have a surjective map Ž Al .F ª Ž Al .F . Let us define a subset M0 of M by M0 s Ž Al .F = Ž AlF . n. We have a surjective map w : M0 ª Ms, N . As a corollary to Theorem 2.16 and Lemma 2.18, we have the following parametrization of Ts, E . COROLLARY 2.21. Assume that Or s ON . Then AlF is isomorphic to ZLŽ l. F rZL1 Ž l. F . There exists a bijection Ts, E l Ms, N satisfying the following. For a pair Ž c, j . g Ms, N , let rŽ c, j . g Ts, E be the character corresponding to Ž c, j .. Then for each Ž c9, j 9. g M0 , ² GŽ c9, j 9. , rŽ c , j . :G F s

½

1 0

if w Ž Ž c9, j 9 . . s Ž c, j . if w Ž Ž c9, j 9 . . / Ž c, j . .

Moreo¨ er, if Ž c9, j 9. g M is not in M0 , we ha¨ e ² GŽ c9, j 9. , r 1 :G F s 0 for any r 1 g Ts, E . Proof. By applying Theorem 2.16 together with Lemma 2.18 to r x g ˜ F . n, we see that there exists a unique character j x g AˆlF such that ŽG ² GŽ c, j . , r x :G F / 0. By Lemma 2.18, j x is trivial on ZL1 Ž l. F rZL0 Ž l. F Žcf. x Ž2.19.2... In particular, Ž c, j x . is in the image of Ms, N . Now, by Ž2.19.3., Tr x is parametrized by the pair Ž c, j x . with c g Ž Al .F . Note that <Ž V s .F 9 < s <Ž Al .F < s < AlF < s t9, and that < ZLŽ l. F rZL1 Ž l. F < F t9. Hence in order to complete the proof we have only to show that Ž2.21.1. j x are all distinct for x g Ž V s .F 9. m

We shall show Ž2.21.1.. We choose m large enough so that s x g T˜* F . ˜ F m . Hence Since p Ž s x . s p Ž s . s s, s x is written as s x s sz x with z x g Z*

208

TOSHIAKI SHOJI m

there exists an F9-stable linear character uˆ of T˜ F and a linear character ˜ F m such that uv ˆ x is ˙xF9-stable. Let M˜ s M˜1 = ??? = M˜t 9 be the v x of G ˜ as in the proof of Lemma 2.16. Then M˜i are all decomposition of M ˜ We choose x as a generator of F9-stable, and x permutes the factors of M. ˜i . s M˜iq1 for i g Zrt9Z. Then T˜ ; M˜ is also decomŽ V s .F 9 such that x Ž M ˜ where posed as T˜ s T˜1 = ??? = T˜t 9 according to the decomposition of M, ˜ ˜ Ti is an F9-stable maximal torus of Mi mutually isomorphic. We compute ˆ x on T˜ F m and its Shintani descent. Now v x is written the restriction of uv m on T˜ F as

v x < T˜ F s v 1 G ??? G v 1 , m

m

m

where v 1 is a linear character of T˜i F , T˜1F . Since uˆ is F9-stable, and ˆ x is ˙xF9-stable, uˆ may be written on T˜ F m as uv

uˆs u 1 G u 1 c G ??? G u 1 c t 9y1 ,

with c t 9 s 1,

Ž 2.21.2.

˜ F , w1 .. Note, since v x is a linear character of G where c s v 1 F9Ž vy1 1 stabilizes v 1. It follows that m

1y q c s v 1 F Ž vy1 . 1 . s v1

Ž 2.21.3.

Moreover, by the same reason, v 1 is expressed as v 1Ž y . s v 1Ždet y . for m y g T˜1F , where v 1 is a homomorphism from FqUm to QUl . Let us denote by u a generator of the multiplicative group FqUm . Since c t 9 s 1, we see, by Ž2.21.3., that v 1t 9 is F-stable. Since Ws is of type b , we see that c, c 2 , . . . , c t 9y1 are all distinct. It follows that Ž2.21.4. v 1Ž u. qy 1 is a primitive t th root of unity. We now consider the restriction S˜ s T˜ l ZL˜Ž l.. Then S˜ is a torus ˜ and for y s Ž y 1 , . . . , y 1 . g S˜F m , contained in the diagonal subgroup of T, m we have v x Ž y . s v 1t 9Ž y 1 .. Since the restriction of v x on ZL˜Ž l. F is F-stable, one can consider the Shintani descent Sh F m r F Ž v x < Z L˜ Ž l . F m ., and similarly Sh F m r F Ž v x < S˜F m .. Put v 0 s Sh F m r F Ž v x < S˜F m .. Then v 0 is a character of S˜F , which is the restriction of Sh F m r F Ž v x < Z L˜ Ž l . F m . on S˜F . We now identify S˜ with the subtorus of T˜1. Since v 1t 9 is F9-stable, we have a relation

v 0 s Sh F m r F Ž v 1t 9 . < S˜F . Now, in view of Ž2.19.2., we can choose as a generator y g Ž Al .F an element y g ZLŽ l. such that y s Ž y 1 , . . . , y 1 . g S˜ with det y 1 a primitive my 1 t th root of unity. Put u 0 s NF m r F Ž u. s u1q qq? ? ?qq . Then u 0 is a generaU tor of the multiplicative group Fq . Since t9 < q y 1, det y 1 g Fq . So we may

SHINTANI DESCENT

209

assume that det y 1 s uŽ0qy1.r t 9. It follows from this that m v 0 Ž y . s v 1t 9 Ž NFy1 rF Ž y. .

s v 1t 9 Ž uŽ qy1.r t 9 . s v 1Ž u .

qy 1

.

Hence, by Ž2.21.4., we see that v 0t 9 s 1 have mutually distinct F restriction on S˜ l ZLŽ l. . In the above discussion if we replace v x by v x i for i such that m 0 F i F t9 y 1, v x i on T˜ F is written as v x i < T˜ F m s v X1 G ??? G v X1 , and by X Ž2.21.2. we see that v 1 F Ž v Xy1 . s c i s v 1Ž1yq.i. Let v X0 s Sh F m r F Ž v x i < S˜F m .. 1 Then the similar argument as before shows that v X0 Ž y . s v 1Ž u. iŽ qy1. s v 0i Ž y .. This implies that the restriction of Sh F m r F Ž v x i < Z L˜ Ž l. F m . on S˜ l ZLŽ l. F are all distinct. It follows, if we denote by w x i the linear character of ZL˜Ž l. F corresponding to s x i as given in Proposition 2.14, that the restrictions of w x i on ZLŽ l. F are all distinct. This shows Ž2.21.1., and the corollary follows.

v 0 , v 02 , . . . ,

2.22. We shall extend the previous results to a more general case. In ˜ and G as this subsection, we consider the groups G

˜ , GLn1 = ??? = GLn r . G ˜ as a subgroup of GLn , with n s Ý n i and put G s G˜ l SL n . We regard G ˜ ˜ of the form Let Gi s GLn i . We consider a Frobenius map F on G ˜ F s f F0 , where F0 is a split Frobenius map on G and f is a permutation ˜ Let T˜i be the F0-split maximal torus of G˜i , and let of the factors in G. ˜ which is maximally T˜ s T˜1 = ??? = T˜r be an F-stable maximal torus of G split with respect to F. The notation given in 2.8 can be applied also to this case. Then W s NG˜ŽT˜.rT˜ , W1 = ??? = Wr with Wi , NG˜iŽT˜i .rT˜i . A simi˜ , G˜U1 = ??? = G˜Ui . Then lar definition works also for the dual group G* s g T˜* is written as s s Ž s1 , . . . , sr . with si g T˜iU , and Ws can be expressed as Ws , W1, s1 = ??? = Wr, s r. We now assume that Wi, s i is of type Ž bi , . . . , bi ., t-times, where t s n irbi is a fixed integer independent of i. We consider an irreducible character E of Ws of the type E , E1 G ??? G Er ,

where Ei , EiX G ??? G EiX g Wi ,ns i , with EiX g Sbni .

Ž 2.22.1. Assume that the class  s4 is F-stable. Then there exists w 1 g Z s as in 2.8 n such that s is Fw 1-stable. We assume that E g ŽWs .ex . Let bi be a X partition of bi corresponding to Ei , and let m i be the partition of n i which is dual to Ž bi , . . . , bi ., t-times. ŽHere we regard Ž bi , . . . , bi . as a partition of n i s bi t..

210

TOSHIAKI SHOJI

Let Ni be a nilpotent element in g l n i corresponding to m i , and let N s Ý Ni g g l n s [g l n i . Hence N is a nilpotent element corresponding to the partition m s Ž m 1 , . . . , m r .. Then we have Or s ON for r s r s, E . Let ˜i and let L ˜ i be its Levi subgroup P˜i be the parabolic subgroup of G associated to Ni as in 2.2. We define a parabolic subgroup P˜ and its Levi ˜ of G˜ by P˜ s P˜1 = ??? = P˜r and L ˜sL ˜1 = ??? = L ˜ r . Put P s P˜ subgroup L ˜ l G and L s L l G. If we denote by l the linear map u 1 ª k corresponding to N as in 2.2, then ZL˜Ž l., ZLŽ l., and ZL0 Ž l. are described in a similar way as in 2.15 for m s Ž1m 1 , 2 m 2 , . . . .. We can write the partition m* s Ž mU1 , . . . , mUr . as m* s Ž a , . . . , a ., ttimes, where a s Ž b 1 , . . . , br ., regarded as a partition of nrt. Then as in the proof of Lemma 2.16, m* can be expressed as m* s Ž a 9, . . . , a 9., t9-copies of a 9, where a 9 is a partition of nrt9 consisting of trt9 copies of ˜ , M˜1 = ??? = M˜t 9 be the decomposition of M˜ with respect to a . Let M ˜0 be the factor of M˜i corresponding to the partition Ž a 9, . . . , a 9., and let M of a as in the proof of Lemma 2.18. We now assume the following property for s.

˜0U be the projection of s g M*. ˜ Then s0 is contained Ž2.22.2. Let s0 g M ˜0U is regarded as a subgroup of GLn r t . in the center of GLn r t when M ˜ ª G*. Note that G* , We consider the natural projection p : G* ˜ ˜ where Z* ˜ is the center of GLn which is regarded as a subgroup of G*rZ*, ˜ Let s s p Ž s . g G*. Then Ws can be written as Ws s Ws V s , where V s G*. 0 Ž . is a cyclic group isomorphic to ZG* Ž s .rZG* s . We now assume that Ž2.22.3. V s s ² w 0 :, where w 0 g W is an element of order t, permuting transitively factors of Wi, s i for each i. Let F9 s w 1 F and let t9 be the order of V Fs 9. We define a subgroup of ZL˜Ž l. as in Ž2.15.2.. Then under the assumption of Ž2.22.2. and Ž2.22.3. the arguments given in 2.16]2.21 can be applied without change. In particular, we have the following. ZL1 Ž l.

COROLLARY 2.23. Let Ts, E be a set of irreducible characters of G F as defined in 2.20. Put Ž Al .F s ZLŽ l.rZL1 Ž l., AlF s ZLŽ l. F rZLX Ž l. F , and let Ms, N s Ž Al .F = Ž AlF . n. Then the set Ts, E is parametrized by Ms, N as gi¨ en in Corollary 2.21.

3. SHINTANI DESCENT IDENTITIES 3.1. In this section we formulate Shintani descent identities, and discuss some consequences in the case of special linear groups. In what follows, we

211

SHINTANI DESCENT

˜ s GLn , for simplicity. However, state the results only for G s SL n and G ˜ similar results hold by an appropriate modification for the groups G and G discussed in 2.22. First we review the notion of twisted induction. Let ˜ s GLn as in Section 2. We follow the notation in 2.8. For G s SL n and G a subset J of P, we consider a standard parabolic subgroup PJ s L J UJ of type J, where L J is the Levi subgroup of PJ containing T, and UJ is the unipotent radical of PJ . We put L s L J . For a w g W such that Fw Ž J . s J, choose a representative w ˙ g NG ŽT .. We consider the variety S s  g g G < gy1 F Ž g . g F Ž wU ˙ J . 4 rUJ l F Ž w˙UJ . . Then G F = LF w˙ acts naturally on S, and we get an induced action of G F = LF w˙ on Hci Ž S . s Hci Ž S, Q l .. For each irreducible LF w˙-module p , a virtual G F -module R GLŽ w˙ .Žp . is defined by R GLŽ w˙ . Ž p . s

Ý Ž y1. i Ž Hci Ž S . m p .

L F w˙

.

iG0

Extending the correspondence p ¬ R GLŽ w˙ .Žp . linearly, we get a linear map R GLŽ w˙ . : C Ž LF w˙r;. ª C Ž G F r;., which is called the twisted induction. ˆ F is partitioned into subsets E Ž G F,  s4. according to the F-stable Now G semisimple classes  s4 in G*. For an F-stable class  s4 , we denote by C Ž s. Ž G F r;. the subspace of C Ž G F r;. spanned by irreducible characters belonging to E Ž G F ,  s4.. If we take an Fw-stable class  s4 in L*, the class ˙  s4 in G* is F-stable. It is known that R GLŽ w˙ . maps the subspace C Ž s. Ž LF w˙r;. into C Ž s. Ž G F r;.. Following wS2x, we shall define a twisted version of R GLŽ w˙ . by a Frobenius map F m. Take a positive integer m such that the map F m stabilizes S. Then it induces a map Ž F m .* on Hci Ž S .. We now assume that m is large enough so that F m acts trivially on LF w˙. Then Ž F m .* stabilizes the Fw ˙ subspace Ž Hci Ž S . m p . L , and one can define a map R ŽLŽm.w˙ . Ž p . Ž x . s

Ý Ž y1. i Tr ž Ž F m . * Ž xy1 . *, Ž Hci Ž S . m p .

iG0

L F w˙

/

for each x g G F. Extending linearly, we have a linear map R ŽLŽm.w˙ . : C Ž LF w˙r; . ª C Ž G F r; . . Note that R ŽLŽm.w˙ . also maps C Ž s. Ž LF w˙r;. to C Ž s. Ž G F r;. for a class  s4 as above.

˜ J UJ be the parabolic subgroup of G˜ of type J such that 3.2. Let P˜J s L ˜ J l G. We put PJ s P˜J l G, L J s L ˜< gy1 F Ž g . g F Ž wU S˜ s  g g G ˙ J . 4 rUJ l F Ž w˙UJ . .

212

TOSHIAKI SHOJI

˜F = L ˜FJ w˙ acts naturally on S, ˜ and we have an isomorphism S˜ , Then G GF F F ˜ ˜ G = S compatible with G -action. It follows that ˜ F x mQ l wG F x Hci Ž S . . Hci Ž S˜. , Q l w G

Ž 3.2.1.

The action of F m on S˜ induces an action Ž F m .* on Hci Ž S˜.. This action is compatible with the isomorphism in Ž3.2.1., where Ž F m .* acts trivially on ˜ F x. Now it is known Že.g., wS2, Proposition 1.6x. that if m is large Q lwG enough, then the eigenvalues of F m on Hci Ž S˜. are integral powers of q m r2 . Hence by Ž3.2.1., a similar result holds also for Hci Ž S ., i.e., Ž3.2.2. There exists an integer m 0 such that if m is divisible by m 0 , then the eigenvalues of F m on Hci Ž S . are integral powers of q m r2 . 3.3. In order to describe Shintani descent identities, we shall review here the definition of the map a F w˙ . For a fixed positive integer m, we m consider an irreducible representation p of LF . We assume that p is m Fw-stable. Let us denote by s the restriction of F on LF . Then the ˙ m restriction of Fw ˙ on LF ism written as s w. ˙ Let p˜ be an extension of p to the semidirect product LF ² s w ˙ :, and we denote by V the representation space of p ˜ . The representation p is naturally lifted to the representation m of PJF , which we denote also by p . Let P be the space of all functions m m f : G F ª V endowed with an G F -module structure by Ž gf .Ž x . s m f Ž xg . Ž g, x g G F .. We define a subspace Pp of P by Pp s  f g P < f Ž pg . s p Ž p . f Ž g . for p g PJF , g g G F m

m G Fm F

m

Then Pp is a G F -submodule of P, which realizes Ind P that Fw ˙ Ž J . s J, we can define a linear map tp , w˙ on P as

tp , w˙ Ž f . Ž x . s < UwFJ
Ý

m

4.

p . Noticing

f Žw ˙y1 yx . .

Ž 3.3.1.

Fm

ygUw J

Let us define F: P ª P by F Ž f .Ž x . s f Ž Fy1 Ž x .., and p ˜ Ž s w˙ .: P ª P F m. Ž .Ž .Ž . Ž . Ž . Ž by p s w f x s p s w f x f g P , x g G . Then it is easily veri˜ ˙ ˜ ˙ Ž . fied that a linear map p s w F t : P ª P leaves P ˜ ˙ p , w˙ p invariant. We m m define a map a F w˙ : C Ž LF r;F w˙ . ª C Ž G F r;F . by a F w˙ Ž p ˜ . Ž x . s Tr Ž xp˜ Ž s w ˙ . Ftp , w˙ , Pp . m

m

Ž x g G F . , Ž 3.3.2.

for each p to the whole ˜ g C Ž LF r;F w˙ . and then extending linearly Fm Ž . space. ŽHere we regard p as an element in C L r; ˜ Fw ˙ by taking its m restriction on LF s w . The collection of such p corresponding to Fw-sta˙ ˜ ˙ m ble irreducible characters gives rise to a basis of C Ž LF r;F w˙ ... Now the following formula can be proved in an entirely similar way as in wS2, S3x.

213

SHINTANI DESCENT

PROPOSITION 3.4 ŽShintani Descent Identity.. Under the notation in 3.3, the following diagram is commutati¨ e. Sh F m r F

C Ž G F r;. 6

6

m

C Ž G F r;F . 6

R ŽLŽm.w˙ .

a F w˙ Sh F m r F w˙

C Ž LF w˙r;..

6

m

C Ž LF r;F w˙ .

m

3.5. Let d be an irreducible cuspidal representation of LF . In order to make the map a F w˙ more comprehensive, we relate it to the Hecke algebra associated to the induced representation of d . For this, we shall describe the structure of the Hecke algebra. Put Wd s  w g W < wJ s J , wd , d 4 . According to Howlett and Lehrer wHLx, Wd can be decomposed as Wd , Wd0 V d , where Wd0 is a normal subgroup of Wd which is a reflection group with a set of simple reflections Sd associated to some root system G ; S Žin the sense of wHL, Sect. 2x., and V d is given by V d s  w g Wd < w Ž Gq . ; Gq 4 , where Gqs G l Sq is the set of positive roots in G. We consider the m induced representation Pd on G F . It was proved in wHL, Theorem 6.1x, for reductive groups G in general, that the endomorphism algebra H Ž d . s End G F m Ž Pd . is isomorphic to the group algebra Q l w Wd x m , twisted by a certain two cocycle m. In the case where G has the connected center, Lusztig wL2x proved that m is trivial. This fact was also verified for G s SL n by Lehrer wLex. Finally, the triviality of m was proved in general by M. Geck wGx by reducing the problem to Lusztig’s results. Let M be the subgroup of NG Ž L. generated by L and w g Wd . Then d m can be extended to a representation of M F . In fact, d can be extended to Fm a projective representation of M , and according to wHLx the corresponding two cocycle is cohomologous to m , which is trivial as remarked above. m We fix an extension d˜ of d to M F . Then according to wHL, Theorem m 4.14x there exists a suitable extension d˜ of d to M F satisfying the

214

TOSHIAKI SHOJI

following. H Ž d . has a basis Tw Ž w g Wd . over Q l with relations

Ž i. Ž ii . Ž iii .

Tw Tx s Tw x Tw Tw 9 s Tw w 9

for w g Wd , x g V d if ˜ l Ž ww9 . s ˜ lŽ w. q ˜ l Ž w9 . Ž w, w9 g Wd0 .

Ž Ts q 1 . Ž Ts y q m lŽ s. . s 0

for s g Sd .

Ž 3.5.1. Here ˜ l is the length function of Wd0 with respect to Sd , and l: Sd ª Z ) 0 is a function which takes constant values under Wd-conjugate. Furthermore, Tw : Pd ª Pd is constructed from td , w˙ Žsee Ž3.3.1.. by the formula Tw s « wŽ m. Ž qw .

mr 2

˜

q lŽ w . m r2d˜Ž w ˙ . td , w˙ ,

where, if w g Wd is written as w s yx Ž y g Wd0 , x g V d ., qw is given as qw s Ł s q lŽ s., s runs through the elements in a reduced expression of y in Wd0 . Furthermore, w ¬ « wŽ m. s "1 is a linear character of Wd which is trivial on V d . m

3.6. We now consider a cuspidal character d of LF more precisely. ˜sL ˜ J is of the form L ˜,L ˜1 = ??? = L ˜ k , where L ˜ i is a Assume that L ˜ Ž standard Levi subgroup such that L i , GLn i = ??? = GLn i d i-times.. Let ˜F m of the following type; d˜ , d˜1 G ??? G d˜r , d˜ be a cuspidal character of L ˜ i such that d˜i , d˜iX G ??? G d˜iX , with a cuspidal where d˜i is a character of L m m X F character d˜i of GLn i . Then the restriction of d˜ on LF is a sum of m cuspidal characters of LF . We denote by d one of the irreducible constituents. Then by Lehrer wLe, Theorem 10x, Wd s Wd0 V d is given as Wd0 , S d1 = ??? = S d k, V d , ZrtZ, for some integer t ) 0, and V d acts on Wd0 as permuting the factors in the direct product. We note also that the ˜ F m ism isomorphic to Wd0 . Let H Ž d˜. be the ramification group Wd˜ of d˜ in G ˜F G ˜F mendomorphism algebra of V s Ind P˜F m d˜. Then the restriction of the G m Fm G Fm Ž ˜< F m . module V to G coincides with V0 s Ind P F d L . Hence any element T g H Ž d˜. gives rise to an element of E s End G F m Ž V0 .. Then we have Ž3.6.1. Let H ŽWd0 . be the subalgebra of H Ž d . generated by Tw Ž w g Wd . Then H ŽWd0 . is isomorphic to H Ž d˜.. In particular, the structure of H Ž d . is completely determined by H Ž d˜. and by Wd . 0.

In fact, by wLe, 4.17x, we see that E , H Ž d .. Hence we have an algebra homomorphism H Ž d˜. ª H Ž d ., which is clearly injective. The definition of ˜ F m implies that the image of td , w˙ in Ž3.3.1., and the similar formula for G 0 this map is equal to H ŽWd .. This shows Ž3.6.1.. We now consider a set Zd s  w g W < F Ž wJ . s J , F wd , d 4 .

215

SHINTANI DESCENT

Then Zd can be written as Zd s wd Wd for some wd g W. We choose wd so that Fwd Ž Gq. ; Gq. But contrast to the case where the center is connected, wd is not unique, and in fact, such wd form a coset of V d . Anyway, ˜ i Ž1 F i F r .. Let gd : Wd ª Wd be the automorphism Fwd permutes L ˜d s Wd ²gd : be the induced by the map Fwd . Then gd stabilizes Wd0 . Let W semidirect product of Wd with the cyclic group generated by gd . Then the Hecke algebra H Ž d . can be extended to an algebra H˜Ž d . with basis ˜d .. We denote by ŽWd .exn the set of isomorphism classes of Tw Ž w g W ˜d-module over Q l . irreducible Wd-modules which are extendable to a W 0 Note, by using the structure of Wd s Wd V d , we see that any irreducible Wd-module E is chosen to be over K, where K is a cyclotomic field QŽ z . with z , a primitive t th root of unity. Let EŽ q m . be the corresponding n irreducible H Ž d .-module. Now, for each E g ŽWd .ex , let E˜ be an extenm. ˜ ˜ Ž sion to Wd . Then corresponding to E, E q can be extended to an H˜Ž d .-module, which we denote by E˜Ž q m .. Note, since gd and V d acts as a permutation on S d1 = ??? = S d k, that one can choose EŽ q m . and E˜Ž q m . in a standard way. It follows from this, for each w g Wd , that Tr Ž Tw , E Ž q m . . g K q m r2 ,

Tr Tgd w , E˜Ž q m . g K q m r2 . Ž 3.6.2.

ž

/

Now, the discussion given in 1.12]1.13 in wS2x works also in our case without change, and we have the following formula for sufficiently large m. m

Ž3.6.3. For each x g G F , y g Wd , we have Tr x d˜Ž s w ˙d . Ftd , w˙d Ty , Pd

ž

s

Ý

n Eg ŽWd .ex

/

˜ qyl Ž wd . m r2 Tr Ž x s , r˜E . Tr Tgd y , E˜Ž q m . .

ž

m

/

Here r˜E is an extension of the irreducible G F -module r E , corresponding n ˜ F m . The extension is uniquely determined by the to E g ŽWd .ex , to G choice of d˜ and E˜Ž q m .. We now extend the length function ˜ l on Wd0 to the whole of Wd by 0 ˜l Ž x¨ . s ˜l Ž x . for x g Wd , ¨ g V d . We note here that the values lŽ s . for s g Sd are constant in our case, which we denote by l. For each E g Wd n , we define Dim E by Dim E s Dim E1 , where E1 is an irreducible character of Wd0 appearing in the restriction of E to Wd0 , and Dim E1 is the corresponding formal dimension of the H Ž d˜.-module E1Ž q m .. Note Dim E1 is independent of the choice of E1. Then the following formula is easily verified using similar arguments given in wL1, Ž1.5.5.x.

216

TOSHIAKI SHOJI n Ž3.6.4. For each E, E9 g ŽWd .ex ,

Ý xgWd

˜ ˜ Ž qm. qym l lŽ x . Tr Tgd x , E˜Ž q m . Tr Txy1 gdy 1 , E9

ž

s

½

/ ž

˜

/

Ý x g Wd q m l lŽ x . dim ErDim E

˜ if E˜ , E9,

0

if E ` E9.

Now it follows from Ž3.6.3. and Ž3.6.4. that TrŽ x s , r˜E . can be expressed as a linear combination of a F w˙d ˙y Ž d˜.Ž x . for various y g Wd . Since the map R LŽ w˙d ˙y . preserves the subspaces corresponding to the class  s4 , Proposition 3.4 implies the following result. PROPOSITION 3.7. Assume that Sh F m r F w˙ 1 ˙y Ž d < LF ms w˙d ˙y . g C Ž s. Ž LF w˙d ˙yr;. for any y g Wd . Then we ha¨ e Sh F m r F Ž r˜E . g C Ž s. Ž G F r;.. 3.8. By making use of the formula Ž3.6.3., one can obtain an explicit description of the map a F w as given in wS2, Lemma 1.15x. Let p be an m Fw-stable irreducible Žnot necessarily cuspidal. representation of LF . ˙ Then there exists a Levi subgroup L K Ž K ; J . and a cuspidal representam tion d of LFK such that p is isomorphic to p E9 , the irreducible represenFm tation of L corresponding to E9, where E9 is an irreducible representation of WdX s ŽWJ .d . Since p is Fw-stable, d is Fww9-stable for some ˙ ˙˙ ˆFKm . We define ZdX ; WJ in w9 g WJ . Let Zd be as before defined for d g L a similar way as Zd , but replacing F by Fw, ˙ and W by WJ . Then one can choose wd and wdX such that Zd s wd Wd , ZdX s wdX WdX , as in 3.6. Accordingly, we have automorphisms gd : Wd ª Wd , gdX : WdX ª WdX . We have an irreducible H˜Ž d .-module E˜Ž q m . as before. On the other hand, the semidirect ˜dX s WdX²gdX : is defined similar to W˜d . Let H9Ž d . be the product group W ˜ Ž d . be the subalgebra of H Ž d . generated by Tw Ž w g WdX ., and let H9 ˜dX . Then E9 g ŽWdX . n is gdX-stable, and extended algebra corresponding to W ˜ Ž d .-module E9 ˜ Ž q m . corresponding to an extenone gets an irreducible H9 X ˜ of E9 to W˜d . Put sion E9 Ž m. ˜ Ž q m . , E˜Ž q m . . . VE9, ˜ E˜ s Hom H 9Ž d . Ž E9

Now w can be written as w s wd ywdXy1 for some y g Wd . We define an Ž m. Ž m. Ž m. Ž m. y1 endomorphism gwŽ m. : VE9, ˜ E˜ ª VE9, ˜ E˜ by gw s Tg y ( f (Tg 9 for f g VE9, ˜ E˜, where g s gd , g 9 s gdX . Then the following lemma is proved in a similar way as in wS3, 1.15x. LEMMA 3.9. Assume that m is large enough. Let p s p E9 be an Fw-stam ble irreducible character of LFJ , and put w s wd ywdXy1 as in 3.8. Then we

217

SHINTANI DESCENT

ha¨ e aF w Žp ˜ E9 < LF ms w . s « yŽ m. Ž q y .

ym r2

˜

˜

˜

X

qyŽ lŽ wd .qlŽ y .ylŽ wd .. m r2

Ý

n Eg ŽWd .ex

ˆF where p ˜ E9m Ž resp. r˜E . is an extension p E9 g L F ˜ ., respecti¨ ely. Ž resp. G

m

Ž m.

Tr gwŽ m. , VE9, ˜E < G F ms , ˜ E˜ r

ž

/

m

ˆ F . to L ˜F Ž resp. r E g G

m

4. MAIN RESULTS 4.1. In this section, we parametrize irreducible characters of SL nŽFq . in terms of generalized Gelfand]Graev characters, and then determine the Shintani descent of irreducible characters. In order to make the induction process smooth, we consider more ˜ and G such as discussed in 2.22. We follow the notation general groups G in the first part of 2.22. Let s g T˜* and Ws be as in 2.22. Concerning the structure of Ws , we pose the same assumption as Ž2.22.2.. Under such a setup, we consider an irreducible character E g Ws n of the following type, which is more general than the one considered in 2.22; E , E1 G ??? G Er , with Ei g Wi,ns i and Ei , Ei1 G ??? G Ei1 G ??? G Ei k i G ??? G Ei k i ,

Ž 4.1.1.

where Ei1 , . . . , Ei k i are distinct irreducible characters of Sb i , and Ei j appears d i j-times. Now assume that s is F9-stable, where F9 s Fw 1 is as in 2.22. We denote by g an automorphism on Ws induced by F9. We n ˜s assume here that E g ŽWs .ex and consider a Levi subgroup L ˜ ˜ ˜ ˜ L1 = ??? = L r ; G according to the decomposition of E, where L i , ˜ i1 = ??? = L ˜ i k i with L ˜ i k i , GLb i d i j . Then clearly Ws coincides with the L ˜ Note that L ˜ is stabilizer WL*, ˜ s of s in the Weyl group WL* ˜ of L*. F9-stable. Replacing s by its conjugate in W if necessary, we may assume ˜ is F-stable. Hence w1 g WL* further that L ˜ . It follows that the map ˜ is regarded as a map defined in 2.8 replacing W by WL* F9 s Fw 1 on L* ˜ . ˜ We consider a standard parabolic subgroup P˜ of G with Levi subgroup L. ˜ l G. For Define F-stable subgroups of G by P s P˜ l G and L s L n ˜ F . n as in 2.8. Let r 9 s r s,L˜ E be the E g ŽWs .ex , we consider r s r s, E g Ž G F ˜ defined in a similar way as in 2.8 for E g irreducible character of L n ŽWL˜*, s .ex . Let us denote by T Žresp. T 9. the set of irreducible characters of G F Žresp. LF . appearing in the decomposition of r < G F Žresp. r 9 < LF ., respectively. Then we have the following lemma.

218

TOSHIAKI SHOJI

LEMMA 4.2. For any r 0 g T 9, we ha¨ e r X0 s Ind GP F r 0 g T. Under the map r 0 ¬ r X0 , the set T 9 is in bijecti¨ e correspondence with T. F

˜F G

Proof. By the transitivity of the twisted induction, we see that Ind P˜F r 9 F s r . It follows that Ind GP F r 0 is a direct summand of r < G F . Also we know that r < G F and r 9 < LF are multiplicity free. Hence to prove the lemma, it is 0 Ž . enough to show that < T < s < T 9 <. Let V s, L s ZL* Ž s .rZL* s , where s de˜ ª L*. Then F9 acts notes the image of s under the natural map L* naturally on V s and V s, L . Let V s Ž E . Žresp. V s, LŽ E .. be the stabilizer of E in V s Žresp. V s, L ., and V Fs 9Ž E . Žresp. V Fs, 9LŽ E .. be the subgroup of F9-fixed points. Then it follows from wL3, Proposition 5.1x that we have < T < s < V Fs 9Ž E .< and < T 9 < s < V Fs, 9LŽ E .<. Now it is easy to see that V s Ž E . , V s, L , ZraZ, where a s gcd d i j 4 . Hence, if we denote by t9 the order of the cyclic group V Fs 9, we have V Fs 9Ž E . , V Fs, 9L , ZrcZ with c s gcd t9, d i j 4 . Since any element in V s, L stabilizes E, we get V Fs, 9LŽ E . , ZrcZ. This shows < T < s < T 9 <, and the lemma is proved.

˜ be as in 4.1. We shall consider a parametrization of 4.3. Let G and G ˜ F . n in advance to that of Gˆ F. By a general theory, Gˆ F is partitioned ŽG into subsets E Ž G F ,  s4., where  s4 runs over F-stable semisimple classes in G*. We fix s g T * for a given F-stable semisimple class  s4 ; G*, and choose s g T˜* such that p Ž s . s s and that the class  s4 is F-stable. One can find w 1 g Z s and an isomorphism g s Fw 1: Ws ª Ws as in 2.8. Now Ws is written as Ws s Ws V s . For each x g V s , we choose s x g ŽT˜*. x F 9 such that p Ž s x . s s. Take z g V s and put y s zxF9Ž zy1 . g V s . Then z ŽT˜*. x F 9 zy1 s ŽT˜*. y F 9, and we may take s y s zs x zy1. Clearly Ws x s Ws y s ˜ we have Ws . Since s x and s y are in a same class in G*, F  4. F  4. ˜ ˜ Ž Ž E G , s x s E G , s y . However, the parameter sets for them are dif˜ F,  s x 4. is parametrized by ŽWs n .g x , the set of ferent. The former set E Ž G gx-stable irreducible characters of Ws , with gx s xF9: Ws ª Ws , and the latter is done by ŽWs n. g y . The relation between these two parametrizations is described as follows. Ž4.3.1. The map E ª E9 sz E gives a bijection ŽWs n. g x , ŽWs n. g y . For each E g ŽWs n .g x , under an appropriate choice of extensions E˜ and ˜ we have E9, r s x , E s r s y , E9 . In fact, the first assertion is clear. We show the equality. Now r s x , E and r s y , E9 are defined by the formula in Ž2.8.1.. We fix an extension E˜ to ˜ to Ws²gy : by the isomorphism ad z: Ws²gx : , Ws²gx : and determine E9 m Ws²gy :. For each w g Ws , we consider the norm maps NF m r x F 9w : T F ª x F 9w Fm y F 9w T and NF m r y F 9w : T ª T for sufficiently large m. Since the m m maps NF m r x F 9w and NF m r y F 9 z w zy 1 commute with ad z: T F ª T F and

219

SHINTANI DESCENT y1

ad z: T x F 9w ª T y F 9 z w z , the verification of the formula is reduced to the following statement: let ad z: T w F , T w 9F with w9 s zwF Ž zy1 . for w, z g W. Take u g Tˆ w F and let zu s u ( zy1 g Tˆ w 9F . Then we have R T wŽ u . s R T w 9Ž zu . .

Ž 4.3.2.

But using the orthogonality relations for R T Ž u ., we can check that the inner products ² R T wŽ u ., R T wŽ u .:, ² R T w 9Žzu ., R T w 9Žzu .:, and ² R T wŽ u ., R T w 9Žzu .: are all the same. This implies Ž4.3.2. and so Ž4.3.1. follows. 4.4. We now give a parametrization of irreducible characters of G F. We fix a set of representatives of V sr;F 9 and identify it with Ž V s .F 9. For each x g Ž V s .F 9 we define s x g ŽT˜*. x F 9 and gx : Ws ª Ws as in 4.3. Let ŽWs n. g xrV gs x s ŽWs n. g xrV Fs 9 be the set of V gs x-orbits in ŽWs n. g x . For ˜ F. Let Tx, E9 be the set of each E9 g ŽWs n .g xrV Fs 9, we consider r s x , E9 g G F irreducible characters of G appearing in the restriction of r s x , E9 to G F. Then by wL3x, the set Tx, E9 are mutually disjoint and E Ž G F ,  s4 . s

@ Tx , E9 ,

Ž 4.4.1.

Ž x , E9.

where Ž x, E9. runs over all x g Ž V s .F 9 and E9 g ŽWs n. g xrV Fs 9. Moreover, the set Tx, E9 is parametrized by V sx F 9Ž E9. ns V Fs 9Ž E9. n. But this correspondence depends on the choice of r 0 g Tx, E9 and so it is not necessarily canonical. We now modify the parameter set of the right hand side of Ž4.4.1.. The following formula is essentially due to C. Bonnafe. ´ The author is very grateful to him for it. Ž4.4.2. There exists a natural bijection f:

@ n

Eg ŽWs rV s

.F9

V s Ž E9 . F 9 ,

@ Ž Ws n .

gx

rV Fs 9 ,

xg Ž V s .F9

where ŽWs n rV s . F 9 means the set of F9-stable V s-orbits in Ws n . We show Ž4.4.2.. We choose an element E g O for each F9-stable V s-orbit O in Ws n and fix them. Then there exists a E g V s such that

 u g V s < u F 9E s E 4 s V s Ž E . a E . We fix such a s a E for each E. Now for each y g V s Ž E ., one can find x g Ž V s .F 9 and z g V s such that ya s zy1 xF9Ž z .. Since y a F 9E s E, we have x F 9Žz E . sz E, and so E9 sz E g ŽWs n. g x . The correspondence y ¬ E9 induces a well-defined map from V s Ž E . to @x ŽWs n. g xrV Fs 9. It is easy to

220

TOSHIAKI SHOJI

see that this gives an injective map V s Ž E . r;F 9 ª

@ Ž Ws n .

gx

rV Fs 9 .

xg Ž V s .F9

The surjectivity of f is also clear from the construction of the map. Hence Ž4.4.2. holds. 4.5. Let E g ŽWs n rV s . F 9. For each y g V s Ž E .F 9 , one gets a pair Ž x, E9. s f Ž y . by Ž4.4.2.. As described in 4.4, the set Tx, E9 is in bijection with V Fs 9Ž E9. n. But since E9 sz E for z g V s , we see that V Fs 9Ž E9. ns V Fs 9Ž E . n. Let Ts, E be the set of unions of various Tx, E9 where Ž x, E9. runs over all the pairs in the image of V s Ž E .F 9 under f. It follows from Ž4.4.1. and Ž4.4.2., we have

@

E Ž G F ,  s4 . s

Ts, E .

Eg ŽWsnrV s . F 9

We define a set Ms, E by Ms, E s V Fs 9Ž E . n= V s Ž E .F 9. Then it follows from Ž4.4.2. that Ts, E is in bijection with the set Ms, E . But this parametrization is not canonical. The set Ts, E has another description. Let a s a E be as in 4.4 for E g ŽWs n rV s . F 9. Since ya s zy1 xF9Ž z . and E9 sz E for some z g V s , we see that r s y , E s r s x , E9 by Ž4.3.1., where s y s zy1 s x z g ŽT˜*. y a F 9 and E g ŽWs n. a F 9. Hence if we define a set Ts y , E as before using r s y , E , the set Ts y , E coincides with Ts x , E9. It follows that Ž4.5.1. Under the correspondence Ts, E l Ms, E , the set Ts , E correy sponds to the set V Fs 9Ž E . n= y4 , and we have Ts, E s

@

Ts y , E .

ygV sŽ E .F9

We shall give a parametrization of Ts, E in terms of generalized Gelfand]Graev characters, i.e., Ž4.5.2. There exists a natural bijective correspondence Ts, E l Ms, E for each E in ŽWs n rV s . F 9 and so we have a natural bijection E Ž G F ,  s4 . ,

@

Ms, E .

Eg ŽWsnrV s . F 9

We construct a bijection Ms, E l Ts, E inductively according to the rank of G. So, we assume that the bijection was constructed for any proper Levi subgroup L of G. Ža. First we consider the case discussed in 2.22, i.e., Ws , W1, s1 = ??? = Wr, s r, where Wi, s i is of type Ž bi , . . . , bi ., t-times, for an integer t s n irbi independent of i, V s , ² w 0 :, where w 0 g Ws is an element of

221

SHINTANI DESCENT

order t, permuting the factors of Wi, s i transitively Žcf. Ž2.22.3... Furthern more, E g ŽWs .ex is given as in Ž2.22.1.. Note, in our setting, that s g T˜* does not necessarily satisfy the condition Ž2.22.2.. However, replacing ˜ F, i.e., by r s r s, E by u m r , where u is a suitable linear character of G ˜ . F, one replacing s by zs, where z is a suitable element in the center ZŽ G* can achieve the condition Ž2.22.2. for zs. We have a natural bijection Tr , Tu mr , and this bijection does not depend on the choice of u . So, for the parametrization we may assume that s satisfies Ž2.22.2.. Now by Ž2.19.3., we can choose an irreducible character r 0 s rŽ c, j . g Tr . Then by 0 making use of r 0 , we get a bijection between Ž Al .F and V Fs 9Ž E . n. This gives rise to a natural bijection between Ms, N in 2.20 Žand in 2.23. and Ms, E given in this section. Hence by Corollary 2.23, we get a natural bijection between Ms, E and Ts, E . Žb. Next we consider the case discussed in Subsection 4.1, i.e., Ws is n similar to case Ža., but E g ŽWs .ex is given as in Ž4.1.1.. We assume that n F9 Ž . E g Ws rV s , and put F0 s a E F9. Then E g ŽWs n. F 0 . We consider an ˜ as in 4.1. Then L ˜ is also F0-stable, and F-stable Levi subgroup L if we write F0 s Fw 2 , we see that w 2 g WL* ˜ as in 4.1. It follows from the proof of Lemma 4.2, that we have V s Ž E . , V s, L . Hence, Ms, E , n F0 Ž V Fs, 0L . n=Ž V s, L .F 0 . On the other hand, E g ŽWL*, satisfies the prop˜ s. Ž . erty in case a , and so we have a natural bijection Ms,LE l Ts,LE , where Ms,LE , Ts,LE are corresponding objects in L, and in particular, Ms,LE s Ž V Fs, 0L . n=Ž V s, L .F 0 which coincides with Ms, E . Now Lemma 4.2 implies that Ts,LE is in bijection with Ts, E under the map r 0 ¬ Ind GP FF r 0 Ž r 0 g Ts,LE .. In view of Ž4.5.1., this gives a bijection Ms, E l Ts, E . Žc. We consider a general case. Let Ws , W1, s = ??? = Wr, s , and 1 r Ws , Ws V s with V s s ² w 0 :. Here we assume that for some i, w 0 permutes the factors of Wi, s i non-transitively. In this case, there exists a standard Levi subgroup L* ; G* such that Ws is contained in WL* and that L* is both F-stable and F9-stable. Then ZG* Ž s . is contained in L*. Under such a condition, it is known that the twisted induction R GLŽ w˙ 1 . induces a bijection between E Ž LF 9,  s4. and E Ž G F ,  s4.. If we denote by Ms,LE and Ts,LE the corresponding objects with respect to L, we have a bijection Ms,LE l Ts,LE by the induction hypothesis. Since V s Ž E . s V s, LŽ E ., we have a natural bijection Ms, E , Ms,LE . Then the twisted induction R GLŽ w˙ 1 . induces a bijection Ms, E l Ts, E compatible with the bijection Ms,LE l Ts,LE . This gives the required bijection, and Ž4.5.2. holds. m

4.6. We shall parametrize F-stable irreducible characters of G F for m sufficiently large m. We consider a set E Ž G F ,  s4. for an F-stable class  s4 . As in 4.4, we fix s g T * for each F-stable class  s4 , and choose s g T˜* ˜ is F-stable. We assume that such that p Ž s . s s and that the class  s4 ; G* F9Ž s . s s for F9 s Fw 1 as before. We take m large enough so that

222

TOSHIAKI SHOJI m

s g T˜* F for all s, and that for such s, F m acts trivially on V s . For each Ž m. E g Ws n , we denote by Ms,Ž m. E and Ts, E the set Ms, E and Ts, E defined m m in 4.5 replacing F9 by F . Then E Ž G F ,  s4. is a disjoint union of Žm. Žm. various Ts, E , and Ts, E is in bijection with Ms,Ž mE .. Note by Ž . n= V s Ž E .. We now define a subset our assumption, that Ms,Ž m. E s Vs E Ž m. n Ž . Ms, E of Ms, E by Ms, E s V s E ex = V s Ž E . F 9. Then the following gives the m description of F-stable irreducible characters in E Ž G F ,  s4.. Ž4.6.1. Let Ž Ts,Ž m. . F be the set of F-stable irreducible characters in E Ž m. Ž m. F Ts, E . Then Ž Ts, E . / B if and only if E g ŽWs n rV s . F 9. For such E, Ž m. Ž Ts,Ž m. . F corresponds to Ms, E under the bijection Ts,Ž m. E E l Ms, E . In particular, we have m

F

E Ž G F ,  s4 . ,

@

Ms, E .

Eg ŽWsnrV s . F 9

We show Ž4.6.1.. For each x g V s Ž E ., we choose s x g ŽT˜*. x F 9 and let ˜ F m defined similar to r s x , E in 4.3. First we note be the character of G that

r sŽxm. ,E

Ž m. F Ž r sŽxm. , E . s r F 9Ž s x ., F 9 E .

Ž 4.6.2.

In fact, since the twisted induction R T Ž w˙ . defined in 3.1 commutes with the action of F, we see easily that Ž m. F Ž r sŽxm. , E . s r F Ž s x ., F E ,

where F E is the character of WF Ž s x . obtained from E under the isomorŽ m. Ž . phism F: Ws x ª WF Ž s x . . But since F9 s Fw 1 , r FŽ m. Ž s x ., F E s r F 9Ž s x ., F 9 E by 4.3.1 . Ž . This shows 4.6.2 . Fm Now F9Ž s x . s sF 9Ž x . , and so the restriction of r sŽxm. is F-stable if , E to G and only if x g V Fs 9Ž E . and E g ŽWs n rV s . F 9. Take E g ŽWs n rV s . F 9. We need to describe F-stable irreducible characters in the restriction of r sŽxm. , E. Ž m. Ž . Ž . Under the construction of the bijection Ts,Ž m. E l Ms, E in 4.5, a , b , and Žc., each step is compatible with the action of F. Hence the description of Ž Ts,Ž m. . F is reduced to case Ža.. But in this case, the bijection is given as in E Corollary 2.23 in terms of generalized Gelfand]Graev characters. If the generalized Gelfand]Graev character is F-stable, the corresponding irreducible character in G F is F-stable. Using the parametrization of F-stable generalized Gelfand]Graev characters Žcf. 2.5 and Lemma 1.7., we see that the characters in Ts,Ž m. E corresponding to Ms, E are actually F-stable. By counting the total number of F-stable irreducible characters corre. F corresponds exactly to Ms, E . sponding to various Ms, E , we see that Ž Ts,Ž m. E This proves Ž4.6.1..

223

SHINTANI DESCENT

Now, under the correspondence Ms, E l Ts, E , we denote by r y the irreducible character in Ts, E corresponding to y g Ms, E . Also under the . F , we denote by r xŽ m. the F-stable irrecorrespondence Ms, E l Ž Ts,Ž m. E Ž m. ducible character in Ts, E corresponding to x g Ms, E . For x s Žh , z . g Ms, E and y s Žh 9, z9. g Ms, E , we define a pairing  x, y4 by

 x, y 4 s < V s Ž E . F 9
Ž 4.6.3.

n ŽHere we regard h g V s Ž E .ex as a linear character of V s Ž E .F 9 in a natural way.. We define a function R x g C Ž s. Ž G F r;. for each x g Ms, E by

Rx s

Ý  x, y 4 r y .

Ž 4.6.4.

yg Ms, E

The following result describes the Shintani descent of irreducible characm ters in G F . m

THEOREM 4.7. Let G be as in 4.1 and let s g T˜* F be an element such that the class  s4 is F-stable. Assume that m is sufficiently di¨ isible. For each m . F Ž x g Ms, E ., we fix an extension r˜xŽ m. of r xŽ m. to G F ² s :. r xŽ m. g Ž Ts,Ž m. E Then we ha¨ e Sh F m r F Ž r˜xŽ m. < G F ms . s m x R x ,

Ž 4.7.1.

where m x is a root of unity depending on the choice of r˜xŽ m.. In particular, m Sh F m r F gi¨ es an isomorphism from C Ž s. Ž G F sr;. to C Ž s. Ž G F r;.. Remark 4.8. In wL2x, Lusztig defined almost characters for connected reductive groups with connected center, and in wS2, S3x, it is shown that the image of the Shintani descent of irreducible characters coincides with almost characters Žunder some permutation of the labeling.. So, in our case, it would be appropriate to call the functions R x* the almost character associated to x g Ms, E , where x* is given by x* s Žhy1 , z . g Ms, E for x s Žh , z . Žcf. see 1.9.. 4.9. The latter statement follows from Ž4.7.1.. The remaining part of this section is devoted to the proof of the formula Ž4.7.1.. By induction on the rank of G, we may assume that the theorem was proved for any proper Levi subgroups L of G. We prove Ž4.7.1. for each F-stable semisimple class  s4 separately. In this subsection we show that the verification of Ž4.7.1. is reduced to the case where ZG* Ž s . is not contained in any proper Levi subgroup. So, assume that ZG* Ž s . is contained in a standard Levi subgroup L*. We may assume that we are in a situation in Žc. of 4.5. Let L be the Levi subgroup considered there. We consider the Shintani descent

224

TOSHIAKI SHOJI

identity given in Proposition 3.4 for F9 s Fw 1. Using the specialization argument as in wS2, S3x, one can convert the diagram of Shintani descent identity into the following commutative diagram, Sh F m r F

6

m

C Ž G F r;F . 6 a w1

RG LŽ w ˙ 1. Sh F m r F w˙ 1

C Ž LF w˙ 1r;..

6

m

C Ž LFF w˙r;.

C Ž G F r;. 6

Here a w 1 is a certain linear map obtained from a F w by specializing the formula in Lemma 3.9 Žcf. wS3, 3.6x.. Now in our case, Žy1. lŽ w 1 . R GLŽ w˙ 1 . sends irreducible characters in E Ž LF w˙ 1 ,  s4. onto irreducible characters in E Ž G F ,  s4.. Hence R GLŽ w˙ 1 . induces an isomorphism from C Ž s. Ž LF w˙ 1r;. to C Ž s. Ž G F r;.. On the other hand, it is verified that a w 1 induces an isomorm m phism from C Ž s. Ž LF r;F w˙ 1 . to C Ž s. Ž G F r;., where a w 1 sends meach ˆF m to its induction Ind GP FF m r 0 , Fw ˙1-stable irreducible character r 0 g L which is left irreducible.m Since the parametrization given in 4.3 and 4.4 are F compatible with Ind GP F m and R GLŽ w˙ 1 . , we see that Ž4.6.1. holds for such a class  s4 . 4.10. We assume that ZG* Ž s . is not contained in any proper Levi subgroup of G*. This implies that Ws satisfies the condition in Ž2.22.3.. We n . F separately. First we now consider, for each E g ŽWs .ex , the set Ž Ts,Ž m. E note the following. n ˜ ; G˜ be as in SubsecŽ4.10.1. Let E g ŽWs .ex be as in Ž4.1.1., and L Ž m. F ˜ ˜ tion 4.1. If L / G, then Ž4.7.1. holds for Ž Ts, E . .

In fact, as discussed in Žb. in 4.5, the sets Ts, E and Ts,LE are in bijection F L, Ž m. via the induction functor Ind GP Fm. Similarly, the sets Ts,Ž m. are in E and Ts, E G Fm .F bijection via the functor Ind P F . This induces a bijection between Ž Ts,Ž m. E L, Ž m. F and Ž Ts, E . . Now the former bijection corresponds to the natural bijecL, Ž m. tion Ms,Ž m. on the parameter sets. Since this bijection is compatiE , Ms, E ble with the action of F9, we see that the latter bijection is realized by the natural bijection Ms, E , Ms,LE . The assertion Ž4.10.1. follows from the fact that the inductions are compatible with Shintani descents for G and L. n . F is as in case Ža. of 4.5, i.e., E g ŽWs .ex 4.11. We now assume that Ž Ts,Ž m. E is in the setup in 2.22. In this case, the set Ts, E is parametrized by Ms, N by Corollary 2.23. Since m is large enough, the set Ts,Ž m. E is parametrized by n Ž m. . F Ž Ms,Ž m. s A = A . Then under this parametrization, T is parametrized N l l s, E F n by Al = Ž Al .ex , which we put Ms, N . For x s Ž c, j . g Ms, N and y s Ž c9, j 9. g Ms, N , we define a pairing  x, y4 by

 x, y 4 s < AlF
225

SHINTANI DESCENT

ŽHere we regard F-stable characters on Al as characters on Ž Al .F .. Note that under the natural bijection Ms, N l Ms, E , Ms, N l Ms, E , the pairing m  x, y4 coincides with the pairing given in Ž4.6.3.. Take s g T˜* F , and let m Ž ˜ F . n. Let Or be the corresponding nilpotent orbit Žsee 2.9.. r s r s,Ž m. E g G . F by using the induction on the codimenWe shall prove Ž4.6.1. for Ž Ts,Ž m. E sion of Or . In this subsection, we show the following. Ž4.11.1. Assume that Ž4.7.1. holds for any Ž Ts,Ž m. . F for r 9 s r s,Ž m. E9 E9 with n E9 g ŽWs .ex such that dim Or - dim Or 9. Assume further that Or is not a . F. regular nilpotent class. Then Ž4.7.1. holds for Ž Ts,Ž m. E m

First we note that if C Ž s. Ž G F sr;. contains a cuspidal character, then the class  s4 is regular semisimple. This implies that Or 9 is a regular m nilpotent class for any r 9 g C Ž s. Ž G F sr;.. Hence in our setting, m C Ž s. Ž G F sr;. does not contain a cuspidal character. By applying Proposition 3.7 Žthe assumption is satisfied by the induction hypothesis., we see m that Sh F m r F sends the space C Ž s. Ž G F sr;. onto C Ž s. Ž G F r;.. Let Vs, O 9 m be the subspace of C Ž s. Ž G F sr;. spanned by irreducible characters of m . F under the condition that Or 9 s O 9 for G F belonging to various Ž Ts,Ž m. E9 Ž m. r 9 s r s, E9. Similarly, we define the subspace Us, O 9 as a subspace of C Ž s. Ž G F r;. spanned by irreducible characters of G F belonging to various Ts, E9 under the condition that Or 9 s O 9 for r 9 s r s, E9. Let O s Or for r given in Ž4.11.1.. We define a subspace Vs, O Žresp. Us, O . as a sum of subspaces Vs, O 9 Žresp. Us, O 9 . for all the nilpotent orbits O 9 such that O 9 ; O . By the assumption in Ž4.11.1., Sh F m r F maps the orthogonal m complement of Vs, O in C Ž s. Ž G F sr;. onto the orthogonal complement of Us, O in C Ž s. Ž G F r;.. Since Sh F m r F is an isometry, this implies that Sh F m r F . F be the gives rise to isomorphisms from Vs, O to Us, O . Let r xŽ m. g Ž Ts,Ž m. E Ž . character corresponding to x s c, j g Ms, N . Then there exists x 1 g M0Ž m. s Al = Aln such that w Ž m. Ž x 1 . s x, where M0Ž m. and w Ž m. are the correm n sponding objects in G F . Let M s A F = A ex be the set defined in 1.6. Since the map AlF ª AlF is surjective by Corollary 2.21, we may choose x 1 in Ž M0Ž m. . F , i.e., x 1 g M . We now consider the corresponding generalized m Gelfand]Graev character GxŽ1m. on G F , which is F-stable. We note that Ž m. Ž m. the multiplicity of r x in Gx 1 is equal to 1 by Corollary 2.23. As was mentioned in 2.5, GxŽ1m. can be written as Fm

GxŽ1m. s Ind GH F m r H m

for some F-stable irreducible character r H of H F . Here H s LU1.5 , and r H is given as in 1.6. Now as discussed in 1.8, we can choose an extension m ˜xŽ1m. s r˜H tom H F ² s : of r H . Thus one can define a character G G F m² s : Ž m. Ž m. ˜F m, Ind H F ² s : r˜H . This determines a unique extension r˜x of r x to G which we fix hereafter.

226

TOSHIAKI SHOJI m

Let p Žresp. p9. be the orthogonal projection from C Ž G F sr;. Žresp. C Ž G F r;.. to Vs, O Žresp. Us, O ., respectively. Note that r s r s,Ž m. E is uniquely determined for a given s by the condition that Or s O . This implies that m . F . It Vs, O is the subspace of C Ž s. Ž G F sr;. spanned by characters in Ž Ts,Ž m. E Ž m. ˜ follows from this, by using Theorem 2.18Ži., that we have pŽ Gx 1 . s r˜xŽ m.. Since Sh F m r F commutes with the projections p, p9, we have

˜xŽ1m. . Sh F m r F Ž r˜xŽ m. . s p9 Sh F m r F G

ž

/

˜xŽ1m., ŽHere and in the discussions below we regard, by abbreviation, r˜xŽ m., G Fm . etc., as functions on G s . Then by Theorem 1.10, we have ˜ Ž m. s Sh F m r F my1 x 1 Gx 1

ž

/

Ý  x 1 , y1 4 Gy . 1

y 1g M

Note, under our assumption on m, that lŽym. s 1 for any y 1 g M . More1 over, since A s Al is abelian, the pairing  x 1 , y 1 4 defined in 1.9 is expressed as

 x 1 , y 1 4 s < AlF
 x, y 4 s Ž < AlF
˜ Ž m. s p9 my1 x 1 Sh F m r F Gx 1

ž

/

Ý  x, y 4 r y . yg Ms, N

The right hand side coincides with R x under the natural bijection Ms, N l . F . So Ž4.11.1. holds. Ms, E , Ms, N l Ms, E . This proves Ž4.7.1. for Ž Ts,Ž m. E . F is as in 4.11, 4.12. It remains to prove Ž4.7.1. in the case where Ž Ts,Ž m. E m F but Or is the regular nilpotent class. If E Ž G ,  s4. does not contain cuspidal characters, then the previous argument in 4.11 can be applied m without change. So, we assume that E Ž G F ,  s4. contains a cuspidal character. Then s is a regular semisimple element, and there exists a . F which contains all of F-stable cuspidal characters of unique set Ž Ts,Ž m. 1 Fm G , where E s 1 is the unit character of Ws s  14 . Then all the cuspidal

SHINTANI DESCENT

227

characters of G F are contained in the set Ts, 1. We consider the parameter . F and Ms, N for Ts, 1. Let GxŽ m. be the generalized sets Ms, N for Ž Ts,Ž m. 1 1 Gelfand]Graev character corresponding to x g Ms, N . In this case, GxŽ1m. is m simply the Gelfand]Graev character of G F . Let V be the orthogonal m complement in C Ž G F sr;. of the space spanned by characters in various Ž Ts9,Ž m.E . F such that  s94 /  s4 under the condition that Or 9 s ON for r 9 s Ž m. Ž F . r s9, E . Similarly, we define a subspace V9 in C G r; as the orthogonal complement of the space spanned by characters in various Ts9, E such that  s94 /  s4 under the condition that Or 9 s ON for r 9 s r s9, E . Let p Žresp. m p9. be the orthogonal projection from C Ž G F sr;. to V Žresp. from ˜xŽ1m. . s r˜xŽ m., C Ž G F r;. to V 9., respectively. Then it is easily verified that pŽ G Ž . and that p9 Gy 1 s r y for x g Ms, N , y g Ms, N . Hence a similar argument . F . This completes the as in 4.11 can be applied to show Ž4.7.1. for Ž Ts,Ž m. 1 proof of the theorem. Remark 4.13. We can deduce some information on the twisted induction R GLŽ w˙ . from Lemma 3.9 by using the specialization argument as employed in wS2, S3x. Assume we are in a setting in 3.8. Let p E9 Žresp. r E . m m Žresp. F-stable . irreducible character of LF Žresp. G F . be an Fw-stable ˙ as in Lemma 3.9. Then by using the parametrization given in 4.6, one can define corresponding functions R x as in Ž4.6.4.. We denote by R E9 Žresp. R E . the function on LF w˙ Žresp. G F . corresponding to p E9 Žresp. r E ., respectively. Then we have R GLŽ w˙ . Ž R E9 . s « y

Ý

Tr Ž gw , VE9, ˜ E˜ . R E ,

n Eg ŽWd .ex

where the endomorphism gw on VE9, ˜ E˜ is obtained from the endomorphism Ž m. gwŽ m. on VE9, in Lemma 3.9 by specializing q m r2 ª 1, and y ¬ « y s "1 ˜ E˜ 0 is a certain linear character of Wd . Since such R E Žresp. R E9 . form a basis of C Ž G F r;. Žresp. C Ž LF w˙r;.., this formula gives a sort of description of the decomposition of the twisted induction R GLŽ w˙ . . However, in order to get precise information from our formula, one needs to know a relationship between two kinds of parametrization: one is given by the endomorphism algebra H Ž d ., and the other by Ms, E which is essentially related to unipotent classes. We hope to consider this problem in another paper.

REFERENCES wAx

T. Asai, Twisting operators on the space of class functions of finite special linear groups, in ‘‘The Arcata Conference on Representations of Finite Groups,’’ Proceedings of Symposia in Pure Math., Vol. 47, pp. 99]148, Amer. Math. Soc., Providence, RI, 1987.

228 wBx wDLMx wGx wHLx wK1x wK2x wK3x wLex wL1x wL2x wL3x wL4x wS1x wS2x wS3x wS4x wSSx

TOSHIAKI SHOJI

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