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A Remark on Gelfand Pairs of Finite Groups of Lie Type
201, 493]500 Ž1998. JA977297
JOURNAL OF ALGEBRA ARTICLE NO.
A Remark on Gelfand Pairs of Finite Groups of Lie Type Ehud Moshe Baruch and Steve Rallis* Department of Mathematics, The Ohio State Uni¨ ersity, Columbus, Ohio 43210 Communicated by Walter Feit Received September 28, 1996
We show how to deduce multiplicity one theorems for cuspidal representations of finite groups of Lie type from analogous results for p-adic groups. We then look at examples where the latter is known. One such example is the restriction of irreducible representations of SO Ž n. to SO Ž n y 1. wS. Rallis, preprintx. We show that the multiplicity of a cuspidal representation of the finite group SO Ž n y 1. in the restriction of a cuspidal representation of SO Ž n. to SO Ž n y 1. is at most one. Q 1998 Academic Press
INTRODUCTION Multiplicity one theorems for Gelfand pairs were established in many cases for p-adic groups and for finite groups of Lie type. Let G be such a group and let H be a closed subgroup of G. In general, one picks a specific representation of H and one tries to prove that the induced representation from H to G is multiplicity free. In many instances, the proof in the finite field case mimics the proof in the p-adic field case or vice versa. However, there are some examples where it is impossible to mimic the p-adic proof, and moreover, the finite field statement may be false. In this paper we show that in these examples, it might still be possible to have multiplicity one theorems for a certain class of irreducible representations of the finite Lie group, namely the cuspidal ones. The idea is to ‘‘embed’’ the finite group representation into a representation of the p-adic group using induction from a compact open subgroup mod center, and to apply the p-adic result for this supercuspidal representation of the p-adic group. * Research of the second author supported in part by NSF grant DMS-9401013. 493 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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Representations of reductive p-adic groups obtained in such a way are called ‘‘level 1’’ representations and were studied extensively in the literature Žsee, for example, w11, 6, 7x.. In our examples we will only consider level 1 representations coming from special parahoric subgroups Žsee w12x 3.1.1 for definition.. We expect similar phenomenon to hold for other level 1 representations. Before stating our main theorem we will look at the example of orthogonal groups mentioned in the abstract. Let F be a finite field and let V be an n-dimensional vector space over F. Let Q be a nondegenerate quadratic form on V. Let G s SO Ž Q . be the group of special linear transformations on V preserving Q. Let F be a p-adic field such that the residue field of F is isomorphic to F. Let O be the ring of integers in F. Let V be an n-dimensional space over F, let Q be a nondegenerate quadratic form on V, and let G s SO Ž Q .. We may choose Q and an embedding of G into GLnŽ F . so that G is quasi-split over F and splits over an unramified extension of F and so that KrK 1 ( G, where K s GŽ O . s GLnŽ O . l G is a hyperspecial maximal compact subgroup of G, Žsee w12x., and K 1 is a congruence subgroup of G. It is clear how to do that when G s SOnqŽ q . is split. ŽHere q is the order of F. See w5x Ž2.5.6. for this notation.. When G s SOnyŽ q ., we can lift the standard basis in Žw5x Proposition 2.5.3. to a basis of V Ž O ., and we can let Q be a lift of Q. Then SO Ž Q . will satisfy the previous requirements. Notice that we can assume that < F < is odd because there are no odd orthogonal groups for even order finite fields. Let j g V be such that q Ž j . / 0. Then V s Ž j . [ W where W s j H . Let H s SO ŽW . and we shall assume that j is chosen so that H is quasi-split and splits over an unramified extension of F. Let K H s K l H, K 1, H s K 1 l H and H s K H rK 1, H . We can identify H with SO ŽW . where W is a nondegenerate subspace of V. THEOREM 1. Let Žp , X . be a cuspidal representation of G and let Ž s , Y . be a cuspidal representation of H. Then dim Ž Hom H Ž p , s . . F 1. Proof. We inflate p and s to K and K H , respectively. ŽWe will use the same notation for the inflated representations. . We let
p s c-ind GK p ,
s s c-ind HK H s .
Because K and K H are their own normalizers it follows from w6x or Žw7x Proposition 6.6. that p and s are irreducible admissible Žsupercuspidal. representations of G and H, respectively.
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We claim that there is an injection, Hom H Ž p , s . ¨ Hom H Ž p , s . . Let l: X ª Y be in Hom H Žp , s .. We define l g Hom H Žp , s . in the following way: Let f g c-ind GK p . Then f : G ª X is a compactly supported smooth function satisfying f Ž kg . s p Ž k . f Ž g .
for all k g K , g g G.
The action of p on such functions is given by right translations. Now define l Ž f . s f 9 where f 9: H ª Y is given by f 9Ž h. s l Ž f Ž h. . ,
h g H.
It is easy to check that f 9 g s , l g Hom H Žp , s . and l ª l is a linear map. To show that it is injective, assume that l s 0. Let X˜ ; p be the space of functions supported on K. It is easy to see that X˜ is a K invariant space, and that it is isomorphic to p as a representation of K. It is also easy to see that l ' 0 on X˜ is equivalent to l ' 0 on X. This proves the claim. By a result of Rallis w9x, we have that dim Ž Hom H Ž p , s . . F 1. Hence the theorem is proved. Our assumption that p and s are cuspidal is justified by the following example. The group SO4qŽ F . is almost the same as the group PGL2 Ž F . = PGL2 Ž F . and SOT3 Ž F . ( PGL2 Ž F .. The embedding of SOT3 Ž F . into SO4qŽ F . is essentially the diagonal embedding of H s PGL2 Ž F . into G s PGL2 Ž F . = PGL2 Ž F .. ŽThe p-adic multiplicity one result in this case was proved by Prasad w8x.. We assume that < F < is odd. Let x be a character of F*. We extend x to the Borel subgroup B of H. Define px s Ind cB x . Let x be such that x 2 / 1. Then px and pxy1 are both irreducible and px m pxy1 is an irreducible representation of G. It is easy to check that dim Ž Hom H D Ž px m pxy 1 , p 1 . . s 3, where H D is the diagonal embedding on H into G. Because p 1 has only two irreducible components, the trivial representation and the Steinberg representation and because dim Ž Hom H D Ž px m pxy 1 , 1 . . s dim Ž Hom H D Ž px , px . . s 1, it follows that the multiplicity of the Steinberg representation in the preceding tensor product is two, hence multiplicity one fails.
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It will be convenient to use the following terminology. Let G be a finite group of Lie type and H a subgroup. We say that Ž G, H . is a Gelfand pair if ind GH 1 is multiplicity free. We say that Ž G, H . is a cuspidal Gelfand pair if the multiplicity of a cuspidal representation of G in ind GH 1 is at most one. Hence Theorem 1 states that Ž SO Ž n. = SO Ž n y 1., SO Ž n y 1. D . is a cuspidal Gelfand pair while the example shows that Ž PGL2 = PGL2 , PGLD2 . is not a Gelfand pair of finite groups.
1. MAIN THEOREM Let F be a p-adic field and let G be the group of rational points of a connected reductive group defined over F. Let P be a maximal parahoric subgroup of G and let Pqs NG Ž P .. Then Pq possesses a canonical filtration Pn , n s 0, 1, . . . , such that P0 s P, and PrP1 is a reductive group defined over the residue field of F. Let ZŽ G . be the center of G. Then ZŽ G . ; Pq and Pq is compact mod ZŽ G .. ŽSee w7x, 6.5.. Let Žp , V . be an irreducible representation of Pq such that p < P1 ' 1 and p < P is a cuspidal representation Žin the finite Chevalley group sense. of the group PrP1. Then by w6x or Žw7x, 6.6.,
p s c-ind GPq p is an irreducible admissible Žsupercuspidal. representation of G. We let vp be the central character of p . Let H be a closed subgroup of G. We denote by PHq s H l Pq, and ZH s ZŽ G . l H. Let x be a character of H such that x < ZH s vp < ZH . We denote by x s x < PHq . THEOREM 2. There is an injection, Hom PqH Ž p , x . ¨ Hom H Ž p , x . . Proof. Let l g Hom PqH Žp , x .. We define l g Hom H Žp , x . as follows, lŽ f . s
y1
HZ _Hl Ž f Ž h . . x
Ž h . dh,
fgp.
H
Because f is compactly supported mod ZŽ G . the integral converges, thus l is well defined and lies in Hom H Žp , x .. To show that the linear map l ª l is injective assume l ' 0. Let X˜ ; p be the subspace of functions with
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support in Pq. For every f g X˜ we have 0 s lŽ f . s
y1
HZ _Hl Ž f Ž h . . x
Ž h . dh s H
H
l Ž f Ž pq . . xy1 Ž pq . dpq
q Z H_PH
s vol Ž ZH _ PHq . l Ž f Ž 1 . . .
˜ Because the space f Ž1.< f g X˜4 Thus l Ž f Ž1.. s 0 for every function f g X. is X we get that l ' 0. 1.1. Application We shall apply Theorem 2 in the following situation. We shall assume G to be quasi-split over F and to be split over an unramified extension of F. We will fix an embedding of G into GLnŽ O . and we will let P s G l GLnŽ O .. Then P is an hyperspecial maximal parahoric subgroup of G Žw12x 3.4.2, 3.4.3, and 2.6.1, see also 3.9.1 for a similar statement ., and using the Cartan decomposition for G we can see that Pqs NG Ž P . s ZŽ G . P. We let G s PrP1. Then G is a finite reductive group. We let H be a closed subgroup of G and define Pi, H s Pi l H. We let H s PH rP1, H . We view H as a subgroup of G. We let p be a cuspidal representation of G and we inflate it to P. We extend p to Pq by extending the central character of p to the whole of ZŽ G .. We let x be a character of H as in Theorem 2 with the additional assumption that x < P1, H ' 1. Then x < P induces a character on H which we denote by x . Theorem 2 in this case reads: COROLLARY 3. There exist an injection, Hom H Ž p , x . ¨ Hom H Ž p , x . . Hence if Ž G, H . is a Gelfand pair of p-adic groups then Ž G, H . is a cuspidal Gelfand pair of finite groups Žas defined in the end of the introduction..
2. TWO EXAMPLES There are several examples where multiplicity one holds for representations of a Gelfand pair of p-adic groups. One such example is the Whittaker model. If G is as in Corollary 3, H s N is a maximal unipotent subgroup of G and x is a suitably chosen nondegenerate character of N then Corollary 3 coupled with the uniqueness of Whittaker models for representations of p-adic groups w10x gives us another proof of this uniqueness for cuspidal representations of finite groups of Lie type. In this case however, there is a stronger result, namely, the uniqueness of a Whittaker model for any representation.
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The following two examples are such that multiplicity one fails for some noncuspidal representations of the finite groups. 2.1. Spherical Models for Cuspidal Representations of Orthogonal Groups Before considering the general spherical models defined in w1x, we would like to give another proof of Theorem 1 using Corollary 3. Let G, H, p , and s be as in Theorem 1. Then Hom H Ž p , s . ( Hom H D Ž p m sˆ , 1 . , where H D s Ž h, h.< h g H 4 is the diagonal embedding of H into G = H, sˆ is the contagredient representation to s and p m sˆ is the corresponding representation of G = H. To prove Theorem 1 we set G s SO Ž n. = SO Ž n y 1., H s SO Ž n y 1. D , and x s 1. Then Corollary 3 gives us an injection, Hom H D Ž p m sˆ , 1 . ¨ Hom H D Ž p m sˆ , 1 . , where p m s is the appropriate representation of G as in Theorem 2. By w9x the dimension of the second Hom is less than or equal to 1, hence Theorem 1 follows. A similar argument can be applied for general spherical models constructed in w1x. For these models we shall choose G to be a product of two orthogonal groups of different dimensions, i.e., G s SO Ž n . = SO Ž k . ,
k - n,
H s Ž Ž SO Ž k . = NR . j UR . = SO Ž k . , and
x s Ž c¨ * m cj m¨ * . m 1. For all these notations and definitions see w1x. From Corollary 3 and w1x we obtain a multiplicity one statement in this case, i.e., a cuspidal representation p of the finite group SO Ž n. will have at most one spherical model associated to the data Ž s , c¨ * , cj m ¨ * ., where s is a cuspidal representation of the finite group SO Ž k . and c¨ * , cj m ¨ * are characters of unipotent subgroups of SO Ž n. defined as in w1x. We can also show that every cuspidal representation of SO Ž n. will have at least one nonzero model for some data as in the foregoing text. The argument follows the same lines as in Žw2x Section 3. and is omitted.
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2.2. Linear Periods for GLn Let Gn s GLnŽ F . and let Gn s GLnŽ F . where F is a p-adic field and F is the residue field of F. Let p G 1, q G 1 be two integers with p q q s n and denote by H s Hp, q , H s Hp, q the subgroups of Gn and Gn , respectively, of matrices of the form, hs
ž
g1 0
0 g2
/
with g 1 g Gp , g 2 g Gq or g 1 g Gp , g 2 g Gq , respectively .
ž
/
Let p be an admissible irreducible admissible representation of Gn . THEOREM 4. Žw4x, Theorem 1.1., dim Ž Hom H Ž p , 1 . . F 1. Let p be a cuspidal representation of Gn . Using Corollary 3 and Theorem 4 we get THEOREM 5. dim Ž Hom H Ž p , 1 . . F 1. Hence Ž Gn , Hk, nyk . is a cuspidal Gelfand pair Žsee end of introduction.. We shall now show that they do not form a Gelfand pair, i.e., that Theorem 5 is not true if p is not assumed to be cuspidal. Let H be as in Theorem 5 and let P be a standard parabolic subgroup of Gn whose Levi factor is H. Let
v s Ind GP n 1. It is well known Žsee for example w3x Ž3.3. and Ž2.7.. that the number of irreducible components of v Žcounting multiplicities. is at most < P _ GnrP < . On the other hand, the dimension of the space of H fixed vectors in v is exactly < P _ GnrH < . It is clear that < P _ GnrP < F < P _ GnrH < .
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It is not difficult to see Žby looking at the big cell., that this is in fact a strict inequality, hence at least one irreducible component of v will have a space of H fixed vectors with dimension greater than 1.
REFERENCES 1. D. Ginzburg, I. Piatetski-Shapiro, and S. Rallis, L-functions for the orthogonal groups, Mem. Amer. Math. Soc. 128, No. 611 Ž1997., 1]218. 2. D. Ginzburg, S. Rallis, and D. Soudry, Self dual GLn automorphic modules Ž2., Construction of a backward lifting from GLn to classical groups, preprint. 3. R. Howe with A. Moy, Harish-Chandra homomorphisms for p-adic groups, in ‘‘Regional Conference Series in Math, No. 59,’’ Amer. Math. Soc., Providence, RI. 4. H. Jacquet and S. Rallis, Uniqueness of linear periods, Comput. Math. 102 Ž1996., 65]123. 5. P. Kleidman and M. Liebeck, ‘‘The Subgroup Structure of the Finite Classical Groups,’’ London Math. Soc. Lecture Note Ser. No. 129, Cambridge Univ. Press, Cambridge, U.K., 1990. 6. L. Morris, P-cuspidal representations of level one, Proc. London Math. Soc. (3) 58 Ž1989., 550]558. 7. A. Moy and G. Prasad, Jacquet functors and unrefined minimal K-types, Comment. Math. Hel¨ . 71 Ž1996., 98]121. 8. D. Prasad, Trilinear forms for representations of GL2 and local epsilon factors, Compositio Math. 75 Ž1990., 1]46. 9. S. Rallis, On certain Gelfand]Graev models which are Gelfand pairs, preprint. 10. J. A. Shalika, The multiplicity one theorem for GLn , Ann. of Math. 100 Ž1974., 171]193. 11. T. Shintani, On certain square integrable unitary representations of some p-adic linear groups, J. Math. Soc. Japan 20 Ž1968., 522]565. 12. J. Tits, Reductive groups over local fields, Proc. Sympos. Pure Math. 33Žpart 1., Ž1979..
JOURNAL OF ALGEBRA ARTICLE NO.
A Remark on Gelfand Pairs of Finite Groups of Lie Type Ehud Moshe Baruch and Steve Rallis* Department of Mathematics, The Ohio State Uni¨ ersity, Columbus, Ohio 43210 Communicated by Walter Feit Received September 28, 1996
We show how to deduce multiplicity one theorems for cuspidal representations of finite groups of Lie type from analogous results for p-adic groups. We then look at examples where the latter is known. One such example is the restriction of irreducible representations of SO Ž n. to SO Ž n y 1. wS. Rallis, preprintx. We show that the multiplicity of a cuspidal representation of the finite group SO Ž n y 1. in the restriction of a cuspidal representation of SO Ž n. to SO Ž n y 1. is at most one. Q 1998 Academic Press
INTRODUCTION Multiplicity one theorems for Gelfand pairs were established in many cases for p-adic groups and for finite groups of Lie type. Let G be such a group and let H be a closed subgroup of G. In general, one picks a specific representation of H and one tries to prove that the induced representation from H to G is multiplicity free. In many instances, the proof in the finite field case mimics the proof in the p-adic field case or vice versa. However, there are some examples where it is impossible to mimic the p-adic proof, and moreover, the finite field statement may be false. In this paper we show that in these examples, it might still be possible to have multiplicity one theorems for a certain class of irreducible representations of the finite Lie group, namely the cuspidal ones. The idea is to ‘‘embed’’ the finite group representation into a representation of the p-adic group using induction from a compact open subgroup mod center, and to apply the p-adic result for this supercuspidal representation of the p-adic group. * Research of the second author supported in part by NSF grant DMS-9401013. 493 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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Representations of reductive p-adic groups obtained in such a way are called ‘‘level 1’’ representations and were studied extensively in the literature Žsee, for example, w11, 6, 7x.. In our examples we will only consider level 1 representations coming from special parahoric subgroups Žsee w12x 3.1.1 for definition.. We expect similar phenomenon to hold for other level 1 representations. Before stating our main theorem we will look at the example of orthogonal groups mentioned in the abstract. Let F be a finite field and let V be an n-dimensional vector space over F. Let Q be a nondegenerate quadratic form on V. Let G s SO Ž Q . be the group of special linear transformations on V preserving Q. Let F be a p-adic field such that the residue field of F is isomorphic to F. Let O be the ring of integers in F. Let V be an n-dimensional space over F, let Q be a nondegenerate quadratic form on V, and let G s SO Ž Q .. We may choose Q and an embedding of G into GLnŽ F . so that G is quasi-split over F and splits over an unramified extension of F and so that KrK 1 ( G, where K s GŽ O . s GLnŽ O . l G is a hyperspecial maximal compact subgroup of G, Žsee w12x., and K 1 is a congruence subgroup of G. It is clear how to do that when G s SOnqŽ q . is split. ŽHere q is the order of F. See w5x Ž2.5.6. for this notation.. When G s SOnyŽ q ., we can lift the standard basis in Žw5x Proposition 2.5.3. to a basis of V Ž O ., and we can let Q be a lift of Q. Then SO Ž Q . will satisfy the previous requirements. Notice that we can assume that < F < is odd because there are no odd orthogonal groups for even order finite fields. Let j g V be such that q Ž j . / 0. Then V s Ž j . [ W where W s j H . Let H s SO ŽW . and we shall assume that j is chosen so that H is quasi-split and splits over an unramified extension of F. Let K H s K l H, K 1, H s K 1 l H and H s K H rK 1, H . We can identify H with SO ŽW . where W is a nondegenerate subspace of V. THEOREM 1. Let Žp , X . be a cuspidal representation of G and let Ž s , Y . be a cuspidal representation of H. Then dim Ž Hom H Ž p , s . . F 1. Proof. We inflate p and s to K and K H , respectively. ŽWe will use the same notation for the inflated representations. . We let
p s c-ind GK p ,
s s c-ind HK H s .
Because K and K H are their own normalizers it follows from w6x or Žw7x Proposition 6.6. that p and s are irreducible admissible Žsupercuspidal. representations of G and H, respectively.
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We claim that there is an injection, Hom H Ž p , s . ¨ Hom H Ž p , s . . Let l: X ª Y be in Hom H Žp , s .. We define l g Hom H Žp , s . in the following way: Let f g c-ind GK p . Then f : G ª X is a compactly supported smooth function satisfying f Ž kg . s p Ž k . f Ž g .
for all k g K , g g G.
The action of p on such functions is given by right translations. Now define l Ž f . s f 9 where f 9: H ª Y is given by f 9Ž h. s l Ž f Ž h. . ,
h g H.
It is easy to check that f 9 g s , l g Hom H Žp , s . and l ª l is a linear map. To show that it is injective, assume that l s 0. Let X˜ ; p be the space of functions supported on K. It is easy to see that X˜ is a K invariant space, and that it is isomorphic to p as a representation of K. It is also easy to see that l ' 0 on X˜ is equivalent to l ' 0 on X. This proves the claim. By a result of Rallis w9x, we have that dim Ž Hom H Ž p , s . . F 1. Hence the theorem is proved. Our assumption that p and s are cuspidal is justified by the following example. The group SO4qŽ F . is almost the same as the group PGL2 Ž F . = PGL2 Ž F . and SOT3 Ž F . ( PGL2 Ž F .. The embedding of SOT3 Ž F . into SO4qŽ F . is essentially the diagonal embedding of H s PGL2 Ž F . into G s PGL2 Ž F . = PGL2 Ž F .. ŽThe p-adic multiplicity one result in this case was proved by Prasad w8x.. We assume that < F < is odd. Let x be a character of F*. We extend x to the Borel subgroup B of H. Define px s Ind cB x . Let x be such that x 2 / 1. Then px and pxy1 are both irreducible and px m pxy1 is an irreducible representation of G. It is easy to check that dim Ž Hom H D Ž px m pxy 1 , p 1 . . s 3, where H D is the diagonal embedding on H into G. Because p 1 has only two irreducible components, the trivial representation and the Steinberg representation and because dim Ž Hom H D Ž px m pxy 1 , 1 . . s dim Ž Hom H D Ž px , px . . s 1, it follows that the multiplicity of the Steinberg representation in the preceding tensor product is two, hence multiplicity one fails.
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It will be convenient to use the following terminology. Let G be a finite group of Lie type and H a subgroup. We say that Ž G, H . is a Gelfand pair if ind GH 1 is multiplicity free. We say that Ž G, H . is a cuspidal Gelfand pair if the multiplicity of a cuspidal representation of G in ind GH 1 is at most one. Hence Theorem 1 states that Ž SO Ž n. = SO Ž n y 1., SO Ž n y 1. D . is a cuspidal Gelfand pair while the example shows that Ž PGL2 = PGL2 , PGLD2 . is not a Gelfand pair of finite groups.
1. MAIN THEOREM Let F be a p-adic field and let G be the group of rational points of a connected reductive group defined over F. Let P be a maximal parahoric subgroup of G and let Pqs NG Ž P .. Then Pq possesses a canonical filtration Pn , n s 0, 1, . . . , such that P0 s P, and PrP1 is a reductive group defined over the residue field of F. Let ZŽ G . be the center of G. Then ZŽ G . ; Pq and Pq is compact mod ZŽ G .. ŽSee w7x, 6.5.. Let Žp , V . be an irreducible representation of Pq such that p < P1 ' 1 and p < P is a cuspidal representation Žin the finite Chevalley group sense. of the group PrP1. Then by w6x or Žw7x, 6.6.,
p s c-ind GPq p is an irreducible admissible Žsupercuspidal. representation of G. We let vp be the central character of p . Let H be a closed subgroup of G. We denote by PHq s H l Pq, and ZH s ZŽ G . l H. Let x be a character of H such that x < ZH s vp < ZH . We denote by x s x < PHq . THEOREM 2. There is an injection, Hom PqH Ž p , x . ¨ Hom H Ž p , x . . Proof. Let l g Hom PqH Žp , x .. We define l g Hom H Žp , x . as follows, lŽ f . s
y1
HZ _Hl Ž f Ž h . . x
Ž h . dh,
fgp.
H
Because f is compactly supported mod ZŽ G . the integral converges, thus l is well defined and lies in Hom H Žp , x .. To show that the linear map l ª l is injective assume l ' 0. Let X˜ ; p be the subspace of functions with
A REMARK ON GELFAND PAIRS
497
support in Pq. For every f g X˜ we have 0 s lŽ f . s
y1
HZ _Hl Ž f Ž h . . x
Ž h . dh s H
H
l Ž f Ž pq . . xy1 Ž pq . dpq
q Z H_PH
s vol Ž ZH _ PHq . l Ž f Ž 1 . . .
˜ Because the space f Ž1.< f g X˜4 Thus l Ž f Ž1.. s 0 for every function f g X. is X we get that l ' 0. 1.1. Application We shall apply Theorem 2 in the following situation. We shall assume G to be quasi-split over F and to be split over an unramified extension of F. We will fix an embedding of G into GLnŽ O . and we will let P s G l GLnŽ O .. Then P is an hyperspecial maximal parahoric subgroup of G Žw12x 3.4.2, 3.4.3, and 2.6.1, see also 3.9.1 for a similar statement ., and using the Cartan decomposition for G we can see that Pqs NG Ž P . s ZŽ G . P. We let G s PrP1. Then G is a finite reductive group. We let H be a closed subgroup of G and define Pi, H s Pi l H. We let H s PH rP1, H . We view H as a subgroup of G. We let p be a cuspidal representation of G and we inflate it to P. We extend p to Pq by extending the central character of p to the whole of ZŽ G .. We let x be a character of H as in Theorem 2 with the additional assumption that x < P1, H ' 1. Then x < P induces a character on H which we denote by x . Theorem 2 in this case reads: COROLLARY 3. There exist an injection, Hom H Ž p , x . ¨ Hom H Ž p , x . . Hence if Ž G, H . is a Gelfand pair of p-adic groups then Ž G, H . is a cuspidal Gelfand pair of finite groups Žas defined in the end of the introduction..
2. TWO EXAMPLES There are several examples where multiplicity one holds for representations of a Gelfand pair of p-adic groups. One such example is the Whittaker model. If G is as in Corollary 3, H s N is a maximal unipotent subgroup of G and x is a suitably chosen nondegenerate character of N then Corollary 3 coupled with the uniqueness of Whittaker models for representations of p-adic groups w10x gives us another proof of this uniqueness for cuspidal representations of finite groups of Lie type. In this case however, there is a stronger result, namely, the uniqueness of a Whittaker model for any representation.
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The following two examples are such that multiplicity one fails for some noncuspidal representations of the finite groups. 2.1. Spherical Models for Cuspidal Representations of Orthogonal Groups Before considering the general spherical models defined in w1x, we would like to give another proof of Theorem 1 using Corollary 3. Let G, H, p , and s be as in Theorem 1. Then Hom H Ž p , s . ( Hom H D Ž p m sˆ , 1 . , where H D s Ž h, h.< h g H 4 is the diagonal embedding of H into G = H, sˆ is the contagredient representation to s and p m sˆ is the corresponding representation of G = H. To prove Theorem 1 we set G s SO Ž n. = SO Ž n y 1., H s SO Ž n y 1. D , and x s 1. Then Corollary 3 gives us an injection, Hom H D Ž p m sˆ , 1 . ¨ Hom H D Ž p m sˆ , 1 . , where p m s is the appropriate representation of G as in Theorem 2. By w9x the dimension of the second Hom is less than or equal to 1, hence Theorem 1 follows. A similar argument can be applied for general spherical models constructed in w1x. For these models we shall choose G to be a product of two orthogonal groups of different dimensions, i.e., G s SO Ž n . = SO Ž k . ,
k - n,
H s Ž Ž SO Ž k . = NR . j UR . = SO Ž k . , and
x s Ž c¨ * m cj m¨ * . m 1. For all these notations and definitions see w1x. From Corollary 3 and w1x we obtain a multiplicity one statement in this case, i.e., a cuspidal representation p of the finite group SO Ž n. will have at most one spherical model associated to the data Ž s , c¨ * , cj m ¨ * ., where s is a cuspidal representation of the finite group SO Ž k . and c¨ * , cj m ¨ * are characters of unipotent subgroups of SO Ž n. defined as in w1x. We can also show that every cuspidal representation of SO Ž n. will have at least one nonzero model for some data as in the foregoing text. The argument follows the same lines as in Žw2x Section 3. and is omitted.
A REMARK ON GELFAND PAIRS
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2.2. Linear Periods for GLn Let Gn s GLnŽ F . and let Gn s GLnŽ F . where F is a p-adic field and F is the residue field of F. Let p G 1, q G 1 be two integers with p q q s n and denote by H s Hp, q , H s Hp, q the subgroups of Gn and Gn , respectively, of matrices of the form, hs
ž
g1 0
0 g2
/
with g 1 g Gp , g 2 g Gq or g 1 g Gp , g 2 g Gq , respectively .
ž
/
Let p be an admissible irreducible admissible representation of Gn . THEOREM 4. Žw4x, Theorem 1.1., dim Ž Hom H Ž p , 1 . . F 1. Let p be a cuspidal representation of Gn . Using Corollary 3 and Theorem 4 we get THEOREM 5. dim Ž Hom H Ž p , 1 . . F 1. Hence Ž Gn , Hk, nyk . is a cuspidal Gelfand pair Žsee end of introduction.. We shall now show that they do not form a Gelfand pair, i.e., that Theorem 5 is not true if p is not assumed to be cuspidal. Let H be as in Theorem 5 and let P be a standard parabolic subgroup of Gn whose Levi factor is H. Let
v s Ind GP n 1. It is well known Žsee for example w3x Ž3.3. and Ž2.7.. that the number of irreducible components of v Žcounting multiplicities. is at most < P _ GnrP < . On the other hand, the dimension of the space of H fixed vectors in v is exactly < P _ GnrH < . It is clear that < P _ GnrP < F < P _ GnrH < .
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It is not difficult to see Žby looking at the big cell., that this is in fact a strict inequality, hence at least one irreducible component of v will have a space of H fixed vectors with dimension greater than 1.
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