Hecke Algebras of Classical Groups over p-adic Fields II

In the previous part of this paper, we constructed a large family of Hecke algebras on some classical groups G de¢ned over p-adic ¢elds in order to understand their admissible representations. EachHecke algebra is associated to a pair …JS; rS†of an open compact subgroup JS and its irreducible representation rSwhich is constructed fromgiven dataS ˆ …G;P0 0; R†. Here, G is a semisimple element in the Lie algebraofG,P0 0 is a parahoric subgroup in the centralizerofG in G, and R is a cuspidal representation on the ¢nite reductive quotient of P0 0. In this paper, we explicitly describe those Hecke algebras when P0 0 is a minimal parahoric subgroup, R is trivial and rS is a character. Mathematics Subject Classi¢cation (2000). 22E50. Key words. p-adic groups, classical groups, Hecke algebras. Introduction Let k be a p-adic ¢eld with odd residue characteristic p, and let G be a connected reductive group over k. In their work on GLn [HM1, 2], Howe and Moy sketch a scheme for understanding the harmonic analysis on G via the harmonic analysis on Hecke algebras associated to open compact data for G. More recently, Bushnell and Kutzko have generalized this scheme to reductive groups via the theory of types [BK2]. Especially, those Hecke algebras should be in a form such that their harmonic analysis is tractable; in fact, they are expected to be generalized af¢ne Hecke algebras. In the stream of this philosophy, in [K1], we constructed a large family of Hecke algebras on some classical groups. Here, we will prove that some of those Hecke algebras are in fact generalized af¢ne Hecke algebras. We recall the basic situation from [K1]; Let k be a p-adic ¢eld with an involution s and let k0 be its s-¢xed sub¢eld of k. Let V be a ¢nite dimensional k-linear space equipped with e-Hermitian form h ; i …e ˆ ‡1 or ÿ 1†. Let G be the connected component of a group of isometries on …V ; h ; i†. In [K1], we constructed a large family of Hecke algebras on G when the residue characteristic of k is big enough *Research partially supported by NSF grant DMS-9970454.The first version of this paper was written while the author was supported by NSF grant DMS-9304580 as a member of the Institute for Advanced Study in 1997±98. Compositio Mathematica 127: 117^167, 2001. 117 # 2001 Kluwer Academic Publishers. Printed in the Netherlands. (see [K1, 3.2.3]). Let S ˆ …G;P0 0; R† be given as in Section 1.5.B in [K1], that is, G is a semisimple element in the Lie algebra g of G as in [K1, 1.3.2], P0 0 is a parahoric subgroup in the centralizer CG…G† of G in G and R is a cuspidal representation of the ¢nite reductive quotient of P0 0. Associated to such a S, we constructed a pair …JS; rS† consisting of an open compact subgroup JS and its irreducible representation rS. Let H ˆ H…G==JS; rS† be the Hecke algebra associated to …JS; rS†. This is the convolution algebra on the space of all compactly supported functions f : Gÿ!End…rS†, which transform via rS under left and right translations by JS. That is, f … jgj0† ˆ rS… j†f …g†rS… j0† for g 2 G and j; j0 2 JS. H also carries a natural involution and an inner product … ; † (see (5.1.2)). Assume that P0 0 is a minimal parahoric subgroup I 0 0 (see Section 1.5.A) and R is a trivial character of I 0 0. Let e W 0 be the af¢ne Weyl group of G0 ˆ CG…G†. Then from Proposition 4.2.6 in [K1], we have Supp…H† ˆ JSG0JS ˆ JS e W 0JS and H is linearly spanned by functions fw whose support is a single double coset JSwJS with w 2 e W 0. In this paper, for the case when rS is a character, we will describe the Hecke algebra H ˆ H…G==JS; rS† by directly ¢nding generators and relations. Moreover, we relate those Hecke algebras to Hecke algebras on G0 ˆ CG…G† by establishing an L2-isomorphism between Hecke algebras: MAIN THEOREM.Let k satisfy the assumption in [K1, 3.2.3] and letG be a classical group considered in [K1]. Let G be a semisimple element in the Lie algebra g as in [K1, 1.3.2] and let I 0 0 be aminimal parahoric subgroup ofG 0 ˆ CG…G†, the centralizer ofG in G. Let S ˆ …G; I 0 0; 1†, where 1 is the trivial character of I 0 0. Let …JS; rS† be a pair consisting of an open compact subgroup JS and its irreducible representation rS associated to S as in Theorem 4.2.9 in [K1]. Suppose rS is a character. Then for some tamely rami¢ed character w of I 0 0, there is a -preserving, support-preserving L2-isomorphism Z:H0 ˆ H…G0==I 0 0; w† ÿ!H…G==JS; rS† ˆ H of C-algebras. In case of GLn, in [HM1, 2], Howe and Moy ¢nd Hecke algebra isomorphisms by going through certain inductive procedures. On the other hand, in [BK1], Bushnell and Kutzko ¢nd them by comparing two Hecke algebras directly. In both cases, the Hecke algebras described are isomorphic to a product of Iwahori Hecke algebras. In our case, we ¢rst ¢nd generators and relations of H…G==JS; rS† directly and then compare it with a Hecke algebra on a related group G0 as in the Main Theorem. Hence the choice of w in the Main Theorem is made so as to match Hecke algebras. In our case, direct computation is possible because our open compact subgroup behaves well with respect to the root space decomposition (see [K1] for details). Unlike the case of GLn, where we had only Hecke algebras of Iwahori types, we now see a twisting by tamely rami¢ed characters. This phenomenon can be already found in the work of A. Moy on GSp4 ([My 2, Cor. 5.8]). Moreover, we ¢nd that scaling G (e.g. replacing G with g G in S where g is an element of an extension ¢eld E over k; see [K1, 1.3.2]) may yield different shapes of Hecke algebras. We have not found any good explanation for this and hope to return to this point. 118 JU-LEE KIM