The Tame Algebra

摘要

The tame subgroup I_t of the Iwahori subgroup I and the tame Hecke algebra H_t = C_c(I_tG/I_t) are introduced. It is shown that the tame algebra has a presentation by means of generators and relations, similar to that of the Iwahori-Hecke algebra H = C_c(IG/I). From this it is deduced that each of the generators of the tame algebra is invertible. This has an application concerning an irreducible admissible representation π of an unramified reduc-tive p-adic group G: π has a nonzero vector fixed by the tame group, and the Iwahori subgroup I acts on this vector by a character X, iff π is a constituent of the representation induced from a character of the minimal parabolic subgroup, denoted XA , which extends X. The proof is an extension to the tame context of an unpublished argument of Bernstein, which he used to prove the follow-ing. An irreducible admissible representation π of a quasisplit reductive p-adic group has a nonzero Iwahori-fixed vector iff it is a constituent of a representa-tion induced from an unramified character of the minimal parabolic subgroup. The invertibility of each generator of Ht is finally used to give a Bernstein-type presentation of H_t, also by means of generators and relations, as an extension of an algebra with generators indexed by the finite Weyl group, by a finite index maximal commutative subalgebra, reflecting more naturally the structure of G and its maximally split torus.