The tame algebra

The tame subgroup I_t of the Iwahori subgroup I and the tame Hecke algebra H_t = C_c(I_t|G/=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(I|G/=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 χ, iff π is a constituent of the representation induced from a character of the minimal parabolic subgroup, denoted χA, which extends χ. The proof is an extension to the tame context of an unpublished argument of Bernstein, which he used to prove the following. 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 representation induced from an unramified character of the minimal parabolic subgroup. The invertibility of each generator of H_t 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, reecting more naturally the structure of G and its maximally split torus.