Induced R-representations of p-adic reductive groups

Abstract

This article concerns the relation between the non cuspidal irreducible smooth representations of the F -points of a reductive connected algebraic group over F over an R-vector space, and the simple modules of affine Hecke algebras with coefficients in R, where R is an algebraically closed field of positive characteristic = p, and F is a local non archimedean field, of finite residual field with q elements and characteristic p. When G = GL(n, F ): 1) We describe a set of unipotent irreducible R-representations of GL(n, F ) (i.e. subquotients of parabolically induced representations from unramified characters of a Borel subgroup), classified by the Deligne-Langlands R-parameters. In particular, we get a set of superunipotent irreducible R-representations of GL(n, F ) (i.e with an Iwahori fixed non zero vector), classified by the Deligne-Langlands R-parameters without cycle (V.6). They are in bijection with a set of simple modules for the affine Hecke R-algebras of type A and parameter the image of q in R (which is invertible in the prime field of R hence a root of unity). 2) We reduce the classification of all irreducible R-representations of GL(n, F ) to the classification of the unipotent irreducible representations and of the supercuspidal irreducible representations, or to the classification of the superunipotent representations and of the cuspidal representations.