Irreducible Modular Representations of a Reductive p-Adic Group and Simple Modules for Hecke Algebras

Abstract

Let F be a local non-archimedean field of residual characteristic p, and let G be the group of rational points of a connected reductive group defined over F. The two main points in the search for a classification of the irreducible complex representations of G is to try to prove that any irreducible cuspidal representation is induced from an open compact subgroup and that the irreducible representations with a given inertial cuspidal support are classified by simple modules for the Hecke algebra of a type. Over a field R of characteristic ≠ p which is not the complex field, new serious difficulties arise and the purpose of this article is to indicate a way to avoid them. The mirabolic trick used when the group is GL(n, F) does not generalize, but our new method is general and we can extend from the complex case to R the results of Morris and Moy—Prasad for level 0 representations.