Smooth representations of GL_m(D), V: Endo-classes

Let F be a locally compact nonarchimedean local field. In this article, we extend to any inner form of GL_n over F, with n>0, the notion of endo-class introduced by Bushnell and Henniart for GL_n(F). We investigate the intertwining relations of simple characters of these groups, in particular their preservation properties under transfer. This allows us to associate to any discrete series representation of an inner form of GL_n(F) an endo-class over F. We conjecture that this endo-class is invariant under the local Jacquet-Langlands correspondence

Unramified -modular representations of GLn(F) and its inner forms

International audienceLet F be a non-Archimedean locally compact field of residue characteristic p and R be an algebraically closed field of characteristic ℓ different from p. Let G be the group GL(n) over F or one of its inner forms. In this article, we classify the unramified irreducible smooth R-representations of G(F): more precisely we prove that they are those representations that are irreducibly induced from an unramified character of a Levi subgroup. We deduce that any smooth irreducible unramified mod ℓ representation of G(F) can be lifted to an ℓ-adic representation, which proves a conjecture by Vignéras