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

Abstract

Let 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 GLn, n ¥ 1, over F or one of its inner forms. In this article, we prove that the unramified irreducible smooth R-representations of G(F) are those representations that are irreducibly induced from an unramified character of a Levi subgroup. We deduce that any irreducible unramified Fl-representation of G(F) can be lifted to Ql, which proves a conjecture by Vignéras.