A classification of irreducible admissible mod p representations of p-adic reductive groups

TL;DR

This paper provides a complete classification of irreducible admissible mod p representations of p-adic reductive groups, linking them to supercuspidal representations of Levi subgroups and parabolic induction.

Contribution

It extends previous work to all p-adic reductive groups, establishing a classification framework based on supersingular and supercuspidal representations.

Findings

Classification in terms of supercuspidal representations

Supersingularity is equivalent to supercuspidality

Unified approach for all reductive groups

Abstract

Let F be a locally compact non-archimedean field, p its residue characteristic, and G a connected reductive group over F. Let C an algebraically closed field of characteristic p. We give a complete classification of irreducible admissible C-representations of G = G(F), in terms of supercuspidal C-representations of the Levi subgroups of G, and parabolic induction. Thus we push to their natural conclusion the ideas of the third-named author, who treated the case G = GL_m, as further expanded by the first-named author, who treated split groups G. As in the split case, we first get a classification in terms of supersingular representations of Levi subgroups, and as a consequence show that supersingularity is the same as supercuspidality.