Repr\'esentations lisses modulo l de GL(m,D)

TL;DR

This paper classifies all smooth irreducible representations of the p-adic group GL(m,D) over algebraically closed fields, extending previous classifications and establishing uniqueness of supercuspidal support.

Contribution

It provides a comprehensive classification of irreducible representations of GL(m,D) using multisegments, generalizing prior work and employing local type theory methods.

Findings

Unique supercuspidal support for each irreducible representation

Classification via supercuspidal and aperiodic multisegments

Extension of Zelevinski, Tadic, and Vignéras' frameworks

Abstract

Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite dimensional central division F-algebra and let R be an algebraically closed field of characteristic different from p. We classify all smooth irreducible representations of GL(m,D) with coefficients in R, in terms of multisegments, generalizing works by Zelevinski, Tadic and Vign\'eras. We prove that any irreducible R-representation of GL(m,D) has a unique supercuspidal support, and thus get two classifications: one by supercuspidal multisegments, classifying representations with a given supercuspidal support, and one by aperiodic multisegments, classifying representations with a given cuspidal support. These constructions are made in a purely local way, with a substantial use of type theory.