Repr\'esentations modulo $p$ de $\mathrm{GL}(m,D)$, $D$ alg\`ebre \`a division sur un corps local

TL;DR

This thesis advances the understanding of mod p representations of inner forms of general linear groups over local fields, providing classifications and irreducibility criteria for various cases including rank 1 and higher.

Contribution

It develops a detailed classification of irreducible admissible smooth mod p representations for inner forms of GL(m,D), extending known results to higher ranks and more general groups.

Findings

Classification of irreducible admissible smooth representations for rank 1 case.

Generalized Steinberg representations valid for any reductive group over local fields.

New classification results for GL(3,D) and higher ranks with technical assumptions.

Abstract

This Ph.D. thesis belongs to the realm of mod $p$ representation theory of $p$-adic groups. The main object of study is the inner form of the general linear group $\mathrm{GL}(m,D)$ where $D$ is a division algebra over a non-Archimedean local field. The first part focuses on the rank $1$ case ($m=2$): it develops in detail the irreducibility criterion for parabolically induced representations and a classification of irreducible admissible smooth representations "\`a la Barthel-Livn\'e". Then one turns to the generalized Steinberg representations: building on ideas of Grosse-Kl\"onne and Herzig, this chapter manages to get a result that is valid for any reductive group over a non-Archimedean local field. Finally the last part exploits the two previous ones, and by overcoming some combinatorial difficulties that arise in the rank $>1$ case, one gets a classification "\`a la Herzig" for $\mathrm{GL}(3,D)$ and $\mathrm{GL}(m,D)$, $m>3$, with some technical assumptions on $D$.