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

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 $D$.