  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 $D$.

Representation Theory