Let $p$ be prime. In the representation theory of $p$-adic groups over algebraically closed fields of characteristic $p$, such as the general linear group $GL_n(F)$ or the special linear group $SL_n(F)$, where the entries vary in a finite extension $F$ of the field $\mathbb{Q}_p$ of $p$-adic numbers, an important concept is that of distinguished representations. Given a subgroup $H$ of $G$ and a character $\chi$ of $H$ (i.e. a group homomorphism from $H$ to $\mathbb{K}^*$), a representation $(\pi, V)$ of $G$ is said to be $(H, \chi)$-distinguished when

\[ Hom_H (\pi, \chi) := \{l: \, V \to \chi \, \text{linear map} : \, \forall h \in H, \, v \in V, l(\pi(h)v) = \chi(h)l(v) \} \ne 0.\]

When $\chi$ is trivial (i.e. $\chi= \mathbb{1}$), we simply say that $(\pi, V)$ is $H$-distinguished. We study this concept of $p$-modular representation of $SL_2(\mathbb{F}_q)$, focusing in this talk on distinction with respect to its standard Borel subgroup of upper-triangular matrices.

This is joint work with Ramla Abdellatif (Université de Picardie Jules Verne) and Jocelyn P. Vilela (Mindanao State University-IIT).