Let $p$ be a prime and $F$ a non-archimedean local field of residue characteristic $p$. In this paper, we study the restriction of smooth irreducible $\bar{ \mathbb{ F}}_p$-representations of $\mathrm{ SL}_2(F)$ to its Borel subgroup. In essence, we show that the action of $\mathrm{ SL}_2(F)$ on its irreducibles is controlled by the action of the Borel subgroup. The results of this paper constitute the $\mathrm{ SL}_2$-analogue of a work of Pa\v{ s}k\=unas\cite{ PaskunasRestriction}.
Categories: math.RT