A restriction problem for mod-$p$ representations of ${\mathrm{S}\mathrm{L}}_2(F)$

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 ${\mathrm{S}\mathrm{L}}_2(F)$ to its Borel subgroup. In essence, we show that the action of ${\mathrm{S}\mathrm{L}}_2(F)$ on its irreducibles is controlled by the action of the Borel subgroup. The results of this paper constitute the ${\mathrm{S}\mathrm{L}}_2$-analogue of a work of Paškūnas\cite{PaskunasRestriction}.