On freeness of compactly induced mod-$p$ representations of $\rm{SL}_{2}(F)$

Abstract

Let $p$ be a prime, and $F$ a non-archimedean local field with residue characteristic $p$ and ring of integers $\mathcal{O}_{F}$. Set $G_{S}:={\rm SL}_{2}(F)$and $K_{0}:={\rm SL}_{2}(\mathcal{O}_{F})$ . For a smooth irreducible $\bar{\mathbb{F}}_{p}$-representation $\sigma$ of $K_{0}$, we study the structure of the compact induction ${\rm ind}_{K_{0}}^{G_{S}}(\sigma)$ as a left module over the standard spherical Hecke algebra ${\rm End}_{G_{S}}\left({\rm ind}_{K_{0}}^{G_{S}}(\sigma)\right)$. We prove that it is free and of infinite rank.