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.
Categories: math.RT