Freeness of spherical Hecke modules of unramified $U(2,1)$ in characteristic $p$

Abstract: Let $F$ be a non-archimedean local field of odd residue characteristic $p$. Let $G$ be the unramified unitary group $U(2, 1)(E/F)$ in three variables, and $K$ be a maximal compact open subgroup of $G$. For an irreducible smooth representation $\sigma$ of $K$ over $\overline{\mathbf{F}}_p$, we prove that the compactly induced representation $\text{ind}^G _K \sigma$ is free of infinite rank over the spherical Hecke algebra $\mathcal{H}(K, \sigma)$.