A short proof of Hironaka's Theorem on freeness of some Hecke modules

Let $E\mathrm{/}F$ be an unramified extension of non-archimedean local fields of residual characteristic different than $2$. We provide a simple geometric proof of a variation of a result of Y. Hironaka. Namely we prove that the module $\mathcal{S}(X{)}^{K_0}$ is free over the Hecke algebra $\mathcal{H}(S{L}_n(E),S{L}_n(O_E))$, where $X$ is the space of unimodular Hermitian forms on $E^n$ and $O_E$ is the ring of integers in $E$.