Let $E/F$ 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 $\mathcal{ H}(SL_{ n}(E),SL_{ n}(O_E))$, where $X$ is the space of unimodular Hermitian forms on $E^n$ and $O_E$ is the ring of integers in $E$.
Categories: math.RT