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

TL;DR

This paper provides a straightforward geometric proof demonstrating that certain Hecke modules associated with unimodular Hermitian forms over unramified extensions are free over their Hecke algebra, extending Hironaka's results.

Contribution

It offers a simplified geometric proof of the freeness of specific Hecke modules, a variation of Hironaka's theorem, in the context of unramified extensions of local fields.

Findings

Proved the module ^{K_0} is free over the Hecke algebra.

Extended Hironaka's theorem to a new setting involving Hermitian forms.

Provided a more accessible proof method for this class of problems.

Abstract

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 $E$.