Hecke operators for curves over non-archimedean local fields and related finite rings

We study Hecke operators associated with curves over a non-archimedean local field $K$ and over the rings $O\mathrm{/}{\mathfrak{m}}^N$, where $O\subset K$ is the ring of integers. Our main result is commutativity of a certain “small” local Hecke algebra over $O\mathrm{/}{\mathfrak{m}}^N$, associated with a connected split reductive group $G$ such that $[G,G]$ is simple and simpy connected. The proof uses a Hecke algebra associated with $G(K(\text{\hspace{0.17em}⁣}(t)\text{\hspace{0.17em}⁣}))$ and a global argument involving $G$-bundles on curves.