Hecke module structure on first and top pro-$p$-Iwahori cohomology

Abstract: Let $p\geq 5$ be a prime number, $G$ a split connected reductive group defined over a $p$-adic field, and $I_1$ a choice of pro-$p$-Iwahori subgroup. Let $C$ be an algebraically closed field of characteristic $p$ and $\mathcal{H}$ the pro-$p$-Iwahori--Hecke algebra over $C$ associated to $I_1$. In this note, we compute the action of $\mathcal{H}$ on $\textrm{H}^1(I_1,C)$ and $\textrm{H}^{\textrm{top}}(I_1,C)$ when the root system of $G$ is irreducible. We also give some partial results in the general case.