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

Let $p\ge 5$ 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 ${\text{H}}^1(I_1,C)$ and ${\text{H}}^{\text{top}}(I_1,C)$ when the root system of $G$ is irreducible. We also give some partial results in the general case.