  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.
