Hecke module structure on first and top pro-$p$-Iwahori cohomology
Volume 186 / 2018
Acta Arithmetica 186 (2018), 349-376
MSC: Primary 20C08, 20J06; Secondary 16E30.
DOI: 10.4064/aa170903-24-3
Published online: 9 November 2018
Abstract
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 $I_1$. We compute the action of $\mathcal{H}$ on ${\rm H}^1(I_1,C)$ and ${\rm H}^{{\rm top}}(I_1,C)$ when the root system of $G$ is irreducible. We also give some partial results in the general case.
