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

TL;DR

This paper investigates the structure of first and top cohomology groups of a pro-$p$-Iwahori subgroup with coefficients in an algebraically closed field, revealing the Hecke algebra action in the case of irreducible root systems.

Contribution

It explicitly computes the Hecke algebra action on the first and top cohomology groups for irreducible root systems, providing partial results for the general case.

Findings

Computed Hecke algebra action on H^1 and H^{top} for irreducible root systems

Established partial results for the general case

Enhanced understanding of cohomology modules in p-adic groups

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.