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

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 $I_1$. 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.

arXiv comment17 pages. Added result about Poincare duality along with other minor changes. To appear in Acta Arithmetica

Cited by in corpus (1)

• The modular pro-$p$ Iwahori-Hecke $\mathrm{Ext}$-algebra