Functorial properties of pro-$p$-Iwahori cohomology

TL;DR

This paper develops spectral sequences connecting pro-$p$-Iwahori cohomology with derived functors on mod-$p$ representations of reductive groups over local fields, providing new tools for understanding their structure.

Contribution

It introduces two spectral sequences linking pro-$p$-Iwahori cohomology to derived functors on mod-$p$ representations, expanding the theoretical framework in this area.

Findings

Established a link between parabolic induction and derived ordinary parts.

Created a Poincaré duality spectral sequence for pro-$p$-Iwahori cohomology.

Computed examples of Hecke modules $ extrm{H}^i(I_1, extrm{pi})$.

Abstract

Suppose $F$ is a finite extension of $\mathbb{Q}_p$, $G$ is the group of $F$-points of a connected reductive $F$-group, and $I_1$ is a pro-$p$-Iwahori subgroup of $G$. We construct two spectral sequences relating derived functors on mod-$p$ representations of $G$ to the analogous functors on Hecke modules coming from pro-$p$-Iwahori cohomology. More specifically: (1) using results of Ollivier--Vign\'eras, we provide a link between the right adjoint of parabolic induction on pro-$p$-Iwahori cohomology and Emerton's functors of derived ordinary parts; and (2) we establish a "Poincar\'e duality spectral sequence" relating duality on pro-$p$-Iwahori cohomology to Kohlhaase's functors of higher smooth duals. As applications, we calculate various examples of the Hecke modules $\textrm{H}^i(I_1,\pi)$.