Homological dimension of simple pro-p-Iwahori--Hecke modules

TL;DR

This paper investigates the projective resolutions of simple modules over pro-$p$-Iwahori--Hecke algebras associated with split reductive groups over nonarchimedean fields, revealing that most simple modules have infinite projective dimension.

Contribution

It classifies simple modules with finite projective dimension under mild conditions on p, and demonstrates that most simple modules have infinite projective dimension.

Findings

Most simple modules have infinite projective dimension.

A classification of simple modules with finite projective dimension is provided.

The results depend on a mild condition on p.

Abstract

Let $G$ be a split connected reductive group defined over a nonarchimedean local field of residual characteristic $p$, and let $\mathcal{H}$ be the pro-$p$-Iwahori--Hecke algebra associated to a fixed choice of pro-$p$-Iwahori subgroup. We explore projective resolutions of simple right $\mathcal{H}$-modules. In particular, subject to a mild condition on $p$, we give a classification of simple right $\mathcal{H}$-modules of finite projective dimension, and consequently show that "most" simple modules have infinite projective dimension.