Model categories and pro-$p$ Iwahori-Hecke modules

TL;DR

This paper explores the relationship between pro-$p$ Iwahori-Hecke modules and smooth representations of topological groups, establishing equivalences and model structures that generalize previous theorems in the context of Gorenstein rings.

Contribution

It introduces a new framework connecting Gorenstein projective modules over Hecke algebras with smooth group representations, extending known results to broader classes of groups.

Findings

Establishes an equivalence between Gorenstein projective modules and certain smooth representations.

Shows the Gorenstein projective model structure admits a right transfer.

Demonstrates the right derived functor of $I$-invariants is an equivalence on a subcategory.

Abstract

Let $G$ denote a possibly discrete topological group admitting an open subgroup $I$ which is pro-$p$. If $H$ denotes the corresponding Hecke algebra over a field $k$ of characteristic $p$ then we study the adjunction between $H$-modules and $k$-linear smooth $G$-representations in terms of various model structures. If $H$ is a Gorenstein ring we single out a full subcategory of smooth $G$-representations which is equivalent to the category of all Gorenstein projective $H$-modules via the functor of $I$-invariants. This applies to groups of rational points of split connected reductive groups over finite and over non-archimedean local fields, thus generalizing a theorem of Cabanes. Moreover, we show that the Gorenstein projective model structure on the category of $H$-modules admits a right transfer. On the homotopy level the right derived functor of $I$-invariants then admits a right inverse and becomes an equivalence when restricted to a suitable subcategory.