A canonical torsion theory for pro-p Iwahori-Hecke modules

TL;DR

This paper develops a torsion theory for modules over pro-$p$-Iwahori Hecke algebras associated with split reductive groups over nonarchimedean fields, linking algebraic modules to smooth group representations and describing their structure explicitly.

Contribution

It introduces a canonical torsion theory for pro-$p$-Iwahori Hecke modules, establishing conditions for embedding into smooth representations and explicitly characterizing torsion and torsionfree classes.

Findings

The torsionfree class embeds fully faithfully into smooth representations generated by Iwahori fixed vectors.

When the field characteristic differs from $p$, the torsionfree class equals the entire module category.

Explicit descriptions of torsion and torsionfree classes for $G=SL_2(rak F)$, including cases where $rak F=b Q_p$.

Abstract

Let $\mathfrak F$ be a locally compact nonarchimedean field with residue characteristic $p$ and $G$ the group of $\mathfrak{F}$-rational points of a connected split reductive group over $\mathfrak{F}$. We define a torsion pair in the category Mod$(H)$ of modules over the pro-$p$-Iwahori Hecke $k$-algebra $H$ of $G$, where $k$ is an arbitrary field. We prove that, under a certain hypothesis, the torsionfree class embeds fully faithfully into the category Mod${}^I(G)$ of smooth $k$-representations of $G$ generated by their pro-$p$-Iwahori fixed vectors. If the characteristic of $k$ is different from $p$ then this hypothesis is always satisfied and the torsionfree class is the whole category Mod$(H)$. If $k$ contains the residue field of $\mathfrak F$ then we study the case $G = \mathbf{SL_2}(\mathfrak F)$. We show that our hypothesis is satisfied, and we describe explicitly the torsionfree and the torsion classes. If $\mathfrak F\neq \mathbb Q_p$ and $p\neq 2$, then an $H$-module is in the torsion class if and only if it is a union of supersingular finite length submodules; it lies in the torsionfree class if and only if it does not contain any nonzero supersingular finite length module. If $\mathfrak{F} = \mathbb{Q}_p$, the torsionfree class is the whole category Mod$(H)$, and we give a new proof of the fact that Mod$(H)$ is equivalent to Mod${}^I(G)$. These results are based on the computation of the $H$-module structure of certain natural cohomology spaces for the pro-$p$-Iwahori subgroup $I$ of $G$.