On semisimple l-modular Bernstein-blocks of a p-adic general linear group

TL;DR

This paper explores the structure of Bernstein blocks in the representation category of p-adic general linear groups over fields of characteristic different from p, extending known complex case results to modular settings.

Contribution

It introduces a replacement for the Iwahori-Hecke algebra in level zero and describes how to relate arbitrary blocks to simpler ones, generalizing complex case techniques.

Findings

Construction of a Morita-equivalence for level zero blocks.

Description of arbitrary blocks in terms of simpler blocks.

Extension of complex case methods to modular representations.

Abstract

Let $G_n=\operatorname{GL}_n(F)$, where $F$ is a non-archimedean local field with residue characteristic $p$. Our starting point is the Bernstein-decomposition of the representation category of $G_n$ over an algebraically closed field of characteristic $\ell \neq p$ into blocks. In level zero, we associate to each block a replacement for the Iwahori-Hecke algebra which provides a Morita-equivalence just as in the complex case. Additionally, we will explain how this gives rise to a description of an arbitrary $G_n$-block in terms of simple $G_m$-blocks (for $m\leq n$), paralleling the approach of Bushnell and Kutzko in the complex setting.