Unipotent representations as a categorical centre

TL;DR

This paper establishes an equivalence between a category of unipotent representations of a finite group and the center of a monoidal category of sheaves, revealing deep structural insights in representation theory.

Contribution

It introduces a novel categorical equivalence linking unipotent representations of G(F_q) to the center of a sheaf-based monoidal category, extending to nonsplit groups.

Findings

Categorical equivalence between unipotent representations and sheaf centers

Extension of results to nonsplit groups

Insight into the structure of unipotent representations

Abstract

Let G(F_q) be the group of rational points of a split connected reductive group G defined over the finite field F_q. In this paper we show that the category of representations of G(F_q) which are finite direct sums of unipotent representations in a fixed two-sided cell is equivalent to the centre of a certain monoidal category of sheaves on the product of two copies of the flag manifold of G. We also consider a version of this for nonsplit groups.