Unipotent representations as a categorical centre

Abstract

Let G(F[subscript q]) be the group of rational points of a split connected reductive group G over the finite field F[subscript q]. In this paper we show that the category of representations of G(F[subscript 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 flag manifold of G x G. We also consider a version of this for nonsplit groups.