Categorical actions on unipotent representations of finite classical groups

TL;DR

This paper explores the categorical representation of Kac-Moody algebras on unipotent representations of finite classical groups, extending previous work and establishing connections with Fock spaces and Broue's conjecture.

Contribution

It extends the categorical framework to types B and C finite reductive groups and links the Harish-Chandra branching graph to crystal graphs of Fock spaces.

Findings

Decategorified representation matches a sum of level 2 Fock spaces

Harish-Chandra branching graph coincides with crystal graph

Derived equivalences support Broue's abelian defect group conjecture

Abstract

We review the categorical representation of a Kac-Moody algebra on unipotent representations of finite unitary groups in non-defining characteristic given by the authors. Then, we extend this construction to finite reductive groups of types B or C, in non-defining characteristic. We show that the decategorified representation is isomorphic to a direct sum of level 2 Fock spaces. We deduce that the Harish-Chandra branching graph coincides with the crystal graph of these Fock spaces. We also obtain derived equivalences between blocks, yielding Broue's abelian defect group conjecture for unipotent l-blocks at linear primes.