On modular Harish-Chandra series of finite unitary groups

TL;DR

This paper advances the understanding of modular representation theory of finite unitary groups by using crystal graphs to describe Harish-Chandra branching and cuspidal support, especially for large primes.

Contribution

It introduces the use of the $rak{sl}_$-crystal on level 2 Fock spaces to fully describe Harish-Chandra branching rules for unipotent representations.

Findings

$rak{sl}_e$-crystal describes Harish-Chandra branching for level 2 Fock spaces.

$rak{sl}_$-crystal completes the description for large primes.

Provides a combinatorial framework for cuspidal support in modular representation theory.

Abstract

In the modular representation theory of finite unitary groups when the characteristic $\ell$ of the ground field is a unitary prime, the $\hat{\mathfrak{sl}}_e$-crystal on level $2$ Fock spaces graphically describes the Harish-Chandra branching of unipotent representations restricted to the tower of unitary groups. However, how to determine the cuspidal support of an arbitrary unipotent representation has remained an open question. We show that for $\ell$ sufficiently large, the $\mathfrak{sl}_\infty$-crystal on the same level $2$ Fock spaces provides the remaining piece of the puzzle for the full Harish-Chandra branching rule.