Categorical action for finite classical groups and its applications: characteristic 0

TL;DR

This paper constructs a categorical double quantum Heisenberg action on the representation categories of finite classical groups over fields of characteristic zero, enabling new character distinctions and explicit Kac-Moody actions.

Contribution

It introduces a novel categorical framework for finite classical groups, allowing complete invariants for characters and explicit Kac-Moody algebra actions.

Findings

Colored weight functions distinguish all irreducible characters.

Complete invariants for quadratic unipotent characters.

Explicit determination of Kac-Moody actions via theta correspondence.

Abstract

In this paper, we construct a categorical double quantum Heisenberg action on the representation category of finite classical groups $\mathrm{O}_{2n+1}(q)$, $\mathrm{Sp}_{2n}(q)$ and $\mathrm{O}^{\pm}_{2n}(q)$ with $q$ odd. Over a field of characteristic zero or characteristic $\ell$ with $\ell\nmid q(q-1)$, we deduce a categorical action of a Kac-Moody algebra $\mathfrak{s}\mathfrak{l}'_{I_+}\oplus\mathfrak{s}\mathfrak{l}'_{I_-}$ on the representation category of finite classical groups. We show that the colored weight functions $\mathbb{O}^+(u)(\bullet)$, $\mathbb{O}^-(v)(\bullet)$ and uniform projection can distinguish all irreducible characters of finite classical groups. In particular, the colored weight functions are complete invariants of quadratic unipotent characters. We also show that using the theta correspondence and extra symmetries of categorical double quantum Heisenberg action, the Kac-Moody action on the Grothendieck group of the whole category can be determined explicitly.