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

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.