Whittaker categories, properly stratified categories and Fock space categorification for Lie superalgebras

TL;DR

This paper explores Whittaker modules over Lie superalgebras, showing they can be described as Serre quotients of known categories and categorify a $q$-symmetrized Fock space, linking algebraic and categorical structures.

Contribution

It introduces a new categorical framework for Whittaker modules over Lie superalgebras and connects these to Fock space categorification, extending prior work to superalgebra contexts.

Findings

Whittaker categories are equivalent to Serre quotients of BGG category $\\mathcal{O}$

The Backelin functor $\\Gamma_\zeta$ realizes Serre quotient functors

These categories categorify the $q$-symmetrized Fock space and bases correspond to tilting and simple objects

Abstract

We study various categories of Whittaker modules over a type I Lie superalgebra realized as cokernel categories that fit into the framework of properly stratified categories. These categories are the target of the Backelin functor $\Gamma_\zeta$. We show that these categories can be described, up to equivalence, as Serre quotients of the BGG category $\mathcal O$ and of certain singular categories of Harish-Chandra $(\mathfrak g,\mathfrak g_{\bar 0})$-bimodules. We also show that $\Gamma_\zeta$ is a realization of the Serre quotient functor. We further investigate a $q$-symmetrized Fock space over a quantum group of type A and prove that, for general linear Lie superalgebras our Whittaker categories, the functor $\Gamma_\zeta$ and various realizations of Serre quotients and Serre quotient functors categorify this $q$-symmetrized Fock space and its $q$-symmetrizer. In this picture, the canonical and dual canonical bases in this $q$-symmetrized Fock space correspond to tilting and simple objects in these Whittaker categories, respectively.