Whittaker Categories, Properly Stratified Categories and Fock Space Categorification for Lie Superalgebras

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.