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.

53 pages

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• Projective and Whittaker functors on category $\mathcal{O}$