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

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 Γ _ζ . We show that these categories can be described, up to equivalence, as Serre quotients of the BGG category 𝒪 and of certain singular categories of Harish-Chandra (𝔤,𝔤_0̅) -bimodules. We also show that Γ _ζ 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 Γ _ζ 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.