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 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, g0)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. Contents 1. Introduction 1 2. Preliminaries 6 3. Quantum groups and canonical bases 9 4. Equivalence of categories 17 5. Stratified structure of O 24 6. Tilting modules and Ringel duality 29 7. Categorification of the q-symmetrized Fock space 33 8. Appendix A. Structural modules in O ν-pres 41 9. Appendix B. A realization of O and its graded version. 43 References 47