Super duality for general linear Lie superalgebras and applications

TL;DR

This paper uses super duality to establish equivalences between module categories of general linear Lie superalgebras, leading to solutions for character problems in certain categories using Kazhdan-Lusztig polynomials.

Contribution

It introduces new category equivalences for modules over general linear Lie superalgebras and solves the irreducible character problem for specific categories.

Findings

Established correspondence of standard, tilting, and simple modules.

Identified u-homology groups under category equivalences.

Solved the irreducible character problem for certain gl(m|n) categories.

Abstract

We apply the super duality formalism recently developed by the authors to obtain new equivalences of various module categories of general linear Lie superalgebras. We establish the correspondence of standard, tilting, and simple modules, as well as the identification of the u-homology groups, under these category equivalences. As an application, we obtain a complete solution of the irreducible character problem for some new parabolic BGG categories of gl(m|n)-modules, including the full BGG category of gl(m|2)-modules, in terms of type A Kazhdan-Lusztig polynomials.