Brundan-Kazhdan-Lusztig and super duality conjectures

TL;DR

This paper proposes a super duality conjecture linking categories of modules over Lie superalgebras and Lie algebras, using Fock space formalism to relate Kazhdan-Lusztig theories, and verifies some special cases of the conjecture.

Contribution

It formulates a general super duality conjecture and demonstrates that BKL polynomials for gl(m|n) match classical parabolic Kazhdan-Lusztig polynomials, supporting the conjecture.

Findings

BKL polynomials for gl(m|n) identified with classical parabolic Kazhdan-Lusztig polynomials

Established special cases of the BKL conjecture for parabolic category O

Supported super duality conjecture through additional results

Abstract

We formulate a general super duality conjecture on connections between parabolic categories O of modules over Lie superalgebras and Lie algebras of type A, based on a Fock space formalism of their Kazhdan-Lusztig theories which was initiated by Brundan. We show that the Brundan-Kazhdan-Lusztig (BKL) polynomials for Lie superalgebra gl(m|n) in our parabolic setup can be identified with the usual parabolic Kazhdan-Lusztig polynomials. We establish some special cases of the BKL conjecture on the parabolic category O of gl(m|n)-modules and additional results which support the BKL conjecture and super duality conjecture.