Brundan–Kazhdan–Lusztig and Super Duality Conjectures

Abstract

We formulate a general super duality conjecture on connections between parabolic categories $\mathcal{O}$ 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 $\mathfrak{gl}(m|n)$ 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 $\mathcal{O}$ of $\mathfrak{gl}(m|n)$-modules and additional results which support the BKL conjecture and super duality conjecture.