Relating signed and classical Kazhdan–Lusztig polynomials

Abstract

Motivated by studying the unitary dual problem, a variation of Kazhdan–Lusztig polynomials was defined in a previous publication by the author which encodes signature information at each level of the Jantzen filtration. These so-called signed Kazhdan–Lusztig polynomials may be used to compute the signatures of invariant Hermitian forms on irreducible highest weight modules. The key result of this paper is a simple relationship between signed Kazhdan–Lusztig polynomials and classical Kazhdan–Lusztig polynomials: signed Kahzdan–Lusztig polynomials are shown to equal classical Kazhdan–Lusztig polynomials evaluated at $-q$ rather than $q$ and multiplied by a sign. A simple signature character inversion formula follows from this relationship. These results have applications to finding the unitary dual for real reductive Lie groups since Harish–Chandra modules may be constructed by applying Zuckerman functors to the highest weight modules.