Relating Signed Kazhdan-Lusztig Polynomials and Classical Kazhdan-Lusztig Polynomials

TL;DR

This paper establishes a direct relationship between signed Kazhdan-Lusztig polynomials and classical Kazhdan-Lusztig polynomials, revealing that the signed versions are essentially classical polynomials evaluated at -q with a sign factor, aiding in the study of unitary duals.

Contribution

It demonstrates that signed Kazhdan-Lusztig polynomials are equal to classical ones evaluated at -q times a sign, simplifying their computation and application in representation theory.

Findings

Signed Kazhdan-Lusztig polynomials equal classical polynomials at -q with a sign

The result simplifies the computation of signatures in representation theory

Applications to the classification of unitary duals for real reductive Lie groups

Abstract

Motivated by studying the Unitary Dual Problem, a variation of Kazhdan-Lusztig polynomials was defined in [Yee08] 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. This result has applications to finding the unitary dual for real reductive Lie groups since Harish-Chandra modules may be constructed by applying Zuckerman functors to highest weight modules.