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arXiv · 1205.5607

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

Wai Ling Yee
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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.

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Wai Ling Yee. 2012-05-25. Relating Signed Kazhdan-Lusztig Polynomials and Classical Kazhdan-Lusztig Polynomials. https://doi.org/10.1215/00127094-2795435

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