Classical and Signed Kazhdan-Lusztig Polynomials: Character Multiplicity Inversion by Induction

TL;DR

This paper proves Kazhdan-Lusztig's character multiplicity inversion formulas for Verma modules and irreducible highest weight modules using induction and coherent continuation functors, extending to signature characters with signed polynomials.

Contribution

It introduces an inductive proof method for Kazhdan-Lusztig's inversion formulas, including for signature characters with signed polynomials.

Findings

Proved character multiplicity inversion formulas using induction.

Extended formulas to signature characters with signed Kazhdan-Lusztig polynomials.

Provided a new approach to understanding unitary representations.

Abstract

The famous Kazhdan-Lusztig Conjecture of the 1970s states that the multiplicity of an irreducible composition factor of a Verma module can be computed by evaluating Kazhdan-Lusztig polynomials at 1. Thus the character of a Verma module is a linear combination of characters of irreducible highest weight modules where the coefficients in the linear combination are Kazhdan-Lusztig polynomials evaluated at 1. Kazhdan-Lusztig showed that inverting and writing the character of an irreducible highest weight module as a linear combination of characters of Verma modules, the coefficients in the linear combination are also Kazhdan-Lusztig polynomials evaluated at 1, up to a sign. In this paper, we show how to prove Kazhdan-Lusztig's character multiplicity inversion formula by induction using coherent continuation functors. Unitary representations may be identified by determining if characters and signature characters are the same. The signature character of a Verma module may be written as a linear combination of signature characters of irreducible highest weight modules where the coefficients in the linear combination are signed Kazhdan-Lusztig polynomials evaluated at 1. An analogous argument by induction using coherent continuation functors proves an analogous multiplicity inversion formula for signature characters: the signature character of an irreducible highest weight module is a linear combination of signature characters of Verma modules where the coefficients, up to a sign, are also signed Kazhdan-Lusztig polynomials evaluated at 1.