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The famous Kazhdan-Lusztig Conjecture of the 1970s states that themultiplicity of an irreducible composition factor of a Verma module can becomputed by evaluating Kazhdan-Lusztig polynomials at 1. Thus the character ofa Verma module is a linear combination of characters of irreducible highestweight modules where the coefficients in the linear combination areKazhdan-Lusztig polynomials evaluated at 1. Kazhdan-Lusztig showed thatinverting and writing the character of an irreducible highest weight module asa linear combination of characters of Verma modules, the coefficients in thelinear combination are also Kazhdan-Lusztig polynomials evaluated at 1, up to asign. In this paper, we show how to prove Kazhdan-Lusztig's charactermultiplicity inversion formula by induction using coherent continuationfunctors. Unitary representations may be identified by determining ifcharacters and signature characters are the same. The signature character of aVerma module may be written as a linear combination of signature characters ofirreducible highest weight modules where the coefficients in the linearcombination are signed Kazhdan-Lusztig polynomials evaluated at 1. An analogousargument by induction using coherent continuation functors proves an analogousmultiplicity inversion formula for signature characters: the signaturecharacter of an irreducible highest weight module is a linear combination ofsignature characters of Verma modules where the coefficients, up to a sign, arealso signed Kazhdan-Lusztig polynomials evaluated at 1.

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