  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.
