  Perhaps the most important problem in representation theory in the 1970s and
early 1980s was the determination of the multiplicity of composition factors in
a Verma module. This problem was settled by the proof of the Kazhdan-Lusztig
Conjecture which states that the multiplicities may be computed via
Kazhdan-Lusztig polynomials. In this paper, we introduce signed Kazhdan-Lusztig
polynomials, a variation of Kazhdan-Lusztig polynomials which encodes signature
information in addition to composition factor multiplicities and Jantzen
filtration level. Careful consideration of Gabber and Joseph's proof of Kazhdan
and Lusztig's inductive formula for computing Kazhdan-Lusztig polynomials leads
to an inductive formula for the signed Kazhdan-Lusztig polynomials. We use
these polynomials to compute the signature of an invariant Hermitian form on an
irreducible highest weight module. Such a formula has applications to unitarity
testing.
