Abstract: One of the most important open problems in mathematics is the Unitary Dual Problem: given a group, classify its irreducible unitary representations. The most common approach to classifying unitary representations is to study representations admitting non-degenerate invariant Hermitian forms (these are classified), compute the signatures of the forms, and then determine when the forms are definite. Signature character formulas are very complicated due to the recursive nature of the formulas. However, it turns out that the complexity may be encoded by well-known combinatorial objects: signature character formulas involve signed Kazhdan-Lusztig polynomials (which turn out to be classical Kazhdan-Lusztig polynomials evaluated at $-q$ times a sign) and pieces of Hall-Littlewood polynomials (called Hall-Littlewood polynomial summands) evaluated at $q=-1$. Some of the material covered in this talk is joint work with Justin Lariviere. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.