Quantum spherical functions and Macdonald polynomials, beyond the trivial character

IMPRS seminar extra talk

A celebrated result by Gail Letzter identifies the zonal spherical functions on quantum symmetric pairs as Macdonald polynomials, her results unify and generalize earlier identifications made by Koornwinder, Noumi and others. One ingredient that goes into this identification is a character, that is chosen to be trivial. In the classical case Heckman shows that, even if the character is nontrivial, the associated spherical functions correspond to a distinguished family of Heckman–Opdam polynomials. The goal of this talk is to present a generalization of Letzter’s result to the case where the character is nontrivial. I will focus on the construction the orthogonal polynomials, taking the classical case as intuition. Additionally, I will outline a method for deriving the orthogonality measure.