Quantum spherical functions of type $\ddot{I}‡$ as Macdonald-Koornwinder polynomials

Abstract

The theory of quantum symmetric pairs is applied to $q$-special functions. Previous work shows the existence of a family $\ddot{I}‡$-spherical functions indexed by the integers for each Hermitian quantum symmetric pair. A distinguished family of such functions, invariant under the Weyl group of the restricted roots, is shown to be a family of Macdonald-Koornwinder polynomials if the restricted root system is reduced or if the Satake diagram is of type $\mathsf{AII}{\mathsf{I}}_{\mathsf{a}}$.

31 pages