Quantum spherical functions of type $\chi$ as Macdonald-Koornwinder polynomials

TL;DR

This paper connects quantum symmetric pair theory with $q$-special functions, demonstrating that certain $ ext{chi}$-spherical functions are equivalent to Macdonald-Koornwinder polynomials under specific conditions.

Contribution

It establishes that $ ext{chi}$-spherical functions associated with quantum symmetric pairs are Macdonald-Koornwinder polynomials when the root system is reduced or of type $ ext{AIII}_a$.

Findings

$ ext{chi}$-spherical functions exist for each Hermitian quantum symmetric pair.

Invariant $ ext{chi}$-spherical functions are identified with Macdonald-Koornwinder polynomials.

The identification holds for reduced root systems and type $ ext{AIII}_a$ Satake diagrams.

Abstract

The theory of quantum symmetric pairs is applied to $q$-special functions. Previous work shows the existence of a family $\chi$-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 $\mathsf{AIII_a}$.