Quantum Matrix Spherical Functions

TL;DR

This paper develops a comprehensive theory of matrix-spherical functions within quantum groups, establishing their properties and linking them to orthogonal polynomials like Macdonald polynomials.

Contribution

It introduces a general framework for matrix-spherical functions in quantum algebra, connecting classical triples to quantum analogs and constructing new families of orthogonal polynomials.

Findings

Established existence and orthogonality of matrix-spherical functions.

Linked classical triples to quantum commutative triples.

Constructed vector-valued orthogonal polynomials related to Macdonald polynomials.

Abstract

A general theory of matrix-spherical functions for dual Hopf algebras and right coideal subalgebras is developed. We establish their existence and define their orthogonality relations. When specialized to Kolb and Letzter's quantum symmetric pair coideal subalgebras, we associate, to each classical commutative triple, a unique corresponding quantum commutative triple. This leads to families of vector-valued orthogonal polynomials, which diagonalize a commutative algebra of difference-reflection operators and are invariant under sending $q \mapsto q^{-1}$. Various examples of these vector-valued orthogonal polynomials are given and identified with Intermediate Macdonald polynomials