Radial part calculations for quantum symmetric pairs with simple generators

TL;DR

This paper introduces quantum symmetric pairs with simple generators and explicitly computes the radial parts of their elements, leading to q-difference equations for matrix-valued spherical functions.

Contribution

It defines a new class of quantum symmetric pairs with simple generators and provides explicit calculations of their radial parts, including for key Casimir elements.

Findings

Explicit formulas for radial parts of quantum symmetric pair elements

Derivation of q-difference equations for matrix-valued spherical functions

Application to quantum analogues of classical groups

Abstract

We introduce the class of quantum symmetric pairs with simple generators. It is proved that the radial part of every element of a quantum symmetric pair with simple generators restricted to the set of regular points of this element can be computed. These computations are done explicitly for the Casimir elements of the quantum analogues of (SU(2), U(1)), (SU(2) x SU(2), diag) and (SU(3), U(2)) and give rise to second order q-difference equations for matrix valued spherical functions in general.