Symmetries for spherical functions of type $\chi$ for quantum symmetric pairs

TL;DR

This paper investigates the properties of spherical functions associated with quantum symmetric pairs, demonstrating their invariance under certain algebraic operators and symmetries, which advances understanding of quantum group symmetries.

Contribution

The paper establishes invariance properties of spherical functions of quantum groups under braid group operators and Weyl group actions, providing new insights into their symmetry structure.

Findings

Invariance under Wang-Zhang braid group operators

Relative Weyl group invariance on quantum torus

Enhanced understanding of quantum symmetric pair symmetries

Abstract

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\mathbf{U}_q(\mathfrak{g})$ denote the associated Drinfel'd Jimbo quantized enveloping algebra. In this paper we study spherical functions of $\mathbf{U}_q(\mathfrak{g})$ related to characters. We show invariance under the Wang-Zhang braid group operators and show relative Weyl group invariance, when restricted to the quantum torus.