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

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.