Symmetries for spherical functions of type for quantum symmetric pairs

Let g be a complex semisimple Lie algebra and let Uq (g) denote the associated Drinfel'd Jimbo quantized enveloping algebra. In this paper we study spherical functions of Uq (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.