Quantum Symmetric Pairs and Their Zonal Spherical Functions

We study the space of biinvariants and zonal spherical functions associated to quantum symmetric pairs in the maximally split case. Under the obvious restriction map, the space of biinvariants is proved isomorphic to the Weyl group invariants of the character group ring associated to the restricted roots. As a consequence, there is either a unique set, or an (almost) unique two-parameter set of Weyl group invariant quantum zonal spherical functions associated to an irreducible symmetric pair. Included is a complete and explicit list of the generators and relations for the left coideal subalgebras of the quantized enveloping algebra used to form quantum symmetric pairs. INTRODUCTION The representation theory of semisimple Lie algebras and Lie groups has been closely intertwined with the theory of symmetric spaces since E. Cartan’s pioneering work in the 1920’s. A beautiful classical result shows that the zonal spherical functions of these spaces can be identified with a family of orthogonal polynomials. With the introduction of quantum groups in the 1980’s, it was natural to look for and study quantum versions of symmetric spaces. This search became especially compelling as q orthogonal polynomials, which are obvious candidates for quantum zonal spherical functions, appeared in the literature (see for example [Ma] and [K2]). However, the theory of quantum symmetric spaces was initially slow to develop because it was not obvious how to form them. supported by NSA grant no. MDA904-01-1-0033. AMS subject classification 17B37