Quantum Symmetric Pairs and Their Zonal Spherical Functions

TL;DR

This paper investigates quantum symmetric pairs, establishing their biinvariant spaces' structure, classifying associated zonal spherical functions, and providing explicit algebraic generators and relations.

Contribution

It characterizes the biinvariant space structure and classifies quantum zonal spherical functions for symmetric pairs, including explicit algebraic descriptions.

Findings

Biinvariant space is isomorphic to Weyl group invariants.

Unique or nearly unique quantum zonal spherical functions are identified.

Complete list of generators and relations for relevant coideal subalgebras provided.

Abstract

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.