Quantum quaternion spheres

For the quantum symplectic group SP _q (2n), we describe the C ^?-algebra of continuous functions on the quotient space S P _q (2n)/S P _q (2n?2) as an universal C ^?-algebra given by a finite set of generators and relations. The proof involves a careful analysis of the relations, and use of the branching rules for representations of the symplectic group due to Zhelobenko. We then exhibit a set of generators of the K-groups of this C ^?-algebra in terms of generators of the C ^?-algebra.