From the introduction: In this paper we carry out the -analogue of harmonic analysis on spheres. Using quantum -matrices of type or , we first construct a quantum analogue of the algebra of differential operators with polynomial coefficients on , the algebra of regular functions on the quantum vector space. This helps us to analyze the algebra of regular functions on quantum sphere . This algebra has the structure of a -module. To investigate the zonal spherical functions on , we introduce two kinds of coideals , corresponding to the left ideal of where . The zonal spherical functions on are defined as -invariant functions in .
They are expressed by two kinds of -orthogonal polynomials associated with discrete and continuous measures, that is, big -Jacobi polynomials and Rogers’ continuous -ultraspherical polynomials , according to the choice of the coideals . Furthermore, their orthogonality relations are also described by the invariant measure on . We remark that big -Jacobi polynomials will be considered only when . These results give a generalization of earlier works to the higher-dimensional quantum spheres, although we only consider the zonal spherical functions. M. Noumi, T. Umeda and M. Wakayama recently studied the quantized spherical harmonics on the -commutative polynomial ring “of type ”, in the sense of a -module [Dual pairs, spherical harmonics and a Capelli identity in quantum group theory, preprint (1993)]. They also obtained an explicit quantum analogue of the Capelli identity related to the dual pair .