*(English)*Zbl 0872.17013

From the introduction: In this paper we carry out the $q$-analogue of harmonic analysis on spheres. Using quantum $R$-matrices of type $B$ or $D$, we first construct a quantum analogue of the algebra $\mathcal{D}$ of differential operators with polynomial coefficients on ${A}_{q}\left(V\right)$, the algebra of regular functions on the quantum vector space. This helps us to analyze the algebra ${A}_{q}\left({S}^{N-1}\right)$ of regular functions on quantum sphere ${S}_{q}^{N-1}$. This algebra ${A}_{1}\left({S}^{N-1}\right)$ has the structure of a ${U}_{q}\left({\mathrm{\U0001d530\U0001d52c}}_{N}\right)$-module. To investigate the zonal spherical functions on ${S}_{q}^{N-1}$, we introduce two kinds of coideals ${J}_{q}$, corresponding to the left ideal $J=U\left({\mathrm{\U0001d530\U0001d52c}}_{N}\right)\xb7\U0001d528$ of $U\left({\mathrm{\U0001d530\U0001d52c}}_{N}\right)$ where $\U0001d528={\mathrm{\U0001d530\U0001d52c}}_{N-1}\subset {\mathrm{\U0001d530\U0001d52c}}_{N}$. The zonal spherical functions on ${S}_{q}^{N-1}$ are defined as ${J}_{q}$-invariant functions in ${A}_{q}\left({S}^{N-1}\right)$.

They are expressed by two kinds of $q$-orthogonal polynomials associated with discrete and continuous measures, that is, big $q$-Jacobi polynomials ${P}_{n}^{(\alpha ,\beta )}(X;q)$ and Rogers’ continuous $q$-ultraspherical polynomials ${C}_{n}^{\lambda}(X;q)$, according to the choice of the coideals ${J}_{q}$. Furthermore, their orthogonality relations are also described by the invariant measure on ${A}_{q}\left({S}^{N-1}\right)$. We remark that big $q$-Jacobi polynomials will be considered only when $N=2n+1\ge 3$. 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 $q$-commutative polynomial ring “of type $A$”, in the sense of a ${U}_{q}\left({\mathrm{\U0001d524\U0001d529}}_{n}\right)$-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 $({\mathrm{\U0001d530\U0001d529}}_{2},{\U0001d52c}_{n})$.