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Harmonic analysis on quantum spheres associated with representations of ${U}_{q}\left({\mathrm{𝔰𝔬}}_{N}\right)$ and $q$-Jacobi polynomials. (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 $𝒟$ 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 ${U}_{q}\left({\mathrm{𝔰𝔬}}_{N}\right)$-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 $J=U\left({\mathrm{𝔰𝔬}}_{N}\right)·𝔨$ of $U\left({\mathrm{𝔰𝔬}}_{N}\right)$ where $𝔨={\mathrm{𝔰𝔬}}_{N-1}\subset {\mathrm{𝔰𝔬}}_{N}$. 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 ${P}_{n}^{\left(\alpha ,\beta \right)}\left(X;q\right)$ and Rogers’ continuous $q$-ultraspherical polynomials ${C}_{n}^{\lambda }\left(X;q\right)$, 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 ${U}_{q}\left({\mathrm{𝔤𝔩}}_{n}\right)$-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 $\left({\mathrm{𝔰𝔩}}_{2},{𝔬}_{n}\right)$.

##### MSC:
 17B37 Quantum groups and related deformations 43A99 Miscellaneous topics in harmonic analysis 33D80 Connections of basic hypergeometric functions with groups, algebras and related topics