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Harmonic analysis on quantum spheres associated with representations of $U\sb q ({\germ {so}}\sb N)$ 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 ${\cal D}$ of differential operators with polynomial coefficients on $A_q(V)$, the algebra of regular functions on the quantum vector space. This helps us to analyze the algebra $A_q(S^{N-1})$ of regular functions on quantum sphere $S_q^{N-1}$. This algebra $A_1(S^{N-1})$ has the structure of a $U_q({\germ{so}}_N)$-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 $J=U({\germ{so}}_N) \cdot{\germ k}$ of $U({\germ{so}}_N)$ where ${\germ k}={\germ {so}}_{N-1}\subset{\germ {so}}_N$. The zonal spherical functions on $S_q^{N-1}$ are defined as $J_q$-invariant functions in $A_q(S^{N-1})$. 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 $A_q(S^{N-1})$. We remark that big $q$-Jacobi polynomials will be considered only when $N=2n+1\geq 3$. These results give a generalization of earlier works to the higher-dimensional quantum spheres, although we only consider the zonal spherical functions. {\it M. Noumi}, {\it T. Umeda} and {\it M. Wakayama} recently studied the quantized spherical harmonics on the $q$-commutative polynomial ring “of type $A$”, in the sense of a $U_q({\germ {gl}}_n)$-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 $({\germ {sl}}_2,{\germ o}_n)$.
17B37Quantum groups and related deformations
43A99Miscellaneous topics in harmonic analysis
33D80Connections of basic hypergeometric functions with groups, algebras and related topics
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