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Harmonic analysis on quantum spheres associated with representations of U q (𝔰𝔬 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 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 (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(𝔰𝔬 N )·𝔨 of U(𝔰𝔬 N ) where 𝔨=𝔰𝔬 N-1 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 (α,β) (X;q) and Rogers’ continuous q-ultraspherical polynomials C n λ (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+13. 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 (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 (𝔰𝔩 2 ,𝔬 n ).

MSC:
17B37Quantum groups and related deformations
43A99Miscellaneous topics in harmonic analysis
33D80Connections of basic hypergeometric functions with groups, algebras and related topics