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Zbl 0760.33009
Noumi, Masatoshi; Mimachi, Katsuhisa
Spherical functions on a family of quantum 3-spheres. (English)
[J] Compos. Math. 83, No.1, 19-42 (1992). ISSN 0010-437X; ISSN 1570-5846/e

The authors define a family of quantum spaces $S\sb q\sp 3(c,d)\equiv S\sb q\sp 3$ called quantum 3-spheres. They are given by two parameters $c$, $d$ and generalize quantum 2-spheres. The structure of $(G,K)$-space over the quantum group $G=SU\sb q(2)$ and $K=U(1)$ is defined on $S\sb q\sp 3(c,d)$. By the action of the group $SU\sb q(2)$, the algebra of functions $A(S\sb q\sp 3)$ on $S\sb q\sp 3(c,d)$ is decomposed into irreducible representations of $SU\sb q(2)$. Spherical functions on $S\sb q\sp 3(c,d)$ are defined which correspond to these representations. Two expressions for spherical functions in terms of $q$-orthogonal polynomials are given; one by the big $q$-Jacobi polynomials and the other by the $q$-Hahn polynomials. The spherical functions are orthogonal with respect to the scalar product determined on $A(S\sb q\sp 3)$. This leads to the orthogonality relations for the corresponding polynomials. The explicit formulas for transition coefficients from the spherical functions to the matrix elements of irreducible representations of $SU\sb q(2)$ are given in terms of Stanton's $q$-Krawtchouk polynomials.
[A.Klimyk (Kiev)]

MSC 2000:: *33D80 Connections with groups, algebras and related topics
33D45 Basic hypergeometric functions and integrals in several variables
17B37 Quantum groups and related deformations
43A90 Spherical functions (abstract harmonic analysis)

Keywords: quantum 3-spheres; quantum group; irreducible representations; spherical functions; big $q$-Jacobi polynomials; $q$-Hahn polynomials

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