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Spherical functions on a family of quantum 3-spheres. (English) Zbl 0760.33009

The authors define a family of quantum spaces \(S_ q^ 3(c,d)\equiv S_ q^ 3\) 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 \(G=SU_ q(2)\) and \(K=U(1)\) is defined on \(S_ q^ 3(c,d)\). By the action of the group \(SU_ q(2)\), the algebra of functions \(A(S_ q^ 3)\) on \(S_ q^ 3(c,d)\) is decomposed into irreducible representations of \(SU_ q(2)\). Spherical functions on \(S_ q^ 3(c,d)\) 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 \(A(S_ q^ 3)\). 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 \(SU_ q(2)\) are given in terms of Stanton’s \(q\)-Krawtchouk polynomials.
Reviewer: A.Klimyk (Kiev)

MSC:

33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
43A90 Harmonic analysis and spherical functions
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References:

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