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Transactions of the American Mathematical Society

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A family of quantum projective spaces
and related $q$-hypergeometric
orthogonal polynomials

Authors: Mathijs S. Dijkhuizen and Masatoshi Noumi
Journal: Trans. Amer. Math. Soc. 350 (1998), 3269-3296
MSC (1991): Primary 33D80, 81R50, 17B37, 33D45
MathSciNet review: 1432197
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Abstract: A one-parameter family of two-sided coideals in $\mathcal{U}_{q} (\mathfrak{g}\mathfrak{l}(n))$ is defined and the corresponding algebras of infinitesimally right invariant functions on the quantum unitary group $U_{q}(n)$ are studied. The Plancherel decomposition of these algebras with respect to the natural transitive $U_{q}(n)$-action is shown to be the same as in the case of a complex projective space. By computing the radial part of a suitable Casimir operator, we identify the zonal spherical functions (i.e. infinitesimally bi-invariant matrix coefficients of finite-dimensional irreducible representations) as Askey-Wilson polynomials containing two continuous and one discrete parameter. In certain limit cases, the zonal spherical functions are expressed as big and little $q$-Jacobi polynomials depending on one discrete parameter.

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Additional Information

Mathijs S. Dijkhuizen
Affiliation: Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe 657, Japan

Masatoshi Noumi
Affiliation: Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe 657, Japan

Keywords: Quantum unitary group, quantum projective space, two-sided coideal, zonal spherical function, Casimir operator, radial part, second-order $q$-difference operator, Askey-Wilson polynomials, big and little $q$-Jacobi polynomials
Received by editor(s): April 28, 1996
Received by editor(s) in revised form: October 1, 1996
Additional Notes: The first author acknowledges financial support by the Japan Society for the Promotion of Science (JSPS) and the Netherlands Organization for Scientific Research (NWO)
Article copyright: © Copyright 1998 American Mathematical Society

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