A family of quantum projective spaces and related $q$-hypergeometric orthogonal polynomials
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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.References
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Additional Information
- Mathijs S. Dijkhuizen
- Affiliation: Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe 657, Japan
- Email: msdz@math.s.kobe-u.ac.jp
- Masatoshi Noumi
- Affiliation: Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe 657, Japan
- MR Author ID: 210204
- Email: noumi@math.s.kobe-u.ac.jp
- 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)
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 3269-3296
- MSC (1991): Primary 33D80, 81R50, 17B37, 33D45
- DOI: https://doi.org/10.1090/S0002-9947-98-01971-0
- MathSciNet review: 1432197