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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A family of quantum projective spaces and related $q$-hypergeometric orthogonal polynomials
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by Mathijs S. Dijkhuizen and Masatoshi Noumi PDF
Trans. Amer. Math. Soc. 350 (1998), 3269-3296 Request permission

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
  • 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