Some limit transitions between BC type orthogonal polynomials interpreted on quantum complex Grassmannians
type: Item
instance of: scholarly article
title: Some limit transitions between BC type orthogonal polynomials interpreted on quantum complex Grassmannians
zbMATH Open document ID: 0960.33010
author: Mathijs S. Dijkhuizen
author: Jasper V. Stokman
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2001-05-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9806123
review text: The authors determine zonal spherical functions associated with the quantum analogue of the complex Grassmannian (denoted as \(U_q/K_q\)) which is defined as the quotient of the quantum unitary group \(U_q(n)\) by the quantum subgroup \(K_q\) corresponding to the classical subgroup \(K=U(n-l)\times U(l)\) of the group \(U(n)\). It is shown that the quantum space \(U_q/K_q\) can be formally obtained from the one-parameter family of quantum Grassmannians (known from the literature) by sending the parameter to infinity. It is shown that this limit transition on quantum Grassmannians is compatible with the limit transition from Koornwinder polynomials to multivariable big and little \(q\)-Jacobi polynomials. These transitions lead to a rigorous computation of the limit of the quantum Schur orthogonality relations for the zonal spherical functions. This in turn leads to a complete description of the harmonic analysis on the quantum Grassmannian \(U_q/K_q\). The authors obtain an identification of the zonal spherical functions on the quantum complex Grassmannian \(U_q/K_q\) with multivariable big or little \(q\)-Jacobi polynomials. It is indicated that such limit arguments can be applied for other quantum complex symmetric spaces.
Mathematics Subject Classification ID: 33D80
Mathematics Subject Classification ID: 33D45
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 1439857
zbMATH Keywords: quantum complex Grassmannian
zbMATH Keywords: zonal spherical function
zbMATH Keywords: \(q\)-Jacobi polynomials
zbMATH Keywords: Koornwinder polynomials
reviewed by: A. U. Klimyk
MaRDI profile type: Publication
OpenAlex ID: W2098267553
arXiv ID: math/9806123
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DOI: 10.2977/PRIMS/1195143610
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description: scientific article