JournalsprimsVol. 35, No. 3pp. 451–500

Some Limit Transitions between <em>BC</em> Type Orthogonal Polynomials Interpreted on Quantum Complex Grassmannians

  • Jasper V. Stokman

    University of Amsterdam, Netherlands
  • Mathijs S. Dijkhuizen

    Kobe University, Japan
Some Limit Transitions between <em>BC</em> Type Orthogonal Polynomials Interpreted on Quantum Complex Grassmannians cover
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Abstract

The quantum complex Grassmannian Uq/Kq of rank l is the quotient of the quantum unitary group Uq = Uq(n) by the quantum subgroup Kq = Uq(n–l) x Uq(l). We show that (Uq, Kq) is a quantum Gelfand pair and we express the zonal spherical functions, i.e. Kq-biinvariant matrix coefficients of finite-dimensional irreducible representations of Uq, as multivariable little q-Jacobi polynomials depending on one discrete parameter. Another type of biinvariant matrix coefficients is identified as multivariable big q-Jacobi polynomials. The proof is based on earlier results by Noumi, Sugitani and the first author relating Koornwinder polynomials to a one-parameter family of quantum complex Grassmannians, and certain limit transitions from Koornwinder polynomials to multivariable big and little q-Jacobi polynomials studied by Koornwinder and the second author.

Cite this article

Jasper V. Stokman, Mathijs S. Dijkhuizen, Some Limit Transitions between <em>BC</em> Type Orthogonal Polynomials Interpreted on Quantum Complex Grassmannians. Publ. Res. Inst. Math. Sci. 35 (1999), no. 3, pp. 451–500

DOI 10.2977/PRIMS/1195143610