Some Limit Transitions between Type Orthogonal Polynomials Interpreted on Quantum Complex Grassmannians

  • Mathijs S. Dijkhuizen

    Kobe University, Japan
  • Jasper V. Stokman

    University of Amsterdam, Netherlands

Abstract

The quantum complex Grassmannian of rank is the quotient of the quantum unitary group by the quantum subgroup . We show that is a quantum Gelfand pair and we express the zonal spherical functions, i.e. -biinvariant matrix coefficients of finite-dimensional irreducible representations of , as multivariable little -Jacobi polynomials depending on one discrete parameter. Another type of biinvariant matrix coefficients is identified as multivariable big -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 -Jacobi polynomials studied by Koornwinder and the second author.

Cite this article

Mathijs S. Dijkhuizen, Jasper V. Stokman, Some Limit Transitions between 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