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

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.