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

Abstract

The quantum complex Grassmannian $U_q/K_q$ of rank $l$ is the quotient of the quantum unitary group $U_q=U_q(n)$ by the quantum subgroup $K_q=U_q(n–l)x{U}_q(l)$. We show that $(U_q,K_q)$ is a quantum Gelfand pair and we express the zonal spherical functions, i.e. $K_q$-biinvariant matrix coefficients of finite-dimensional irreducible representations of $U_q$, 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.