  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)xU_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.
