Some limit transitions between BC type orthogonal polynomials interpreted on quantum complex Grassmannians
The authors determine zonal spherical functions associated with the quantum analogue of the complex Grassmannian (denoted as \(U_q/K_q\)) which is defined as the quotient of the quantum unitary group \(U_q(n)\) by the quantum subgroup \(K_q\) corresponding to the classical subgroup \(K=U(n-l)\times U(l)\) of the group \(U(n)\). It is shown that the quantum space \(U_q/K_q\) can be formally obtained from the one-parameter family of quantum Grassmannians (known from the literature) by sending the parameter to infinity. It is shown that this limit transition on quantum Grassmannians is compatible with the limit transition from Koornwinder polynomials to multivariable big and little \(q\)-Jacobi polynomials. These transitions lead to a rigorous computation of the limit of the quantum Schur orthogonality relations for the zonal spherical functions. This in turn leads to a complete description of the harmonic analysis on the quantum Grassmannian \(U_q/K_q\). The authors obtain an identification of the zonal spherical functions on the quantum complex Grassmannian \(U_q/K_q\) with multivariable big or little \(q\)-Jacobi polynomials. It is indicated that such limit arguments can be applied for other quantum complex symmetric spaces.
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