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Some limit transitions between BC type orthogonal polynomials interpreted on quantum complex Grassmannians

Mathijs S. Dijkhuizen·Jasper V. Stokman
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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)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.

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Mathijs S. Dijkhuizen, Jasper V. Stokman. 1998-06-23. Some limit transitions between BC type orthogonal polynomials interpreted on quantum complex Grassmannians. https://arxiv.org/abs/math/9806123

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