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A family of quantum projective spaces and related q-hypergeometric orthogonal polynomials

M. S. Dijkhuizen·M. Noumi
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Abstract

We define a one-parameter family of two-sided coideals in U_q(gl(n)) and study the corresponding algebras of infinitesimally right invariant functions on the quantum unitary group U_q(n). The Plancherel decomposition of these algebras with respect to the natural transitive U_q(n)-action is shown to be the same as in the case of a complex projective space. By computing the radial part of a suitable Casimir operator, we identify the zonal spherical functions (i.e. infinitesimally bi-invariant matrix coefficients of finite-dimensional irreducible representations) as Askey-Wilson polynomials containing two continuous and one discrete parameter. In certain limit cases, the zonal spherical functions are expressed as big and little q-Jacobi polynomials depending on one discrete parameter.

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M. S. Dijkhuizen, M. Noumi. 1996-05-13. A family of quantum projective spaces and related q-hypergeometric orthogonal polynomials. https://arxiv.org/abs/q-alg/9605017

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