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

Dijkhuizen, Mathijs; Noumi, Masatoshi

doi:10.1090/S0002-9947-98-01971-0

A family of quantum projective spaces and related $q$-hypergeometric orthogonal polynomials
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Trans. Amer. Math. Soc. 350 (1998), 3269-3296 Request permission

Abstract:

A one-parameter family of two-sided coideals in $\mathcal {U}_{q} (\mathfrak {g}\mathfrak {l}(n))$ is defined and the corresponding algebras of infinitesimally right invariant functions on the quantum unitary group $U_{q}(n)$ are studied. 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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