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

A one-parameter family of two-sided coideals in U-q(gl(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.