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 func tions 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 pro jective space. By computing the radial part of a suitable Casimir oper ator, we identify the zonal spherical functions (i.e. infinitesimally bi-invariant matrix coefficients of finite-dimensional irreducible rep resentations) as Askey-Wilson polynomials containing two continuous an d one discrete parameter. In;;certain limit cases, the zonal spherical functions are expressed as big and little q-Jacobi polynomials depend ing on one discrete parameter.