A family of quantum projective spaces and related q-hypergeometric orthogonal polynomials
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.
Keywords
Cite
@article{arxiv.q-alg/9605017,
title = {A family of quantum projective spaces and related q-hypergeometric orthogonal polynomials},
author = {M. S. Dijkhuizen and M. Noumi},
journal= {arXiv preprint arXiv:q-alg/9605017},
year = {2008}
}
Comments
31 pages, AMS-TeX 2.1, no figures