An orthogonal family of polynomials on the generalized unit disk and ladder representations of 𝑈(𝑝,𝑞)

Abstract

Inner product structures are given for realizations of the positive spin ladder representations over the generalized unit disk D p , q = U ( p , q ) / K \textbf {D}_{p,q} =U(p,q)/K . This is accomplished by combining previous results of the author with the construction of a family of holomorphic polynomials on D p , q \textbf {D}_{p,q} . These polynomials, which play a crucial role in the present work, are shown to be orthogonal with respect to Lebesgue measure, and their norms are computed. The orthogonal family is then used to invert a certain integral transform, giving the desired inner product structures.