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

Lorch, John

doi:10.1090/S0002-9939-98-04506-7

An orthogonal family of polynomials
on the generalized unit disk
and ladder representations of

Author: John D. Lorch
Journal: Proc. Amer. Math. Soc. 126 (1998), 3755-3762
MSC (1991): Primary 22E45, 22E70; Secondary 32L25, 32M15, 58G05, 81R05, 81R25
DOI: https://doi.org/10.1090/S0002-9939-98-04506-7
MathSciNet review: 1458255
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Abstract: Inner product structures are given for realizations of the positive spin ladder representations over the generalized unit disk ${\bf D}_{p,q} =U(p,q)/K$ . This is accomplished by combining previous results of the author with the construction of a family of holomorphic polynomials on ${\bf D}_{p,q}$ . 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.

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