2002/06/26 by Mark Davidson, Gestur Ólafsson, Davidson, Mark +5
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Waves and Solitons #Representation Theory (math.RT) #math.FA #math.RT
paper · pdf · doi:10.48550/arxiv.math/0206275
arxiv created 2002/06/26 · openalex publication_date 2002/06/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let D=G/K be a complex bounded symmetric domain of tube type in a complex Jordan algebra V and let Dℝ=H/L⊂ D be its real form in a formally real Euclidean Jordan algebra J⊂ V. We consider representations of H that are gotten by the generalized Segal-Bargmann transform from a unitary G-space of holomorphic functions on D to an L2-space on \mathcalDR. We prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the Laplace transform. We find the extension to D of the spherical functions on Dℝ and find the expansion in terms of the L-spherical polynomials on D, which are Jack symmetric polynomials. We prove that the coefficients are orthogonal polynomials in an L2-space, the measure being the Harish-Chandra Plancherel measure multiplied by the symbol of the Berezin transform. We prove the difference equation and recurrence relation for those polynomials by considering the action of the Lie algebra and the Cayley transform on the polynomials on \mathcal D.