Laplace and Segal-Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials |
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Authors: | Mark Davidson Genkai Zhang |
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Institution: | a Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA 70803, USA b Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96, Göteborg, Sweden |
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Abstract: | Let be a complex bounded symmetric domain of tube type in a complex Jordan algebra V and let be its real form in a formally real Euclidean Jordan algebra J⊂V; is a bounded realization of the symmetric cone in J. We consider representations of H that are gotten by the generalized Segal-Bargmann transform from a unitary G-space of holomorphic functions on to an L2-space on . 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 of the spherical functions on and find their expansion in terms of the L-spherical polynomials on , 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 . Finally, we use the Laplace transform to study generalized Laguerre functions on symmetric cones. |
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Keywords: | Holomorphic discrete series Highest weight representations Branching rule Bounded symmetric domains Real bounded symmetric domains Jordan pairs Jack symmetric polynomials Orthogonal polynomials Laplace transform |
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