Laplace and Segal–Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials

Let D=G/K be a complex bounded symmetric domain of tube type in a complex Jordan algebra V and let DR=J∩D⊂D be its real form in a formally real Euclidean Jordan algebra J⊂V; DR=H/L 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 D to an L2-space on DR. 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 their 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 D. Finally, we use the Laplace transform to study generalized Laguerre functions on symmetric cones