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Laplace and Segal–Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials. (English) Zbl 1035.32014

Summary: Let \(\mathcal D=G/K\) be a complex bounded symmetric domain of tube type in a complex Jordan algebra \(V\) and let \(\mathcal D_{\mathbb R}=J \cap \mathcal D \subset \mathcal D\) be its real form in a formally real Euclidean Jordan algebra \(J\subset V; \mathcal D_{\mathbb R}=H/L\) 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 \(\mathcal D\) to an \(L^2\)-space on \(\mathcal D_{\mathbb{R}}\). 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 \(\mathcal D\) of the spherical functions on \(\mathcal D_{\mathbb R}\) and find their expansion in terms of the \(L\)-spherical polynomials on \(\mathcal D\), which are Jack symmetric polynomials. We prove that the coefficients are orthogonal polynomials in an \(L^2\)-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 \(\mathcal D\). Finally, we use the Laplace transform to study generalized Laguerre functions on symmetric cones.

MSC:

32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
44A10 Laplace transform
43A85 Harmonic analysis on homogeneous spaces
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
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