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Branching coefficients of holomorphic representations and Segal-Bargmann transform. (English) Zbl 1019.22006
Author’s abstract: “Let 𝐃=G/K be a complex bounded symmetric domain of tube type in a Jordan algebra V C and let D=H/L=𝐃V be its real form in a Jordan algebra VV C . The analytic continuation of the holomorphic discrete series on 𝐃 forms a family of interesting representations of G. We consider the restriction on D and the branching rule under H of the scalar holomorphic representations. The unitary part of the restriction map gives then a generalization of the Segal-Bargmann transform. The group L is a spherical subgroup of K and we find a canonical basis of L-invariant polynomials in the components of the Schmid decomposition and we express them in terms of the Jack symmetric polynomials. We prove that the Segal-Bargmann transforms of those L-invariant polynomials are, under the spherical transform on D, multi-variable Wilson-type polynomials and we give a simple alternative proof of their orthogonality relation. We find the expansion of the spherical functions on D, when extended to a holomorphic function in a neighborhood of 0𝐃, in terms of the L-spherical holomorphic polynomials on 𝐃, the coefficients being the Wilson polynomials”.
22E30Analysis on real and complex Lie groups