Branching Coefficients of Holomorphic Representations and Segal–Bargmann Transform

Let D=G/K be a complex bounded symmetric domain of tube type in a Jordan algebra VC, and let D=H/L=D∩V be its real form in a Jordan algebra V⊂VC. The analytic continuation of the holomorphic discrete series on Dopf; 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 transform 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 0∈D, in terms of the L-spherical holomorphic polynomials on D, the coefficients being the Wilson polynomials.