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 D forms a family of interesting representations of G. We consider the restriction on D of the scalar holomorphic representations of G, as a representation of H . 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 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 neighborhood in D, in terms of the L-spherical holomorphic polynomials on D, the coefficients being the Wilson polynomials.