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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 $D=H/L=𝐃\cap V$ be its real form in a Jordan algebra $V\subset {V}_{C}$. 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 $0\in 𝐃$, in terms of the $L$-spherical holomorphic polynomials on $𝐃$, the coefficients being the Wilson polynomials”.
##### MSC:
 2.2e+31 Analysis on real and complex Lie groups