| Using a polarization of a suitable restriction map, and heat-kernel analysis, we construct a generalized Segal-Bargmann transform associated with each finite Coxeter group $G$ on $\R^N.$ We find the integral representation of this transform, and we prove its unitarity. To define the Segal-Bargmann transform, we introduce a Hilbert space ${\mathcal F}_k(\C^N)$ of holomorphic functions on $\C^N $ with reproducing kernel equal to the Dunkl-kernel. The definition and properties of $\mathcal F_k(\C^N)$ extend naturally those of the well-known classical Fock space. The generalized Segal-Bargmann transform allows to exhibit some relationships between the Dunkl theory in the Schrödinger model and in the Fock model. Further, we prove a branching decomposition of $\mathcal F_k(\C^N)$ as a unitary $G\times \widetilde{SL(2,\R)}$-module and a general version of Hecke's formula for the Dunkl transform. |
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