Segal-Bargmann transforms associated with finite Coxeter groups

Using a polarization of a suitable restriction map, and heat-kernel analysis, we construct a generalized Segal-Bargmann transform associated with every finite Coxeter group G on RN. We find the integral representation of this transform, and we prove its unitarity. To define the Segal-Bargmann transform, we introduce a Hilbert space F-k(C-N) of holomorphic functions on C-N with reproducing kernel equal to the Dunkl-kernel. The definition and properties of 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 Schrodinger model and in the Fock model. Further, we prove a branching decomposition of F-k(C-N) as a unitary G x SL(2, R)-module and a general version of Hecke's formula for the Dunkl transform.