, Volume 334, Issue 2, pp 281-323
Date: 14 Dec 2005

Segal-Bargmann transforms associated with finite Coxeter groups

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Abstract

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 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 of holomorphic functions on with reproducing kernel equal to the Dunkl-kernel. The definition and properties of 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 as a unitary -module and a general version of Hecke's formula for the Dunkl transform.