Mathematische Annalen

, 334:281

Segal-Bargmann transforms associated with finite Coxeter groups


DOI: 10.1007/s00208-005-0718-3

Cite this article as:
Saïd, S. & Ørsted, B. Math. Ann. (2006) 334: 281. doi:10.1007/s00208-005-0718-3


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

Mathematics Subject Classification (2000)

33C52 43A85 44A15 

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Département de MathématiquesInstitut Elie Cartan, Université Henri Poincaré–Nancy 1CedexFrance
  2. 2.Department of Mathematical SciencesAarhus UniversityAarhus CDenmark

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