Mathematische Annalen

, Volume 334, Issue 2, pp 281-323

First online:

Segal-Bargmann transforms associated with finite Coxeter groups

  • Salem Ben SaïdAffiliated withDépartement de Mathématiques, Institut Elie Cartan, Université Henri Poincaré–Nancy 1 Email author 
  • , Bent ØrstedAffiliated withDepartment of Mathematical Sciences, Aarhus University

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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.

Mathematics Subject Classification (2000)

33C52 43A85 44A15