TY - JOUR
T1 - Segal-Bargmann transforms associated with finite Coxeter groups
AU - Saïd, Salem Ben
AU - Ørsted, Bent
PY - 2006/2
Y1 - 2006/2
N2 - 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 [InlineMediaObject not available: see fulltext.] of holomorphic functions on [InlineMediaObject not available: see fulltext.] with reproducing kernel equal to the Dunkl-kernel. The definition and properties of [InlineMediaObject not available: see fulltext.] 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 [InlineMediaObject not available: see fulltext.] as a unitary [InlineMediaObject not available: see fulltext.]-module and a general version of Hecke's formula for the Dunkl transform.
AB - 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 [InlineMediaObject not available: see fulltext.] of holomorphic functions on [InlineMediaObject not available: see fulltext.] with reproducing kernel equal to the Dunkl-kernel. The definition and properties of [InlineMediaObject not available: see fulltext.] 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 [InlineMediaObject not available: see fulltext.] as a unitary [InlineMediaObject not available: see fulltext.]-module and a general version of Hecke's formula for the Dunkl transform.
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U2 - 10.1007/s00208-005-0718-3
DO - 10.1007/s00208-005-0718-3
M3 - Article
AN - SCOPUS:30444449690
SN - 0025-5831
VL - 334
SP - 281
EP - 323
JO - Mathematische Annalen
JF - Mathematische Annalen
IS - 2
ER -