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Segal-Bargmann transforms associated with finite Coxeter groups. (English) Zbl 1109.33015
It is well known that the classical Segal-Bergmann transform maps unitarily from the Schrödinger model to the Fock model intertwining the action of the Heisenberg group. The authors in this paper use the restriction principle, i.e. polarization of a suitable restriction map to construct the Segal-Bergmann transform associated with finite Coxeter groups. A new class of Fock-type spaces have been introduced and studied. The definition and properties of this class of Hilbert spaceces generalize naturally those of the well-known classical Fock spaces. Rösler’s results on the heat-kernel associated with reflection groups [{\it M. Rösler}, Commun. Math. Phys. 192, 519--542 (1998; Zbl 0908.33005)] have been used to obtain explicitly the integral representation of the Segal-Bergmann transform. The generalized Segal- Bergmann transform allows to exhibit some relationships between the Dunkl theory in the Schrödinger model and in the Fock model. Further the branching decomposition of the generalized Fock spaces under the action of $G\times\text{SL}( 2,\Bbb R)$, where $G$ is the Coxeter group and SL$( 2, \Bbb R)$ is the universal covering of the group SL$( 2,\Bbb R)$.

##### MSC:
 33C52 Orthogonal polynomials and functions associated with root systems 43A85 Analysis on homogeneous spaces 44A15 Special transforms (Legendre, Hilbert, etc.)
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