Copyright © 2011 Stephen Bruce Sontz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We apply a special case, the restriction principle (for which we give a definition simpler than the usual one), of a basic result in functional analysis (the polar decomposition of an operator) in order to define , the -version of the Segal-Bargmann transform, associated with a finite Coxeter group acting in and a given value of Planck's constant, where is a multiplicity function on the roots defining the Coxeter group. Then we immediately prove that is a unitary isomorphism. To accomplish this we identify the reproducing kernel function of the appropriate Hilbert space of holomorphic functions. As a consequence we prove that the Segal-Bargmann transforms for Versions , , and are also unitary isomorphisms though not by a direct application of the restriction principle. The point is that the -version is the only version where a restriction principle, in our definition of this method, applies directly. This reinforces the idea that the -version is the most fundamental, most natural version of the Segal-Bargmann transform.