The C-version Segal-Bargmann transform for finite Coxeter groups defined by the restriction principle

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 $C_{\mu, t}$, the $C$-version of the Segal-Bargmann transform, associated to a finite Coxeter group acting in $\mathbb{R}^N$ and a given value $t>0$ of Planck's constant, where $\mu$ is a multiplicity function on the roots defining the Coxeter group. Then we immediately prove that $C_{\mu, t}$ is a unitary isomorphism. To accomplish this we identify the reproducing kernel function of the appropriate Hilbert space of holomorphic functions. As consequences we prove that the Segal-Bargmann transforms for Versions $A$, $B$ and $D$ are also unitary isomorphisms, though not by a direct application of the restriction principle. The point is that the $C$-version is the the only version where a restriction principle, in our definition of this method, applies directly. This reinforces the idea that the $C$-version is the most fundamental, most natural version of the Segal-Bargmann transform.