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We apply a special case, the restriction principle (for which we give adefinition simpler than the usual one), of a basic result in functionalanalysis (the polar decomposition of an operator) in order to define $C_{\mu,t}$, the $C$-version of the Segal-Bargmann transform, associated to a finiteCoxeter group acting in $\mathbb{R}^N$ and a given value $t>0$ of Planck'sconstant, where $\mu$ is a multiplicity function on the roots defining theCoxeter group. Then we immediately prove that $C_{\mu, t}$ is a unitaryisomorphism. To accomplish this we identify the reproducing kernel function ofthe appropriate Hilbert space of holomorphic functions. As consequences weprove that the Segal-Bargmann transforms for Versions $A$, $B$ and $D$ are alsounitary isomorphisms, though not by a direct application of the restrictionprinciple. The point is that the $C$-version is the the only version where arestriction principle, in our definition of this method, applies directly. Thisreinforces the idea that the $C$-version is the most fundamental, most naturalversion of the Segal-Bargmann transform.

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