The C-Version Segal-Bargmann Transform for Finite Coxeter Groups Defined by the Restriction Principle

Abstract EN

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 Cμ,t, the C-version of the Segal-Bargmann transform, associated with a finite Coxeter group acting in ℝN and a given value t>0 of Planck's constant, where μ is a multiplicity function on the roots defining the Coxeter group.

Then we immediately prove that Cμ,t 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 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 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.