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Advances in Mathematical Physics
Volume 2011 (2011), Article ID 365085, 23 pages
http://dx.doi.org/10.1155/2011/365085
Research Article

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

Centro de Investigación en Matemáticas, A.C. (CIMAT), 36240 Guanajuato, GTO, Mexico

Received 24 March 2011; Revised 22 July 2011; Accepted 22 July 2011

Academic Editor: N. Kamran

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 𝐢 πœ‡ , 𝑑 , the 𝐢 -version of the Segal-Bargmann transform, associated with a finite Coxeter group acting in ℝ 𝑁 and a given value 𝑑 > 0 of Planck's constant, where πœ‡ is a multiplicity function on the roots defining the Coxeter group. Then we immediately prove that 𝐢 πœ‡ , 𝑑 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 , 𝐡 , and 𝐷 are also unitary isomorphisms though not by a direct application of the restriction principle. The point is that the 𝐢 -version is the only version where a restriction principle, in our definition of this method, applies directly. This reinforces the idea that the 𝐢 -version is the most fundamental, most natural version of the Segal-Bargmann transform.