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

Stephen Bruce Sontz

Advances in Mathematical Physics, 2011, vol. 2011, issue 1

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 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.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlamp:v:2011:y:2011:i:1:n:365085

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