On Segal-Bargmann analysis for finite Coxeter groups and its heat kernel

TL;DR

This paper establishes identities and relations among various Segal-Bargmann transforms linked to finite Coxeter groups and Dunkl theory, generalizing classical results and introducing new transforms and spaces.

Contribution

It introduces new identities and a generalized Segal-Bargmann transform for Coxeter groups, connecting Dunkl heat kernels with analytic continuation and defining novel Segal-Bargmann spaces.

Findings

Identities relating different Segal-Bargmann kernels

Definition of a new Segal-Bargmann space for Dunkl theory

A relation between Versions A and C kernels

Abstract

We prove identities involving the integral kernels of three versions (two being introduced here) of the Segal-Bargmann transform associated to a finite Coxeter group acting on a finite dimensional, real Euclidean space (the first version essentially having been introduced around the same time by Ben Sa\"id and {\O}rsted and independently by Soltani) and the Dunkl heat kernel, due to R\"osler, of the Dunkl Laplacian associated with the same Coxeter group. All but one of our relations are originally due to Hall in the context of standard Segal-Bargmann analysis on Euclidean space. Hall's results (trivial Dunkl structure and arbitrary finite dimension) as well as our own results in $\mu$-deformed quantum mechanics (non-trivial Dunkl structure, dimension one) are particular cases of the results proved here. So we can understand all of these versions of the Segal-Bargmann transform associated to a Coxeter group as Hall type transforms. In particular, we define an analogue of Hall's Version C generalized Segal-Bargmann transform which is then shown to be Dunkl convolution with the Dunkl heat kernel followed by analytic continuation. In the context of Version C we also introduce a new Segal-Bargmann space and a new transform associated to the Dunkl theory. Also we have what appears to be a new relation in this context between the Segal-Bargmann kernels for Versions A and C.