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Mathematische Zeitschrift

, Volume 269, Issue 1–2, pp 9–28 | Cite as

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

  • Stephen Bruce Sontz
Article

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 Saïd and Ørsted and independently by Soltani) and the Dunkl heat kernel, due to Rösler, 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 μ-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.

Keywords

Segal–Bargmann analysis Heat kernel analysis Coxeter group Dunkl operator 

Mathematics Subject Classification (2000)

Primary 33C52 45H05 Secondary 46E15 81S99 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Centro de Investigación en Matemáticas, A.C. (CIMAT)GuanajuatoMexico

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