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On SegalBargmann analysis for finite Coxeter groups and its heat kernel
, 903
"... We prove identities involving the integral kernels of three versions (two being introduced here) of the SegalBargmann 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 ..."
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We prove identities involving the integral kernels of three versions (two being introduced here) of the SegalBargmann 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 SegalBargmann analysis on Euclidean space. Hall’s results (trivial Dunkl structure and arbitrary finite dimension) as well as our own results in µdeformed quantum mechanics (nontrivial Dunkl structure, dimension one) are particular cases of the results proved here. So we can understand all of these versions of the SegalBargmann transform associated to a Coxeter group as Hall type transforms. In particular, we define an analogue of Hall’s Version C generalized SegalBargmann 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 SegalBargmann 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 SegalBargmann kernels for
the µdeformed SegalBargmann space gets two measures, to appear in
 Proceedings of the 11th Workshop: Noncommutative Harmonic Analysis and Applications to Probability
, 2008
"... This note explains how the two measures used to define the µdeformed SegalBargmann space are natural and essentially unique structures. As is well known, the density with respect to Lebesgue measure of each of these measures involves a Macdonald function. Our primary result is that these densities ..."
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This note explains how the two measures used to define the µdeformed SegalBargmann space are natural and essentially unique structures. As is well known, the density with respect to Lebesgue measure of each of these measures involves a Macdonald function. Our primary result is that these densities are the solution of a system of ordinary differential equations which is naturally associated with this theory. We then solve this system and find the known densities as well as a “spurious ” solution which only leads to a trivial holomorphic Hilbert space. This explains how the Macdonald functions arise in this theory. Also we comment on why it is plausible that only one measure will not work. We follow Bargmann’s approach by imposing a condition sufficient for the µdeformed creation and annihilation operators to be adjoints of each other. While this note uses elementary techniques, it reveals in a new way basic aspects of the structure of the µdeformed SegalBargmann space. Keywords: SegalBargmann analysis, µdeformed quantum mechanics. 1