Asymptotic analysis and the link to symmetric cones in Dunkl theory
Final Report Abstract
The results of this project contribute to Dunkl theory, mostly but not exclusively in the rational setting. A particular focus was on the analogies between Dunkl theory for root systems of type A and the analysis on symmetric cones. Indeed, many results of radial analysis on symmetric cones turned out to have natural generalizations to the Dunkl setting, but due to the lack of the group context, they often required completely new proofs. A key tool in these studies was the Laplace transform in the Dunkl setting of type A studied by Rösler (2020) and going back, at a formal level, to unpublished notes by I.G. Macdonald from the 1980ies. The first part of this project was motivated by Macdonald’s notes and the work of Baker and Forrester related to Calogero-Moser models. We obtained a Laplace transform identity for non-symmetric Jack polynomials by means of the raising operators of Knop and Sahi. This generalizes the fundamental Laplace transform identity of power functions on a symmetric cone. It led to Laplace transform identities for Heckman-Opdam hypergeometric functions and the Cherednik kernel, and also for for Jack-hypergeometric series. As a further consequence, we obtained a Post-Widder inversion theorem for the Dunkl-type Laplace transform. Moreover, K-Bessel functions in the Dunkl setting as well as related Zeta distributions were studied, in analogy to classical results by Clerc and Rubin on symmetric cones. In this part of the project, interesting connections between the root systems of type A and type B emerged. Further parts of the project were devoted to asymptotic harmonic analysis in the Dunkl setting, both for fixed rank with transitions between non-symmetric Heckman-Opdam polynomials and Cherednik kernels for root systems of type BC and those of type A, as well as for the situation where the rank tends to infinity, where we precisely characterized the possible limit functions both for root systems of type A and type B. These results are related to the harmonic analysis on infinite dimensional symmetric spaces and to so-called beta-ensembles in random matrix theory and integrable probability. Concerning Dunkl theory for arbitrary root systems, a Helgason- Johnson type theorem was obtained for the Cherednik kernel, which characterizes the spactral parameters for which the kernel is bounded. This generalizes results of Narayanan et al. (2014) to the non-symmetric setting. As a consequence, a Riemann-Lebesgue Lemma for the Cherednik transform was obtained. Moreover, the multitemporal wave equation in the rational Dunkl setting was studied in the spirit of work for non-compact symmetric spaces by Philips, Shahshahani and Helgason, and finally a full elliptic regularity theorem for rational Dunkl operators was proven, based on recently obtained support properties of the Dunkl translation due to Dziubanski and Hejna.
Publications
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The Dunkl-Laplace transform and Macdonald’s hypergeometric series. Transactions of the American Mathematical Society.
Brennecken, Dominik & Rösler, Margit
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Contributions to Dunkl theory. PhD thesis, Paderborn University, July 2024. 193 pages.
D. Brennecken
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Dunkl convolution and elliptic regularity for Dunkl operators. Mathematische Nachrichten, 297(12), 4416-4436.
Brennecken, Dominik
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Hankel transform, K-Bessel functions and zeta distributions in the Dunkl setting. Journal of Mathematical Analysis and Applications, 535(2), 128125.
Brennecken, Dominik
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The Laplace Transform in Dunkl Theory. Trends in Mathematics, 77-87. Springer Nature Switzerland.
Brennecken, Dominik & Rösler, Margit
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Boundedness of the Cherednik kernel and its limit transition from type BC to type A. Indagationes Mathematicae, 36(6), 1717-1744.
Brennecken, Dominik
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Limits of Bessel functions for root systems as the rank tends to infinity. Indagationes Mathematicae, 36(1), 245-269.
Brennecken, Dominik & Rösler, Margit
