Asymptotic analysis and the link to symmetric cones in Dunkl theory

This project is located in the area of harmonic analysis and special functions associated with root systems, with a focus on the interaction with the analysis on symmetric spaces. First, the interface of Dunkl theory and the analysis on symmetric cones shall be elaborated. This will continue a program recently started by the applicant, which is motivated by ideas of I.G. Macdonald. The analytic structures emerging here are for example of interest in the study of quantum models of Calogero-Moser type. We expect that many results from the harmonic analysis on symmetric cones have analogues in the Dunkl setting of type A, but due to the lack of the geometric background, the structures in the Dunkl setting are more rigid, and different proofs will be needed, including methods from algebraic combinatorics. In particular, the Dunkl-type Laplace transform of Heckman-Opdam hypergeometric functions as well as Zeta distributions shall be studied. Among the consequences, we expect to obtain interesting results for Macdonald's hypergeometric Jack polynomial series, which will generalize classical results for special functions of matrix argument. In a second part of the project, asymptotic properties of Heckman-Opdam hypergeometric functions of type A and type B will be studied, in particular as the rank (i.e. the number of variables) goes to infinity. The central aim is to extend results of Okounkov and Olshanksi for the polynomial case to the non-polynomial setting. The desired limits should generalize known results for infinite-dimensional symmetric spaces which are obtained as inductive limits of non-compact symmetric spaces of type A and type BC. Finally, it is our hope that the expertise and results gained in the course of the project will also help to achieve some progress towards longstanding open conjectures in Dunkl-Cherednik theory concerning the existence of positive product formulas and convolution structures. This would have important consequences in harmonic analysis.

DFG Programme: Research Grants