An introduction to Dunkl theory and its analytic aspects

TL;DR

This paper provides an updated overview of Dunkl theory, a generalization of Fourier analysis linked to root systems, highlighting its analytic aspects and connections to symmetric spaces and special functions.

Contribution

It offers a comprehensive introduction to Dunkl theory, emphasizing recent developments and its analytic framework, bridging classical harmonic analysis with modern algebraic structures.

Findings

Unified framework for harmonic analysis on symmetric spaces

Connections between Dunkl operators and special functions

Enhanced understanding of analytic properties of Dunkl theory

Abstract

Dunkl theory is a far reaching generalization of Fourier analysis and special function theory related to root systems. During the sixties and seventies, it became gradually clear that radial Fourier analysis on rank one symmetric spaces was closely connected with certain classes of special functions in one variable. During the eighties, several attempts were made, mainly by the Dutch school, to extend these results in higher rank (i.e. in several variables), until the discovery of Dunkl operators in the rational case and Cherednik operators in the trigonometric case. Together with q-special functions introduced by Macdonald, this has led to a beautiful theory, developed by several authors, which encompasses in a unified way harmonic analysis on all Riemannian symmetric spaces and spherical functions thereon.In this series of lectures, delivered at the Summer School AAGADE 2015 (Analytic, Algebraic and Geometric Aspects of Differential Equations, Mathematical Research and Conference Center, Bedlewo, Poland, September 2015), we aim at giving an updated overview of Dunkl theory, with an emphasis on its analytic aspects.