Dunkl Operators, Random Matrices and Hurwitz Numbers

Explore the probabilistic aspects of Dunkl operators in this 49-minute talk by Nizar Demni from Aix-Marseille Université. Begin with a revisit of Cepa and Lepingle's study of particles on the real line and its extension to radial Dunkl processes associated with reduced root systems. Delve into the reflected Brownian motion in Weyl chambers, examining its construction using folding operators and the Tanaka-type formula it satisfies. Conclude by investigating the intriguing appearance of simple Hurwitz numbers in the expression of the Dunkl intertwining operator, particularly in the generalized Bessel function (HCIZ integral). Gain insights into the connections between Dunkl operators, random matrices, and Hurwitz numbers in this advanced mathematical presentation delivered at the Institut des Hautes Etudes Scientifiques (IHES).