Dunkl operator

Summary

A Dunkl operator is a partial derivative plus a sum of difference terms, one for each reflection of a finite reflection group. Charles Dunkl introduced these operators and proved that, for a fixed group and multiplicity function, they commute with each other. They give a theory of special functions attached to root systems. In it, Dunkl harmonics, the Dunkl transform and Dunkl kernels generalize spherical harmonics, the Fourier transform and the exponential. They are also used to study the Calogero–Moser–Sutherland quantum systems, whose eigenfunctions in type $A$ are related to the Jack functions.