In mathematics, particularly the study of Lie groups, a Dunkl operator is a certain kind of mathematical operator, involving differential operators but also reflections in an underlying space.
Formally, let G be a Coxeter group with reduced root system R and k<sub>v</sub> an arbitrary "multiplicity" function on R (so k<sub>u</sub> = k<sub>v</sub> whenever the reflections ÃÂ<sub>u</sub> and ÃÂ<sub>v</sub> corresponding to the roots u and v are conjugate in G). Then, the Dunkl operator is defined by:
where is the i-th component of v, 1 ⤠i ⤠N, x in R<sup>N</sup>, and f a smooth function on R<sup>N</sup>.
Dunkl operators were introduced by . One of Dunkl's major results was that Dunkl operators "commute," that is, they satisfy just as partial derivatives do. Thus Dunkl operators represent a meaningful generalization of partial derivatives.