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 σ<sub>u</sub> and σ<sub>v</sub> corresponding to the roots u and v are conjugate in G). Then, the Dunkl operator is defined by:
:<math>T_i f(x) = \frac{\partial}{\partial x_i} f(x) + \sum_{v\in R_+} k_v \frac{f(x) - f(x \sigma_v)}{\left\langle x, v\right\rangle} v_i</math>
where <math>v_i </math> 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 <math>T_i (T_j f(x)) = T_j (T_i f(x))</math> just as partial derivatives do. Thus Dunkl operators represent a meaningful generalization of partial derivatives.