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 kv an arbitrary "multiplicity" function on R (so ku = kv whenever the reflections σu and σv corresponding to the roots u and v are conjugate in G).
Then, the Dunkl operator is defined by: where
One of Dunkl's major results was that Dunkl operators "commute," that is, they satisfy
Thus Dunkl operators represent a meaningful generalization of partial derivatives.