
Dunkl operator

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""v" an arbitrary "multiplicity" function on "R" (so "k""u" = "k""v" whenever the reflections σ"u" and σ"v" corresponding to the roots "u" and "v" are conjugate in "G"). Then, the Dunkl operator is defined by:
where formula_2 is the "i"-th component of "v", 1 ≤ "i" ≤ "N", "x" in "R""N", and "f" a smooth function on "R""N".
Dunkl operators were introduced by . One of Dunkl's major results was that Dunkl operators "commute," that is, they satisfy formula_3 just as partial derivatives do. Thus Dunkl operators represent a meaningful generalization of partial derivatives.