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 a 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 v i 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 Charles Dunkl (1989). One of Dunkl's major results was that Dunkl operators "commute," that is, they satisfy T i ( T j f ( x ) ) = T j ( T i f ( x ) ) just as partial derivatives do. Thus Dunkl operators represent a meaningful generalization of partial derivatives.