Dunkl operator explained

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:

T _{i} f(x) = (\partial)/(\partial x _{i}) f(x) + \sum _{v\in R _{+}} k _{v} (f(x) - f(x \sigma _{v)})/(\left\langle x, v\right\rangle) v _{i}

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 . 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