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:

${\displaystyle T_{i}f(x)={\frac {\partial }{\partial x_{i}}}f(x)+\sum _{v\in R_{+}}k_{v}{\frac {f(x)-f(x\sigma _{v})}{\left\langle x,v\right\rangle }}v_{i}}$

where ${\displaystyle v_{i}}$ 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 ${\displaystyle T_{i}(T_{j}f(x))=T_{j}(T_{i}f(x))}$ just as partial derivatives do. Thus Dunkl operators represent a meaningful generalization of partial derivatives.

References

• Dunkl, Charles F. (1989), "Differential-difference operators associated to reflection groups", Transactions of the American Mathematical Society, 311 (1): 167–183, doi:10.2307/2001022, ISSN 0002-9947, JSTOR 2001022, MR 0951883