Dunkl operator

{{Short description|Mathematical 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:

:$T\_i\; f(x)\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_i\}\; f(x)\; +\; \backslash sum\_\{v\backslash in\; R\_+\}\; k\_v\; \backslash frac\{f(x)\; -\; f(x\; \backslash sigma\_v)\}\{\backslash left\backslash langle\; x,\; v\backslash right\backslash rangle\}\; v\_i$

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 {{harvs|txt|authorlink=Charles F. Dunkl|first=Charles |last=Dunkl|year=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.