Dunkl operator

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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 kv a multiplicity function on R (so ku = kv whenever the reflections σu and σv corresponding to the roots u and v are conjugate in G). Then, the Dunkl operator is defined by:

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 v_i is the i-th component of v, 1 ≤ iN, x in RN, and f a smooth function on RN.

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.