Concept

# Dunkl operator

Summary
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 an arbitrary "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: where is the i-th component of v, 1 ≤ i ≤ N, x in RN, and f a smooth function on RN. Dunkl operators were introduced by . One of Dunkl's major results was that Dunkl operators "commute," that is, they satisfy just as partial derivatives do. Thus Dunkl operators represent a meaningful generalization of partial derivatives.
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