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. 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: where v i {displaystyle v_{i}} 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 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 ) ) {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.