Let be a Coxeter group with a reduced root system . A multiplicity function on satifies by definition property iff the corresponding reflections and are conjugate each to another. A Dunkl operator is defined on smooth functions in by the formula
There are also many variants and generalizations of this definition to various setups. Dunkl operators appear in the theory of Caloger-Moser systems and of the Cherednik (= double affine Hecke) algebras.
C. Dunkl, Differential-difference operators associated to reflection groups, Trans. AMS 311 (1989), 167–183.