nLab
Dunkl operator

Let $G$ be a Coxeter group with a reduced root system $R$. A multiplicity function $k$ on $R$ satifies by definition property $k(\lambda )=k(\mu )$ iff the corresponding reflections $s_{\lambda }$ and $s_{\mu }$ are conjugate each to another. A Dunkl operator is defined on smooth functions in ${ℝ}^N$ by the formula

$$T_{i}f(x)=({\partial }_{i}f)(x)-{\Sigma }_{\lambda \in R^{+}}k(\lambda )\frac{f(x)-f(s_{\lambda }x)}{⟨x,\lambda ⟩}{\lambda }_i$$T_i f(x) = (\partial_i f)(x) - \Sigma_{\lambda\in R^+} k(\lambda)\frac{f(x)-f(s_\lambda x)}{\langle x,\lambda\rangle} \lambda_i

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.
• Ivan Cherednik, Introduction to double Hecke algebras, arXiv/math/0404307
• Pavel Etingof, Lectures on Calogero-Moser systems, pdf
• P. Etingof, X. Ma, On elliptic Dunkl operators, arXiv/0706.2152
• Yu. Berest, P. Etingof, V. Ginzburg, Finite-dimensional representations of rational Cherednik algebras. Int. Math. Res. Not. 2003, no. 19, 1053-1088.
• C. Dunkl, E. Opdam, Dunkl operators for complex reflection groups, Proc. London Math. Soc. 86 (2003), 70–108.

Dunkl operators are named after Charles Dunkl.