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 $\mathbb{R}^N$ by the formula

$$T_i f(x) = (\partial_i f)(x) - \Sigma_{\lambda\in R^+} k(\lambda)\frac{f(x)-f(s_\lambda x)}{\langle x,\lambda\rangle} \lambda_i$$

There are also many variants and generalizations of this definition to various setups. Dunkl operators appear in the theory of Calogero model 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.
• A.N. Sergeev, A.P. Veselov, Dunkl operators at infinity and Calogero-Moser systems, arxiv/1311.0853

There is also a braided/$q$-deformation:

• Yuri Bazlov, Arkady Berenstein, Noncommutative Dunkl operators and braided Cherednik algebras, Selecta Math. (N.S.) 14 (2009), no. 3-4, 325–372, pdf, MR2010k:16044 doi

Dunkl operators are named after Charles Dunkl.