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.

Last revised on September 22, 2022 at 09:57:30. See the history of this page for a list of all contributions to it.