nLab Dunkl operator

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

T if(x)=( if)(x)Σ λR +k(λ)f(x)f(s λx)x,λλ 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 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/qq-deformation:

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.