Compositio Mathematica

Research Article

Quasi-invariants of complex reflection groups

Yuri Beresta1 and Oleg Chalykha2

a1 Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA (email:

a2 School of Mathematics, University of Leeds, Leeds LS2 9JT, UK (email:


We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space of quasi-invariants of a given multiplicity is not, in general, an algebra but a module Qk over the coordinate ring of a (singular) affine variety Xk. We extend the main results of Berest et al. [Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), 279–337] to this setting: in particular, we show that the variety Xk and the module Qk are Cohen–Macaulay, and the rings of differential operators on Xk and Qk are simple rings, Morita equivalent to the Weyl algebra An(xs2102) , where n=dim Xk. Our approach relies on representation theory of complex Cherednik algebras introduced by Dunkl and Opdam [Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), 70–108] and is parallel to that of Berest et al. As an application, we prove the existence of shift operators for an arbitrary complex reflection group, confirming a conjecture of Dunkl and Opdam. Another result is a proof of a conjecture of Opdam, concerning certain operations (KZ twists) on the set of irreducible representations of W.

(Received February 09 2010)

(Accepted May 05 2010)

(Online publication September 27 2010)

2000 Mathematics Subject Classification

  • 16S38 (primary);
  • 14A22;
  • 17B45 (secondary)


  • complex reflection group;
  • Coxeter group;
  • rational Cherednik algebra;
  • Dunkl operator;
  • Hecke algebra;
  • ring of differential operators;
  • Weyl algebra


The first author was partially supported by the NSF grants DMS 04-07502 and DMS 09-01570.