Quasi-Invariants of Complex Reflection Groups

Abstract

We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space Q_k of quasi-invariants of a given multiplicity is not, in general, an algebra but a module over the coordinate ring of some (singular) affine variety X_k. We extend the main results of Etingof, Ginzburg and the first author (see [BEG]) to this setting: in particular, we show that the variety X_k and the module Q_k are Cohen-Macaulay, and the rings of differential operators on X_k and Q_k are simple rings, Morita equivalent to the Weyl algebra A_n(C), where n = dim X_k . Our approach relies on representation theory of complex Cherednik algebras and is parallel to that of [BEG]. As a by-product, 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.

38 pages, final version, to appear in Compositio Math

References in corpus (3)

• On the category O for rational Cherednik algebras
• On the quotient ring by diagonal harmonics
• Graded cofinite rings of differential operators

Cited by in corpus (7)

• Double affine Hecke algebras and generalized Jones polynomials
• Derived equivalences for Rational Cherednik algebras
• Deformed Calogero--Moser operators and ideals of rational Cherednik algebras
• Harish-Chandra bimodules over rational Cherednik algebras
• Dunkl and Cherednik operators
• Semi-simplicity of the category of admissible D-modules
• The diagonal coinvariant ring of a complex reflection group