Quasi-Invariants of Complex Reflection Groups

Original · EN

We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space Qₖ 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 Xₖ. We extend the main results of Etingof, Ginzburg and the first author (see [BEG]) 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(C), where n = dim Xₖ. 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.