Quasi-Invariants of Complex Reflection Groups

TL;DR

This paper generalizes the theory of quasi-invariants for complex reflection groups, establishing their algebraic and geometric properties, and confirms conjectures related to shift operators and representation operations.

Contribution

It extends the concept of quasi-invariants to all complex reflection groups, proving their Cohen-Macaulay property and the simplicity of associated differential operator rings.

Findings

X_k and Q_k are Cohen-Macaulay.

Rings of differential operators are simple and Morita equivalent to Weyl algebra.

Existence of shift operators for all 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.