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Quasi-invariants of complex reflection groups

Part of: Algebraic geometry: Foundations Rings and algebras arising under various constructions Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  27 September 2010

Yuri Berest  and
Oleg Chalykh
Yuri Berest
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA (email: [email protected])
Oleg Chalykh
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK (email: [email protected])
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Abstract

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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(ℂ) , 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.

Keywords

complex reflection groupCoxeter grouprational Cherednik algebraDunkl operatorHecke algebraring of differential operatorsWeyl algebra

MSC classification

Secondary: 16S38: Rings arising from non-commutative algebraic geometry 14A22: Noncommutative algebraic geometry 17B45: Lie algebras of linear algebraic groups
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 3 , May 2011 , pp. 965 - 1002
Copyright
Copyright © Foundation Compositio Mathematica 2010

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