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Quasi-invariants of complex reflection groups

Berest, Y and Chalykh, O (2011) Quasi-invariants of complex reflection groups. Compositio Mathematica, 147 (3). 965 - 1002 . ISSN 0010-437X

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Abstract

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() , 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.

Item Type: Article
Copyright, Publisher and Additional Information: © 2011, Cambridge University Press. This is an author produced version of a paper published in Compositio Mathematica. Uploaded in accordance with the publisher's self-archiving policy.
Keywords: complex reflection group, Coxeter group, rational Cherednik algebra, Dunkl operator, Hecke algebra, ring of differential operators, Weyl algebra, rational Cherednik algebras, differential operators, Dunkl operators, Braid-groups, Category-O, polynomials, rings
Institution: The University of Leeds
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds)
Depositing User: Symplectic Publications
Date Deposited: 02 Jul 2012 09:04
Last Modified: 05 Jun 2014 03:01
Published Version: http://dx.doi.org/10.1112/S0010437X10005063
Status: Published
Publisher: Cambridge University Press
Identification Number: 10.1112/S0010437X10005063
URI: http://eprints.whiterose.ac.uk/id/eprint/74389

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