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Quasi-invariants of complex reflection groups

  • Yuri Berest (a1) and Oleg Chalykh (a2)
Abstract
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(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.

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[5] Yu. Berest and G. Wilson , Differential isomorphism and equivalence of algebraic verieties, in Topology, geometry and quantum field theory, London Mathematical Society Lecture Note Series, vol. 308 (Cambridge University Press, Cambridge, 2004), 98126.

[28] G. J. Heckman , A remark on the Dunkl differential-difference operators, in Harmonic analysis on reductive groups, Progress in Mathematics, vol. 101 (Birkhäuser, Boston, 1991), 181191.

[33] S. Montgomery , Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Mathematics, vol. 818 (Springer, Berlin, 1980).

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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