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On elliptic Calogero-Moser systems for complex crystallographic reflection groups

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journal contribution
posted on 2014-07-16, 09:07 authored by Pavel Etingof, G. Felder, Xiaoguang Ma, Alexander VeselovAlexander Veselov
To every irreducible nite crystallographic reflection group (i.e., an irreducible finite reflection group G acting faithfully on an abelian variety X), we attach a family of classical and quantum integrable systems on X (with meromorphic coefficients). These families are parametrized by G-invariant functions of pairs (T; s), where T is a hypertorus in X (of codimension 1), and s 2 G is a reflection acting trivially on T. If G is a real reflection group, these families reduce to the known generalizations of elliptic Calogero-Moser systems, but in the non-real case they appear to be new. We give two constructions of the integrals of these systems - an explicit construction as limits of classical Calogero-Moser Hamiltonians of elliptic Dunkl operators as the dynamical parameter goes to 0 (implementing an idea of [BFV]), and a geometric construction as global sections of sheaves of elliptic Cherednik algebras for the critical value of the twisting parameter. We also prove algebraic integrability of these systems for values of parameters satisfying certain integrality conditions.


The work of P.E. and X.M. was partially supported by the NSF grants DMS-0504847 and DMS-0854764. The work of G.F. was partially supported by SNF grant 200020-122126.



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  • Mathematical Sciences

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Journal of Algebra




107 - 129


ETINGOF, P. ... et al., 2011. On elliptic Calogero-Moser systems for complex crystallographic reflection groups. Journal of Algebra, 329, pp. 107-129.


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NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Algebra. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published at http://dx.doi.org/10.1016/j.jalgebra.2010.04.011.




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