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The Quarterly Journal of Mathematics Advance Access published online on May 15, 2009
The Quarterly Journal of Mathematics, doi:10.1093/qmath/hap017
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© 2009. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

DIFFERENTIAL OPERATORS ON AN AFFINE CURVE: IDEAL CLASSES AND PICARD GROUPS

Yuri Berest {dagger}

Department of Mathematics, Cornell University, Ithaca, NY 14853, USA

George Wilson{ddagger}

Mathematical Institute, 24–29 St Giles, Oxford OX1 3LB, UK

{ddagger} Corresponding author. E-mail: wilsong{at}maths.ox.ac.uk

Received 4 November 2008;
  Abstract

Let X be a smooth complex affine curve, and let R be the space of right ideal classes in the ring D of differential operators on X. We introduce and study a fibration {gamma} : R -> Pic X. We relate this fibration to the corresponding one in the classical limit, and derive an integer invariant n which indexes the decomposition of the fibres of {gamma} into Calogero–Moser spaces. We also study the action of the group Pic D on our fibration; and we explain how to define {gamma} in the framework of the Grassmannian description of R due to Cannings and Holland.

{dagger} E-mail: berest{at}math.cornell.edu


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