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Homogeneous vector bundles over abelian varieties via representation theory


Author: Michel Brion
Journal: Represent. Theory 24 (2020), 85-114
MSC (2010): Primary 14J60, 14K05; Secondary 14L15, 20G05
DOI: https://doi.org/10.1090/ert/537
Published electronically: February 3, 2020
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Abstract: Let $ A$ be an abelian variety over a field. The homogeneous (or translation-invariant) vector bundles over $ A$ form an abelian category $ {\rm HVec}_A$; the Fourier-Mukai transform yields an equivalence of $ {\rm HVec}_A$ with the category of coherent sheaves with finite support on the dual abelian variety. In this paper, we develop an alternative approach to homogeneous vector bundles, based on the equivalence of $ {\rm HVec}_A$ with the category of finite-dimensional representations of a commutative affine group scheme (the ``affine fundamental group'' of $ A$). This displays remarkable analogies between homogeneous vector bundles over abelian varieties and representations of split reductive algebraic groups.


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Additional Information

Michel Brion
Affiliation: Institut Fourier, Université de Grenoble, 100 rue des Mathématiques, 38610 Gières, France
Email: Michel.Brion@univ-grenoble-alpes.fr

DOI: https://doi.org/10.1090/ert/537
Received by editor(s): June 12, 2018
Received by editor(s) in revised form: December 5, 2019
Published electronically: February 3, 2020
Article copyright: © Copyright 2020 American Mathematical Society