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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Homogeneous vector bundles over abelian varieties via representation theory
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by Michel Brion PDF
Represent. Theory 24 (2020), 85-114 Request permission


Let $A$ be an abelian variety over a field. The homogeneous (or translation-invariant) vector bundles over $A$ form an abelian category $\textrm {HVec}_A$; the Fourier-Mukai transform yields an equivalence of $\textrm {HVec}_A$ 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 $\textrm {HVec}_A$ 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
  • MR Author ID: 41725
  • Email:
  • Received by editor(s): June 12, 2018
  • Received by editor(s) in revised form: December 5, 2019
  • Published electronically: February 3, 2020
  • © Copyright 2020 American Mathematical Society
  • Journal: Represent. Theory 24 (2020), 85-114
  • MSC (2010): Primary 14J60, 14K05; Secondary 14L15, 20G05
  • DOI:
  • MathSciNet review: 4058941