Homogeneous vector bundles over abelian varieties via representation theory

TL;DR

This paper explores an alternative approach to classifying homogeneous vector bundles over abelian varieties by linking them to finite-dimensional representations of a commutative affine group scheme, revealing deep analogies with reductive algebraic groups.

Contribution

It introduces a new perspective by connecting homogeneous vector bundles over abelian varieties with representations of the affine fundamental group, expanding understanding of their structure.

Findings

Establishes an equivalence between homogeneous vector bundles and affine group scheme representations.

Highlights analogies between vector bundles over abelian varieties and reductive algebraic group representations.

Provides a framework for analyzing vector bundles via representation theory.

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.