  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.
