Flat principal bundles over an abelian variety (Q557215)

Let \(G\) be a complex connected reductive linear algebraic group and \(A\) a complex abelian variety. Let \(E_G\) be a principal \(G\)-bundle over the abelian variety \(A\). It is well known that if \(E_G\) admits a flat holomorphic connection then \(E_G\) is invariant under translations. The author proves in this paper the reciprocal statement: if the principal bundle \(E_G\) is invariant under translations (that is: \(\tau^*_x E_G\simeq E_G\) for any \(x\in A)\) then \(E_G\) admits a flat holomorphic connection. In this proof of this result it is also proved that the translation-invariant \(G\)-bundles are semistable.