Hermitian-einstein connections on principal bundles over flat affine manifolds

Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal G-bundle E _G over M admits a HermitianEinstein structure if and only if E _G is polystable. A polystable flat principal G-bundle over M admits a unique HermitianEinstein connection. We also prove the existence and uniqueness of a HarderNarasimhan filtration for flat vector bundles over M. We prove a Bogomolov type inequality for semistable vector bundles under the assumption that the Gauduchon metric g is astheno-Kähler.