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 EG over M admits a Hermitian–Einstein structure if and only if EG is polystable. A polystable flat principal G–bundle over M admits a unique Hermitian–Einstein connection. We also prove the existence and uniqueness of a Harder–Narasimhan 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.