Connections and Higgs fields on a principal bundle

Let M be a compact connected Kahler manifold and G a connected linear algebraic group defined over . A Higgs field on a holomorphic principal G-bundle &#949;<SUB>G</SUB> over M is a holomorphic section &#952; of ad(&#949;<SUB>G</SUB>)&#8855;&#937;<SUP>1</SUP><SUB>M</SUB> such that &#952;&#8743; &#952; = 0. Let L(G) be the Levi quotient of G and (&#949;<SUB>G</SUB> (L(G)), &#952;<SUB>l</SUB> ) the Higgs L(G)-bundle associated with (&#949;<SUB>G</SUB> , &#952;). The Higgs bundle (&#949;<SUB>G</SUB> , &#952;) will be called semistable (respectively, stable) if (&#949;<SUB>G</SUB> (L(G)), &#952; l ) is semistable (respectively, stable). A semistable Higgs G-bundle (&#949;G , &#952;) will be called pseudostable if the adjoint vector bundle ad(&#949;<SUB>G</SUB> (L(G))) admits a filtration by subbundles, compatible with &#952;, such that the associated graded object is a polystable Higgs vector bundle. We construct an equivalence of categories between the category of flat G-bundles over M and the category of pseudostable Higgs G-bundles over M with vanishing characteristic classes of degree one and degree two. This equivalence is actually constructed in the more general equivariant set-up where a finite group acts on the Kahler manifold. As an application, we give various equivalent conditions for a holomorphic G-bundle over a complex torus to admit a flat holomorphic connection