On semistable principal bundles on complex projective manifolds, II

Let (X, ω) be a compact connected Kähler manifold of complex dimension d and EG a holomorphic principal G-bundle, where G is a connected reductive linear algebraic group defined over C. Let Z(G) denote the center of G. We prove that the following three statements are equivalent: (1) There is a parabolic subgroup P ⊂ G and a holomorphic reduction of structure group EP ⊂ EG to P, such that the corresponding L(P)/Z(G)-bundle EL(P)/Z(G):= EP(L(P)/Z(G)) → X admits a unitary flat connection, where L(P) is the Levi quotient of P. (2) The adjoint vector bundle ad(EG) is numerically flat. (3) The principal G-bundle EG is pseudostable, and If X is a complex projective manifold, and ω represents a rational cohomology class, then the third statement is equivalent to the statement that EG is semistable with c2(ad(EG)) = 0.