Let (X, ω) be a compact connected Kähler manifold of complex dimension d and \({E_G\,\longrightarrow\,X}\) a holomorphic principal G–bundle, where G is a connected reductive linear algebraic group defined over \({\mathbb{C}}\). Let Z(G) denote the center of G. We prove that the following three statements are equivalent: (1) There is a parabolic subgroup \({P\,\subset\,G}\) and a holomorphic reduction of structure group \({E_P\,\subset\,E_G}\) to P, such that the corresponding L(P)/Z(G)–bundle $$E_{L(P)/Z(G)}\,:=\,E_P(L(P)/Z(G))\,\longrightarrow\,X$$admits a unitary flat connection, where L(P) is the Levi quotient of P. (2) The adjoint vector bundle ad(E G ) is numerically flat. (3) The principal G–bundle E G is pseudostable, and $$\int\limits_X c_2({\rm ad}(E_G))\omega^{d-2}\,=\,0.$$ If X is a complex projective manifold, and ω represents a rational cohomology class, then the third statement is equivalent to the statement that E G is semistable with c 2(ad(E G )) = 0.