Let (X, ω) be a compact connected Kähler manifold of complex dimension d and \documentclass[12pt]{minimal}
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\begin{document}$${E_G\,\longrightarrow\,X}$$\end{document} a holomorphic principal G–bundle, where G is a connected reductive linear algebraic group defined over \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{C}}$$\end{document}. Let Z(G) denote the center of G. We prove that the following three statements are equivalent: There is a parabolic subgroup \documentclass[12pt]{minimal}
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\begin{document}$${P\,\subset\,G}$$\end{document} and a holomorphic reduction of structure group \documentclass[12pt]{minimal}
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\begin{document}$${E_P\,\subset\,E_G}$$\end{document} to P, such that the corresponding L(P)/Z(G)–bundle
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\begin{document}$$E_{L(P)/Z(G)}\,:=\,E_P(L(P)/Z(G))\,\longrightarrow\,X$$\end{document}admits a unitary flat connection, where L(P) is the Levi quotient of P. The adjoint vector bundle ad(EG) is numerically flat.The principal G–bundle EG is pseudostable, and \documentclass[12pt]{minimal}
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\begin{document}$$\int\limits_X c_2({\rm ad}(E_G))\omega^{d-2}\,=\,0.$$\end{document}
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.
