Holomorphic principal bundles over a compact Kähler manifold

Let p : G → H be a homomorphism between connected reductive algebraic groups over ℂ such that the center of the Lie algebra g is sent to the center of h. If E_G is a holomorphic principal G-bundle over a compact connected Kähler manifold M, and E_G is semistable (resp. polystable), then the principal H -bundle E_G X_G H is also semistable (resp. polystable). A G-bundle over M is polystable if and only if it admits an Einstein-Hermitian connection; this is an analog of a theorem of Uhlenbeck and Yau for G-bundles. Two different formulations of the G-bundle analog of the Harder-Narasimhan reduction have been established. The equivalence of the two formulations is a consequence of a group theoretic result.