Semistability of Principal Bundles on a Kähler Manifold with a Non-Connected Structure Group

We investigate principal $G$-bundles on a compact Kähler manifold, where $G$ is a complex algebraic group such that the connected component of it containing the identity element is reductive. Defining (semi)stability of such bundles, it is shown that a principal $G$-bundle $E_G$ admits an Einstein–Hermitian connection if and only if $E_G$ is polystable. We give an equivalent formulation of the (semi)stability condition. A question is to compare this definition with that of [Gómez T. L., Langer A., Schmitt A. H. W., Sols I., Ramanujan Math. Soc. Lect. Notes Ser., Vol. 10, Ramanujan Math. Soc., Mysore, 2010, 281–371].