  We investigate principal $G$-bundles on a compact K\"ahler 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 math.AG/0506511.
