Cohomogeneity one Kähler and Kähler-Einstein manifolds with one singular orbit I

Let M be a cohomogeneity one manifold of a compact semisimple Lie group G with one singular orbit (Formula presented.). Then M is G-diffeomorphic to the total space (Formula presented.) of the homogeneous vector bundle over (Formula presented.) defined by a sphere transitive representation of G in a vector space V. We describe all such manifolds M which admit an invariant Kähler structure of standard type. This means that the restriction (Formula presented.) of the moment map of M to a regular orbit (Formula presented.) is a holomorphic map of S with the induced CR structure onto a flag manifold (Formula presented.), where (Formula presented.), endowed with an invariant complex structure (Formula presented.). We describe all such standard Kähler cohomogeneity one manifolds in terms of the painted Dynkin diagram associated with (Formula presented.) and a parameterized interval in some T-Weyl chamber. We determine which of these manifolds admit invariant Kähler–Einstein metrics.