Cohomogeneity one Kahler and Kahler-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 $S_0=G\mathrm{/}H$. Then $M$ is $G$- diffeomorphic to the total space $G{\times }_{H}V$ of the homogeneous vector bundle over $S_0$ defined by a sphere transitive representation of $G$ in a vector space $V$. We describe all such manifolds $M$ which admit an invariant Kahler structure of standard type. This means that the restriction $\mu :S=Gx=G\mathrm{/}L\to F=G\mathrm{/}K$ of the moment map of $M$ to a regular orbit $S=G\mathrm{/}L$ is a holomorphic map of $S$ with the induced CR structure onto a flag manifold $F=G\mathrm{/}K$, where $K = N_G(L)$,endowedwithaninvariantcomplexstructure$J^F$ . We describe all such standard Kahler cohomogeneity one manifolds in terms of the painted Dynkin diagram associated with $(F=G\mathrm{/}K;J^F)$ and a parametrized interval in some T-Weyl chamber. We determine which of these manifolds admit invariant Kahler-Einstein metrics.