In this comprehensive [answer](https://mathoverflow.net/questions/45004/kahler-structure-on-flag-manifolds) to an old question, it is stated that 



> 
> Flag manifolds exhaust all compact homogeneous Kähler manifolds corresponding to a compact connected semi-simple Lie group.
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> 


Firstly, where can I find a proof of this? Secondly, if I replace **compact** in the assumptions with **complex** is the result still true? Finally if one removes connected in the assumptions on the Lie group what happens.



Classifying compact homogeneous Kähler manifolds
The comments make your second question moot unless reformulated, right?


For the first, this goes back to [Borel (1954, Thms 1 & 2)](http://www.pnas.org/content/40/12/1147.citation); more details in e.g. [Serre (1954, Thms 1,2,3 and remark following Thm 1)](http://www.numdam.org/item?id=SB_1951-1954__2__447_0), [Matsushima (1957, Thm 2)](http://ams.org/mathscinet-getitem?mr=87177), or [Besse (1987, Thm 8.89)](http://ams.org/mathscinet-getitem?mr=867684).


Third question: a bit too vague, but you may be interested in what [Duistermaat-Kolk (2000)](http://ams.org/mathscinet-getitem?mr=1738431) say on p. 316.


