I am looking for a proof or counterexample for following assertion
In 1950s A. Borel, R. Bott, J. L. Koszul, F. Hirzebruch et al. investigated the coadjoint orbits as complex homogeneous manifolds. It was proven that each coadjoint orbit of a compact connected Lie group $G$ admits a canonical G-invariant complex structure and the only (within homotopies) $G$-invariant Kählerian metrics.
Reference: Bott R. The Geometry and Representation Theory of Compact Lie Groups, In: Representation Theory of Lie Groups, London Mathematical Society Lecture Note Series, Cambridge Univ. Press, 34 (1979) 65–90.