I have heard for many times that the coadjoint orbits of a compact semi-simple Lie group are Kahler. While I know that the symplectic structure on a coadjoint orbit can be given by the symplectic reduction of $T^*G$, I wonder if the Kahler structure can be given in this way as well, namely by Kahler reduction.
However, on page 5 of the notes https://math.berkeley.edu/~alanw/277papers00/zhu.pdf about symplectic reductions, Prof. Zhu remarked that one cannot expect to do so and referred this issue to the PhD thesis by R. S. Filippini. Since I have no access to this thesis, I would like to ask what we can say about this approach. For example, does she mean that $T^*G$ does not possess (or it’s hard to find) an integrable almost complex structure compatible with the standard symplectic form such that it is invariant under the action by $G$?
Thank you!
