Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. It is well-known that each orbit for the coadjoint representation of $G$ on $\mathfrak{g}^*$ carries a canonical symplectic structure, known as the Kirillov-Kostant-Souriau symplectic form.
Moreover, I've read at a few different places that the coadjoint orbits are also Kähler manifolds:
Theorem. Let $G$ be a compact Lie group, $\mathcal{O}$ a coadjoint orbit and $\omega$ its Kirillov-Kostant-Souriau symplectic form. Then, there exists a unique $G$-invariant Kähler metric on $\mathcal{O}$ that is compatible with $\omega$.
For example, this result is mentioned in Robert Bryant's lecture notes An Introduction to Lie Groups and Symplectic Geometry on page 150, and at the beginning of this paper by Kronheimer.
However, I didn't find any proof of that theorem. Does someone know how to prove it or can point a good reference?
According to Bryant, it is "not hard" to prove it "using roots and weights". But I wasn't able to do so.
Best Answer
Suppose that $G$ is compact and semi-simple, let $c\in {\cal G}^*$ the dual of ${\cal G}$, consider ${\cal G}_c$ the stabilizer of $c$, and ${\cal,G}'_c$ its orthogonal for the Killing metric, you can identify the tangent space at $c$ of the coadjoint orbit to ${\cal G}'_c$. If you restrict -Killing to ${\cal G}'_c$ you obtain a metric on the adjoint orbit, is this metric does not define the Kahler structure with the Konstant Kirillov Souriau form?