Kähler metrics on the coadjoint orbits of a compact Lie group 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.

In fact if $M$ be a compact Kähler manifold, then its symplectic quotient is also Kähler manifold. So, because $T^∗G$ is Kähler manifold, so coadjoint orbit is also Kähler due to the following reason,
We have this useful decomposition $G^{\mathbb C}\cong G\times \mathfrak g^{*}\cong T^*G$
If we take $\mu:T^*G\to \mathfrak g^*$ as momentum map then $\mu^{-1}(\lambda)=G$, so
$$\mu^{-1}(\lambda)/G_\lambda\cong G/G_\lambda\cong \mathcal O_\lambda$$
so we need to take $M=T^*G$
As far as I know this method is from Mostow 
