Is there a simple condition when the co-adjoint orbit of a semi-simple Lie group $\mathcal{G}$ is a Kähler manifold?
I am particularly interested in the symplectic group, so I do not want to require compactness!
Let us recall that a dual vector $\beta\in\mathfrak{g}$ of dual Lie algebra generates a coadjoint orbit as $O=\{\mathrm{Ad}_g^* \beta|g\in\mathcal G\}\subset\mathfrak{g}^*$. The Lie bracket naturally equips $O$ with a non-degenerate, closed 2-form, this is to say it $O$ must be even dimensional and is a symplectic manifold.
However, I also studied several examples where $O$ is naturally a Kähler manifold with compatible inner product induced by the Killing form. More precisely, if we define \begin{align} \mathfrak{s}=\{A\in\mathfrak{g}|\mathrm{ad}_A^*(\beta)=0\}\,, \end{align} we can use the non-degenerate Killing form to define the orthogonal complement $\mathfrak{s}_\perp$ and its dual $\mathfrak{s}_\perp^*$ (using again the Killing form to provide an isomorphism between $\mathfrak{g}$ and $\mathfrak{g}^*$).
In the examples, I studied, it turned out that minus the inverse Killing form restricted to $\mathfrak{s}_\perp^*$ was positive definite and compatible with the symlectic form on $\mathfrak{s}_\perp^*$. Using the group action, we can move this everywhere in $O$, which proves that it is a Kähler manifold. Note that I used here that $\mathfrak{s}^*_\perp\subset\mathfrak{g}^*$ can be naturally understood as the tangent space of $O$ at the point $\beta\in\mathfrak{g}^*$.
I would like to know how general this is and if there are simple conditions onto $\beta$ to ensure that the resulting metric (constructed from the Killing form) gives rise to a Kähler manifold?
I invite you to read: [A] Della Vedova A.. Special homogeneous almost complex structures on symplectic manifolds. Journal of Symplectic Geometry, 17(5), 1251-1295, 2019 [B] Della Vedova A., Gatti A., Almost Kaehler geometry of adjoint orbits of semisimple Lie groups, arXiv:1811.06958, 2018 [C] Gatti A., Special almost-Kähler geometry of some homogeneous manifolds, PhD of Universi-tà degli Studi di Pavia, supervised by Dr. Alberto Della Vedova, December 2019