Let $\mathfrak{g}$ be a real semisimple Lie algebra. We know that a coadjoint orbit $\mathcal{O} \hookrightarrow \mathfrak{g}^*$ carries a natural symplectic form $\omega$, namely the Kirillov-Kostant-Souriau form. Is there something tying $\omega$ to the Killing form $\kappa$ of $\mathfrak{g}$? It seems to me like there should be something like that, since both objects are intrinsic to $\mathfrak{g}$. I'm thinking something along the lines of, "if $\mathcal{O}$ is a semi-Riemannian submanifold of $\mathfrak{g}^*$, then the Levi-Civita connection of $\kappa$ (defined on the dual $\mathfrak{g}^*$ in the obvious way) induces a symplectic connection on $\mathcal{O}$", or something.

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in addition to being a coadjoint orbit, $\mathcal O$ is …"? I thought you were introducing a new meaning of $\mathcal O$. $\endgroup$ – LSpice Feb 12 at 0:17