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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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  • $\begingroup$ Your body deals with an arbitrary semi-Riemannian submanifold, whereas your title deals with a coadjoint orbit. Surely this is too much generality, since, for example, a submanifold could have odd dimension. $\endgroup$ – LSpice Feb 11 at 23:43
  • $\begingroup$ I'm not sure I follow. My body deals with a coadjoint orbit that happens to be a semi-Riemannian submanifold of the dual of the Lie algebra, i.e. the pullback of the Killing form on it is non-degenerate. $\endgroup$ – user18063 Feb 12 at 0:05
  • $\begingroup$ Oh, I see. So "if $\mathcal O$ is a semi-Riemannian submanifold of $\mathfrak g^*$" means "if, 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
  • $\begingroup$ A obvious observation: Since the map $\mathfrak{g} \to \mathfrak{g}^*$ induced by the Killing form is an Ad-equivariant isomorphism, you can use it transport the symplectic structure from the coadjoint orbits to a symplectic structure on the adjoint orbits. That's the only construction involving the KKS symplectic structure and the Killing form that I'm aware of. $\endgroup$ – Tobias Diez Feb 12 at 20:35

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