# Compatibility of Kirillov-Kostant-Souriau form and Killing form

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.

• 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. – LSpice Feb 11 at 23:43
• 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. – user18063 Feb 12 at 0:05
• 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 $\mathcal O$. – LSpice Feb 12 at 0:17
• 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. – Tobias Diez Feb 12 at 20:35