Suppose we have some Lie group $G$, and we consider a co-adjoint orbit $\Omega$, which is a symplectic manifold. Does $G$ act on $\Omega$ by symplectomorphisms? As in, is $G$ a symmetry of $\Omega$, (an action that preserves symplectic structure)?
2026-03-25 22:09:27.1774476567
coadjoint orbits and symmetry
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Yes, the action is even Hamiltonian, with moment map given by the inclusion of $\Omega$ into $g^*$. You can check this directly.