 Coadjoint orbit

A coadjoint orbit is the orbit of $\ell \in {\mathfrak{g}}^{\ast }$ under ${\mathrm{Ad}}_g^{\ast }\ell$, defined by $⟨{\mathrm{Ad}}_g^{\ast }\ell ,\xi ⟩=⟨\ell ,{\mathrm{Ad}}_{g^{-1}}\xi ⟩$. For a connected Lie group these orbits are the symplectic leaves of its Lie-Poisson manifold. Their tangent vectors are infinitesimal coadjoint actions, and their natural two-form is the Kirillov–Kostant–Souriau symplectic form.