Coadjoint orbit
= Coadjoint orbit
{wiki}
A <coadjoint orbit> is the orbit of $\ell\in\mathfrak g^*$ under $\operatorname{Ad}^*_g\ell$, defined by $\langle\operatorname{Ad}^*_g\ell,\xi\rangle=\langle\ell,\operatorname{Ad}_{g^{-1}}\xi\rangle$. 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>.