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.

With $⟨{\mathrm{ad}}_{\xi }^{\ast }\ell ,\eta ⟩=-⟨\ell ,[\xi ,\eta ]⟩$, the displayed form on a coadjoint orbit is well-defined: changing a generator by an element of the stabilizer changes no pairing. It is nondegenerate on orbit tangents, and its closedness follows from the Jacobi identity. This sign agrees with the positive Lie-Poisson bracket when Hamiltonian vector fields act by $X_f(g)=\{g,f\}$. A negative Lie-Poisson convention reverses the orbit form as well.