The Kirillov--Kostant--Souriau form on a coadjoint orbit

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $G$ be a finite-dimensional real Lie group with Lie algebra $\mathfrak{g}$ and let $\mathcal{O}=G\cdot \alpha \subseteq {\mathfrak{g}}^{\ast }$ be the coadjoint orbit of $\alpha$ under the coadjoint action (The coadjoint representation, action and orbits). Give $\mathcal{O}$ its canonical injectively immersed homogeneous-space structure, transported from $G\mathrm{/}{G}_{\alpha }$ (Every orbit is an injectively immersed homogeneous space); thus $\mathcal{O}$ is the orbit of a smooth action and each tangent space $T_{\beta }\mathcal{O}$ consists exactly of the values ${\xi }_{\mathcal{O}}(\beta )$ of the fundamental vector fields of that action on $\mathcal{O}$, with ${\xi }_{\mathcal{O}}(\beta )=0$ exactly for $\xi$ in the stabilizer Lie algebra ${\mathfrak{g}}_{\beta }$ (Kernel of the infinitesimal orbit map). Here ${\xi }_{\mathcal{O}}$ denotes the restriction to $\mathcal{O}$ of the fundamental vector field ${\xi }_{{\mathfrak{g}}^{\ast }}$ of the coadjoint action, ${\xi }_{{\mathfrak{g}}^{\ast }}(\beta )(\eta )=\beta ([\xi ,\eta ])$ for $\eta \in \mathfrak{g}$ (The coadjoint representation, action and orbits, Fundamental vector fields for a left action).

The Kirillov--Kostant--Souriau form (KKS form) $\omega$ on $\mathcal{O}$ is defined pointwise by its values on fundamental fields:

${\omega }_{\beta }({\xi }_{\mathcal{O}}(\beta ),{\eta }_{\mathcal{O}}(\beta )):=\beta ([\xi ,\eta ])(\beta \in \mathcal{O},\xi ,\eta \in \mathfrak{g})\mathrm{.}$

The following lemma proves that this prescription is independent of the chosen Lie-algebra representatives, so that it defines an alternating bilinear form on each tangent space $T_{\beta }\mathcal{O}$; the next theorem proves that the resulting family of forms is smooth, nondegenerate and closed, and that it is $G$-invariant. The sign is chosen so that the inclusion $\mathcal{O}\hookrightarrow {\mathfrak{g}}^{\ast }$ satisfies the library moment equation $d⟨\mathrm{\Phi },\xi ⟩=-{\iota }_{{\xi }_{\mathcal{O}}}\omega$; the opposite sign would produce $+\omega$ in that identity and is not used here. The form is alternating because the Lie bracket is alternating, and the definition makes no freeness, compactness or regularity assumption: the orbit of $\alpha =0$ is the singleton $\{0\}$, on which the zero form is symplectic. ${\mathrm{A}\mathrm{C}}_{\omega }$ is countable choice, used only through the orbit structure and fundamental-field suppliers.