Kirillov–Kostant–Souriau symplectic form

If $X^\#_\mu=0$, then $X$ lies in the stabilizer Lie algebra $\mathfrak g_\mu$, so $\langle\mu,[X,Y]\rangle=0$ for every $Y$. The displayed value therefore depends only on the tangent vectors, not on their representatives in $\mathfrak g$. The same observation shows that a tangent vector pairing to zero with all others must itself vanish.

Equivariance of the coadjoint action and invariance of the Lie bracket imply $G$-invariance of $\omega_{\mathrm{KKS}}$. Evaluating the exterior derivative on fundamental vector fields reduces $d\omega_{\mathrm{KKS}}=0$ to the Jacobi identity. These properties are part of the canonical orbit construction described in Kostant, pp. 87–208.

With the convention displayed in the core and $d\mu^\xi=\iota_{\xi_{\mathcal O}}\omega$, the inclusion $\mathcal O\hookrightarrow\mathfrak g^*$ is an equivariant moment map. Reversing the definition of fundamental vector fields or using $\iota_{X_H}\omega=-dH$ reverses the KKS sign in many texts. The convention must therefore be checked before comparing formulas.

1. A. A. Kirillov, Elements of the Theory of Representations, Springer, 1976. DOI record. Relevant: Chapter 1, coadjoint orbits and their canonical two-form.
2. Bertram Kostant, “Quantization and Unitary Representations,” in Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics 170, Springer, 1970, pp. 87–208. Volume DOI record. Relevant: coadjoint orbits, symplectic structure, and moment maps.