Let $G$ be a Lie group, and $\fr g$ its Lie algebra. Then $G$ has a natural action on $\fr g^*$ called the coadjoint action, since it is dual to the adjoint action of $G$ on $\fr g$ . The orbits of this action are submanifolds of $\fr g^*$ which carry a natural symplectic structure, and are in a certain sense, the minimal symplectic manifolds on which $G$ acts. The orbit through a point $\lambda\in\fr g^*$ is typically denoted $\O_\lambda$ .

The tangent space $T_\lambda\O_\lambda$ is naturally idenified by the action with $\fr g/\fr r_\lambda$ , where $\fr r_\lambda$ is the Lie algebra of the stabilizer of $\lambda$ . The symplectic form on $\O_\lambda$ is given by $\omega_\lambda(X,Y)=\lambda([X,Y])$ . This is obviously anti-symmetric and non-degenerate since $\lambda([X,Y])=0$ for all $Y\in\fr g$ if and only if $X\in \fr r_\lambda$ . This also shows that the form is well-defined.

There is a close association between coadoint orbits and the representation theory of $G$ , with irreducible representations being realized as the space of sections of line bundles on coadjoint orbits. For example, if $\fr g$ is compact, coadjoint orbits are partial flag manifolds, and this follows from the Borel-Bott-Weil theorem.