coadjoint orbit

Let $G$ be a Lie group, and $𝔤$ its Lie algebra. Then $G$ has a natural action on ${𝔤}^{*}$ called the coadjoint action, since it is dual to the adjoint action of $G$ on $𝔤$. The orbits of this action are submanifolds of ${𝔤}^{*}$ 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 $\lambda \in {𝔤}^{*}$ is typically denoted ${𝒪}_{\lambda }$.

The tangent space $T_{\lambda }{𝒪}_{\lambda }$ is naturally idenified by the action with $𝔤/{𝔯}_{\lambda }$, where ${𝔯}_{\lambda }$ 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 $Y\in 𝔤$ if and only if $X\in {𝔯}_{\lambda }$. 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 $𝔤$ is compact, coadjoint orbits are partial flag manifolds, and this follows from the Borel-Bott-Weil theorem.