nLab
coadjoint orbit

Given a Lie group $G$, it acts smoothly on the dual $g^{*}$ of its Lie algebra $g$ by the coadjoint action. The orbits of that action are called coadjoint orbits. They are especially important in the orbit method of representation theory or, more generally, geometric quantization.

The dual $g^{*}$ of a (say finite-dimensional real) Lie algebra has a structure of a Poisson manifold with the Poisson structure due to A. Kirillov and Souriau, namely for any $a\in g^{*}$,

$$\{f,g\}(a):=⟨[d{f}_a,d{g}_a],a⟩$$\{ f, g\}(a) := \langle [d f_a, d g_a],a\rangle

The coadjoint orbits are the symplectic leaves of that structure; hence each orbit is a symplectic manifold.

Sometimes coadjoint orbits are studied in the infinite-dimensional case (for example in study of Virasoro algebra).