nLab coadjoint action

Definition

For $G$ a Lie group with Lie algebra $\mathfrak{g}$, the coadjoint action or coadjoint representation of $G$ is the action/representation on the dual vector space $\mathfrak{g}^*$ which is dual to the adjoint action of $G$ on $\mathfrak{g}$.

An orbit of the coadjoint action is accordingly called a coadjoint orbit.

Related concepts

• orbit method

• Borel-Weil-Bott theorem