Coadjoint action of a Lie group

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ . The adjoint action assigns to each $g\in G$ an automorphism $\mathrm{Ad}_g:\mathfrak{g}\to\mathfrak{g}$ . The coadjoint action is the action of $G$ on the dual vector space $\mathfrak{g}^*$ (canonically analogous to a fiber of a cotangent bundle) defined by

\mathrm{Ad}^*_g:\mathfrak{g}^*\to\mathfrak{g}^*, \qquad \mathrm{Ad}^*_g(\lambda) := \lambda\circ \mathrm{Ad}_{g^{-1}}.

Equivalent characterizations

Equivalently, $\mathrm{Ad}^*_g$ is characterized by the pairing identity

\langle \mathrm{Ad}^*_g\lambda,\, X\rangle = \langle \lambda,\, \mathrm{Ad}_{g^{-1}}X\rangle \quad (\lambda\in\mathfrak{g}^*,\, X\in\mathfrak{g}).

The assignment $g\mapsto \mathrm{Ad}^*_g$ is a smooth group homomorphism $G\to \mathrm{GL}(\mathfrak{g}^*)$ .

Examples

Abelian groups. If $G$ is abelian, then $\mathrm{Ad}$ is trivial, hence $\mathrm{Ad}^*$ is also trivial: $\mathrm{Ad}^*_g=\mathrm{id}$ for all $g$ .
Compact rotations. For $G=\mathrm{SO}(3)$ , using the standard inner product to identify $\mathfrak{so}(3)\cong\mathfrak{so}(3)^*\cong\mathbb{R}^3$ , the coadjoint action corresponds to the usual rotation action on $\mathbb{R}^3$ .
Matrix groups with trace pairing. For $G=\mathrm{GL}(n,\mathbb{R})$ , identify $\mathfrak{gl}(n,\mathbb{R})^*$ with matrices by $\langle A,B\rangle=\operatorname{tr}(A^\mathsf{T}B)$ . Under this identification, $\operatorname{Ad}_g^*A=g^{-\mathsf{T}}Ag^\mathsf{T}.$