Conjugation action of a Lie group

Let $G$ be a Lie group .

Definition. The conjugation action is the map

This is a smooth action of $G$ on the manifold $G$.

Orbits and stabilizers.

• The orbit of $h\in G$ is its conjugacy class, an example of an orbit of a Lie group action.
• The stabilizer of $h$ is its centralizer $C_G(h)=\{g\in G:gh=hg\}$, a closed subgroup; compare stabilizers and the closed subgroup theorem .
• The kernel of the action is the center $Z(G)$.

Infinitesimal picture. Differentiating conjugation at the identity yields the adjoint action $\mathrm{Ad}:G\to \mathrm{Aut}(\mathfrak{g})$, so conjugation is the global geometric source of the adjoint representation.