Stabilizer (isotropy subgroup) in a Lie group action

For a smooth action , the stabilizer fixes a point and is a closed Lie subgroup.

Stabilizer (isotropy subgroup) in a Lie group action

Definition

Let $G$ be a Lie group acting smoothly on a manifold $M$ via an action map $a:G\times M\to M$ (see smooth actions of Lie groups ). For a point $x\in M$ , the stabilizer (or isotropy subgroup) at $x$ is

G_x=\{g\in G\mid g\cdot x=x\}.

It is a subgroup of $G$ . Since it is the preimage of the closed set $\{x\}$ under the continuous map $g\mapsto g\cdot x$ , it is closed in $G;$ hence $G_x$ is a closed subgroup , and by the closed subgroup theorem it is automatically an embedded Lie subgroup.

Lie algebra of the stabilizer

Let $\mathfrak g$ be the Lie algebra of $G$ . For $X\in\mathfrak g$ , let $X_M$ denote the fundamental vector field on $M$ generated by $X$ (equivalently, $X_M(x)=\frac{d}{dt}\big|_{t=0}\exp(tX)\cdot x$ ). Then the Lie algebra of $G_x$ is

\mathfrak g_x=\{X\in\mathfrak g\mid X_M(x)=0\}.

Orbit–stabilizer geometry

The orbit $G\cdot x$ is an immersed submanifold (see orbits of Lie group actions ), and the natural map

G/G_x \to G\cdot x,\quad gG_x\mapsto g\cdot x

is a smooth bijection; under mild hypotheses (e.g. proper actions), it is a diffeomorphism. In the transitive case (see transitive actions ) this identifies $M$ with a homogeneous space of the form $G/G_x$ (compare coset spaces ).