orbit type stratification

Definition - Orbit type/ points of orbit type

Let be a Lie group that acts on a smooth manifold . For a subgroup , the orbit type is the conjugation class of .

For any (closed) subgroup we define the subset

This is the points for which is the stabilizer subgroup. Note that this is different from the fixed point set , where . For then is the maximal stabilizing group.

The set

is the set of points of orbit type (H). Said another way, is the set of points with stabilizers conjugate to .

Note that

This is because orbits of are related to stabilizer subgroups via conjugation:

Definition - Orbit type decomposition

Given a manifold , and a connected Lie group , we have that for closed subgroups and conjugacy class of the subgroups

Since each subset is a union of orbits, then it is -invariant and the decomposition descends to the orbit space

These decompositions are called the orbit type decomposition of and the orbit space respectively.

Note if is not connected then the piece of may not be connected. In that case, we can decompose further taking to be connected components of . Then, we can take the corresponding decomposition of to be all the preimages of .

As a stratification

The orbit type decomposition of and is a stratified topological space.

That is, the decompositions are locally finite and satisfy the frontier condition. Furthermore, each is a smooth embedded submanifold of and inherits a unique manifold structure such that the quotient map is a smooth submersion.

Proof

todo

Importance

By the orbit stabilizer theorem, this means that all points of a given orbit type strata will have orbits of the same dimension. Thus, the orbit type stratification indexes or “sorts” elements by how the action moves around that point. Namely, it indexes on the dimensions (roughly since there can be multiple conjugacy classes of stabilizers of the same dimension) of the submanifold that the orbit forms (assuming a proper action).

Resources

• Dr. Meinrenken’s lecture notes “Group actions on manifolds”