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: