Importance

For a manifold that is a space, we can look at all the subgroups of that appear as stabilizers for points . There is a “smallest” stabilizer subgroup (up to conjugation). By the orbit stabilizer theorem this will give the larges orbits. This theorem says that most elements on will have orbits of this size, and there is one stratum that is open and dense in that contains these biggest orbits.

Statement

Let be a group, and a manifold. Let be a proper group action, with connected orbit space . There is a unique conjugacy class of a stabilizer subgroup such that . That is, a subgroup of is conjugate to a subgroup of for any other stabilizer subgroup for .

The corresponding orbit type stratum (called the principal orbit type stratum) is open and dense in , and it’s quotient is open, dense and connected.

Proof

The most important idea of the proof is that the principal stratum is the unique orbit type stratum of depth 0. #todo

Compact connected Lie groups

Let be a compact, connected Lie group. acts on via the coadjoint action. The principal orbit type stratum of is related to the Weyl chamber , and the regular elements.