# Compact linear group orbit equivalent to linear compact group orbit

A corollary of the Mostow-Palais theorem is that every homogeneous space for a compact group is a linear group orbit. In other words, if $$H$$ is a closed subgroup of a compact group $$K$$ then there exists some representation $$\pi: K \to GL(V)$$ and $$v \in V$$ such that the orbit $$\mathcal{O}_v=\{ \pi(k)v: k \in K\}$$ is diffeomorphic to $$K/H$$ (the stabilizer of $$v$$ is exactly $$H$$).

Is the converse true; does every compact linear group orbit admit a transitive action by a compact group? In other words, if $$G$$ is a group, $$\pi: G \to GL(V)$$ a representation, $$v \in V$$, and the stabilizer $$\mathcal{O}_v=\{ \pi(g)v: g \in G \}$$ is compact, then must there exist some compact group $$K$$ acting transitively on the manifold $$\mathcal{O}_v$$?

The answer of your question is true if the orbit $$O_v$$ is simply connected.

Montgomery, D. Zippin, L. Topological transformations groups p. 226

• Hey sorry should have clarified in my question that I already know the answer to my question is yes when $O_v$ is simply connected (or even if it has finite fundamental group). I'll edit my post to reflect this :) Nov 29, 2021 at 2:17

Answer is yes. By theorem of Mostow mentioned in this question

Homogeneous manifold deformation retracts onto compact submanifold

["Covariant Fiberings of Klein spaces" Mostow 1955]

If G and G' both have finitely many connected components (for example if they are algebraic groups) then G/G' is a vector bundle over K/K' where K and K' are maximal compacts.

This is the case here because the image of a representation is always an algebraic group and the stabilizer of a vector is always Zariski closed and thus always an algebraic group.

So if G/G' is a linear group orbit which is compact then the vector bundle part is trivial and we just have that linear group orbit is K/K'.

Just a note here that a similar result to the Mostow result is corollary 2 of "Simply Connected Homogeneous Spaces" 1950 by Montgomery which states that if G is connected and G' has finitely many connected components and G/G' compact then maximal compact K acts transitively.