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 $?



Compact linear group orbit equivalent to linear compact group orbit
Answer is yes. By theorem of Mostow mentioned in this question


[Homogeneous manifold deformation retracts onto compact submanifold](https://mathoverflow.net/questions/345905/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.


