2
$\begingroup$

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 $?

$\endgroup$

2 Answers 2

2
$\begingroup$

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

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

$\endgroup$
1
  • 2
    $\begingroup$ 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 :) $\endgroup$ Nov 29, 2021 at 2:17
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.