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Let $X$ and $Y$ be real manifolds. Under what conditions is there a Lie group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of a point in $Y$ and both are themselves Lie groups?

Equivalently, what conditions need to be satisfied by $X$ and $Y$ for it to be possible to realise $X$ and $Y$ as simultaneous coset spaces $G/H_1$ and $G/H_2$ of $G$ with $H_1\leq H_2$?

This question is related: Simultaneous coset spaces.

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    $\begingroup$ Do you want $G$ to be finite-dimensional? If not, I suspect that the condition is simply that $X$ be a fiber bundle over $Y$, with $G$ the group of diffeomorphisms of $X$ taking fibers to fibers. $\endgroup$
    – Allen Knutson
    Commented Dec 22, 2015 at 16:09
  • $\begingroup$ I think I do need $G$ to be finite-dimensional, but your idea is interesting nonetheless. Do you know if the fibre bundle condition is necessary as well as sufficient? $\endgroup$
    – octopus
    Commented Dec 22, 2015 at 23:31
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    $\begingroup$ If your assertion is satisfied, then you automatically get a map $X=G/H_1\twoheadrightarrow G/H_2=Y$ sending $gx$ to $gy$, where $x$ has stabiliser $H_1$ and $y$ has stabiliser $H_2$. In the finite-dimensional case, this map is automatically locally trivial, that is, a fibre bundle. On the other hand, there are manifolds that do not admit transitive actions by finite-dimensional Lie groups, so the fibre bundle condition alone is not sufficient. $\endgroup$
    – Sebastian Goette
    Commented Dec 23, 2015 at 10:03
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    $\begingroup$ The existence a $G$-equivariant map $F:X\to Y$, namely, one such that $F(gx)=gF(x)$ for all $g\in G$, $x\in X$, is a necessary and sufficient condition. $\endgroup$
    – Claudio Gorodski
    Commented Dec 24, 2015 at 1:38
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    $\begingroup$ After thinking about the motivation of this question for a while, I guess that among the various equivalent conditions that there might be, the one that $H_1\le H_2$ is probably the easiest to check in practice. Of course I assume that we know what $G$ is and how $G$ acts. If this is not the case, please give some motivation, so one can understand what kind of description you are looking for. $\endgroup$
    – Sebastian Goette
    Commented Dec 24, 2015 at 10:23

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