Suppose G is a Lie group (or algebraic group) acting on a manifold (or scheme) X, and H⊆G is a subgroup. Let x,y∈X be points such that x is in the closure of the orbit H⋅y (but not in H⋅y itself). Then obviously x is in the closure of G⋅y, but can it happen that x is actually in the orbit G⋅y (not just in the closure)?


**Background:** I got stuck on this point when trying to understand the very last line of the proof of Theorem 0.3.1 of Kapranov's [Chow quotients of Grassmannian I](http://arxiv.org/abs/alg-geom/9210002), which states that every irreducible component of an algebraic cycle corresponding to a point in the Chow quotient X//G is the closure of a single G-orbit. In this case, H⊆G is a torus and X is a smooth projective variety.



Do subgroups respect the orbit-closure relation?
Sure, that can happen. G = PGL\_2, H is the torus, X is the Riemann sphere, x is the north pole, y is some point other than the two poles.


I've read that Kapranov paper recently, I'll see if I can find something more useful to say.


