Question

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, 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.

Answer 1

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.