Take the 2-minute tour ×  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.  share|improve this
question I guess you mean H is a closed subgroup? Otherwise,
think of G=the reals, H=the rationals, X=the reals, x=0, y=pi.
–  Andrew Critch Nov 2 '09 at 22:24 Yes, I'm really thinking
about algebraic groups. –  Anton Geraschenko Nov 3 '09 at 0:28
1 Answer 1  up vote 3 do