I asked the following question over at [math.stackexchange](https://math.stackexchange.com/questions/191032/when-is-an-orbit-spherical), but got no answers. Maybe it's less well-known than I thought, but I still wanted to ask here:


Let's assume we have an affine, reductive, algebraic group $G$ acting algebraically on a variety $X$, everything over an algebraically closed field of characteristic zero. Let $x\in X$ be some point with reductive stabilizer $H:=G\_x$. Under what conditions on $x$ or $H$ is the orbit $G.x\cong G\newcommand{\qq}{/\hspace{-.8ex}/}\qq H$ a *spherical variety*? Let me briefly recall that a spherical variety is a homogeneous space $G\qq H$ satisfying one of the following, equivalent properties:


1. Any Borel subgroup $B\subseteq G$ has an open orbit in $G\qq H$.
2. Every equivariant completion of $G\qq H$ contains only finitely many orbits.
3. For every irreducible $G$-module $V$ and any character $\chi$ of $H$,
$$\dim\left\{~v\in V \mid \forall h\in H: h.v = \chi(h)v ~\right\}\le 1.$$


I was hoping that this is well-known, but I cannot find any direct statements of that kind. Searching for the keywords "orbit" and "spherical" is quite fruitless because of property 1. 


**Edit:** In the cases of interest to me, the orbit $G.x$ is affine. 



When is an orbit spherical?
This was an active problem 20 years ago. I don't know if it's completely resolved, but more can be said for *affine* spherical homogeneous spaces. (Also---in general, a spherical variety need not be homogeneous, so searching for "spherical homogeneous variety might yield better results.) Try Brion, "Classification des espaces homogènes sphériques" (Compositio, 1987) and more recent work of Knop et al, e.g., <http://arxiv.org/abs/math/0505102> . The latter has convenient tables at the end.


