Let's assume we have an affine, reductive, algebraic, complex, generally friendly and cooperative group $G$ acting algebraically on a variety $X$. Let $x\in X$ be some point. Under what conditions on $x$ or its stabilizer $H:=G_x$ is the orbit $G.x\cong G\newcommand{\qq}{/\hspace{-.8ex}/}\qq H$ a spherical variety? If you wonder, a spherical variety is a homogeneous space $G\qq H$ satisfying one of the following, equivalent properties:
- Any Borel subgroup $B\subseteq G$ has an open orbit in $G\qq H$.
- Every equivariant completion of $G\qq H$ contains only finitely many orbits.
- 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. Googling for the keywords "orbit" and "spherical" is quite fruitless because of property 1.
