Let $G$ be a connected reductive group over an algebraic closed field $k$. By Bruhat decomposition, $P \backslash G/P \cong W_P \backslash W / W_P$ is a finite set for any parabolic subgroup $P$ of $G$, so there are some connections between combinatorics of the Weyl group and geometry of the orbits.

Now consider two closed subgroup $H_i$ of $G$, when is $ H_1 \backslash G / H_2$ finite? In such case, can we describe the $H_1$-orbits in $G/H_2$ using some group theoretic datas or combinatoric datas in general?

If $H_1$ is a Borel subgroup, the finiteness is equivalent to that $G/H_2$ is a spherical variety. And things become trivial if $H_1=G$. If $H_1=H_2$, is the finiteness equivalent to that $H_1=H_2$ is a parabolic subgroup? For low rank $G$, can we classify all such pair $(H_1,H_2)$?

Let $G$ be a connected reductive group over an algebraically closed field $k$. By the Bruhat decomposition, $P \backslash G/P \cong W_P \backslash W / W_P$ is a finite set for any parabolic subgroup $P$ of $G$. So there is some connection between the combinatorics of the Weyl group and the geometry of the orbits.

Now consider two closed subgroups $H_i$ of $G$.

When is $ H_1 \backslash G / H_2$ finite? In such cases, can we describe the $H_1$-orbits in $G/H_2$ using some group theoretic data or combinatoric data in general?

If $H_1$ is a Borel subgroup, the finiteness is equivalent to $G/H_2$ being a spherical variety. And things become trivial if $H_1=G$.

If $H_1=H_2$, is the finiteness equivalent to $H_1=H_2$ being a parabolic subgroup? For low rank $G$, can we classify all such pair $(H_1,H_2)$?

Let $G$ be a connected reductive group over an algebraic closed field $k$. By Bruhat decomposition, $P \backslash G/P \cong W_P \backslash W / W_P$ is a finite set for any parabolic subgroup $P$ of $G$, so there are some connections between combinatorics of the Weyl group and geometry of the orbits.

Now consider two closed subgroup $H_i$ of $G$, when is $ H_1 \backslash G / H_2$ finite? In such case, can we describe the $H_1$-orbits in $G/H_2$ using some group theoretic datas or combinatoric datas in general?

If $H_1$ is a Borel subgroup, the finiteness is equivalent to that $G/H_2$ is a spherical variety. And things become trivial if $H_1=G$. If $H_1=H_2$, is the finiteness equivalent to that $H_1=H_2$ is a parabolic subgroup? For lower rank $G$, can we classify all such pair $(H_1,H_2)$?

Let $G$ be a connected reductive group over an algebraic closed field $k$. By Bruhat decomposition, $P \backslash G/P \cong W_P \backslash W / W_P$ is a finite set for any parabolic subgroup $P$ of $G$, so there are some connections between combinatorics of the Weyl group and geometry of the orbits.

Now consider two closed subgroup $H_i$ of $G$, when is $ H_1 \backslash G / H_2$ finite? In such case, can we describe the $H_1$-orbits in $G/H_2$ using some group theoretic datas or combinatoric datas in general?

If $H_1$ is a Borel subgroup, the finiteness is equivalent to that $G/H_2$ is a spherical variety. And things become trivial if $H_1=G$. If $H_1=H_2$, is the finiteness equivalent to that $H_1=H_2$ is a parabolic subgroup? For low rank $G$, can we classify all such pair $(H_1,H_2)$?

Let $G$ be a connected reductive group over an algebraic closed field $k$. By Bruhat decomposition, $P \backslash G/P \cong W_P \backslash W / W_P$ is a finite set for any parabolic subgroup $P$ of $G$, so there are some connections between combinatorics of the Weyl group and geometry of the orbits.

Now consider two closed subgroup $H_i$ of $G$, when is $ H_1 \backslash G / H_2$ finite? In such case, can we describe the $H_1$-orbits in $G/H_2$ using some group theoretic datas or combinatoric datas in general?

If $H_1$ is a Borel subgroup, the finiteness is equivalent to that $G/H_2$ is a spherical variety. If $H_1=H_2$, is the finiteness equivalent to $H_1=H_2$ is a parabolic subgroup? For lower rank $G$, can we classify all such pair $(H_1,H_2)$?

Let $G$ be a connected reductive group over an algebraic closed field $k$. By Bruhat decomposition, $P \backslash G/P \cong W_P \backslash W / W_P$ is a finite set for any parabolic subgroup $P$ of $G$, so there are some connections between combinatorics of the Weyl group and geometry of the orbits.

Now consider two closed subgroup $H_i$ of $G$, when is $ H_1 \backslash G / H_2$ finite? In such case, can we describe the $H_1$-orbits in $G/H_2$ using some group theoretic datas or combinatoric datas in general?

If $H_1$ is a Borel subgroup, the finiteness is equivalent to that $G/H_2$ is a spherical variety. And things become trivial if $H_1=G$. If $H_1=H_2$, is the finiteness equivalent to that $H_1=H_2$ is a parabolic subgroup? For lower rank $G$, can we classify all such pair $(H_1,H_2)$?