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 $(H_1,H_2)$?

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    $\begingroup$ Is there an example where non of $H_1$ and $H_2$ is parabolic? $\endgroup$ – Wille Liou Jun 7 '19 at 8:49
  • $\begingroup$ @WilleLiou, try taking $H_1=H_2$ to be diagonal matrices in $G=GL_2$... $\endgroup$ – paul garrett Jun 8 '19 at 2:36
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    $\begingroup$ @paulgarrett I believe $T_2 \backslash GL_2 /T_2$ is not finite, as $a_{11}a_{22}/a_{12}a_{21}$ is invariant. $\endgroup$ – sawdada Jun 8 '19 at 5:21
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    $\begingroup$ @sawdada you can also seeing this for dimension reasons, since scalar matrices act the same on both sides, so the quotient is 1-dim’l $\endgroup$ – Daniel Litt Jun 8 '19 at 14:22
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    $\begingroup$ @sawdada Sorry, seems that I was wrong and it is always infinite except the trivial case $K=G$. $\endgroup$ – Victor Petrov Jun 13 '19 at 10:32

This doesn't answer all your questions, but it seems relevant enough to post.

In The orbits of affine symmetric spaces under the action of minimal parabolic subgroups by T. Matsuki(1977), the double coset spaces $H\backslash G/P$ are described and shown to be finite where $G$ is a real semisimple Lie group, $P$ is a minimal parabolic, and $G^{\sigma}_0\subset H\subset G^\sigma$ where $G^{\sigma}$ denotes the closed subgroup of $G$ consisting of all the elements fixed by an involution $\sigma$.

When $H$ is the maximal compact, the quotient is trivial because of the Iwasawa decomposition. When $H$ is a real form of a complex semisimple Lie group $G$, this is a result of Aomoto from 1965 (in that case $P=B$, the Borel). And in the case where $G$ is a product of a semisimple Lie group with itself and $H$ is the diagonal, then this reduces to the Bruhat decomposition you mention in your question.


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