# Finiteness of $H_1 \backslash G / H_2$ and the geometry of the orbits

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 $$(H_1,H_2)$$?

• Is there an example where non of $H_1$ and $H_2$ is parabolic? – Wille Liou Jun 7 '19 at 8:49
• @WilleLiou, try taking $H_1=H_2$ to be diagonal matrices in $G=GL_2$... – paul garrett Jun 8 '19 at 2:36
• @paulgarrett I believe $T_2 \backslash GL_2 /T_2$ is not finite, as $a_{11}a_{22}/a_{12}a_{21}$ is invariant. – sawdada Jun 8 '19 at 5:21
• @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 – Daniel Litt Jun 8 '19 at 14:22
• @sawdada Sorry, seems that I was wrong and it is always infinite except the trivial case $K=G$. – Victor Petrov Jun 13 '19 at 10:32

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.