Let $G$ be a reductive group over a number field $k$, with center $Z$. Let $P$ be a parabolic subgroup. Let $H$ be a reductive subgroup of $G$. To what extent can we understand the double coset space $P\_k\backslash G\_k/H\_k$? I'll give some examples, then my motivation for the question, and then I'll refine the question.


Let's look at some simple examples. 


Example 1: $G=GL\_2$, $P$ is the standard Borel subgroup, $H=Z\cdot GL\_1$, embedded as the Levi component of $P$. Using the Bruhat decomposition, $P\_k\backslash G\_k/H\_k=1\cup w\cup wn$, where $n$ is any nontrivial element of the unipotent radical of $P$, and $w$ is the nontrivial element of the Weyl group of $G$. 


Example 2: $G=GL\_2$, $P$ is the standard Borel subgroup, $H$ is the multiplicative group of a quadratic extension $k\_1$ of $k$. Identifying $P\_k\backslash G\_k$ with $k^2-(0,0)=k\_1^\times$, we see that $P\_k\backslash G\_k/H\_k=1$. 


Example 3: $G=GL\_2\times GL\_2$, $P$ is $B\times B$, the product of the Borel subgroups of $GL\_2$, $H$ is $GL\_2$ embedded diagonally in $G$. Then 
$P\_k\backslash G\_k/H\_k=B\_k\backslash H\_k/B\_k=1\cup w$.


Example 4: $G=GL\_2$ over a quadratic extension $k\_1$ of $k$ (considered as a $k$-group), $P$ the standard Borel subgroup, $H=GL\_2$ over $k$. Identifying $P\_k\backslash G\_k$ with $k\_1^2-(0,0)$, we see that $P\_k\backslash G\_k/H\_k=1\cup wn\_\alpha$, where $n\_\alpha=\bigg(\matrix{1&\alpha\cr 0&1}\bigg)$ and $\alpha$ generates $k\_1$ over $k$. 




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So why do I care?  

One of the primary sources of integral representations of automorphic $L$-functions is integrating the restriction of an Eisenstein series on a group $G$, associated to a parabolic subgroup $P$, against a cusp form on a subgroup $H$, i.e. 
$$Z(s,f)=\int\_{Z\_{\mathbb A} H\_k\backslash H\_{\mathbb A} } E\_s(h)f(h)\ dh$$
In order for this to converge, we need $Z\backslash Z\_H$ to be anisotropic (so, e.g., Example 1 won't work without modification). Unwinding the Eisenstein series, we have 
$$Z(s,f)=\sum\_{\xi\in P\_k\backslash G\_k/H\_k}\int\_{Z\_{\mathbb A} \Theta^\xi\_k\backslash H\_{\mathbb A} }\varepsilon\_s(\xi h)f(h)\ dh$$
where $\Theta^\xi=\xi^{-1}P\xi\cap H$. Let's consider a fixed term in the sum. 
$$\int\_{Z\_{\mathbb A} \Theta^\xi\_k\backslash H\_{\mathbb A} }\varepsilon(\xi h)f(h)\ dh=\int\_{Z\_{\mathbb A} \Theta^\xi\_{\mathbb A} \backslash H\_{\mathbb A} }\int\_{Z\_{\mathbb A} \Theta^\xi\_k\backslash\Theta^\xi\_{\mathbb A} }\varepsilon\_s(\xi\theta h)f(\theta h)\delta\_{\Theta\_{\mathbb A} }(\theta)^{-1}\ d\theta\ dh$$
$$=\int\_{Z\_{\mathbb A} \Theta^\xi\_{\mathbb A} \backslash H\_{\mathbb A} }\int\_{Z\_{\mathbb A} \Theta^\xi\_k\backslash\Theta^\xi\_{\mathbb A} }\varepsilon\_s(\xi\theta\xi^{-1}\xi h)f(\theta h)\delta\_{\Theta\_{\mathbb A} }(\theta)^{-1}\ d\theta\ dh$$
$$=\int\_{Z\_{\mathbb A} \Theta^\xi\_{\mathbb A} \backslash H\_{\mathbb A} }\varepsilon\_s(\xi h)\int\_{Z\_{\mathbb A} \Theta^\xi\_k\backslash\Theta^\xi\_{\mathbb A} }\varepsilon\_s(\xi\theta\xi^{-1})f(\theta h)\delta\_{\Theta\_{\mathbb A} }(\theta)^{-1}\ d\theta\ dh$$
Note that if $\varepsilon\_s(\xi\theta\xi^{-1})$ is trivial on $\Theta^\xi\_{\mathbb A} $ and $\Theta^\xi$ has a normal subgroup that is a unipotent radical of a parabolic subgroup of $H$, the term vanishes by the cuspidality of $f$. 


Under certain conditions, the inner integral (and hence the entire integral) factors into a product of local integrals, which we can hopefully compute to be local factors of an automorphic $L$-function. But we must pass over this in silence. (See [Garrett's](http://math.umn.edu/~garrett/m/v/) Euler Factorization of Global Integrals for more.) 




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In order to make further progress, we would need to understand the nature of $\Theta^\xi$, as well. Let's calculate it for the above examples.


Example 1: $\Theta^1=H$, $\Theta^w=H$, $\Theta^{wn}=Z$.


Example 2: $\Theta^1=Z$.


Example 3: $\Theta^1=B$, $\Theta^w=w^{-1}Bw$ (the "opposite" Borel subgroup).


Example 4: $\Theta^1$ is the Borel subgroup of $H$, $\Theta^{wn\_\alpha}=k\_1^\times$.


Another very interesting example is the set-up for the triple product $L$-function, found in either section 3.10 of Bump's book or in [PS-Rallis](http://www.numdam.org/numdam-bin/item?id=CM_1987__64_1_31_0), Rankin triple L functions.


It would be really nice to know, given $G$, $P$, and $H$, what automorphic $L$-function (if any, since most such triples won't produce anything interesting) is represented by the above integral. Hypothetically, we could make a list of the $L$-functions, like there is for the Langlands-Shahidi method. Understanding the double coset decomposition is the first step. 


This is probably way too much to hope for; I'm interested in understanding why it is so hard.
It seems that, over $\mathbb C$, we know when the double coset is finite (at least when $H\_{\mathbb C}$ is an open subgroup of the fixed points of an involution of $G\_{\mathbb C}$). I can't find a good statement of what is known over $\mathbb Q$. Is anything known if $G$ and $H$ are split? Or if $G$ is split and $H$ is anisotropic? Or if $G=GL\_n$? Etc . . . 



Double coset spaces of reductive groups and integral representations of L-functions
This is a very nice question! 


All of your examples are spherical quotients. Please take a look at 
<http://andromeda.rutgers.edu/~sakellar/rs.pdf> and Yiannis' other papers 
for connections between spherical quotients and integral representations for L functions. 


