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 $k$.
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 Euler Factorization of Global Integrals for more.)
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, 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 . . .
