Double coset spaces of reductive groups and integral representations of L-functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:12:12Z http://mathoverflow.net/feeds/question/60039 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60039/double-coset-spaces-of-reductive-groups-and-integral-representations-of-l-functio Double coset spaces of reductive groups and integral representations of L-functions BR 2011-03-30T06:24:42Z 2011-03-30T14:59:11Z <p>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.</p> <p>Let's look at some simple examples. </p> <p>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 $G$. </p> <p>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 $P_k\backslash G_k/H_k=1$. </p> <p>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 $P_k\backslash G_k/H_k=B_k\backslash H_k/B_k=1\cup w$.</p> <p>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 $n_\alpha=\bigg(\matrix{1&amp;\alpha\cr 0&amp;1}\bigg)$ and $\alpha$ generates $k_1$ over $k$. </p> <hr> <p>So why do I care?<br> 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 $f$. </p> <p>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 <a href="http://math.umn.edu/~garrett/m/v/" rel="nofollow">Garrett's</a> Euler Factorization of Global Integrals for more.) </p> <hr> <p>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.</p> <p>Example 1: $\Theta^1=H$, $\Theta^w=H$, $\Theta^{wn}=Z$.</p> <p>Example 2: $\Theta^1=Z$.</p> <p>Example 3: $\Theta^1=B$, $\Theta^w=w^{-1}Bw$ (the "opposite" Borel subgroup).</p> <p>Example 4: $\Theta^1$ is the Borel subgroup of $H$, $\Theta^{wn_\alpha}=k_1^\times$.</p> <p>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 <a href="http://www.numdam.org/numdam-bin/item?id=CM_1987__64_1_31_0" rel="nofollow">PS-Rallis</a>, Rankin triple L functions.</p> <p>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. </p> <p>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 . . . </p> http://mathoverflow.net/questions/60039/double-coset-spaces-of-reductive-groups-and-integral-representations-of-l-functio/60077#60077 Answer by Ramin for Double coset spaces of reductive groups and integral representations of L-functions Ramin 2011-03-30T14:59:11Z 2011-03-30T14:59:11Z <p>This is a very nice question! </p> <p>All of your examples are spherical quotients. Please take a look at <a href="http://andromeda.rutgers.edu/~sakellar/rs.pdf" rel="nofollow">http://andromeda.rutgers.edu/~sakellar/rs.pdf</a> and Yiannis' other papers for connections between spherical quotients and integral representations for L functions. </p>