Let \((\pi, V_\pi)\) and \((\pi^\prime, V_{\pi^\prime})\) be cuspidal, unitary, irreducible automorphic representations of \(GL_n(\mathbb{A})\) and \(GL_m(\mathbb{A})\) respectively. We assume \(m < n\). Let \(\varphi \in V_\pi\) and \(\varphi \in V_{\pi^\prime}\). To pair \(\varphi\) and \(\varphi^\prime\) suitably together, we first need to project \(\varphi\) correspondingly.

Let \(\psi\) be a additive continuous automorphic character of \(\mathbb{A}\). We can extend it to a character of \(N_n(\mathbb{A})\), the standard Borel subgroup of \(GL_n(\mathbb{A})\), in the standard way: \[\psi(u) = \psi\left(\sum_{i=1}^{n-1}u_{i,i+1}\right),\] for \(u = (u_{i,j}) \in N_n(\mathbb{A})\). Let \(Y=Y_{n, m}\) be the standard unipotent radical associated to the partition \((m+1, 1, \dots, 1)\) of \(GL_n\), i.e., \[Y=\left\{\left(\begin{array}{cc}I_{m+1}&*\\ 0&u\end{array}\right): u \in N_{n-m-1}\right\} \subset N_n.\] Let \(P_{m+1}\) be the mirabolic subgroup of \(GL_{m+1}\), then for \(p \in P_{m+1}(\mathbb{A})\) define \[\mathbb{P}^n_m \varphi(p) = |\det p|^{-\frac{n-m-1}{2}}\int_{Y(k)\backslash Y(\mathbb{A})} \varphi\left(y\left(\begin{array}{cc} p& \\ & I_{n-m-1} \end{array}\right)\right)\psi^{-1}(y) dy.\]

Now we can pair \(\varphi\) and \(\varphi^{\prime}\) in the following way: \[I(s; \varphi, \varphi^\prime) = \int_{GL_m(k)\backslash GL_m(\mathbb{A})} \mathbb{P}^n_m \varphi\left(\begin{array}{cc} h& \\ &1\end{array}\right)\varphi^\prime(h)|\det h|^{s-\frac{1}{2}}dh.\] The above integral converges absolutely for \(s \in \C\). For any matrix \(x\), let \(x^\iota\) be the transpose inverse of \(x\). If we do the change of variables \(h \mapsto h^\iota\), then we can get the global functional equation \[\begin{equation} I(s; \varphi, \varphi^\prime) = \widetilde{I}(1-s; \tilde{\varphi}, \tilde{\varphi^\prime}), \label{20150712-1} \end{equation}\]

where \(\tilde{\varphi}(g) = \varphi(g^\iota)\) and \[\widetilde{I}(s; \varphi, \varphi^\prime) = \int_{GL_m(k)\backslash GL_m(\mathbb{A})} \widetilde{\mathbb{P}}^n_m \varphi\left(\begin{array}{cc} h& \\ &1\end{array}\right)\varphi^\prime(h)|\det h|^{s-\frac{1}{2}}dh,\] with \(\tilde{\mathbb{P}}^n_m = \iota \circ \mathbb{P}^n_m \circ \iota\).

The unfolding of the left hand side of \(\eqref{20150712-1}\) can be found in James Cogdell’s note on \(L\)-functions and Converse Theorem for \(GL_n\), with additional details to what have been told above. This post will do the unfolding of the right hand side of \(\eqref{20150712-1}\). I learned the unfolding from James Cogdell.

Notations: I will write \(G_n\) for \(GL_n\) and denote \(G(k) \backslash G(\mathbb{A})\) as \(\underline{G}\) for various \(G\). Inside a matrix, I will use “\([\;]\)” to group blocks together to form one single block. Let \(M=M_{n-m-1,m}\) denote the group of all \((n-m-1) \times m\) matrices. Dimensions of zero vectors will be omitted. Also, let \[\begin{equation*} w_n = \left(\begin{array}{ccc} &&1 \\ &\unicode{x22F0}& \\ 1&&\end{array}\right),\;\; w_{n, m} = \left(\begin{array}{cc} I_m& \\ &w_{n-m}\end{array}\right) \end{equation*}\] Let \(\rho\) be the regular action. Let \(W_{\varphi} \in \mathcal{W}(\pi, \psi^{-1})\) be the Fourier expansion of \(\varphi\) with respect to \(\psi^{-1}\), i.e., \[W_\varphi(g) = \int_{\underline{N_n}} \varphi(ug)\psi(u)du.\] Similarly, one can define \(W_{\varphi^\prime} \in \mathcal{W}(\pi^\prime, \psi)\).

Using the above notations, the main theorem of the post is the following unfolding.

Theorem 1.\(\displaystyle \widetilde{I}(s; \varphi, \varphi^\prime) = \widetilde{\Psi}(s; \rho(w_{n, m})W_{\varphi}, W_{\varphi^\prime})\), where \[\widetilde{\Psi}(s; W, W^\prime) = \int_{N_m(\mathbb{A})\backslash G_m(\mathbb{A})} \int_{M(\mathbb{A})}W\left(\begin{array}{cc} h & \\ \left[\begin{array}{cc}x\\0\end{array}\right]&I_{n-m}\end{array}\right) dx W^\prime(h) |\det h|^{s-\frac{n-m}{2}}dh.\]

In order to obtain the above unfolding, we need to understand \(\widetilde{\mathbb{P}}^n_m \varphi\left(\begin{array}{cc} h& \\ &1\end{array}\right)\) first. To this end, we have

Theorem 2.\[\widetilde{\mathbb{P}}^n_m \varphi\left(\begin{array}{cc} h& \\ &1\end{array}\right) = |\det h|^{-\frac{n-m-1}{2}}\sum_{\gamma\in N_m(k)\backslash G_m(k)}\int_{M(\mathbb{A})}\rho(w_{n,m})W_\varphi\left(\begin{array}{cc} \gamma h & \\ \left[\begin{array}{cc}x\\0\end{array}\right]&I_{n-m}\end{array}\right) dx.\]

Proof. By definition of \(\widetilde{\mathbb{P}}^n_m\), \[\widetilde{\mathbb{P}}^n_m \varphi\left(\begin{array}{cc} h& \\ &1\end{array}\right) = |\det h|^{\frac{n-m-1}{2}}\int_{\underline{Y}}\varphi\left(y^\iota \left(\begin{array}{cc} h& \\ &I_{n-m} \end{array}\right) \right)\psi^{-1}(y)dy.\] For \(y \in \underline{Y}\), we have \[y=\left(\begin{array}{cc} I_m&[0 \; x] \\ &y_1\end{array}\right),\] where \(x \in {^t\underline{M}}\) and \(y_1 \in \underline{N_{n-m}}\). Then \[y^\iota=\left(\begin{array}{cc} I_m& \\ -y_1^\iota\left[\begin{array}{cc}0\\{^tx}\end{array}\right] &y_1^\iota \end{array}\right).\] If we replace the integral over \(\underline{Y}\) by the double integrals over \(\underline{N_{n-m}}\) and \(\underline{^tM}\), then with the change of variables \[x \mapsto -{y_1^\prime} \cdot {^tx}, \text{where } {^ty_1} = \left(\begin{array}{c} [1 \; 0] \\ y_1^\prime\end{array}\right),\] one can get \[\begin{equation} \begin{split} \widetilde{\mathbb{P}}^n_m \varphi & \left(\begin{array}{cc} h& \\ &1\end{array}\right) = |\det h|^{\frac{n-m-1}{2}} \cdot\\ &\int_{\underline{N_{n-m}}}\int_{\underline{M}} \varphi\left(\left(\begin{array}{cc} I_m& \\ \left[\begin{array}{cc}0\\x\end{array}\right]&y^\iota\end{array}\right)\left(\begin{array}{cc} h& \\ &I_{n-m}\end{array}\right)\right)\psi^{-1}(y)dxdy. \end{split} \label{20150712-2} \end{equation}\]

Since \(\varphi\) is automorphic, the argument inside \(\varphi\) in \(\eqref{20150712-2}\) can be replaced by \[\begin{equation*} \begin{split} &w_{n,m}\left(\begin{array}{cc} I_m& \\ \left[\begin{array}{cc}0\\x\end{array}\right]&y^\iota\end{array}\right)\left(\begin{array}{cc} h& \\ &I_{n-m}\end{array}\right) \\ &= \left(\begin{array}{cc} I_m& \\ \left[\begin{array}{cc}w_{n-m-1}x\\0\end{array}\right]&w_{n-m}y^\iota w_{n-m}\end{array}\right)\left(\begin{array}{cc} h& \\ &I_{n-m}\end{array}\right)w_{n,m} \end{split} \end{equation*}\] Therefore, from \(\eqref{20150712-2}\) together with the change of variables \[x \mapsto w_{n-m-1}x, \;\; y \mapsto w_{n-m}y^\iota w_{n-m},\] and noticing that \(\psi^{-1}(w_{n-m}y^\iota w_{n-m}) = \psi(y)\), it can be deduced that \[\begin{equation*} \begin{split} \widetilde{\mathbb{P}}^n_m \varphi & \left(\begin{array}{cc} h& \\ &1\end{array}\right) = |\det h|^{\frac{n-m-1}{2}} \cdot\\ &\int_{\underline{N_{n-m}}}\int_{\underline{M}} \varphi\left(\left(\begin{array}{cc} I_m& \\ \left[\begin{array}{cc}x\\0\end{array}\right]&y\end{array}\right)\left(\begin{array}{cc} h& \\ &I_{n-m}\end{array}\right)w_{n,m}\right)\psi(y)dxdy. \end{split} \end{equation*}\] Replacing \(\varphi\) by its Fourier expansion \[\varphi(g) = \sum_{\alpha \in N_{n-1}(k) \backslash G_{n-1}(k)}W_{\varphi}\left(\left(\begin{array}{cc} \alpha& \\ &1\end{array}\right)g\right),\] and switching orders, we can get \[\begin{equation*} \begin{split} \widetilde{\mathbb{P}}^n_m \varphi & \left(\begin{array}{cc} h& \\ &1\end{array}\right) = |\det h|^{\frac{n-m-1}{2}} \int_{\underline{N_{n-m}}}\sum_{\alpha \in N_{n-1}(k) \backslash G_{n-1}(k)} \int_{\underline{M}}\\ &W_{\varphi}\left(\left(\begin{array}{cc} \alpha& \\ &1\end{array}\right)\left(\begin{array}{cc} I_m& \\ \left[\begin{array}{cc}x\\0\end{array}\right]&y\end{array}\right)\left(\begin{array}{cc} h& \\ &I_{n-m}\end{array}\right)w_{n,m}\right)\psi(y)dxdy. \end{split} \end{equation*}\] Now, \[\left(\begin{array}{cc} I_m& \\ \left[\begin{array}{cc}x\\0\end{array}\right]&y\end{array}\right) = \left(\begin{array}{cc} I_m& \\ &y\end{array}\right)\left(\begin{array}{cc} I_m& \\ y^{-1}\left[\begin{array}{cc}x\\0\end{array}\right]&I_{n-m}\end{array}\right),\] so if we do the change of variables \[x \mapsto y^\prime x, \text{ where } y = \left(\begin{array}{cc} y^\prime & * \\ & 1 \end{array}\right),\] and change orders, we can see \[\begin{equation} \widetilde{\mathbb{P}}^n_m \varphi \left(\begin{array}{cc} h& \\ &1\end{array}\right) = |\det h|^{\frac{n-m-1}{2}} \int_{\underline{M}} \sum_{\alpha \in N_{n-1}(k) \backslash G_{n-1}(k)} A(\alpha; h, x)dx, \label{20150712-3} \end{equation}\]

where \[\begin{equation*} \begin{split} A(\alpha; h, x) &= \int_{\underline{N_{n-m}}} W_{\varphi}\left(\left(\begin{array}{cc} \alpha& \\ &1\end{array}\right)\left(\begin{array}{cc} I_m& \\ &y\end{array}\right) \right.\cdot \\ &\left.\left(\begin{array}{cc} I_m& \\ \left[\begin{array}{cc}x\\0\end{array}\right]&I_{n-m}\end{array}\right)\left(\begin{array}{cc} h& \\ &I_{n-m}\end{array}\right)w_{n,m}\right)\psi(y)dy. \end{split} \end{equation*}\] For \(y \in \underline{N_{n-m}}\), let’s write \[y=[y_1 \, \dots \, y_{n-m}],\] with each \(y_i\) being column vector such that the \(i\)-th entry is \(1\) and \(j\)-th entry is \(0\) if \(j > i\). With this notation, \[\left(\begin{array}{cc} I_m& \\ &y\end{array}\right)=\left(\begin{array}{cc} I_m& \\ &[e_1 \, \dots \, e_{n-m-1} \, y_{n-m}]\end{array}\right) \cdots \left(\begin{array}{cc} I_m& \\ &[y_1 \, e_2 \, \dots \, e_{n-m}]\end{array}\right),\] where entries of the column vector \(e_i\) are all zeros except being \(1\) at \(i\)-th entry. Therefore, we can rewrite the integral over \(y\) in \(A(h; \alpha, x)\) into multiple integrals over \(y_{n-m}, \dots, y_2\) (Note that \(y_1\) is just constant \(e_1\)): \[\begin{equation*} \begin{split} A(\alpha; &h, x) = \int_{\underline{N_{n-m}}} W_{\varphi}\left(\left(\begin{array}{cc} \alpha& \\ &1\end{array}\right)\left(\begin{array}{cc} I_m& \\ &[e_1 \, \dots \, e_{n-m-1} \, y_{n-m}]\end{array}\right) \cdots \right. \\ &\left(\begin{array}{cc} I_m& \\ &[y_1 \, y_2\, e_3 \, \dots \, e_{n-m}]\end{array}\right) \left. \left(\begin{array}{cc} I_m& \\ \left[\begin{array}{cc}x\\0\end{array}\right]&I_{n-m}\end{array}\right)\left(\begin{array}{cc} h& \\ &I_{n-m}\end{array}\right)w_{n,m}\right)\\ &\prod_{i=2}^{n-m}\psi(y_{i,i-1})dy_{n-m} \cdots dy_{2}. \end{split} \end{equation*}\] If we write \(\alpha = (\alpha_{i, j})\) and \(y_{n-m} = (y_{n-m, i}) = \left(\begin{array}{cc}y^\prime_{n-m} \\ 1\end{array}\right)\), then \[\begin{equation*} \begin{split} & \left(\begin{array}{cc} \alpha& \\ &1\end{array}\right)\left(\begin{array}{cc} I_m& \\ &[e_1 \, \dots \, e_{n-m-1} \, y_{n-m}]\end{array}\right) \\ &= \left(\begin{array}{cc} I_{n-m-1}&\alpha \left[\begin{array}{cc}0\\y^\prime_{n-m}\end{array}\right] \\ &1\end{array}\right)\left(\begin{array}{cc} \alpha& \\ &1\end{array}\right). \end{split} \end{equation*}\] For convenience, let \(g\) be the product of matrices in the arguments of \(W_\varphi\) not relating to \(\alpha\) and \(y_{n-m}\). Then if we focus on the integral over \(y_{n-m}\), and use the fact that \(W_\varphi(ug) = \psi^{-1}(u)W_\varphi(g)\) for \(u \in N_n(\mathbb{A})\), we can get \[\begin{equation*} \begin{split} A(\alpha; & h, x) = \int \int W_\varphi\left(\left(\begin{array}{cc} \alpha& \\ &1\end{array}\right)g\right) \\ & \psi(-\alpha_{n-1,m+1}y_{n-m,1}-\cdots-(\alpha_{n-1, n-1}-1)y_{n-m,n-m-1}) dy_{n-m} \\ & \prod_{i=2}^{n-m-1}\psi(y_{i, i-1}) dy_{n-m-1} \dots dy_2 \end{split} \end{equation*}\] The inner integral is zero, expect when \(\alpha_{n-1, m+1} = \cdots = \alpha_{n-1, n-2} = 0\) and \(\alpha_{n-1, n-1} = 1\), in which case the inner integral equals \[W_{\varphi}\left(\left(\begin{array}{cc} \alpha& \\ &1\end{array}\right)g\right).\] In this case, \(\alpha\) can be written as \[\left(\begin{array}{cc} *&* \\ {[* \;0]}&1 \end{array}\right) = \left(\begin{array}{cc} I_{n-2}&* \\ &1 \end{array}\right) \left(\begin{array}{cc} *&0 \\ {[* \;0]}&1 \end{array}\right).\] Since \(\alpha \in N_{n-1}(k) \backslash G_{n-1}(k)\), we can choose \(\alpha\) to be in the form \[\left(\begin{array}{cc} *&0 \\ {[* \;0]}&1 \end{array}\right).\] Now repeating the above process again for \(y_{n-m-1}, \dots, y_2\) in order, we can conclude that \(A(\alpha; h, x) = 0\) unless \(\alpha\) has the form \[\alpha = \left(\begin{array}{cc} \gamma&0 \\ \beta&I_{n-m-1} \end{array}\right),\] where \(\gamma \in N_m(k) \backslash G_m(k)\) and \(\beta \in M(k)\). When \(\alpha\) is in this form, \(A(\alpha; h, x)\) equals \[W_{\varphi}\left(\left(\begin{array}{cc} \gamma& \\ \left[\begin{array}{cc}\beta\\0\end{array}\right]&I_{n-m}\end{array}\right)\left(\begin{array}{cc} I_m& \\ \left[\begin{array}{cc}x\\0\end{array}\right]&I_{n-m}\end{array}\right)\left(\begin{array}{cc} h& \\ &I_{n-m}\end{array}\right)w_{n,m}\right),\] which is the same as \[W_\varphi\left(\left(\begin{array}{cc} \gamma h & \\ \left[\begin{array}{cc}(\beta+x)\\0\end{array}\right]h&I_{n-m}\end{array}\right)w_{n,m}\right).\] Gathering the analysis of \(A(\alpha; h, x)\) into \(\eqref{20150712-3}\), \[\begin{equation*} \begin{split} \widetilde{\mathbb{P}}^n_m \varphi & \left(\begin{array}{cc} h& \\ &1\end{array}\right) = |\det h|^{\frac{n-m-1}{2}} \int_{\underline{M}} \sum_{\beta \in M(k)}\sum_{\gamma \in N_{m}(k) \backslash G_{m}(k)} \\ & W_\varphi\left(\left(\begin{array}{cc} \gamma h & \\ \left[\begin{array}{cc}(\beta+x)\\0\end{array}\right]h&I_{n-m}\end{array}\right)w_{n,m}\right)dx. \end{split} \end{equation*}\] Combining $ \int_{\underline{M}}$ and \(\sum_{\beta \in M(k)}\) and then switching orders, \[\begin{equation*} \begin{split} \widetilde{\mathbb{P}}^n_m \varphi & \left(\begin{array}{cc} h& \\ &1\end{array}\right) = |\det h|^{\frac{n-m-1}{2}} \sum_{\gamma \in N_{m}(k) \backslash G_{m}(k)} \int_{M(\mathbb{A})}\\ & W_\varphi\left(\left(\begin{array}{cc} \gamma h & \\ \left[\begin{array}{cc}xh\\0\end{array}\right]&I_{n-m}\end{array}\right)w_{n,m}\right)dx. \end{split} \end{equation*}\] Let’s now do the change of variable \(x \mapsto xh^{-1}\) with \[d(xh^{-1}) = |\det h|^{-(n-m-1)}dx,\] then \[\begin{equation*} \begin{split} \widetilde{\mathbb{P}}^n_m \varphi & \left(\begin{array}{cc} h& \\ &1\end{array}\right) = |\det h|^{-\frac{n-m-1}{2}} \sum_{\gamma \in N_{m}(k) \backslash G_{m}(k)} \int_{M(\mathbb{A})}\\ & W_\varphi\left(\left(\begin{array}{cc} \gamma h & \\ \left[\begin{array}{cc}x\\0\end{array}\right]&I_{n-m}\end{array}\right)w_{n,m}\right)dx. \end{split} \end{equation*}\] This finishes the proof of Theorem 2. \(\square\)

With Theorem 2 at hand, now it’s easy to prove Theorem 1.

Proof of Theorem 1. By Theorem 2, \[\begin{equation*} \begin{split} \widetilde{I}(s; &\varphi, \varphi^\prime) = \int_{\underline{G_m}} \widetilde{\mathbb{P}}^n_m \varphi\left(\begin{array}{cc} h& \\ &1\end{array}\right)\varphi^\prime(h)|\det h|^{s-\frac{1}{2}}dh \\ &=\int_{\underline{G_m}}|\det h|^{s-\frac{n-m}{2}} \sum_{\gamma \in N_{m}(k) \backslash G_{m}(k)} \int_{M(\mathbb{A})} \\ & \;\;\;\;\;\;\;\; \rho(w_{n,m})W_\varphi\left(\begin{array}{cc} \gamma h & \\ \left[\begin{array}{cc}x\\0\end{array}\right]&I_{n-m}\end{array}\right)dx\varphi^\prime(h)dh. \end{split} \end{equation*}\] Since \(\varphi^\prime\) is automorphic and \(|\det \gamma^{-1}|=1\), if we do the change of variables \(h \mapsto \gamma^{-1}h\), then \[\begin{equation*} \begin{split} \widetilde{I}(s; \varphi, \varphi^\prime) = \int_{N_m(k)\backslash G_m(\mathbb{A})} & |\det h|^{s-\frac{n-m}{2}} \int_{M(\mathbb{A})} \\ & \rho(w_{n,m})W_\varphi\left(\begin{array}{cc} h & \\ \left[\begin{array}{cc}x\\0\end{array}\right]&I_{n-m}\end{array}\right)dx\varphi^\prime(h)dh. \end{split} \end{equation*}\] Noticing that \(|\det u| = 1\) for \(u \in \underline{N_m}\), we can rewrite the integral over \(N_m(k)\backslash G_m(\mathbb{A})\) into double integrals over \(N_m(\mathbb{A})\backslash G_m(\mathbb{A})\) and \(\underline{N_m}\) as follow. \[\begin{equation} \begin{split} \widetilde{I}(s; \varphi, \varphi^\prime) = \int_{N_m(\mathbb{A})\backslash G_m(\mathbb{A})} & |\det h|^{s-\frac{n-m}{2}} \int_{\underline{N_m}} \int_{M(\mathbb{A})} \\ & \rho(w_{n,m})W_\varphi\left(\begin{array}{cc} uh & \\ \left[\begin{array}{cc}x\\0\end{array}\right]&I_{n-m}\end{array}\right)dx\varphi^\prime(uh)dh. \end{split} \label{20150712-4} \end{equation}\]

Since \[\left(\begin{array}{cc} uh & \\ \left[\begin{array}{cc}x\\0\end{array}\right]&I_{n-m}\end{array}\right) = \left(\begin{array}{cc} u & \\ &I_{n-m}\end{array}\right)\left(\begin{array}{cc} h & \\ \left[\begin{array}{cc}x\\0\end{array}\right]&I_{n-m}\end{array}\right),\] so \[W_\varphi\left(\begin{array}{cc} uh & \\ \left[\begin{array}{cc}x\\0\end{array}\right]&I_{n-m}\end{array}\right) = \psi^{-1}(u)W_\varphi\left(\begin{array}{cc} h & \\ \left[\begin{array}{cc}x\\0\end{array}\right]&I_{n-m}\end{array}\right).\] Now if we separate the integral over \(u\) in \(\eqref{20150712-4}\), we get \[\int_{\underline{N_m}}\varphi^\prime(uh)\psi^{-1}(u)du = W_{\varphi^\prime}(h).\] Hence, from \(\eqref{20150712-4}\), we obtain \[\widetilde{I}(s; \varphi, \varphi^\prime) = \widetilde{\Psi}(s; \rho(w_{n, m})W_{\varphi}, W_{\varphi^\prime}),\] as desired. \(\square\)