I am familiar with $L$-functions for $\operatorname{GL}_2\times \operatorname{GL}_1$ (e.g. Bump 'Automorphic Forms and Representations'), and currently reading on $\operatorname{GL}_n\times \operatorname{GL}_m$ $L$-functions. I am following Cogdell's articles in "Lectures on Automorphic $L$-functions" and also "$L$-functions and Converse Theorems for $GL_n$". Assume $m<n$. We have the operator $$\Bbb P\varphi(p) = |\det p|^{-\frac{n-m-1}2}\int_{Y(k)\setminus Y(\Bbb A)}\varphi\left(y\begin{pmatrix} p&0\\0&I_{n-m-1}\end{pmatrix}\right)\psi^{-1}(y)\mathrm dy$$ Here $\varphi$ is a cusp form on $\operatorname{GL}_n$, $Y$ is the unipotent of the parabolic subgroup of $\operatorname{GL}_n$ radical corresponding to the partition $(m+1,1,\dots,1)$, $\psi$ a character of $N_n(\Bbb A)$, and $p$ is in the mirabolic subgroup of $\operatorname{GL}_{m+1}$. Then for $\varphi\in V_\pi,\varphi'\in V_{\pi'}$ where $\pi,\pi'$ are cuspidal representations of $\operatorname{GL}_n,\operatorname{GL}_m$ resp., we introduce the global zeta integral $$I(s,\varphi,\varphi')=\int_{\operatorname{GL}_m(k)\backslash \operatorname{GL}_m(\Bbb A)}\Bbb P\varphi\begin{pmatrix}h&0\\0&1\end{pmatrix}\varphi'(h)|\det h|^{s-\frac12}\mathrm dh.$$
Then it is proven that $$I(s,\varphi,\varphi')=\Psi(s,W_\varphi, W'_{\varphi'}):=\int_{N_m(\Bbb A)\backslash \operatorname{GL}_m(\Bbb A)}W_\varphi\begin{pmatrix}h&0\\0&I_{n-m}\end{pmatrix}W'_{\varphi'}(h)|\det h|^{s-\frac{n-m}2}\mathrm dh,$$ which is good because this is Eulerian if $\varphi,\varphi'$ are decomposable in $\pi,\pi'$. Here $W,W'$ are the Whittaker functions corresponding to $\psi,\psi^{-1}$.
Consider the involution $g^\iota={}^tg^{-1}$. Then it follows easily that we have the functional equation $I(s,\varphi,\varphi')=\widetilde I(1-s, \widetilde \varphi,\widetilde\varphi')$ where $\widetilde\varphi(g)=\varphi(g^\iota),\widetilde\varphi'(g)=\varphi'(g^\iota)$, and $$\widetilde I(s,\varphi,\varphi')=\int_{\operatorname{GL}_m(k)\backslash \operatorname{GL}_m(\Bbb A)}\widetilde{\Bbb P}\varphi\begin{pmatrix}h&0\\0&1\end{pmatrix}\varphi'(h)|\det h|^{s-\frac12}\mathrm dh$$ with $\widetilde{\Bbb P}=\iota\circ\Bbb P\circ\iota$.
Now it is claimed that the Eulerian integral representation for $\widetilde I$ is a little different, namely $$\widetilde I(1-s, \widetilde \varphi,\widetilde\varphi') = \widetilde \Psi(1-s;\rho(w_{n,m})\widetilde W_\varphi, \widetilde W'_{\varphi'}),$$ where $$\widetilde \Psi(s;W,W') = \int_{N_m(\Bbb A)\backslash \operatorname{GL}_m(\Bbb A)}\int_{M_{n-m-1,m}(\Bbb A)}W\begin{pmatrix}h&0&0\\x&I_{n-m-1}&0\\0&0&1\end{pmatrix}\mathrm dx\,W'(h)|\det h|^{s-(n-m)/2}\mathrm dh,$$ and $w_{n,m}=\begin{pmatrix}I_m&0\\0&w_{n-m}\end{pmatrix}$ with $w_{n-m}$ being the standard long Weyl element in $\operatorname{GL}_{n-m}$. Also $\widetilde W(g)=W(w_ng^\iota)$. It is completely unclear to me where this expression comes from. He just says that the inner integral is a ''remnant of $\widetilde{\Bbb P}$''. I don't see why $\widetilde{\Bbb P}$ should change that much, it seems to me that the same argument that showed $I(s,\varphi,\varphi')=\Psi(s,W_\varphi, W'_{\varphi'})$ before now shows that \begin{align*} \widetilde I(1-s, \widetilde \varphi,\widetilde\varphi') &= \int_{N_m(\Bbb A)\backslash \operatorname{GL}_m(\Bbb A)}W_{\varphi}\begin{pmatrix}h^\iota&0\\0&I_{n-m}\end{pmatrix}W'_{\widetilde\varphi'}(h)|\det h|^{1-s-\frac{n-m}2}\mathrm dh\\ &=\Psi(1-s, W_{\varphi}\circ\iota, W'_{\widetilde\varphi'}) \end{align*} is Eulerian. What is the need for this different $\widetilde\Psi$?
