Why are Rankin-Selberg convolutions different when $n=m$? Let $\pi$ an automorphic cuspidal representation on $GL_n(\mathbb{A})$ (and similarly $\pi'$ on $GL_m$). For $m=n-1$, it is standard to introduce the Rankin-Selberg L-function as (the gcd of the integrals)
$$I(s,\phi, \phi') = \int_{[GL_m]} \phi \begin{pmatrix} h & \\ & 1 \end{pmatrix} \phi'(h) |\det h|^{s-1/2} dh$$
where $\phi$ is here understood as a certain, explicit, projection of $\phi$ on $GL_{m+1}$. In particular, we recover the classical L-function for $GL_2$ in the case $ m+1 = n =2$, where classically
$$L(s,f) = \int_0^\infty f(iy) y^s \frac{dy}{y}.$$
I would like to understand what breaks when $n=m$, why we don't consider
$$\int_{[GL_n]} \phi(h) \phi'(h) |\det h|^{s-1/2} dh$$
but rather a twist by a certain Eisenstein series
$$\int_{[GL_n]} \phi(g) \phi'(g) E(g,s, \Phi) dg$$
for a certain Schwartz function $\Phi$. Is it due to converge issue or is it there a deeper reason to twist this case compared to the previous one? And how natural is it to do so with an Eisenstein series?
 A: When dealing with Rankin-Selberg integrals involving integrals of automorphic forms, we want these integrals to be Eulerian. That is, such an integral should factorise as a product over all places $v$ of the underlying number field $F$. This factorisation should involve integrals of the local components of this automorphic form, and each of these local integrals should represent the local component of the associated Rankin-Selberg $L$-function.

Let us see how this works for the $\mathrm{GL}_n \times \mathrm{GL}_n$ Rankin-Selberg integral. The integral of interest is
$$\int\limits_{\mathrm{Z}_n(\mathbb{A}) \mathrm{GL}_n(F) \backslash \mathrm{GL}_n(\mathbb{A})} \varphi_{\pi}(g) \varphi_{\sigma}(g) E(g,s;\Phi,\omega_{\pi}\omega_{\sigma}) \, dg.$$
Here $\mathrm{Z}_n$ denotes the centre of $\mathrm{GL}_n$, $\varphi_{\pi}$ and $\varphi_{\sigma}$ are automorphic forms lying in automorphic representations $\pi$ and $\sigma$ of $\mathrm{GL}_n(\mathbb{A})$ with central characters $\omega_{\pi}$ and $\omega_{\sigma}$, and $E(g,s;\Phi,\omega)$ is the Eisenstein series on $\mathrm{GL}_n(\mathbb{A})$ associated to a Schwartz function $\Phi \in \mathscr{S}(\mathbb{A}^n)$ and a character $\omega$.
Note that as stated, the integral cannot immediately be written as a product of local integrals, since it is not over the adelic points $G(\mathbb{A})$ of some group $G$ due to the fact that we are modding out on the left by the discrete subgroup $\mathrm{GL}_n(F)$.
Nonetheless, by unfolding the integral via the fact that Eisenstein series are averages over $\mathrm{GL}_n(F)$, we find that this integral is equal to
$$\int\limits_{\mathrm{N}_n(\mathbb{A}) \backslash \mathrm{GL}_n(\mathbb{A})} W_{\varphi_{\pi}}(g) W_{\varphi_{\sigma}}(g) \Phi(e_n g) \left|\det g\right|_{\mathbb{A}}^s \, dg.$$
Here $\mathrm{N}_n$ is the upper triangular unipotent subgroup of $\mathrm{GL}_n$, while
$$W_{\varphi}(g) = \int\limits_{\mathrm{N}_n(F) \backslash \mathrm{N}_n(\mathbb{A})} \varphi(ug) \overline{\psi_n}(u) \, du$$
is the Whittaker function associated to an automorphic form $\varphi$. The payoff is that the global integral is over the adelic points $G(\mathbb{A})$ of $G = \mathrm{N}_n \backslash \mathrm{GL}_n$. Moreover, the integrand is factorisable if the Whittaker functions $W_{\varphi_{\pi}}$ and $W_{\varphi_{\sigma}}$ and the Schwartz function $\Phi$ are chosen to be factorisable:
$$W_{\varphi_{\pi}}(g) = \prod_v W_{\varphi_{\pi},v}(g_v), \qquad W_{\varphi_{\sigma}}(g) = \prod_v W_{\varphi_{\sigma},v}(g_v), \qquad \Phi(x) = \prod_v \Phi_v(x_v).$$
Here $W_{\varphi_{\pi},v}$ and $W_{\varphi_{\sigma},v}$ are local Whittaker functions from $\mathrm{GL}_n(F_v)$ to $\mathbb{C}$, while $\Phi_v$ is a local Schwartz function from $F_v^n$ to $\mathbb{C}$. (Whittaker functions and Schwartz functions that are factorisable in this way are called pure tensors, and pure tensors span the space of all such Whittaker functions or Schwartz functions.)
Thus the global $\mathrm{GL}_n \times \mathrm{GL}_n$ Rankin-Selberg integral has the factorisation
$$\prod_v \int\limits_{\mathrm{N}_n(F_v) \backslash \mathrm{GL}_n(F_v)} W_{\varphi_{\pi},v}(g_v) W_{\varphi_{\sigma},v}(g_v) \Phi_v(e_n g_v) \left|\det g_v\right|_v^s \, dg_v.$$
Each local integral represents $L(s,\pi_v \times \sigma_v)$, the local component of the global (completed) Rankin-Selberg $L$-function $\Lambda(s,\pi \times \sigma)$.

What about other integrals of automorphic forms? If we instead study something like
$$\int\limits_{\mathrm{GL}_n(F) \backslash \mathrm{GL}_n(\mathbb{A})} \varphi_{\pi}(g) \varphi_{\sigma}(g) \left|\det g\right|_{\mathbb{A}}^s \, dg,$$
then several things go wrong. Firstly, there is no reason for this to be Eulerian: this cannot be written as an integral over the adelic points of a group $G$ of factorisable Whittaker functions. Secondly, it turns out that this integral need not converge. Thirdly, it also turns out that even if this integral converges, it is zero unless $\widetilde{\pi} \cong \sigma \otimes \left|\det\right|_{\mathbb{A}}^s$, since it defines an invariant pairing.
So you can think of the presence of the Eisenstein series in the $\mathrm{GL}_n \times \mathrm{GL}_n$ Rankin-Selberg integral as a mitigating factor that ensures that the integral is Eulerian. For $\mathrm{GL}_n \times \mathrm{GL}_{n - 1}$ Rankin-Selberg integrals, you don't need this, since you can just insert the Fourier-Whittaker expansions of the underlying automorphic forms to get an Eulerian integral. However, this approach doesn't work for the integral
$$\int\limits_{\mathrm{GL}_n(F) \backslash \mathrm{GL}_n(\mathbb{A})} \varphi_{\pi}(g) \varphi_{\sigma}(g) \left|\det g\right|_{\mathbb{A}}^s \, dg,$$
as can easily be checked.
