A question on the period integral of Rankin-Selberg $L$-function

Question

$\DeclareMathOperator\GL{GL}$Let $\Pi$ and $\pi$ be irreducible automorphic representations of $\GL_{n+1}(\mathbb{A}_F)$ and $\GL_n(\mathbb{A}_F)$ respectively, where $n \geq 2$, $F$ is a number field and $\mathbb{A}_F$ is the corresponding adele ring. Let $\Phi \in \Pi$ and $\phi \in \pi$ be automorphic forms. For every $s \in \mathbb{C}$, we may consider the following period integral formally $$ I(s,\Phi,\phi):= \int_{\GL_n(F) \backslash \GL_n(\mathbb{A}_F)} \Phi \begin{pmatrix} g & \\ & 1 \end{pmatrix} \phi(g) \vert \text{det} g \rvert^{s-\frac{1}{2}} dg.$$ This period integral $I(s,\Phi,\phi)$ is closely related to the $\GL(n+1) \times \GL(n)$ Rankin-Selberg $L$-function $L(s, \Pi \times \pi)$. If we further assume that $\Phi$ is a cusp form and therefore $\Phi \begin{pmatrix} g & \\ & 1 \end{pmatrix} $ is of rapid decay, we know that the period integral $I(s,\Phi,\phi)$ is absolutely convergent and defines an entire function of every $s$. Now my question is the following: If we let $\Phi$ be an automorphic form and we further assume that $\phi$ is a cusp form, I am wondering whether the period integral $I(s,\Phi,\phi)$ is still absolutely convergent for every $s \in \mathbb{C}$ or not. Thanks a lot!

Answer 1

The answer is no in general: for example, if $\phi$ is a cuspidal automorphic form in a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n(\mathbb{A}_F)$ and $\Phi$ is an Eisenstein series in the noncuspidal automorphic representation $\Pi = \widetilde{\pi} \boxplus \omega$ of $\mathrm{GL}_n(\mathbb{A}_F)$, where $\omega$ is a Hecke character, then were the integral $I(s,\Phi,\phi)$ to converge absolutely, it would represent $L(s,\Pi \times \pi) = L(s,\widetilde{\pi} \times \pi) L(s,\pi \otimes \omega)$. However, this has a pole at $s = 1$.

On the other hand, if $\Phi$ is cuspidal, then the integral $I(s,\Phi,\phi)$ converges absolutely even if $\phi$ is an Eisenstein series. More generally, in order to relate $I(s,\Phi,\phi)$ to $L$-functions, you need to use a regularisation process, which is due to Ichino and Yamana:

https://doi.org/10.1112/S0010437X14007362

The idea is to use a truncation operator on these automorphic forms that truncates up to height $T$, and then view the integral involving these truncated automorphic forms as a polynomial in $T$. The constant term in this polynomial is precisely what one would hope for, namely $$\int\limits_{\mathrm{N}_n(\mathbb{A}_F) \backslash \mathrm{GL}_n(\mathbb{A}_F)} W_{\Phi}\begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix} W_{\phi}(g) \left|\det g\right|^{s - \frac{1}{2}} \, dg,$$ where $W_{\Phi} \in \mathcal{W}(\Pi,\psi)$ is the Whittaker function associated to $\Phi$ and $W_{\phi} \in \mathcal{W}(\pi,\overline{\psi})$ is the Whittaker function associated to $\phi$.

Indeed, if $\Phi$ is cuspidal, then $I(s,\Phi,\phi)$ is equal to this integral involving Whittaker functions, which is proven by inserting the Whittaker expansion of $\Phi$ and unfolding. When $\Phi$ is not cuspidal, however, this unfolding process will lead to the presence of certain degenerate integrals, and these integrals need not converge (and in fact do not converge due to exponential growth, as discussed in the introduction of Ichino and Yamana's paper).

Finally, if the Whittaker functions $W_{\Phi}$ and $W_{\phi}$ are both pure tensors, then this global integral involving these Whittaker functions factorises as a product of local integrals, each of which represents the local component of the completed $\mathrm{GL}_{n + 1} \times \mathrm{GL}_n$ Rankin-Selberg $L$-function $\Lambda(s,\Pi \times \pi)$.