A remark on a converse theorem of Cogdell and Piatetski-Shapiro (Q2303071)

Let \(\pi\) be an irreducible generic representation of \(\mathrm{GL}_n(\mathbb{A})\) whose central character is invariant by \(F^\times\) (\(F\) is a global field) with a fixed a Whittaker model. For any \(\xi\in \pi\), let \(W_\xi\) be the corresponding element in the Whittaker model. Let \[U_\xi(g)= \sum_{\gamma\in U_{n-1}(F)\backslash \mathrm{GL}_{n-1}(F)} W_{\xi}( \begin{pmatrix} \gamma & 0\\ 0& 1 \end{pmatrix} g) \] where \(U_{n}\) is the group of upper triangular matrix with identity diagonal in \(\mathrm{GL}_n\), and let \[V_\xi(g)= \sum_{\gamma\in U_{n-1}(F)\backslash \mathrm{GL}_{n-1}(F)} W_{\xi}( \begin{pmatrix} 1 & 0\\ 0& \gamma \end{pmatrix}g). \] The autormorphy of \(\pi\) is equivalent to that \(U_\xi=V_\xi\) for any \(\xi\in \pi\). Let \(\mathcal{A}(n,m)\) be the condition that the complete \(L\)-function \(L(\pi\times \tau, s)\) has ``nice'' properties (analytic continuation, bounded in vertical strip and functional equation) for any cuspidal automorphic representation \(\tau\) of \(\mathrm{GL}_m(\mathbb{A})\). The main result proved in this article is the following statement. Let \(n>3\), if for any \(1\leq m\leq n-3\), the condition \(\mathcal{A}(n,m)\) holds, then for any \(\xi\in {\pi}\) we have \[\int_{F\backslash \mathbb{A}} (U_\xi-V_\xi)\begin{pmatrix}1 & 0 & z\\ 0 & I_{n-2} & 0 \\ 0 & 0 & 1 \end{pmatrix} \mathrm{d} z =0. \] The authors prove this by studying the properties of separable functions, then they reduce the identity to a known identity of Cogdell and Piatetski-Shapiro. The reduction step carried out in this article does not rely on the conditions \(\mathcal{A}(n,m)\). As a corollary, the converse theorem of \textit{J. W. Cogdell} and \textit{I. I. Piatetski-Shapiro} [Publ. Math., Inst. Hautes Étud. Sci. 79, 157--214 (1994; Zbl 0814.11033); J. Reine Angew. Math. 507, 165--188 (1999; Zbl 0912.11022)] has a purely global proof and there is an improvement to it: to check the automorphy of \(\pi\), it's sufficient to have \(\mathcal{A}(n,m)\) for \(1\leq m\leq n-2\), instead of \(1\leq m\leq n-1\).