A Siegel-Weil identity for \(G_2\) and poles of \(L\)-functions
The authors study two examples of the relation among the following three properties: (1) \(\pi\), an irreducible, cuspidal, automorphic representation on \(G_{\mathbb{A}}\) is an endoscopic functorial lift from a (similar) representation \(\tau\) on \(H_{\mathbb{A}}\). (2) A certain (partial) \(L\)-function \(L^S(\pi, \rho, s)\) has a pole at \(s=1\). (3) A certain period is nontrivial on the space of \(\pi\). In the first example \(G= \text{SO}_7\) (split), \(\pi\) is generic and \(\rho\) is the second fundamental representation of \(\text{Sp}_6 (\mathbb{C})\) \((\dim\rho= 14)\), \(H= \text{SO}_3\times \text{SO}_3\) (split) and the period is taken along the subgroup \[ \left\{ u= \begin{pmatrix} 1 &x &y &z &*\\ && \vcenter{\vtop{=1em \line{\hss\hbox to 4.4em{\hrulefill}\hss} \line{\strut\hss\vrule\(\begin{matrix} g&0&t\\&1&0\\&&g^*\end{matrix}\)\vrule\hss\strut} \line{\hss\hbox to 4.4em{\hrulefill}\hss}}} &&\vcenter{\vtop{=1em\line{\strut\hss\(\begin{matrix}*\\*\\*\end{matrix}\)\hss\strut}}} \\ &&&&1 \end{pmatrix}\in \text{SO}_7\mid g\in \text{SL}_2 \right\} \] and the character \(u\mapsto \psi(y+ t_{11})\). The implication \((1) \Rightarrow (3)\) is clear. The authors conjecture that this is a special case of a more general set-up, where \(G= \text{SO}_{2n+1}\) (split), \(\rho\) is the second fundamental representation of \(\text{Sp}_{2n} (\mathbb{C})\), \(H= \text{SO}_3 \times \cdots\times \text{SO}_3\) (\(n\) copies of split \(\text{SO}_3)\), \(\tau= \bigotimes_{i=1}^n \tau_i\), where \(\tau_1,\dots, \tau_n\) are pairwise distinct cusp forms, and the period subgroups is the product of the above marked middle block (embedded in \(\text{SO}_{2n+1}\)) and the unipotent radical of the standard parabolic subgroup whose Levi part is isomorphic to \(\text{GL}_1^{n-2} \times \text{SO}_5\). The character is an ``extension'' of the above character by the standard nondegenerate character. In case \(n=2\), all three properties above are equivalent for generic \(\pi\) [\textit{D. Ginzburg, S. Rallis} and \textit{D. Soudry}, J. Reine Angew. Math. 487, 85-114 (1997; Zbl 0928.11025)]. The authors deal with the case \(n=3\). The idea of the proof (that \((2) \Rightarrow (3)\)) is to find a Rankin-Selberg integral which represents \(L^S(\pi, \rho,s)\). It has the form \[ I(s)= \int_{G_2(F)\setminus G_2(\mathbb{A})} \varphi(g) E^\alpha (g, \phi_{\frac{s+1}{3}}) dg. \] Here \(\varphi\) is a cusp form in the space of \(\pi\), and \(E^\alpha\) is the Eisenstein series on \(G_2(\mathbb{A})\) induced from the degenerate principal series of \(G_2\), associated to the standard parabolic subgroup, whose unipotent radical contains the long simple root \(\alpha\). (Similarly, denote, for the short simple root \(\beta\), by \(E^\beta\), the corresponding degenerate Eisenstein series.) \(L^S(\pi, \rho,s)\) may have a double pole at \(s=1\), and the leading term in the Laurent expansion at \(s=1\) is \[ \bigwedge_{s=1}^2 L^S (\pi,\rho,s)= \int_{G_2(F) \setminus G_2(\mathbb{A})} \varphi(g) \bigwedge_{s=2/3}^2 E^\alpha (g,\phi_s) dg \] for appropriate data. One of the key ingredients of the paper is a Siegel-Weil type identity which relates \(\bigwedge_{s=2/3}^2 E^\alpha (g,\phi_s)\) with \(\text{Res}_{s=3/5} E^\beta (g,f_s)\) for appropriate sections, and then it follows that \[ \int_{G_2(F) \setminus G_2(\mathbb{A})} \varphi(g) E^\beta (g,f_s) dg\neq 0 \] for \(\text{Re}(s)\gg 0\). Unfolding the last integral, one obtains the desired period as an inner integral and hence this period is nontrivial. Let me add here that by recent results of Ginzburg, Rallis and Soudry, it follows that the endoscopic lift from (split) \(\text{SO}_3\times \cdots\times \text{SO}_3\) (\(n\) copies) to \(\text{SO}_{2n+1}\) (split) exists as expected. The authors also use the above Siegel-Weil identity to prove that the Rankin-Selberg integral which represents the adjoint \(L\)-function of \(\text{GL}_3\) (studied by Ginzburg) is holomorphic for \(\text{Re}(s)> \frac 12\), except possibly of a simple pole at \(s=2/3\). The second example studied by the authors is \(G= F_4\), \(\pi\)-generic, \(\rho\) the 26-dimensional representation of \(F_4 (\mathbb{C})\), \(H= \text{SO}_8\) (split). The period subgroup is also of the form \(\text{SL}_2\cdot U\), where \(U\) is a certain unipotent subgroup of \(G\). The appropriate character is first defined on \(U\), so that it is stabilized by \(\text{SL}_2\). The idea is similar, and it is interesting to note that the Rankin-Selberg integral representing \(L^S(\pi, \rho,s)\) uses the same Eisenstein series \(E^\alpha\) on \(G_2\), as in the first example.
scientific article; zbMATH DE number 1466187
scientific article