Real and global lifts from \(\text{PGL}_3\) to \(G_2\).

*(English)*Zbl 1037.22033Let \(F\) be a number field and let \(\pi=\otimes \pi_v\) be a cuspidal automorphic representation of \(\text{PGL}_3({\mathbb A}_F)\). The Langlands functoriality conjectures predict the existence of local and global \(L\)-packets \({\mathcal L}(\pi_v)\), \({\mathcal L}(\pi)\) of \(G_2\) corresponding to the natural inclusion of dual groups \(\text{SL}_3({\mathbb C})\hookrightarrow G_2({\mathbb C})\). Moreover, each local packet should contain a unique generic element \(\sigma(\pi_v)\) and the element \(\sigma(\pi)=\otimes \sigma(\pi_v)\) should be cuspidal automorphic.

This paper is part of a series by the authors, whose goal is to establish these conjectures. The authors seek to construct the lifting \(\pi\mapsto \sigma(\pi)\) by means of the theta correspondence obtained by restricting the minimal representation of \(E_6\) to the subgroup \(\text{PGL}_3\times G_2\). The nonarchimedean local theta lift was studied by the authors in [Endoscopic lifts from \(\text{PGL}_3\) to \(G_2\), Compos. Math. 140, No. 3, 793–808 (2004; Zbl 1071.22007)]; there the representations \(\sigma(\pi_v)\) were constructed for nonarchimedean places \(v\), and the lift was shown to be functorial to the extent to which the parametrization of the representations of \(G_2(F_v)\) is known. At the archimedean places less is known. J.-S. Huang, P. Pandžić and G. Savin [Duke Math. J. 82, 447–471 (1996; Zbl 0865.22009)] showed that the theta correspondence is functorial on the level of infinitesimal characters.

In this paper the authors first obtain sharper information at the real places. By controlling the minimal \(K\)-type of the principal series representations in the theta correspondence, they show that the real local correspondence is functorial for spherical generic unitary representations and for generic unitary representations with integral infinitesimal character. They then establish the nonvanishing of the global theta lift. Putting these together, they show that for \(F\) a totally real field and \(\pi=\otimes \pi_v\) an irreducible cuspidal representation of \(\text{PGL}_3({\mathbb A}_F)\) such that at each archimedean place \(\infty\) the representation \(\pi_\infty\) either is spherical or has integral infinitesimal character, the generic lift \(\sigma(\pi)\) occurs in the space of globally generic cusp forms on \(G_2({\mathbb A}_F)\), and moreover that if \(\sigma=\otimes \sigma_v\) is a globally generic cuspidal representation of \(G_2({\mathbb A}_F)\) which is nearly equivalent to \(\sigma(\pi)\), then in fact \(\sigma\) is equal to \(\sigma(\pi)\) as a subspace of the space of globally generic cusp forms. The proof of this uniqueness result relies on a theorem of D. Ginzburg and D. Jiang [Isr. J. Math. 123, 29–59 (2001; Zbl 1005.11023)] which characterizes the generic cuspidal representations of \(G_2({\mathbb A}_F)\) which are global theta lifts in terms of the existence of a pole at \(s=1\) of their partial standard \(L\)-functions.

This paper is part of a series by the authors, whose goal is to establish these conjectures. The authors seek to construct the lifting \(\pi\mapsto \sigma(\pi)\) by means of the theta correspondence obtained by restricting the minimal representation of \(E_6\) to the subgroup \(\text{PGL}_3\times G_2\). The nonarchimedean local theta lift was studied by the authors in [Endoscopic lifts from \(\text{PGL}_3\) to \(G_2\), Compos. Math. 140, No. 3, 793–808 (2004; Zbl 1071.22007)]; there the representations \(\sigma(\pi_v)\) were constructed for nonarchimedean places \(v\), and the lift was shown to be functorial to the extent to which the parametrization of the representations of \(G_2(F_v)\) is known. At the archimedean places less is known. J.-S. Huang, P. Pandžić and G. Savin [Duke Math. J. 82, 447–471 (1996; Zbl 0865.22009)] showed that the theta correspondence is functorial on the level of infinitesimal characters.

In this paper the authors first obtain sharper information at the real places. By controlling the minimal \(K\)-type of the principal series representations in the theta correspondence, they show that the real local correspondence is functorial for spherical generic unitary representations and for generic unitary representations with integral infinitesimal character. They then establish the nonvanishing of the global theta lift. Putting these together, they show that for \(F\) a totally real field and \(\pi=\otimes \pi_v\) an irreducible cuspidal representation of \(\text{PGL}_3({\mathbb A}_F)\) such that at each archimedean place \(\infty\) the representation \(\pi_\infty\) either is spherical or has integral infinitesimal character, the generic lift \(\sigma(\pi)\) occurs in the space of globally generic cusp forms on \(G_2({\mathbb A}_F)\), and moreover that if \(\sigma=\otimes \sigma_v\) is a globally generic cuspidal representation of \(G_2({\mathbb A}_F)\) which is nearly equivalent to \(\sigma(\pi)\), then in fact \(\sigma\) is equal to \(\sigma(\pi)\) as a subspace of the space of globally generic cusp forms. The proof of this uniqueness result relies on a theorem of D. Ginzburg and D. Jiang [Isr. J. Math. 123, 29–59 (2001; Zbl 1005.11023)] which characterizes the generic cuspidal representations of \(G_2({\mathbb A}_F)\) which are global theta lifts in terms of the existence of a pole at \(s=1\) of their partial standard \(L\)-functions.

Reviewer: Solomon Friedberg (Chestnut Hill)

##### MSC:

22E50 | Representations of Lie and linear algebraic groups over local fields |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

11F27 | Theta series; Weil representation; theta correspondences |

11F55 | Other groups and their modular and automorphic forms (several variables) |

22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |

22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |