On lifting from classical groups to \(\text{GL}_N\)
Langlands' functorial lift is one of the central questions in the theory of automorphic forms. In this paper, the four authors unify their methods into a powerful machinery and prove the existence of a weak functorial lift of a generic cuspidal representation of \(\text{SO}_{2n+1}\) to \(\text{GL}_{2n}\). In the follow-u paper ``Functoriality for the classical groups'', the authors establish functoriality from the split classical groups \(\text{SO}_{2n+1}\), \(\text{SO}_{2n}\) and \(\text{Sp}_{2n}\) to \(\text{GL}_{2n}\), for globally generic cuspidal representations. Their approach is based on the Converse Theorem of Cogdell and Piatetski-Shapiro and the Langlands-Shahidi method, which studies automorphic \(L\)-functions via the constant and non-constant terms of the Eisenstein series. Functoriality is a part of the Langlands program. It is a manifestation of the (conjectural) Langlands correspondence. The principle of functoriality states that any \(L\)-homomorphism of \(L\)-groups \( ^L H \to ^L G\) should determine a lifting of admissible or automorphic representations of \(H\) to admissible or automorphic representations of \(G\). If \(H=\text{SO}_{2n+1}\) and \(G=\text{GL}_{2n}\), the connected components of the \(L\)-groups are \(^L H^0=\text{Sp}_{2n}(\mathbb{C})\) and \(^L G^0=\text{GL}_{2n}(\mathbb{C})\), and there is a natural embedding \(^L H^0 \hookrightarrow ^L G^0\). The philosophy of Langlands then says that associated to this map of dual groups there should exist a lifting of automorphic forms on \(\text{SO}_{2n+1}(\mathbb{A})\) to \(\text{GL}_{2n}(\mathbb{A})\). Let \(\pi = \otimes' \pi_v\) be an irreducible automorphic representation of \(\text{SO}_{2n+1}(\mathbb{A})\). A weak lift of \(\pi\) is an automorphic representation \(\Pi = \otimes' \Pi_v\) of \(\text{GL}_{2n}(\mathbb{A})\) such that, at almost all places, \(\Pi_v\) is a local lift of \(\pi_v\). More precisely, for every Archimedean place \(v\) and for almost all non-Archimedean places \(v\) where \(\pi_v\) is unramified, \(\Pi_v\) is a local lift of \(\pi_v\). The main result is the following: Let \(\pi\) be an irreducible globally generic cuspidal automorphic representation of the split \(\text{SO}_{2n+1}(\mathbb{A})\). Then \(\pi\) has a weak lift to \(\text{GL}_{2n}(\mathbb{A})\).
generic cuspidal automorphic representations
scientific article; zbMATH DE number 1874405