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Converse theorems for \(\text{GL}_n\). (English) Zbl 0814.11033
The authors prove a useful \(L\)-function criterion for an irreducible representation \(\Pi\) of \(\text{GL}_ n(\mathbb A)\) to be automorphic. Namely, for \(n\geq 3\) and \(\pi\) an irreducible admissible representation of \(\text{GL}_ n(\mathbb A)\) whose central character is invariant under \(k\) and whose \(L\)-function \(L(\pi,s)\) is absolutely convergent in some half-plane, let \(S\) be a non-empty finite set of places of \(k\) containing all archimedean places and such that the ring of \(S\)-integers has class number 1. For each integer \(m\), let \(\Omega^ 0_ S(m)\) denote the set of irreducible generic cuspidal automorphic representations of \(\text{GL}_ m(\mathbb A)\) which are unramified outside \(S\). Then if for every \(1 \leq m \leq n-1\) and \(\tau\) in \(\Omega^ 0_ m(S)\) the Rankin-Selberg \(L\)-function \(L(\pi\times \tau,s)\) is “nice”, it must follow that there is some irreducible automorphic representation \(\pi'\) of \(\text{GL}_ n(\mathbb A)\) such that \(\pi'_ v \simeq \pi_ v\) for all non-archimedean \(v\) where \(\pi_ v\) is unramified. This “converse theorem” should have crucial application to the problem of Langlands’ lifting of automorphic representations from classical groups such as \(\text{SO}\) and \(\text{Sp}\) to \(\text{GL}_ n\). The authors also conjecture that it should not be necessary to have control of so many twists to be able to draw conclusions about the automorphic nature of \(\pi\).
In particular, they conjecture that if \(L(\pi \otimes \omega,s)\) is “nice” for all characters \(\omega\) of \(k^ x \setminus \mathbb A^ x\), then a similar conclusion should hold; this is known in case \(n = 2\) [H. Jacquet, R. Langlands, Automorphic forms on \(\text{GL}(2)\). Lect. Notes Math. 114 (1970; Zbl 0236.12010)] and \(n = 3\) [H. Jacquet, I. Piatetski-Shapiro and J. Shalika, Ann. Math. (2) 109, 169–258 (1979; Zbl 0401.10037)]. For general \(n\), the authors can also prove that \(\pi\) is itself automorphic cuspidal, provided \(L(\pi \times \tau,s)\) is nice for all \(\tau\) cuspidal on \(\text{GL}_ m(\mathbb A)\); however, this hypothesis is too strong to have immediate applications to Langlands’ functoriality.

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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