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Automorphic induction for \(GL(n)\) (over local nonarchimedean fields). (English) Zbl 0849.11092

Let \(F\) be a nonarchimedean local field. By the Langlands conjectures there should be a canonical correspondence between equivalence classes of complex representations of degree \(n\) of the Galois group of \(F\) and equivalence classes of irreducible admissible representations of \(GL (n, F)\). To induction of Galois representations then should correspond a map which assigns a representation \(\pi\) of \(GL (n, F)\) to a representation \(\tau\) of \(GL (m, E)\), when \(E\) is an extension of \(F\) of degree \(d\) and \(n= md\). The paper treats the case where \(E/F\) is cyclic. Let \(\kappa\) be a character of \(F^*\) defining \(E\). The map \(\tau \mapsto \pi\) is defined by a relation between the character of \(\tau\) and the \(\kappa\)-twisted character of \(\pi\). It is proved that the lift \(\pi\) exists when \(\tau\) is tempered. Then \(\pi\) is tempered and \(\kappa \pi\sim \pi\), and any such \(\pi\) is a lift. The relations between \(L\)- and \(\varepsilon\)-factors for \(\pi\) and those for \(\tau\) are also proved. (The results are even proved for more general representations than tempered ones). The proofs are by global means. The Deligne-Kazhdan trace formula is used in a new manner. The paper is very well written.

MSC:

11S37 Langlands-Weil conjectures, nonabelian class field theory
22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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