Automorphic induction for \(GL(n)\) (over local nonarchimedean fields)
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.
scientific article; zbMATH DE number 784728
scientific article