Automorphic induction for \(GL(n)\) (over local nonarchimedean fields)
type: Item
instance of: scholarly article
title: Automorphic induction for \(GL(n)\) (over local nonarchimedean fields)
zbMATH Open document ID: 0849.11092
DOI: 10.1215/S0012-7094-95-07807-7
author: Guy Henniart
author: Rebecca A. Herb
published in: Duke Mathematical Journal
publication date: 1995-09-17T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11S37
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 784728
zbMATH Keywords: cyclic extension
zbMATH Keywords: tempered character
zbMATH Keywords: tempered representation
zbMATH Keywords: \(L\)-factors
zbMATH Keywords: \(\epsilon\)-factors
zbMATH Keywords: nonarchimedean local field
zbMATH Keywords: Langlands conjectures
zbMATH Keywords: Galois representations
zbMATH Keywords: Deligne-Kazhdan trace formula
reviewed by: J. G. M. Mars
MaRDI profile type: Publication
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full work available at URL: s0012-7094-95-07807-7

Automorphic induction for \(GL(n)\) (over local nonarchimedean fields) (OpenAlex ID)
OpenAlex ID: W1972038645