Local tame lifting for \(GL(N)\). I: Simple characters
type: Item
instance of: scholarly article
title: Local tame lifting for \(GL(N)\). I: Simple characters
zbMATH Open document ID: 0878.11042
DOI: 10.1007/BF02698646
author: Colin J. Bushnell
author: Guy Henniart
published in: Publications Mathématiques
publication date: 1998-01-05T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104111
review text: Let \(K/F\) be a finite separable extension of nonarchimedean local fields and \(N\) a natural number. The base change conjecture of Langlands demands the existence of a process \(\pi\mapsto l_{K/F}(\pi)\) attaching to an irreducible smooth representation \(\pi\) of \(GL_N(F)\) a representation \(l_{K/F}(\pi)\) of the same sort of the group \(GL_N(K)\). As the authors show, the base change map \(l_{K/F}\) can be constructed for \(\text{char}(F)=0\) and \(K/F\) tamely ramified, we assume this from now on. The aim of the present paper is to give a first step towards presenting the base change in explicit local terms. The authors refer to the classification of the irreducible smooth representations of \(GL_N(F)\) by \textit{C. Bushnell} and \textit{P. Kutzko} [in: The admissible dual of \(GL(N)\) via compact open subgroups, Ann. Math. Stud. 129, Princeton University Press, Princeton (1993; Zbl 0787.22016)]. In this book it is shown that every supercuspidal representation contains a simple type. A simple type is a pair \((K,\tau)\) consisting of a compact open subgroup \(K\) of \(GL_N(F)\) and an irreducible representation \(\tau\) of \(K\), both of a very special kind. An admissible representation \(\pi\) of \(GL_N(F)\) is then said to contain the simple type \((K,\tau)\) if its restriction \(\pi|_K\) contains \(\tau\). In that case the \(K\)-isotype of \(\tau\) in \(\pi\) forms a finite-dimensional module under the corresponding Hecke algebra which is irreducible if and only if \(\pi\) is. It is shown that the corresponding Hecke algebra is naturally isomorphic to the Iwahori-Hecke algebra, which reduces the classification problem to the representations with Iwahori-fixed vector. The latter are classified by \textit{D. Kazhdan} and \textit{G. Lusztig} in their proof of the Deligne-Langlands conjecture [Invent. Math. 87, 153-215 (1987; Zbl 0613.22004)]. One main ingredient of the construction of simple types in Bushnell-Kutzko (loc. cit.) is a simple character, which i…
Mathematics Subject Classification ID: 11S37
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 1019349
zbMATH Keywords: tamely ramified extensions
zbMATH Keywords: local tame lifting
zbMATH Keywords: simple type
zbMATH Keywords: supercuspidal representations
zbMATH Keywords: general linear group
zbMATH Keywords: irreducible representations
zbMATH Keywords: Hecke algebra
zbMATH Keywords: Iwahori-fixed vector
zbMATH Keywords: base change conjecture
zbMATH Keywords: base change map
zbMATH Keywords: lifting construction of simple characters
reviewed by: Anton Deitmar
MaRDI profile type: Publication
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OpenAlex ID: W2024183937
description: scientific article; zbMATH DE number 1019349