The essentially tame Jacquet-Langlands correspondence for inner forms of \(\text{GL}(n)\)
type: Item
instance of: scholarly article
title: The essentially tame Jacquet-Langlands correspondence for inner forms of \(\text{GL}(n)\)
zbMATH Open document ID: 1244.11053
author: Colin J. Bushnell
author: Guy Henniart
published in: Pure and Applied Mathematics Quarterly
publication date: 2011-08-24T00:00:00Z
review text: The objective of the paper under review is to render the Jacquet--Langlands correspondence for inner forms of \(\text{GL}(n)\) explicit -- for the class of representations that can be explicitly described through the theory of admissible pairs. This is the class of irreducible essentially tame, cuspidal representations of parametric degree \(n\). Let \(F\) be a non-archimedean local field of finite residue characteristic and \(D\) a central division algebra over \(F\) of index \(d\). For \(m\in\mathbb{N}\), let \(\mathcal{A}^\square_m(D)\) be the set of equivalence classes of irreducible smooth representations of \(\text{GL}_m(D)\) essentially square integrable modulo the centre. If \(n=md\), the Jacquet--Langlands correspondence provides a canonical bijection \(\mathcal{JL}:\mathcal{A}^\square_n(F)\longrightarrow\mathcal{A}^\square_m(D)\). The parametric degree \(\delta(\pi)\) of \(\pi\in\mathcal{A}^\square_m(D)\) is a positive integer dividing \(n\). For \(m\neq1,n\), the authors base their definition on the classification of supercuspidal representations in terms of types given in [\textit{S. Stevens} and \textit{V. Sécherre}, J. Inst. Math. Jussieu 7, 527--574 (2008; Zbl 1140.22014)]. As a matter of fact, a representation \(\pi\) of parametric degree \(n\) is cuspidal. Next, for a representation \(\pi\in\mathcal{A}^\square_m(D)\), let \(t(\pi)\) be the number of unramified characters of \(F^\times\) such that \((\chi\circ\text{Nrd})\pi\cong\pi\). Then \(t(\pi)\mid\delta(\pi)\) and \(\pi\) is called essentially tame, if the residue characteristic does not divide \(\delta(\pi)/t(\pi)\). Let \(\mathcal{A}^{\text{et}}_m(D)\subset\mathcal{A}^\square_m(D)\) denote the set of equivalence classes of irreducible essentially tame representations of \(\text{GL}_m(D)\) of parametric degree \(n\). The Jacquet--Langlands correspondence preserves both the parametric degree and the \(t\)-invariant, hence induces a canonical bijection \(\mathcal{JL}:\mathcal{A}^{\text{et}}_n(F)\…
reviewed by: Gerald Gotsbacher
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11S37
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 5942223
zbMATH Keywords: explicit local Jacquet-Langlands correspondence
zbMATH Keywords: admissible pair
zbMATH Keywords: inner form
zbMATH Keywords: essentially tame representation
zbMATH Keywords: parametric degree
zbMATH Keywords: character formula
MaRDI profile type: Publication
DOI: 10.4310/PAMQ.2011.V7.N3.A2
description: scientific article; zbMATH DE number 5942223