Some results on multiplicities for SL(n)
type: Item
instance of: scholarly article
title: Some results on multiplicities for SL(n)
zbMATH Open document ID: 0937.22002
DOI: 10.1007/BF02773481
author: Erez Lapid
published in: Israel Journal of Mathematics
publication date: 2000-06-13T00:00:00Z
review text: Let \(F\) be a number field; the author considers the multiplicity of \(L\)-packets of automorphic representations of \(\text{SL}(n)/F\). The main point of this paper is to present a conjecture that the multiplicity of an \(L\)-packet of a cuspidal automorphic representation is bounded -- perhaps even by \(\varphi(n)\), where \(\varphi\) denotes Euler's function. Here the author defines multiplicity as follows: the \(L\)-packet is defined by a cuspidal automorphic representation \(\widetilde{\pi}\) of \(\text{GL}(n)\); two representations \(\widetilde{\pi}_1\) and \(\widetilde{\pi}_2\) generate the \(L\)-packet if and only if \(\widetilde{\pi}_1\cong \widetilde{\pi}_2\otimes \omega\), where \(\omega\) is a Hecke character; the multiplicity is the number of equivalence classes of such cuspidal \(\widetilde{\pi}\) inside the equivalence class of such \(\pi\) under the analogous equivalence relation holding locally almost everywhere. The author gives a heuristic basis for this conjecture based on Langlands' Tannakian philosophy. He then analyzes, when \(n\) is prime, the case of cuspidal representations associated to a Hecke character of a cyclic extension \(E/F\). Here he gives a complete description.
Mathematics Subject Classification ID: 22E35
zbMATH DE Number: 1358186
zbMATH Keywords: endoscopy
zbMATH Keywords: Tannakian formalism
zbMATH Keywords: \(L\)-packets
zbMATH Keywords: automorphic representations
zbMATH Keywords: cuspidal representations
zbMATH Keywords: Hecke character
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
cites work: Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. (AM-120)
cites work: Q3332956
cites work: On multiplicities for \(\text{SL}(n)\)
cites work: Structure of finite subgroups of simple algebraic groups
cites work: Q5340151
cites work: Automorphic Forms on Adele Groups. (AM-83)
cites work: Q4278078
cites work: On the local Langlands conjecture for GL(n): the cyclic case
cites work: Automorphic induction for \(GL(n)\) (over local nonarchimedean fields)
cites work: Q5329179
cites work: Automorphic forms on GL (2)
cites work: On Euler Products and the Classification of Automorphic Forms II
cites work: Q3324141
cites work: Stable trace formula: Cuspidal tempered terms
cites work: Q4847234
cites work: L-Indistinguishability For SL (2)
cites work: A note on the global Langlands conjecture.
cites work: Q3892334
cites work: Stable Conjugacy: Definitions and Lemmas
cites work: Q3314082
cites work: On the conjugacy of element-conjugate homomorphisms
cites work: Modularity of the Rankin-Selberg \(L\)-series, and multiplicity one for \(\mathrm{SL}(2)\)
cites work: Conjugacy classes in Lie algebras and algebraic groups
cites work: Representations of \(GL(n)\) and division algebras over a p-adic field
cites work: Automorphic Representation of Unitary Groups in Three Variables. (AM-123)
cites work: The multiplicity one theorem for \(\mathrm{GL}_n\)
cites work: Q5341863
cites work: Quelques applications du théorème de densité de Chebotarev
cites work: Q3926778
cites work: Notes on representation theory of non-Archimedean \(SL(n)\)
cites work: Induced representations of reductive ${\germ p}$-adic groups. II. On irreducible representations of ${\rm GL}(n)$
full work available at URL: https://doi.org/10.1007/bf02773481
OpenAlex ID: W2161605663
Recommended article: On multiplicities for \(\text{SL}(n)\)
Recommended article: A strong multiplicity one theorem for \(\mathrm{SL}_2\)
Recommended article: Q3518041
Recommended article: On multiplicities of maximal weights of \(\widehat{sl}(n)\)-modules
Recommended article: Q3701631
Recommended article: On the analytic strong multiplicity one for SL(2)
Recommended article: The Schur multiplicator of SL(2,Z/mZ) and the congruence subgroup property
Recommended article: On the \(P = W\) conjecture for \(\mathrm{SL}_n\)
Recommended article: A vector partition function for the multiplicities of \(\mathfrak{sl}_k\mathbb C\)
Recommended article: A note on some representations of SL(\(V\))
description: scientific article; zbMATH DE number 1358186