Locally analytic vectors of unitary principal series of \(\mathrm{GL}_2(\mathbb Q_p)\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1320.11112
author: Ruochuan Liu
author: Yuancao Zhang
author: Bingyong Xie
publication date: 2012-05-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1103.2543
full work available at URL: ens ann-sc 45 167-190.php
Mathematics Subject Classification ID: 11S37
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 6037744
zbMATH Keywords: local Langlands correspondence
zbMATH Keywords: unitary principal series
zbMATH Keywords: trianguline representations
zbMATH Keywords: locally analytic vectors
zbMATH Keywords: \((\varphi,\Gamma)\)-modules
MaRDI profile type: Publication
OpenAlex ID: W2598714725
arXiv ID: 1103.2543
DOI: 10.24033/ASENS.2163
title: Locally analytic vectors of unitary principal series of \(\mathrm{GL}_2(\mathbb Q_p)\)
published in: Annales Scientifiques de l'École Normale Supérieure. Quatrième Série
review text: Authors' introduction: ``Let \(F\) be a finite extension of \(\mathbb Q_p\). The aim on the \(p\)-adic local Langlands programme initiated by Breuil is to look for a `natural' correspondence between certain \(n\)-dimensional \(p\)-adic representations of \(\text{GL}(\overline{\mathbb Q}_p/F)\) and certain Banach space representations of \(\text{GL}_n(F)\). Thanks to much recent work, especially that of Colmez and Paskunas, we have gained a fairly clear picture in the case \(F=\mathbb Q_p\) and \(n=2\) which is the so-called \(p\)-adic local Langlands correspondence for \(\text{GL}_2(\mathbb Q_p)\) established a functorial bijection between two-dimensional irreducible \(p\)-adic representations of \(\text{GL}(\overline{\mathbb Q}_p/\mathbb Q_p)\) and non-ordinary irreducible admissible unitary representations of \(\text{GL}_2(\mathbb Q_p)\).''NEWLINENEWLINE Locally analytic vectors of unitary principal series representations of \(\text{GL}_2(\mathbb Q_p)\) play an important role in the \(p\)-adic local Langlands correspondence -- they correspond to trianguline representations of \(\text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\). Let \({\mathcal S}_{\text{irr}}\) be the parametrizing space of two-dimensional irreducible trianguline representations of \(\text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\). For \(s\in{\mathcal S}_{\text{irr}}\), let \(V(s)\) be the corresponding trianguline representation, and let \(\Pi(V(s))\) be the admissible unitary representation of \(\text{GL}_2(\mathbb Q_p)\) attached to \(V(s)\) by the \(p\)-adic local Langlands correspondence.NEWLINENEWLINE ``\textit{M. Emerton} [Pure Appl. Math. Q. 2, No. 2, 279--393 (2006; Zbl 1254.11106)] made a conjectural description of the subspace of locally analytic vectors \(\Pi(V)_{\mathrm{an}}\) for all unitary principal series \(\Pi(V)\)'' (Conjecture 1.1).NEWLINENEWLINE NEWLINEIn this paper, the authors prove Emerton's conjecture for \(p>2\) and \(V(s)\) non-exceptional. The proof builds on \textit{P.…
reviewed by: Andrzej Dąbrowski
description: scientific article; zbMATH DE number 6037744