Vol. 8, No. 6, 2014

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Complétés universels de représentations de $\mathrm{GL}_2({\mathbb Q}_p)$

Pierre Colmez and Gabriel Dospinescu

Vol. 8 (2014), No. 6, 1447–1519
Abstract

Soit Π une représentation unitaire de GL2(p), topologiquement de longueur finie. Nous décrivons la sous-représentation Πan de ses vecteurs localement analytiques, et sa filtration par rayon d’analyticité, en termes du (φ,Γ)-module qui lui est associé via la correspondance de Langlands locale p-adique, et nous en déduisons que le complété universel de Πan n’est autre que Π.

Let Π be a unitary representation of GL2(p), topologically of finite length. We describe the subrepresentation Πan made of its locally analytic vectors, and its filtration by radius of analyticity, in terms of the (φ,Γ)-module attached to Π via the p-adic local Langlands correspondence, and we deduce that the universal completion of Πan is Π itself.

Keywords
$p$-adic representations, local Langlands correspondence, universal completion
Mathematical Subject Classification 2010
Primary: 11SXX
Milestones
Received: 10 March 2013
Revised: 23 May 2013
Accepted: 24 July 2013
Published: 2 October 2014
Authors
Pierre Colmez
C.N.R.S.
Université Pierre et Marie Curie
Institut de Mathématiques de Jussieu
4 Place Jussieu
75005 Paris
France
Gabriel Dospinescu
C.N.R.S.
UMPA
École Normale Supérieure de Lyon
46 allée d’Italie
69007 Lyon
France