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Locally analytic vectors of some crystabelian representations of GL2(ℚp)

Part of: Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  20 December 2011

Ruochuan Liu
Ruochuan Liu*
Affiliation:
University of Michigan, Ann Arbor, Michigan, USA (email: ruochuan@umich.edu)
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Abstract

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For V a two-dimensional p-adic representation of Gℚp, we denote by B(V ) the admissible unitary representation of GL2(ℚp) attached to V under the p-adic local Langlands correspondence of GL2(ℚp) initiated by Breuil. In this paper, building on the works of Berger–Breuil and Colmez, we determine the locally analytic vectors B(V )an of B (V ) when V is irreducible, crystabelian and Frobenius semisimple with distinct Hodge–Tate weights; this proves a conjecture of Breuil. Using this result, we verify Emerton’s conjecture that dim Ref η⊗ψ (V )=dim Exp η∣⋅∣⊗xψ (B (V )an ⊗(x∣⋅∣∘det )) for those V which are irreducible, crystabelian and Frobenius semisimple.

Keywords

p-adic local Langlands correspondence(φ,Γ)-modulesGalois representations

MSC classification

Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F80: Galois representations 11F85: $p$-adic theory, local fields 22E35: Analysis on $p$-adic Lie groups 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 1 , January 2012 , pp. 28 - 64
Copyright
Copyright © Foundation Compositio Mathematica 2011

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