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

For V a two-dimensional p-adic representation of G(Qp), we denote by B(V) the admissible unitary representation of GL(2)(Q(p)) attached to V under the p-adic local Langlands correspondence of GL(2)(Q(p)) 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(eta circle times psi)(V) = dim Exp(eta vertical bar.vertical bar circle times x psi)(B(V)(an) circle times (x vertical bar .vertical bar circle det)) for those V which are irreducible, crystabelian and Frobenius semisimple.