Actions infinitésimales dans la correspondance de Langlands locale p-adique

Abstract : The topic of this thesis is the p-adic Langlands correspondence, imagined by Breuil and established by Colmez for GL_2(Q_p). Let L be a finite extension of Q_p and let V be an irreducible, two-dimensional L-representation of the absolute Galois group of Q_p. Using Fontaine's theory of (phi,Gamma)-modules, Colmez associates to V a GL_2(Q_p)-Banach space representation Pi(V), which is unitary, admissible and topologically irreducible. We give a new proof, much easier, of a theorem of Colmez, which describes the locally analytic vectors Pi(V)^an of Pi(V) in terms of the overconvergent (phi,Gamma)-module attached to V. The main result of this thesis is a simple description of the infinitesimal action of GL_2(Q_p) on Pi(V)^an. In particular, we show that Pi(V)^an has an infinitesimal character, which can be computed in terms of the Hodge-Tate weights of V, answering therefore a question of Harris. We show that there is no p-adic analogue of a classical theorem of Saito and Tunnell, answering another question of Harris. We extend results of Colmez concerning the Kirillov model of the U-finite vectors of Pi(V) (U is the upper unipotent of GL_2(Q_p)). Combining this study with the description of the infinitesimal action, we obtain a simple proof of one of the main results of Colmez, characterizing the representations V such that Pi(V) has nonzero locally algebraic vectors. This result is the first step in making the connection with the classical Langlands correspondence, and it is also a key ingredient in Emerton's proof of the Fontaine-Mazur conjecture in dimension two. We extend our methods to prove the analogous result for infinitesimal deformations of V. This answers a question of Paskunas and has applications to the Breuil-Mézard conjecture. We apply differential methods to study the Jacquet module of Pi(V)^an, proving for instance that it is nonzero if and only if V is trianguline and giving a new and direct proof of conjectures of Berger, Breuil and Emerton. Finally, in joint work with Benjamin Schraen we prove Schur's lemma for topologically irreducible Banach and locally analytic representations of p-adic Lie groups. This basic result was previously known only for commutative Lie groups and for GL_2(Q_p).
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Gabriel Dospinescu. Actions infinitésimales dans la correspondance de Langlands locale p-adique. Théorie des représentations [math.RT]. Ecole Polytechnique X, 2012. Français. ⟨NNT : 0g417e03nq 4⟩. ⟨pastel-00725370⟩

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