EIGENVARIETIES AND INVARIANT NORMS

We give a proof of the Breuil-Schneider conjecture in a large number of cases, which complement the indecomposable case, which we dealt with earlier. In this paper, we view the conjecture from a broader global perspective. If U-/F is any definite unitary group, which is an inner form of GL(n) over kappa, we point out how the eigenvariety X(K-p) parametrizes a global p-adic Langlands correspondence between certain n-dimensional p-adic semisimple representations rho of Gal((Q) over bar/kappa) (or what amounts to the same, pseudorepresentations) and certain Banach-Hecke modules B with an admissible unitary action of U(F circle times Q(p)), when p splits. We express the locally regular-algebraic vectors of B in terms of the Breuil-Schneider representation of rho. As an application, we give a weak form of local-global compatibility in the crystalline case, showing that the Banach space representations B-xi,B-zeta of Schneider and Teitelbaum fit the picture as predicted.