Eigenvarieties and invariant norms: Towards p-adic Langlands for U(n)

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 [Sor]. In some sense, only the Steinberg representation lies at the intersection of the two approaches. 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 $\K$, 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 $\Gal(\bar{\Q}|\K)$ (or what amounts to the same, pseudo-representations) and certain Banach-Hecke modules $\mathcal{B}$ with an admissible unitary action of $U(F\otimes \Q_p)$, when $p$ splits. We express the locally regular-algebraic vectors of $\mathcal{B}$ in terms of the Breuil-Schneider representation of $\rho$. Upon completion, this produces a candidate for the $p$-adic local Langlands correspondence in this context. As an application, we give a weak form of local-global compatibility in the crystalline case, showing that the Banach space representations $B_{\xi,\zeta}$ of Schneider-Teitelbaum [ScTe] fit the picture as predicted. There is a compatible global mod $p$ (semisimple) Langlands correspondence parametrized by $\X(K^p)$. We introduce a natural notion of refined Serre weights, and link them to the existence of crystalline lifts of prescribed Hodge type and Frobenius eigenvalues. At the end, we give a rough candidate for a local mod $p$ correspondence, formulate a local-global compatibility conjecture, and explain how it implies the conjectural Ihara lemma in [CHT].