Simple $\mathcal{L}$-invariants for ${\mathrm{GL}}_n$

Abstract

Let $L$ be a finite extension of ${\mathbb{Q}}_p$, and ${\rho }_L$ be an $n$-dimensional semi-stable non crystalline $p$-adic representation of ${\mathrm{Gal}}_L$ with full monodromy rank. Via a study of Breuil's (simple) $\mathcal{L}$-invariants, we attach to ${\rho }_L$ a locally ${\mathbb{Q}}_p$-analytic representation $\Pi ({\rho }_L)$ of ${\mathrm{GL}}_n(L)$, which carries the exact information of the Fontaine-Mazur simple $\mathcal{L}$-invariants of ${\rho }_L$. When ${\rho }_L$ comes from an automorphic representation of $G({\mathbb{A}}_{F^{+}})$ (for a unitary group $G$ over a totally real filed $F^{+}$ which is compact at infinite places and ${\mathrm{GL}}_n$ at $p$-adic places), we prove under mild hypothesis that $\Pi ({\rho }_L)$ is a subrerpresentation of the associated Hecke-isotypic subspaces of the Banach spaces of $p$-adic automorphic forms on $G({\mathbb{A}}_{F^{+}})$. In other words, we prove the equality of Breuil's simple $\mathcal{L}$-invariants and Fontaine-Mazur simple $\mathcal{L}$-invariants.

Final version