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

Abstract:

Let $L$ be a finite extension of ${\mathbb Q}_p$, and $\rho _L$ be an $n$-dimensional semistable noncrystalline $p$-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 field $F^+$ 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 subrepresentation 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 $L$-invariants.