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

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.