Parabolic Simple $\mathcal{L}$-Invariants

Let $L$ be a finite extension of ${\mathbf{Q}}_p$. Let ${\rho }_L$ be a potentially semi-stable non-crystalline $p$-adic Galois representation such that the associated $F$-semisimple Weil-Deligne representation is absolutely indecomposable. In this paper, we study Fontaine-Mazur parabolic simple $\mathcal{L}$-invariants of ${\rho }_L$, which was previously only known in the trianguline case. Based on the previous work on Breuil’s parabolic simple $\mathcal{L}$-invariants, we attach to ${\rho }_L$ a locally ${\mathbf{Q}}_p$-analytic representation $\mathrm{\Pi }({\rho }_L)$ of ${\mathrm{G}\mathrm{L}}_n(L)$, which carries the information of parabolic simple $\mathcal{L}$-invariants of ${\rho }_L$. When ${\rho }_L$ comes from a patched automorphic representation of $\mathbf{G}({\mathbb{A}}_{F^{+}})$ (for a define unitary group $\mathbf{G}$ over a totally real field $F^{+}$ which is compact at infinite places and ${\mathrm{G}\mathrm{L}}_n$ at $p$-adic places), we prove under mild hypothesis that $\mathrm{\Pi }({\rho }_L)$ is a subrepresentation of the associated Hecke-isotypic subspace of the Banach spaces of (patched) $p$-adic automophic forms on $\mathbf{G}({\mathbb{A}}_{F^{+}})$, this is equivalent to say that the Breuil’s parabolic simple $\mathcal{L}$-invariants are equal to Fontaine-Mazur parabolic simple $\mathcal{L}$-invariants.