Parabolic Simple \(\mathscr{L}\)-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 \(\mathscr{L}\)-invariants of \(\rho_L\), which was previously only known in the trianguline case. Based on the previous work on Breuil's parabolic simple \(\mathscr{L}\)-invariants, we attach to \(\rho_L\) a locally \(\mathbf{Q}_p\)-analytic representation \(\Pi(\rho_L)\) of \(\mathrm{GL}_{n}(L)\), which carries the information of parabolic simple \(\mathscr{L}\)-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 \(\mathrm{GL}_n\) at \(p\)-adic places), we prove under mild hypothesis that \(\Pi(\rho_L)\) 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 \(\mathscr{L}\)-invariants are equal to Fontaine-Mazur parabolic simple \(\mathscr{L}\)-invariants.