Simple ℒ-invariants for GL_{ }

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