Parabolic Simple $\mathcal{L}$-invariants and Local-global Compatibility
Time: 2022-11-11
Published By: He Liu
Speaker(s): Yiqin He (PKU)
Time: 11:00-12:00 November 14, 2022
Venue: Room 77201, Jingchunyuan 78, BICMR
Let $L$ be a finite extension of $\mathbb{Q}_p$ and $\rho_L$ be a potentially semistable noncrystalline $p$-adic representation of $\mathrm{Gal}_L$ such that the associated $F$-semisimple Weil-Deligne representation is absolutely indecomposable. Via a study of Breuil's parabolic 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 parabolic simple $\mathcal{L}$-invariants of $\rho_L$. When $\rho_L$ comes from a patched 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 subspace of the Banach spaces of (patched) $p$-adic automophic forms on $G(\mathbb{A}_{F^+})$.
