Higher $\mathcal{L}$-invariants for ${\mathrm{GL}}_3({\mathbb{Q}}_p)$ and local-global compatibility

Abstract

Let ${\rho }_p$ be a $3$-dimensional $p$-adic semi-stable representation of $\mathrm{Gal}(\overline{{\mathbb{Q}}_p}/{\mathbb{Q}}_p)$ with Hodge-Tate weights $(0,1,2)$ (up to shift) and such that $N^2\ne 0$ on $D_{\mathrm{st}}({\rho }_p)$. When ${\rho }_p$ comes from an automorphic representation $\pi$ of $G({\mathbb{A}}_{F^{+}})$ (for a unitary group $G$ over a totally real field $F^{+}$ which is compact at infinite places and ${\mathrm{GL}}_3$ at $p$-adic places), we show under mild genericity assumptions that the associated Hecke-isotypic subspaces of the Banach spaces of $p$-adic automorphic forms on $G({\mathbb{A}}_{F^{+}}^{\infty })$ of arbitrary fixed tame level contain (copies of) a unique admissible finite length locally analytic representation of ${\mathrm{GL}}_3({\mathbb{Q}}_p)$ which only depends on and completely determines ${\rho }_p$.

139 pages