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

Let $\rho_p$ be a 3-dimensional semi-stable representation of $\operatorname{Gal} (\overline{\mathbb{Q}_p} / \mathbb{Q}_p)$ with Hodge–Tate weights $(0, 1, 2)$ (up to shift) and such that $N^2 \neq 0$ 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 $G(\mathbb{A}^{\infty}_{F^+})$ of arbitrary fixed tame level contain (copies of) a unique admissible finite length locally analytic representation of $\mathrm{GL}_3 (\mathbb{Q}_p)$ of the form considered in [4] which only depends on and completely determines $\rho_p$.

Keywords

$p$-adic Langlands programme, $\mathcal{L}$-invariants, local-global compatibility

2010 Mathematics Subject Classification

11S23, 22E35, 22E50

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C. B. was supported by the C.N.R.S. and by Université Paris-Saclay.

Y. D. was supported by E.P.S.R.C. Grant EP/L025485/1 and by Grant No. 8102600240 from B.I.C.M.R.

Received 1 October 2018

Published 11 December 2020