$p$-adic Hodge parameters in the crystabelline representations of $\mathrm{GL}_n$

Abstract

Let $K$ be a finite extension of $\mathbb{Q}_p$, and $ρ$ be an $n$-dimensional (non-critical generic) crystabelline representation of the absolute Galois group of $K$ of regular Hodge-Tate weights. We associate to $ρ$ an explicit locally $\mathbb{Q}_p$-analytic representation $π_1(ρ)$ of $\mathrm{GL}_n(K)$, which encodes some $p$-adic Hodge parameters of $ρ$. When $K=\mathbb{Q}_p$, it encodes the full information hence reciprocally determines $ρ$. When $ρ$ is associated to $p$-adic automorphic representations, we show under mild hypotheses that $π_1(ρ)$ is a subrepresentation of the $\mathrm{GL}_n(K)$-representation globally associated to $ρ$.