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

Abstract

Let $K$ be a finite extension of ${\mathbb{Q}}_p$, and $\rho$ be an $n$-dimensional (non-critical generic) crystabelline representation of the absolute Galois group of $K$ of regular Hodge-Tate weights. We associate to $\rho$ an explicit locally ${\mathbb{Q}}_p$-analytic representation ${\pi }_1(\rho )$ of ${\mathrm{GL}}_n(K)$, which encodes some $p$-adic Hodge parameters of $\rho$. When $K={\mathbb{Q}}_p$, it encodes the full information hence reciprocally determines $\rho$. When $\rho$ is associated to $p$-adic automorphic representations, we show under mild hypotheses that ${\pi }_1(\rho )$ is a subrepresentation of the ${\mathrm{GL}}_n(K)$-representation globally associated to $\rho$.

final version