Let $K$ be a finite extension of $\mathbf {Q}_{p}$ , 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 $\mathbf {Q}_{p}$ -analytic representation $\pi _{1}(\rho )$ of $\operatorname{\mathrm {G}L}_{n}(K)$ , which encodes some $p$ -adic Hodge parameters of $\rho $ . When $K=\mathbf {Q}_{p}$ , 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 $\operatorname{\mathrm {G}L}_{n}(K)$ -representation globally associated to $\rho $ .
