  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 $\rho$.
