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

Let $p$ be a prime number, $n$ an integer $\geq 2$, and $L$ a finite extension of $\mathrm{Q}_p$. Let $\rho_L$ be an $n$-dimensional (non-critical but not necessary generic) potentially crystalline $p$-adic Galois representation of the absolute Galois groups of $L$ of regular Hodge-Tate weights. By generalizing the previous results and strategy for the crystabelline case of Ding and the recent work of Breuil-Ding, we construct an explicit locally analytic representation $\pi_{1}(\rho_L)$, and describe explicitly the information of Hodge filtration of $\rho_L$ it determines. When $\rho_L$ comes from a patched $p$-adic automorphic representation, we show that $\pi_{1}(\rho_L)$ is a subrepresentation of the $\mathrm{GL}_n(L)$-representation globally associated to $\rho_L$, under some mild hypothesis.