Towards the $p$-adic Hodge parameters in semistable representations of $\mathrm{GL}_n(\mathrm{Q}_p)$

Let $\rho_p$ be an $n$-dimensional non-critical semistable $p$-adic Galois representation of the absolute Galois group of $\mathrm{Q}_p$ with regular Hodge--Tate weights. Let $\mathrm{D}$ be the associated $(\varphi,\Gamma)$-module over the Robba ring. By combining Ding's and Breuil--Ding's methods for the crystalline case with Qian's computation of higher extension groups of locally analytic generalized Steinberg representations, we capture the full information of the $p$-adic Hodge parameters of $\rho_p$ on the automorphic side by considering several Steinberg subquotients of $\mathrm{D}$ and the "crystalline" Hodge parameters between them. These results also admit geometric and Lie-algebraic reformulations on flag varieties related to the moduli space of Hodge parameters. We then construct an explicit locally analytic representation $\pi_{1}(\rho_p)$ and explicitly describe which Hodge-parameters information of $\rho_p$ it determines. In particular, if the monodromy rank of $\rho_p$ is at most $1$, $\pi_{1}(\rho_p)$ determines $\rho_p$. When $\rho_p$ comes from a $p$-adic automorphic representation, we show that $\pi_{1}(\rho_p)$ is a subrepresentation of the $\mathrm{GL}_n(\mathrm{Q}_p)$-representation globally associated to $\rho_p$, under mild hypotheses. Although it is still difficult to construct an explicit representation $\pi_{1}(\rho_p)$ that determines $\rho_p$, our results provide new evidence for the $p$-adic Langlands program in general semistable cases and demonstrate the broad applicability of Ding's, Breuil--Ding's, and Qian's methods.