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

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 $(\phi ,\mathrm{\Gamma })$-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 ${\mathrm{G}\mathrm{L}}_n({\mathrm{Q}}_p)$-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.