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

Abstract

Let $ρ_p$ be an $n$-dimensional non-critical semistable $p$-adic Galois representation of the absolute Galois group of $\mathbf{Q}_p$ with regular Hodge--Tate weights. Let $\mathbf{D}$ be the associated $(φ,Γ)$-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 $ρ_p$ on the automorphic side by considering several Steinberg subquotients of $\mathbf{D}$ 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 $π_{1}(ρ_p)$ and explicitly describe which Hodge-parameters information of $ρ_p$ it determines. In particular, if the monodromy rank of $ρ_p$ is at most $1$, $π_{1}(ρ_p)$ determines $ρ_p$. When $ρ_p$ comes from a $p$-adic automorphic representation, we show that $π_{1}(ρ_p)$ is a subrepresentation of the $\mathrm{GL}_n(\mathbf{Q}_p)$-representation globally associated to $ρ_p$, under mild hypotheses. Although it is still difficult to construct an explicit representation $π_{1}(ρ_p)$ that determines $ρ_p$, 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.\;