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

Abstract

Let $p$ be a prime number, $n\geq 2$, and let $L$ be a finite extension of $\mathbf{Q}_p$. Let $\rho_L$ be an $n$-dimensional non-critical generic potentially crystalline $p$-adic representation of the absolute Galois group of $L$ with regular Hodge--Tate weights. Building on Ding's results and strategy in the crystabelline case, and on the recent work of Breuil--Ding in the critical crystalline case, we construct an explicit locally analytic representation $\pi_{1}(\rho_L)$ and describe explicitly the Hodge-filtration data of $\rho_L$ that it determines. When $\rho_L$ arises from a patched $p$-adic automorphic representation, we show, under mild hypotheses, that $\pi_{1}(\rho_L)$ is a subrepresentation of the $\mathrm{GL}_n(L)$-representation globally associated with $\rho_L$ by using the framework of Bernstein eigenvarieties that were developed by Breuil-Ding.