Hodge filtration and crystalline representations of $\mathrm{GL}_n$

Abstract

Let $p$ be a prime number, $n$ an integer $\geq 2$ and $\rho$ an $n$-dimensional automorphic $p$-adic Galois representation (for a compact unitary group) such that $r:=\rho\vert_{\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)}$ is crystalline. Under a mild assumption on the Frobenius eigenvalues of $D:=D_{\mathrm{cris}}(r)$ and under the usual Taylor-Wiles conditions, we show that the locally analytic representation of $\mathrm{GL}_n(\mathbb{Q}_p)$ associated to $\rho$ in the corresponding Hecke eigenspace of the completed $H^0$ contains an explicit finite length subrepresentation which determines and only depends on $r$. This generalizes previous results of the second author which assumed that the Hodge filtration on $D$ was as generic as possible. Our approach provides a much more explicit link to this Hodge filtration (in all cases), which allows to study the internal structure of this finite length locally analytic subrepresentation.