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

TL;DR

This paper links the Hodge filtration of crystalline Galois representations to the structure of associated locally analytic representations of GL_n, generalizing previous generic cases and providing explicit structural insights.

Contribution

It establishes a precise connection between the Hodge filtration of crystalline Galois representations and the structure of the corresponding locally analytic GL_n representations, extending prior generic results.

Findings

Explicit finite length subrepresentation determined by the Galois representation

Generalization beyond generic Hodge filtration cases

Enhanced understanding of the internal structure of the subrepresentation

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.