A group-theoretic generalization of the p-adic local monodromy theorem

Abstract

Abstract Let G be a connected reductive group over a p-adic number field F. We propose and study the notions of G- $\varphi $ -modules and G- $(\varphi ,\nabla )$ -modules over the Robba ring, which are exact faithful F-linear tensor functors from the category of G-representations on finite-dimensional F-vector spaces to the categories of $\varphi $ -modules and $(\varphi ,\nabla )$ -modules over the Robba ring, respectively, commuting with the respective fiber functors. We study Kedlaya’s slope filtration theorem in this context, and show that G- $(\varphi ,\nabla )$ -modules over the Robba ring are “G-quasi-unipotent,” which is a generalization of the p-adic local monodromy theorem proved independently by Y. André, K. S. Kedlaya, and Z. Mebkhout.