A group-theoretic generalization of thep-adic local monodromy theorem

Abstract

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