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

Let $G$ be a connected reductive group over a $p$-adic local field $F$. We propose and study the notions of $G$-$varphi$-modules and $G$-$(varphi, abla)$-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 $(varphi, abla)$-modules over the Robba ring, respectively, commuting with the respective fiber functors. We study Kedlayas slope filtration theorem in this context, and show that $G$-$(varphi, abla)$-modules over the Robba ring are $G$-quasi-unipotent, which is a generalization of the $p$-adic local monodromy theorem proven independently by Y. Andre, K. S. Kedlaya, and Z. Mebkhout.