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 $G$-$\phi$-modules and $G$-$(\phi ,\mathrm{\nabla })$-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 $\phi$-modules and $(\phi ,\mathrm{\nabla })$-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 $G$-$(\phi ,\mathrm{\nabla })$-modules over the Robba ring are “$G$-quasi-unipotent”, which is a generalization of the $p$-adic local monodromy theorem proven independently by Y. Andr'e, K. S. Kedlaya, and Z. Mebkhout.