$G$-valued crystalline representations with minuscule $p$-adic Hodge type

Abstract

We study $G$-valued semistable Galois deformation rings, where $G$ is a reductive group. We develop a theory of Kisin modules with $G$-structure and use this to identify the connected components of crystalline deformation rings of minuscule $p$-adic Hodge type with the connected components of moduli of “finite flat models with $G$-structure”. The main ingredients are a construction in integral $p$-adic Hodge theory using Liu’s theory of $\left(\phi ,Ĝ\right)$-modules and the local models constructed by Pappas and Zhu.