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

We study G-valued semi-stable 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 $(\varphi, \widehat{G})$-modules and the local models constructed by Pappas and Zhu