G-valued crystalline deformation rings in the Fontaine–Laffaille range

Abstract

Let$G$be a split reductive group over the ring of integers in a$p$-adic field with residue field$\mathbf {F}$. Fix a representation$\overline {\rho }$of the absolute Galois group of an unramified extension of$\mathbf {Q}_p$, valued in$G(\mathbf {F})$. We study the crystalline deformation ring for$\overline {\rho }$with a fixed$p$-adic Hodge type that satisfies an analog of the Fontaine–Laffaille condition for$G$-valued representations. In particular, we give a root theoretic condition on the$p$-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups.