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

Abstract

Let be a split reductive group over the ring of integers in a -adic field with residue field. Fix a representation of the absolute Galois group of an unramified extension of, valued in. We study the crystalline deformation ring for with a fixed -adic Hodge type that satisfies an analog of the Fontaine-Laffaille condition for -valued representations. In particular, we give a root theoretic condition on the -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. © 2023 The Author(s).