Crystalline lifts and a variant of the Steinberg–Winter theorem

Abstract

Let $K/{\mathbb{Q}}_p$ be a finite extension. For all irreducible representations $\bar{\rho }:G_K\to G({\bar{\mathbb{F}}}_p)$ valued in a general reductive group $G$, we construct crystalline lifts of $\bar{\rho }$ which are Hodge–Tate regular. We also discuss rationality questions. We prove a variant of the Steinberg-Winter theorem along the way.