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.