Crystalline lifts of semisimple $G$-valued Galois representations with fixed determinant

Abstract

For a finite extension $K/\mathbb{Q}_p$ and a split reductive group $G$ over $\mathcal{O}_K$, let $\overlineρ \colon \mathrm{Gal}_K \to G(\overline{\mathbb{F}}_p)$ be a continuous quasi-semisimple mod $p$ $G$-valued representation of the absolute Galois group $\mathrm{Gal}_K$. Let $\overlineρ^{\mathrm{ab}}$ be the abelianization of $\overlineρ$ and fix a crystalline lift $ψ$ of $\overlineρ^{\mathrm{ab}}$. We show the existence of a crystalline lift $ρ$ of $\overlineρ$ with regular Hodge-Tate weights such that the abelianization of $ρ$ coincides with $ψ$. We also show analogous results in the case that $G$ is a quasi-split tame group and $\overlineρ \colon \mathrm{Gal}_K \to {^L}G(\overline{\mathbb{F}}_p)$ is a semisimple mod $p$ $L$-parameter. These theorems are generalizations of those of Lin and Böckle-Iyengar-Paškūnas.