For a finite extension $K/\mathbb{ Q}_p$ and a split reductive group $G$ over $\mathcal{ O}_K$, let $\overline{ \rho} \colon \mathrm{ Gal}_K \to G(\overline{ \mathbb{ F}}_p)$ be a continuous quasi-semisimple mod $p$ $G$-valued representation of the absolute Galois group $\mathrm{ Gal}_K$. Let $\overline{ \rho}^{ \mathrm{ ab}}$ be the abelianization of $\overline{ \rho}$ and fix a crystalline lift $\psi$ of $\overline{ \rho}^{ \mathrm{ ab}}$. We show the existence of a crystalline lift $\rho$ of $\overline{ \rho}$ with regular Hodge-Tate weights such that the abelianization of $\rho$ coincides with $\psi$. We also show analogous results in the case that $G$ is a quasi-split tame group and $\overline{ \rho} \colon \mathrm{ Gal}_K \to { ^L}G(\overline{ \mathbb{ F}}_p)$ is a semisimple mod $p$ $L$-parameter. These theorems are generalizations of those of Lin and B\"ockle-Iyengar-Pa\v{ s}k\={ u}nas.
Categories: math.NT