We study irreducible odd mod $p$ Galois representations $\bar{ \rho} \colon \mathrm{ Gal}(\overline{ F}/F) \to G(\overline{ \mathbb{ F}}_p)$, for $F$ a totally real number field and $G$ a general reductive group. For $p \gg_{ G, F} 0$, we show that any $\bar{ \rho}$ that lifts locally, and at places above $p$ to de Rham and Hodge-Tate regular representations, has a geometric $p$-adic lift. We also prove non-geometric lifting results without any oddness assumption.
Categories: math.NT