Relative deformation theory, relative Selmer groups, and lifting irreducible Galois representations

We study irreducible odd mod $p$ Galois representations $\bar{\rho} \colon $\mathrm{G}\mathrm{a}\mathrm{l}$(\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.