Lifting \(G\)-irreducible but \(\mathrm{GL}_n\)-reducible Galois representations (Q1996412)

In the present paper, the authors study the lifting \(G\)-irreducible odd Galois representations \(\text{Gal}(\overline{F}/F) \rightarrow G(\overline{F}_{\ell})\), with \(F\) a totally real number field and \(G\) a reductive group, to geometric \(\ell\)-adic representations. They discuss examples when \(G\) is a classical group and the Galois-theoretic methods apply, but current potential automorphy methods do not. In particular, they treat an example arising from orthogonal representations \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{SO}_N(\overline{\mathbb{F}}_{\ell})\) constructed in \textit{D. Zywina}'s work on the inverse Galois problem [``The inverse Galois problem for orthogonal groups'', Preprint, \url{arXiv:1409.1151}].