Lifting G-irreducible but GLn-reducible Galois representations

Abstract

In recent work, the authors proved a general result on lifting Girreducible odd Galois representations Gal(F/F) → G(Fℓ), with F a totally real number field and G a reductive group, to geometric ℓ- adic representations. In this note we take G to be a classical group and construct many examples of G-irreducible representations to which these new lifting methods apply, but to which the lifting methods currently provided by potential automorphy theorems do not.