Seminar des SFB/TRR 326 GAUSMonodromy of subrepresentations and irreducibility of low degree automorphic Galois representations

Given a compatible system {ρ_λ : Gal_K → GL_n(E_λ)}_λ of semisimple λ-adic representations of a number field K satisfying mild local conditions, we prove that for almost all λ any type A irreducible subrepresentation of ρ_λ ⊗ Q_l is residually irreducible. We apply this result and some potential automorphy theorem to prove that ρ_λ ⊗ Q_l is residually irreducible for almost all λ if the compatible system is attached to a regular algebraic, polarized, cuspidal automorphic representation of GL_n(A_Q) and n ≤ 6.