Irreducibility of three dimensional automorphic Galois representations over totally real fields

Let ρ_ be a semisimple -adic representation of a number field K that is unramified almost everywhere. We introduce a new notion called weak abelian direct summands of ρ_ and completely characterize them, for example, if the algebraic monodromy of ρ_ is connected. If ρ_ is in addition E-rational for some number field E, we prove that the weak abelian direct summands are locally algebraic (and thus de Rham). We also show that the weak abelian parts of a connected semisimple Serre compatible system form again such a system. Using our results on weak abelian direct summands, when K is totally real and ρ_ is the three-dimensional -adic representation attached to a regular algebraic cuspidal automorphic, not necessarily polarizable representation π of GL₃(AK) together with an isomorphism C Q̄_, we prove that ρ_ is irreducible. We deduce in this case also some -adic Hodge theoretic properties of ρ_ if belongs to a Dirichlet density one set of primes.