Weak abelian direct summands and irreducibility of Galois representations

Abstract

Let $ρ_\ell$ be a semisimple $\ell$-adic representation of a number field $K$ that is unramified almost everywhere. We introduce a new notion called weak abelian direct summands of $ρ_\ell$ and completely characterize them, for example, if the algebraic monodromy of $ρ_\ell$ is connected. If $ρ_\ell$ 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 $ρ_\ell$ is the three-dimensional $\ell$-adic representation attached to a regular algebraic cuspidal automorphic, not necessarily polarizable representation $π$ of $\mathrm{GL}_3(\mathbb{A}_K)$ together with an isomorphism $\mathbb{C}\simeq \overline{\mathbb{Q}}_\ell$, we prove that $ρ_\ell$ is irreducible. We deduce in this case also some $\ell$-adic Hodge theoretic properties of $ρ_\ell$ if $\ell$ belongs to a Dirichlet density one set of primes.