Weak abelian direct summands and irreducibility of Galois representations

Abstract

Let $\rho_\ell$ 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 $\rho_\ell$ and completely characterize them, for example, if the algebraic monodromy of $\rho_\ell$ is connected. If $\rho_\ell$ 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 $\rho_\ell$ is the three-dimensional $\ell$-adic representation attached to a regular algebraic cuspidal automorphic, not necessarily polarizable representation $\pi$ of $\mathrm{GL}_3(\mathbb{A}_K)$ together with an isomorphism $\mathbb{C}\simeq \overline{\mathbb{Q}}_\ell$, we prove that $\rho_\ell$ is irreducible. We deduce in this case also some $\ell$-adic Hodge theoretic properties of $\rho_\ell$ if $\ell$ belongs to a Dirichlet density one set of primes.