Monodromy of subrepresentations and irreducibility of low degree automorphic Galois representations

Abstract

Let $X$ be a smooth, separated, geometrically connected scheme defined over a number field $K$ and $\{ρ_λ\}_λ$ a system of n-dimensional semisimple $λ$-adic representations of the étale fundamental group of $X$ such that for each closed point $x$ of $X$, the specialization $\{ρ_{λ,x}\}_λ$ is a compatible system of Galois representations under mild local conditions. For almost all $λ$, we prove that any type A irreducible subrepresentation of $ρ_λ\otimes \bar{\mathbb{Q}}_\ell$ is residually irreducible. When $K$ is totally real or CM, $n\leq 6$, and $\{ρ_λ\}_λ$ is the compatible system of Galois representations of $K$ attached to a regular algebraic, polarized, cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_K)$, for almost all $λ$ we prove that $ρ_λ\otimes\bar{\mathbb{Q}}_\ell$ is (i) irreducible and (ii) residually irreducible if in addition $K=\mathbb{Q}$.