Let $X$ be a smooth, separated, geometrically connected scheme defined over a number field $K$ and $\{ \rho_\lambda\}_\lambda$ a system of n-dimensional semisimple $\lambda$-adic representations of the \'etale fundamental group of $X$ such that for each closed point $x$ of $X$, the specialization $\{ \rho_{ \lambda,x}\}_\lambda$ is a compatible system of Galois representations under mild local conditions. For almost all $\lambda$, we prove that any type A irreducible subrepresentation of $\rho_\lambda\otimes \bar{ \mathbb{ Q}}_\ell$ is residually irreducible. When $K$ is totally real or CM, $n\leq 6$, and $\{ \rho_\lambda\}_\lambda$ 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 $\lambda$ we prove that $\rho_\lambda\otimes\bar{ \mathbb{ Q}}_\ell$ is (i) irreducible and (ii) residually irreducible if in addition $K=\mathbb{ Q}$.
Categories: math.NT math.AG math.GR math.RT