Monodromy of subrepresentations and irreducibility of lowdegree automorphic Galois representations

Let X$X$ be a smooth, separated, geometrically connected scheme defined over a number field K$K$ and {& rho;& lambda;:& pi;1(X)& RARR;GLn(E & lambda;)}& lambda;$\lbrace \rho _\lambda :\pi _1(X)\rightarrow \mathrm{GL}_n(E_\lambda )\rbrace _\lambda$ a system of semisimple & lambda;$\lambda$-adic representations of the etale fundamental group of X$X$ such that for each closed point x$x$ of X$X$, the specialization {& rho;& lambda;,x}& lambda;$\lbrace \rho _{\lambda ,x}\rbrace _\lambda$ is a compatible system of Galois representations under mild local conditions. For almost all & lambda;$\lambda$, we prove that any type A irreducible subrepresentation of & rho;& lambda;& OTIMES;E & lambda;Q over bar l$\rho _\lambda \otimes _{E_\lambda } \overline{\mathbb {Q}}_\ell$ is residually irreducible. When K$K$ is totally real or CM, n & LE;6$n\leqslant 6$, and {& rho;& lambda;}& lambda;$\lbrace \rho _\lambda \rbrace _\lambda$ is the compatible system of Galois representations of K$K$ attached to a regular algebraic, polarized, cuspidal automorphic representation of GLn(AK)$\mathrm{GL}_n(\mathbb {A}_K)$, for almost all & lambda;$\lambda$, we prove that & rho;& lambda;& OTIMES;E & lambda;Q over bar l$\rho _\lambda \otimes _{E_\lambda }\overline{\mathbb {Q}}_\ell$ is (i) irreducible and (ii) residually irreducible if in addition K=Q$K=\mathbb {Q}$.