Monodromy of subrepresentations and irreducibility of lowdegree automorphic Galois representations

Abstract

Let (Formula presented.) be a smooth, separated, geometrically connected scheme defined over a number field (Formula presented.) and (Formula presented.) a system of semisimple (Formula presented.) -adic representations of the étale fundamental group of (Formula presented.) such that for each closed point (Formula presented.) of (Formula presented.), the specialization (Formula presented.) is a compatible system of Galois representations under mild local conditions. For almost all (Formula presented.), we prove that any type A irreducible subrepresentation of (Formula presented.) is residually irreducible. When (Formula presented.) is totally real or CM, (Formula presented.), and (Formula presented.) is the compatible system of Galois representations of (Formula presented.) attached to a regular algebraic, polarized, cuspidal automorphic representation of (Formula presented.), for almost all (Formula presented.), we prove that (Formula presented.) is (i) irreducible and (ii) residually irreducible if in addition (Formula presented.).