Irreducibility of Automorphic Galois Representations of Low Dimensions

Abstract

Let π be a polarizable, regular algebraic, cuspidal automorphic representation of \TextGLn(AF), where F is a CM field. We show that for n≤6, there is a Dirichlet density 1 set L of rational primes, such that for all l∈L, the l-adic Galois representations associated to π are irreducible. We also show that for any integer n≥1, in order to show the existence of the aforementioned set L, it suffices to show that for all but finitely many finite primes λ in a number field determined by π, all the irreducible constituents of the restriction of the corresponding Galois representation \rep to the derived subgroup of the identity component of the Zariski closure of the image, are conjugate self-dual.