Irreducibility of Automorphic Galois Representations of Low Dimensions

Abstract

Let $\pi$ be a polarizable, regular algebraic, cuspidal automorphic representation of $\Text{GL}_n(\mathbb{A}_F)$, where $F$ is a CM field.

We show that for $n\leq 6$, there is a Dirichlet density 1 set $\mathfrak{L}$ of rational primes, such that for all $l\in\mathfrak{L}$, the $l$-adic Galois representations associated to $\pi$ are irreducible.

We also show that for any integer $n\geq 1$, in order to show the existence of the aforementioned set $\mathfrak{L}$, it suffices to show that for all but finitely many finite primes $\lambda$ in a number field determined by $\pi$, 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.