On irreducibility of certain low dimensional automorphic Galois representations

Abstract

We study irreducibility of Galois representations ${\rho }_{\pi ,\lambda }$ associated to a $n=7$ or 8-dimensional regular algebraic essentially self-dual cuspidal automorphic representation $\pi$ of ${\text{GL}}_n({\mathbb{A}}_{\mathbb{Q}})$. We show ${\rho }_{\pi ,\lambda }$ is irreducible for all but finitely many $\lambda$ under the following extra conditions. (i) If $n=7$, and there exists no $\lambda$ such that the Lie type of ${\rho }_{\pi ,\lambda }$ is the standard representation of exceptional group ${\text{G}}_2$. (ii) If $n=8$, and when there exist infinitely many $\lambda$ such that the Lie type of ${\rho }_{\pi ,\lambda }$ is the spin representation of ${\text{SO}}_7$, we assume there exist no three distinct Hodge-Tate weights form a 3-term arithmetic progression.

16 pages