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 $\textbf{G}_2$. (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.