Monodromy and irreducibility of type $A_1$ automorphic Galois representations

Abstract

Let $K$ be a totally real field and $\pi$ be a regular algebraic polarized cuspidal automorphic representation of ${\mathrm{GL}}_n({\mathbb{A}}_K)$. Let $\{{\rho }_{\pi ,\lambda }:{\mathrm{Gal}}_K\to {\mathrm{GL}}_n({\overline{E}}_{\lambda }){\}}_{\lambda }$ be the compatible system of Galois representations attached to $\pi$ and denote by ${\mathbf{G}}_{\lambda }$ the algebraic monodromy group of ${\rho }_{\pi ,\lambda }$. Suppose there exists ${\lambda }_0$ such that (a) ${\rho }_{\pi ,{\lambda }_0}$ is irreducible; (b) ${\mathbf{G}}_{{\lambda }_0}$ is connected and of type $A_1$; and (c) the tautological representation of ${\mathbf{G}}_{{\lambda }_0}$ is of a certain type. We prove that $\bullet$ ${\mathbf{G}}_{\lambda ,\mathbb{C}}\subset {\mathrm{GL}}_{n,\mathbb{C}}$ is independent of $\lambda$; $\bullet$ ${\rho }_{\pi ,\lambda }$ is irreducible for all $\lambda$, and residually irreducible for almost all $\lambda$. Moreover, if $K=\mathbb{Q}$ or $n$ is odd, we prove that the same conclusions hold without the assumption that $\pi$ is polarized. We also prove that if $K=\mathbb{Q}$, then the compatible system $\{{\rho }_{\pi ,\lambda }{\}}_{\lambda }$ is constructed from certain two-dimensional modular compatible systems up to twist.

14 pages