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.