Monodromy and irreducibility of type $A_1$ automorphic Galois representations (2407.12566v2)

Abstract: Let $K$ be a totally real field and ${\rho_{\pi,\lambda}:\mathrm{Gal}K\to\mathrm{GL}_n(\overline E\lambda)}\lambda$ the strictly compatible system of $K$ defined over $E$ attached to a regular algebraic polarized cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n(\mathbb A_K)$. Let $\mathbf G\lambda$ be the algebraic monodromy group of $\rho_{\pi,\lambda}$. If 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) either at most one basic factor in the exterior tensor decomposition of the tautological representation of $\mathbf G_{\lambda_0}^{\mathrm{der}}$ is $(\mathrm{SL}2,\mathrm{std})$ or the tautological representation of $\mathbf G{\lambda_0}^{\mathrm{der}}$ is $(\mathrm{SO}4,\mathrm{std})$, we prove that $\mathbf G{\lambda,\mathbb C}\subset\mathrm{GL}{n, \mathbb C}$ is independent of $\lambda$, $\rho{\pi,\lambda}$ is irreducible for all $\lambda$ and residually irreducible for almost all $\lambda$. If moreover $K=\mathbb Q$, we prove that the compatible system ${\rho_{\pi,\lambda}}_\lambda$ is up to twist constructed from some two-dimensional modular compatible systems.