On irreducibility of certain low dimensional automorphic Galois representations
Boyi Dai
TL;DR
The paper addresses irreducibility of automorphic Galois representations $ ho_{ ho, ho}$ attached to regular algebraic essentially self-dual cuspidal representations of $ ext{GL}_n$ with $n=7$ or $8$. It develops a framework of $oldsymbol{ ext{ L }}$-independent compatible systems, leverages big-image and p-adic Hodge lift techniques, and reduces the problem to the Lie-irreducible case via a Xia-style reduction. Through a detailed, case-by-case elimination of possible Lie-type decompositions of the derived monodromy groups, the authors prove that $ ho_{ ho, ho}$ is irreducible for all but finitely many primes $ ho$ under the stated extra conditions (no $ ext{G}_2$-standard type for $n=7$, and a no-three-term-AP condition on Hodge–Tate weights when infinitely many $ ho_{ ho, ho}$ are of spin type for $n=8$). The results advance understanding of irreducibility patterns for low-dimensional automorphic Galois representations and showcase a robust methodology combining compatible-systems techniques, Lie-theoretic classifications, and Goursat-type arguments. The work has potential implications for automorphy lifting, potential automorphy, and the structure of algebraic monodromy groups in small dimensions.
Abstract
We study irreducibility of Galois representations $ρ_{π,λ}$ associated to a $n=7$ or 8-dimensional regular algebraic essentially self-dual cuspidal automorphic representation $π$ of $\text{GL}_n(\mathbb{A}_\mathbb{Q})$. We show $ρ_{π,λ}$ is irreducible for all but finitely many $λ$ under the following extra conditions. (i) If $n=7$, and there exists no $λ$ such that the Lie type of $ρ_{π,λ}$ is the standard representation of exceptional group $\textbf{G}_2$. (ii) If $n=8$, and when there exist infinitely many $λ$ such that the Lie type of $ρ_{π,λ}$ is the spin representation of $\text{SO}_7$, we assume there exist no three distinct Hodge-Tate weights form a 3-term arithmetic progression.