Irreducibility of polarized automorphic Galois representations in infinitely many dimensions
Zachary Feng, Dmitri Whitmore
TL;DR
This work establishes that for polarized regular algebraic automorphic representations π of GL_n over CM or totally real fields, the associated ℓ-adic Galois representations form a weakly compatible system whose members are irreducible for a Dirichlet density one set of primes, under the divisibility constraints 7 ∤ n and, when 4 | n, n = 4p. The authors develop a multi-step strategy: reduce to a CM setting via Xia’s method, factor the Galois representation into tensor pieces with simple Lie algebras at a density-one prime, and then apply potential automorphy theorems to lift each tensor factor to automorphic representations. The formal-character and complex-conjugation analysis of the tensor factors, together with type-A reductions, constrains the possible monodromy and enables a density-one irreducibility conclusion, while also yielding residual irreducibility results. The paper also clarifies the method’s limitations and the precise algebraic constraints needed for the approach to succeed, contributing significant progress toward a broad irreducibility statement in the Langlands program. Overall, the work advances understanding of when Galois representations attached to polarized automorphic forms are irreducible in infinitely many dimensions and sharpened by potential automorphy and Lie-theoretic techniques.
Abstract
Let $π$ be a polarized, regular algebraic, cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb{A}_F)$ where $F$ is totally real or imaginary CM, and let $(ρ_λ)_λ$ be its associated compatible system of Galois representations. Suppose that $7\nmid n$ and, if $4\mid n$, then $n = 4p$ for some prime number $p$. We prove that there is a Dirichlet density $1$ set of rational primes $\mathcal{L}$ such that whenever $λ\mid \ell$ for some $\ell\in \mathcal{L}$, then $ρ_λ$ is irreducible.