Irreducibility of polarized automorphic Galois representations in infinitely many dimensions

TL;DR

This paper proves that for a broad class of automorphic representations, the associated Galois representations are irreducible for a density-one set of primes, extending understanding of their structural properties in infinitely many dimensions.

Contribution

It establishes irreducibility of Galois representations associated to polarized automorphic forms for almost all primes under specific dimension conditions.

Findings

Irreducibility holds for a density-one set of primes.

Conditions on the dimension n are crucial for the result.

The result applies to automorphic representations over totally real or CM fields.

Abstract

Let $\pi$ 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 $(\rho_\lambda)_\lambda$ 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 $\lambda\mid \ell$ for some $\ell\in \mathcal{L}$, then $\rho_\lambda$ is irreducible.