Irreducibility of polarized automorphic Galois representations in infinitely many dimensions

Abstract

Let \( \pi \) be a polarized, regular algebraic, cuspidal automorphic representation of \( \GL_n(\bb{A}_F) \) where \( F \) is totally real or imaginary CM, and let \( (\rho_\lambda)_\lambda \) be its associated compatible system of Galois representations. We prove that if \( 7\nmid n \) and \( 4 \nmid n \) then there is a Dirichlet density \( 1 \) set of rational primes \( \mc{L} \) such that whenever \( \lambda\mid \ell \) for some \( \ell\in \mc{L} \), then \( \rho_\lambda \) is irreducible.

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