(ρ,ρ)-Galois representations attached to automorphic forms on GL_n

Abstract

We study the local reducibility at ρ of the ρ-adic Galois representation attached to a cuspidal automorphic representation of GL_n(A_Q). In the case that the underlying Weil–Deligne representation is Frobenius semisimple and indecomposable, we analyze the reducibility completely. We use methods from ρ-adic Hodge theory, and work under a transversality assumption on the Hodge and Newton filtrations in the corresponding filtered module.