Galois representations attached to representations of GU(3)

Abstract

For a fixed prime \(\ell\in{\mathbf Z}\) we compute the \(\ell\)-adic Lie algebra of the image of the \(\ell\)-adic Galois representation \(\rho\) attached to a stable cuspidal automorphic representation \(\pi\) of the unitary similitude group GU(3). This result depends on whether \(\pi\) admits extra twists in the sense defined below. Two cases emerge: orthogonal image and non-orthogonal image. We show that in the orthogonal case there exists a character \(\nu\) such that \(\rho\otimes\nu\) is the Galois representation attached to the unitary adjoint lift of a cuspidal representation of GL(2).