An exercise concerning the selfdual cusp forms on GL(3)

Abstract

Using L-functions and various known results, we provide a proof of the following Let F be a number field and II a cuspidal automorphic form on GL(3)/F which is selfdual. Then, up to replacing II by a quadratic twist, it can be realized as the adjoint of a cusp form π on GL(2)/F, with π unramified at any prime where II is. We also investigate the properties of π when II is regular and algebraic.