Irreducibility and cuspidality

Irreducible representations are the building blocks of general, semisimple Galois representations ρ, and cuspidal representations are the building blocks of automorphic forms π of the general linear group. It is expected that when an object of the former type is associated to one of the latter type, usually in terms of an identity of L-functions, the irreducibility of the former should imply the cuspidality of the latter, and vice-versa. It is not a simple matter at all to prove this expectation, and nothing much is known in dimensions > 2. We will start from the beginning and explain the problem below, and indicate a result (in one direction) at the end of the Introduction, which summarizes what one can do at this point. The remainder of the paper will be devoted to showing how to deduce this result by a synthesis of known theorems and some new ideas. We will be concerned here only with the so called easier direction of showing the cuspidality of π given the irreducibility of ρ, and refer to [Ra5] for a more difficult result going the other way, which uses crystalline representations as well as a refinement of certain deep modularity results of Taylor, Skinner-Wiles, et al. Needless to say, easier does not mean easy, and the significance of the problem stems from the fact that it does arise (in this direction) naturally. For example, π could be a functorial, automorphic image r(η), for η a cuspidal automorphic representation of a product of smaller general linear groups: H(A) = ∏ j GL(mj ,A), with an associated Galois representation σ such that ρ = r(σ) is irreducible. If the automorphy of π has been established by using a flexible converse theorem ([CoPS]),