Irreducible representations are the building blocks of general, semisimple Galois representations ��, and cuspidal representations are the building blocks of automorphic forms ��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 in either direction, and nothing much is known in dimensions >2. The main result of this article shows for n < 6, in particular, that the cuspidality of a regular algebraic ��is implied by the irreducibility of ��.

36 pages; three references and a remark added in the replacement; to appear in "Representation Theory and Automorphic Forms", edited by T.Kobayashi, W.Schmid and J.H.Yang, Birkh\"auser