The Multiplicity One Theorem for GL n

The purpose of this paper is to establish preliminary results in the theory of representations in the direction of a systematic treatment of Hecke-theory for the group GLn. In particular, special functions, which may be called Whittaker functions, play a central role. Whether or not the further study of Whittaker functions will lead to definitive results concerning cusp forms for general groups is very much an open question at present. Needless to say, a development of Hecke-theory would necessarily follow the logical outline of the work of H. Jacquet and R. P. Langlands [161. It now seems possible that many of their results can be extended to GLU. This is also indicated by the recent work of I. M. Gelfand and D. A. Kajdan [7]. In the present paper we will be primarily concerned with the study of certain induced representations of the group G = GLn(k) where k is a local field. These representation are of the form