Problems in the theory of automorphic forms

1. There has recently been much interest, if not a tremendous amount of progress, in the arithmetic theory of automorphic forms. In this lecture I would like to present the views not of a number theorist but of a student of group representations on those of its problems that he finds most fascinating. To be more precise, I want to formulate a series of questions which the reader may, if he likes, take as conjectures. I prefer to regard them as working hypotheses. They have already led to some interesting facts. Although they have stood up for a fair length of time to the most careful scrutiny I could give, I am still not entirely easy about them. Indeed even at the beginning in the course of the definitions, which I want to make in complete generality, I am forced, for lack of time and technical competence, to make various assumptions. I should perhaps apologize for such a speculative lecture. However, there are some in-teresting facts scattered amongst the questions. Moreover, the unsolved problems in group representations arising from the theory of automorphic forms are much less technical than the solved ones, and their significance can perhaps be more easily appreciated by the outsider. Suppose G is a connected reductive algebraic group defined over a global field F, whic