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 interesting 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 , which is then an algebraic number field or a function field in one variable over a finite field. Let A(F ) be the adèle ring of F . The topological group GA(F ) is then locally compact with GF as a discrete subgroup. The group GA(F ) acts on the functions on GF\GA(F ). In particular, it acts on L(GF\GA(F )). It should be possible, although I have not done so and it is not important at this stage, to attach a precise meaning to the assertion that a given irreducible representation π of GA(F ) occurs in L (GF\GA(F )). If G is abelian it would mean that π is a character of GF\GA(F ). If G is not abelian it would be true for at least those representations which act on an irreducible invariant subspace of L(GF )\GA(F ). If G is GL(1) then to each such π one, following Hecke, associates an L-function. If G is GL(2) then Hecke has also introduced, without explicitly mentioning group representations, some L-functions. The problems I want to discuss center about the possibility of defining L-functions for all such π and proving that they have the analytic properties we have grown used to expecting of such functions. I shall also comment on the possible relations of these new functions to Artin L-functions and the L-functions attached to algebraic varieties. Given G I am going to introduce the complex analytic group ĜF . To each complex analytic representation σ of ĜF and each π I want to attach an L-function L(s, σ, π). Let me say a few words about the general way in which I want to form the function. The group GA(F ) is a restricted direct product ∐ pGFp . The product is taken over the primes, finite and infinite,