Representations of SL 2 over rings of integers of local fields, and over arithmetic Dedekind domains.

Abstract

In various arithmetic-geometric applications and in the theory of automorphic forms there are open problemswhose answer can be reduced to a question about finite dimensional representations of SL(2, O), where O is a3maximal order in a number field or, more generally, an arithmetic Dedekind domain. It is amazing that evennatural questions like for the group of linear characters of such groups did until recently not have a satisfactoryanswer.In the present talk we describe recent progress in the theory of finite dimensional representations of SL(2, O)for a fairly large class of rings O comprising the rings of integers of local fields and arithmetic Dedekind Dedekinddomains. Amongst other things we describe all linear characters of these groups SL(2, O). We show how to usethe general theory of Weil representations to construct finite dimensional representations of these SL(2, O). Weindicate why these so constructed families of representations possibly contain all finite dimensional representa-tions with finite image of these SL(2, O) (except for certain O). We finish with some open questions concerningthe classification of the central extensions of these SL(2, O) by the cyclic group of order 2.