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 theoryof automorphic forms there are open problems whose answer canbe reduced to a question about finite dimensional representations ofSL(2, O), where O is a maximal order in a number field or, more gen-erally, an arithmetic Dedekind domain. It is amazing that even nat-ural questions like for the group of linear characters of such groupsdid until recently not have a satisfactory answer.In the present talk we describe recent progress in the theoryof finite dimensional representations of SL(2, O) for a fairly largeclass of rings O comprising the rings of integers of local fields andarithmetic Dedekind Dedekind domains. Amongst other things wedescribe all linear characters of these groups SL(2, O). We showhow to use the general theory of Weil representations to constructfinite dimensional representations of these SL(2, O). We indicatewhy these so constructed families of representations possibly con-tain all finite dimensional representations with finite image of theseSL(2, O) (except for certain O). We finish with some open ques-tions concerning the classification of the central extensions of theseSL(2, O) by the cyclic group of order 2.