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

Antalya Algebra Days XVIII, İzmir, Türkiye, 18 - 22 Mayıs 2016, ss.8

In various arithmetic-geometric applications and in the theory

of automorphic forms there are open problems whose answer can

be reduced to a question about finite dimensional representations of

SL(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 groups

did until recently not have a satisfactory answer.

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 Dedekind domains. Amongst other things we

describe all linear characters of these groups SL(2, O). We show

how to use the general theory of Weil representations to construct

finite dimensional representations of these SL(2, O). We indicate

why these so constructed families of representations possibly con-

tain all finite dimensional representations with finite image of these

SL(2, O) (except for certain O). We finish with some open ques-

tions concerning the classification of the central extensions of these

SL(2, O) by the cyclic group of order 2.