Antalya Algebra Days XVIII, İzmir, Turkey, 18 - 22 May 2016, pp.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.