Appears in collection : Groupes, géométrie et analyse : conférence en l'honneur des 60 ans d'Alain Valette

Let G be a countable infinite group. Unless G is virtually abelian, a description of the unitary dual of G (that is, the equivalence classes of irreducible unitary representations of G) is hopeless, as a consequence of theorems of Glimm and Thoma. A sensible substitute for the unitary dual is the set Char(G) of characters of traceable factorial representations of G. The set Char(G) contains in general both finite and infinite characters. The finite characters are given by central positive definite functions on G and have been determined for some classes of discrete groups. In contrast, the set of infinite characters of G is much more mysterious. We will give an overview of recent results about the description of finite as well as of infinite characters for some examples of groups, including the special linear groups SL(n,Z) over the integers.