Abstract:

Some generalities about group representations by isometries on the complex hyperbolic space of infinite dimensions will be introduced. For this purpose the functions of hyperbolic type, which play a role analogous to that of positive type functions with respect to unitary representations, will be discussed.

In particular the case of SL_2 will be treated in detail. A new family of representations will be described as well as the questions it raises about the structure of the set of hyperbolic type functions defined on a given group.