Notes on functions of hyperbolic type

Functions of hyperbolic type encode representations on real or complex hyperbolic spaces, usually infinite-dimensional. These notes set up the complex case. As applications, we prove the existence of a non-trivial deformation family of representations of SU(1,n) and of its infinite-dimensional kin Isom(H^\infty_C). We further classify all the self-representations of Isom(H^\infty_C) that satisfy a compatibility condition for the subgroup Isom(H^\infty_R). It turns out in particular that translation lengths and Cartan arguments determine each other for these representations. In the real case, we revisit earlier results and propose some further constructions.