  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.
