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 Is(H-C(infinity)). We further classify all the self-representations of Is(H-C(infinity)) that satisfy a compatibility condition for the subgroup Is(H-R(infinity)). 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.