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(HC). We further classify all the self-representations of Is(HC) that satisfy a compatibility condition for the subgroup Is(HR). 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.