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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.

Notes on functions of hyperbolic type