Notes on functions of hyperbolic type

Abstract

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 $\mathbf{SU}(1,n)$ and of its infinite-dimensional kin ${\rm Is}(\mathbf{H}_{\mathbf{C}}^{\infty})$. We further classify all the self-representations of ${\rm Is}(\mathbf{H}_{\mathbf{C}}^{\infty})$ that satisfy a compatibility condition for the subgroup ${\rm Is}(\mathbf{H}_{\mathbf{R}}^{\infty})$. 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.