Complex hyperbolic representations of \(\textrm{PSL}_2(\textbf{R})\)

We construct a two-parameter family of irreducible representations of \(\textrm{PSL}_2(\textbf{R})\) ${\text{PSL}}_2(\mathbf{R})$ in the isometry group of the infinite-dimensional complex hyperbolic space. To this end, we introduce the notion of horospherical combination of two representations. Our family then appears as horospherical combinations of two known one-parameter families.