Maximal representations of uniform complex hyperbolic lattices

Abstract

Let $p$ be a maximal representation of a uniform lattice $\Gamma \subset \mathrm{SU}(n,1)$, $n\ge 2$, in a classical Lie group of Hermitian type $G$. We prove that necessarily $G=\mathrm{SU}(p,q)$ withp $p\ge qn$ there exists a holomorphic or antiholomorphic $\rho$-equivariant map from the complex hyperbolic $n$-space to thesymmetric space associated to $\mathrm{SU}(p,q)$. This map is moreover a totally geodesic homothetic embedding. In particular, up to a representation in a compact subgroup of $\mathrm{SU}(p,q)$, the representation $\rho$ extends to a representation of $\mathrm{SU}(n,1)$ in $\mathrm{SU}(p,q)$.