# Maximal representations of uniform complex hyperbolic lattices

### Abstract

Let $\rho$ be a maximal representation of a uniform lattice $\Gamma\subset\mathrm{SU}(n,1)$, $n\geq 2$, in a classical Lie group of Hermitian type $G$. We prove that necessarily $G=\mathrm{SU}(p,q)$ with $p\geq qn$ and there exists a holomorphic or antiholomorphic $\rho$-equivariant map from the complex hyperbolic $n$-space to the symmetric 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 $\mathrm{SU}(p,q)$.

## Authors

Vincent Koziarz

Univ. Bordeaux, IMB, CNRS, UMR 5251, F-33400 Talence, France

Julien Maubon

Université de Lorraine, CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France