Maximal representations of uniform complex hyperbolic lattices

Vincent Koziarz (1) , Julien Maubon (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 IECL - Institut Élie Cartan de Lorraine

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

Domaines: Géométrie différentielle [math.DG]; Géométrie algébrique [math.AG]

Vincent Koziarz, Julien Maubon. Maximal representations of uniform complex hyperbolic lattices. 2016. ⟨hal-01166954v2⟩