Maximal representations of uniform complex hyperbolic lattices in exceptional Hermitian Lie groups
We complete the classification of maximal representations of uniform complex 
hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional 
groups ${\rm E}_6$ and ${\rm E}_7$. We prove that if $\rho$ is a maximal 
representation of a uniform complex hyperbolic lattice $\Gamma\subset{\rm 
SU}(1,n)$, $n>1$, in an exceptional Hermitian group $G$, then $n=2$ and $G={\rm 
E}_6$, and we describe completely the representation $\rho$. The case of 
classical Hermitian target groups was treated by Vincent Koziarz and the second 
named author (arxiv:1506.07274). However we do not focus immediately on the 
exceptional cases and instead we provide a more unified perspective, as 
independent as possible of the classification of the simple Hermitian Lie 
groups. This relies on the study of the cominuscule representation of the 
complexification of the target group. As a by product of our methods, when the 
target Hermitian group $G$ has tube type, we obtain an inequality on the Toledo 
invariant of the representation $\rho:\Gamma\rightarrow G$ which is stronger 
than the Milnor-Wood inequality (thereby excluding maximal representations in 
such groups).