# Maximal representations of uniform complex hyperbolic lattices in exceptional Hermitian Lie groups

## TL;DR

This paper completes the classification of maximal representations of uniform complex hyperbolic lattices into exceptional Hermitian Lie groups, specifically E6 and E7, revealing new structural insights and inequalities related to the Toledo invariant.

## Contribution

It extends the classification of maximal representations to exceptional groups E6 and E7, providing a unified approach and new inequalities for the Toledo invariant.

## Key findings

- Maximal representations in E6 occur only when n=2.
- Complete description of the representation in the E6 case.
- Derived a stronger inequality on the Toledo invariant for tube type groups.

## Abstract

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).