Maximal representations of complex hyperbolic lattices in SU(m,n)

TL;DR

This paper proves that for lattices in $SU(1,p)$ with $p>1$, there are no Zariski dense maximal representations into $SU(m,n)$ when $n>m>1$, using geometric rigidity properties of isotropic subspaces.

Contribution

It establishes a non-existence result for certain maximal representations of complex hyperbolic lattices into $SU(m,n)$, based on geometric and rigidity analysis.

Findings

No Zariski dense maximal representations exist for $n>m>1$

The proof uses geometric properties of isotropic subspaces

Rigidity properties of the associated geometry are key

Abstract

Let $\Gamma$ denote a lattice in $SU(1,p)$, with $p$ greater than 1. We show that there exists no Zariski dense maximal representation with target $SU(m,n)$ if $n>m>1$. The proof is geometric and is based on the study of the rigidity properties of the geometry whose points are isotropic $m$-subspaces of a complex vector space $V$ endowed with a Hermitian metric $h$ of signature $(m,n)$ and whose lines correspond to the $2m$ dimensional subspaces of $V$ on which the restriction of $h$ has signature $(m,m)$.