Maximal representations of uniform complex hyperbolic lattices

TL;DR

This paper characterizes maximal representations of uniform complex hyperbolic lattices into Hermitian Lie groups, showing they must be of a specific form involving totally geodesic embeddings into ${ m SU}(p,q)$.

Contribution

It proves that such maximal representations necessarily target ${ m SU}(p,q)$ with specific bounds and admit holomorphic or antiholomorphic equivariant maps, extending the understanding of their geometric structure.

Findings

H = SU(p,q) with p ≥ qn for maximal representations

Existence of a holomorphic or antiholomorphic equivariant map

The map is a totally geodesic homothetic embedding

Abstract

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 ${\rm SU}(p,q)$.