Maximal representations of uniform complex hyperbolic lattices

Vincent Koziarz; Julien Maubon

doi:https://doi.org/10.4007/annals.2017.185.2.3

Abstract

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

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Maximal representations of uniform complex hyperbolic lattices

Abstract

Authors