Abstract
Let $p$ be a maximal representation of a uniform lattice $\Gamma \subset \mathrm{SU}(n,1)$, $n\ge 2$, in a classical Lie group of Hermitian type $G$. We prove that necessarily $G=\mathrm{SU}(p,q)$ withp $p\ge qn$ there exists a holomorphic or antiholomorphic $\rho$-equivariant map from the complex hyperbolic $n$-space to thesymmetric 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 $\mathrm{SU}(p,q)$.
Citation
Vincent Koziarz. Julien Maubon. "Maximal representations of uniform complex hyperbolic lattices." Ann. of Math. (2) 185 (2) 493 - 540, March 2017. https://doi.org/10.4007/annals.2017.185.2.3
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