Maximal representations of uniform complex hyperbolic lattices

Vincent Koziarz (1) , Julien Maubon (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 Géométrie

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

Domaines: Géométrie différentielle [math.DG]

Vincent Koziarz, Julien Maubon. Maximal representations of uniform complex hyperbolic lattices. 2015. ⟨hal-01166954v1⟩