Maximal representations of uniform complex hyperbolic lattices

2015/06/24 by Koziarz, Vincent, Maubon, Julien
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1506.07274

Abstract

Let ρ be a maximal representation of a uniform lattice Γ⊂\rm SU(n,1), n≥ 2, in a classical Lie group of Hermitian type H. We prove that necessarily H=\rm 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 \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 ρ extends to a representation of \rm SU(n,1) in \rm SU(p,q).

Related

• Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type
• Maximal representations of uniform complex hyperbolic lattices in exceptional Hermitian Lie groups
• Maximal representations of complex hyperbolic lattices in SU(m,n)
• Harmonic maps and representations of non-uniform lattices of PU(m,1)
• Rigidity of maximal holomorphic representations of Kähler groups