LOCAL QUATERNIONIC RIGIDITY FOR COMPLEX HYPERBOLIC LATTICES

Let Gamma i hooked right arrow L be a lattice in the real simple Lie group L. If L is of rank at least 2 (respectively locally isomorphic to Sp(n, 1)) any unbounded morphism rho : Gamma -> G into a simple real Lie group G essentially extends to a Lie morphism rho(L) : L -> G (Margulis's superrigidity theorem, respectively Corlette's theorem). In particular any such morphism is infinitesimally, thus locally, rigid. On the other hand, for L = SU(n, 1) even morphisms of the form rho : Gamma i hooked right arrow L -> G are not infinitesimally rigid in general. Almost nothing is known about their local rigidity. In this paper we prove that any cocompact lattice Gamma in SU(n, 1) is essentially locally rigid (while in general not infinitesimally rigid) in the quaternionic groups Sp(n, 1), SU(2n, 2) or SO(4n, 4) (for the natural sequence of embeddings SU(n, 1) subset of Sp(n, 1) subset of SU(2n, 2) subset of SO(4n, 4)).