On the trivializability of rank-one cocycles with an invariant field of projective measures

Let G be S O ∘ ( n , 1 ) for n ⩾ 3 and consider a lattice Γ < G . Given a standard Borel probability Γ -space ( Ω , μ ) , consider a measurable cocycle σ : Γ × Ω → H ( κ ) , where H is a connected algebraic κ -group over a local field κ . Under the assumption of compatibility between G and the pair ( H , κ ) , we show that if σ admits an equivariant field of probability measures on a suitable projective space, then σ is trivializable. An analogous result holds in the complex hyperbolic case.