Cocycle Superrigidity for Ergodic Actions of Non-semisimple Lie Groups

Suppose L is a semisimple Levi subgroup of a connected Lie group G, X is a Borel G-space with finite invariant measure, and α : X × G → GLn(R) is a Borel cocycle. Assume L has finite center, and that the real rank of every simple factor of L is at least two. We show that if L is ergodic on X, and the restriction of α to X × L is cohomologous to a homomorphism (modulo a compact group), then, after passing to a finite cover of X, the cocycle α itself is cohomologous to a homomorphism (modulo a compact group).