Rigidity for group actions on homogeneous spaces by affine transformations

We give a criterion for the rigidity of actions on homogeneous spaces. Let $G$ be a real Lie group, $\Lambda$ a lattice in $G$, and $\Gamma$ a subgroup of the affine group Aff$(G)$ stabilizing $\Lambda$. Then the action of $\Gamma$ on $G/\Lambda$ has the rigidity property in the sense of S. Popa, if and only if the induced action of $\Gamma$ on $\mathbb{P}(\frak{g})$ admits no $\Gamma$-invariant probability measure, where $\frak{g}$ is the Lie algebra of $G$. This generalizes results of M. Burger, and A. Ioana and Y. Shalom. As an application, we establish rigidity for the action of a class of groups acting by automorphisms on nilmanifolds associated to step 2 nilpotent Lie groups.