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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 ofthe 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 ifthe 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 anapplication, we establish rigidity for the action of a class of groups actingby automorphisms on nilmanifolds associated to step 2 nilpotent Lie groups.

Rigidity for group actions on homogeneous spaces by affine transformations