Multi-invariant measures and subsets on nilmanifolds

Given a $\mathbb Z^r$-action $\alpha$ on a nilmanifold $X$ by automorphisms and an ergodic $\alpha$-invariant probability measure $\mu$, we show that $\mu$ is the uniform measure on $X$, unless modulo finite index modification, one of the following obstructions occurs for an algebraic factor action: 1. The factor measure has zero entropy under every element of the action; 2. The factor action is virtually cyclic. We also deduce a rigidity property for invariant closed subsets.