Homogeneous higher-rank nilmanifold conjecture

Let $\rho$ be a homogeneous $\mathbb R^k$ action on a finite-volume space $G/D$, where $G$ is a connected Lie group and $D$ is a closed subgroup of $G$. An $r$-step nilmanifold is a compact quotient of a nilpotent Lie group whose lower central series has length $r$. Homogeneous higher-rank nilmanifold conjecture. If $\rho$ is globally hypoelliptic, then it is smoothly conjugate to a homogeneous action on an $r$-step nilmanifold with $k\ge r$. This is the paper’s homogeneous classification conjecture for higher-rank globally hypoelliptic actions; its general validity is not established.