Classification of abelian actions with globally hypoelliptic orbitwise laplacian I: The Greenfield-Wallach conjecture on nilmanifolds

For a ${\mathbb{R}}^k-$action generated by vector fields $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_k$ we define an operator $-(X_1^2+\mathrm{.}\mathrm{.}\mathrm{.}+X_k^2)$, the orbitwise laplacian. In this paper, we study and classify ${\mathbb{R}}^k-$actions whose orbitwise laplacian is globally hypoelliptic (GH). In three different settings we prove that any such action is given by a translation action on some compact nilmanifold, (i) when the space is a compact nilmanifold, (ii) when the first Betti number of the manifold is sufficiently large, (iii) when the codimension of the orbitfoliation of the action is $1$. As a consequence, we prove the Greenfield-Wallach conjecture on all nilmanifolds. Along the way, we also calculate the cohomology of GH ${\mathbb{R}}^k-$actions, proving, in particular, that it is always finite dimensional.