On globally hypoelliptic abelian actions and their existence on homogeneous spaces

Abstract

We define globally hypoelliptic smooth ${\mathbb{R}}^k$ actions as actions whose leafwise Laplacian along the orbit foliation is a globally hypoelliptic differential operator. When $k=1$, strong global rigidity is conjectured for such actions by Greenfield-Wallach and Katok: every such action is smoothly conjugate to a Diophantine flow on the torus. The conjecture has been confirmed for all homogeneous flows on homogeneous spaces \cite{FFRH}. In this paper we conjecture that among homogeneous ${\mathbb{R}}^k$ actions ($k\ge 2$) on homogeneous spaces globally hypoelliptic actions exist only on nilmanifolds. We obtain a partial result towards this conjecture: we show non-existence of globally hypoelliptic ${\mathbb{R}}^2$ actions on homogeneous spaces $G/\Gamma$, with at least one quasi-unipotent generator, where $G=SL(n,\mathbb{R})$. We also show that the same type of actions on solvmanifolds are smoothly conjugate to homogeneous actions on nilmanifolds.