Remarks on minimal sets and conjectures of Cassels, Swinnerton-Dyer, and Margulis

We prove that a hypothesis of Cassels, Swinnerton-Dyer, recast by Margulis as statement on the action of the diagonal group $A$ on the space of unimodular lattices, is equivalent to several assertions about minimal sets for this action. More generally, for a maximal $\mathbb{R}$-diagonalizable subgroup $A$ of a reductive group $G$ and a lattice $\Gamma$ in $G$, we give a sufficient condition for a compact $A$-minimal subset $Y$ of $G/\Gamma$ to be of a simple form, which is also necessary if $G$ is $\mathbb{R}$-split. We also show that the stabilizer of $Y$ has no nontrivial connected unipotent subgroups.