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

We prove that a hypothesis of Cassels, Swìnnerton-Dyer, recast by Margulis as a 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 R-diagonaliz-able subgroup A of a reductive group G and a lattice r in G, we give a sufficient condition for a compact A-minimal subset Y of G/T to be of a simple form, which is also necessary if G is R-split. We also show that the stabilizer of Y has no nontrivial connected unipotent subgroups.