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

TL;DR

This paper explores the equivalence of certain conjectures related to minimal sets in the action of diagonal groups on lattices, providing new conditions and structural insights into these sets within the context of algebraic groups.

Contribution

It establishes the equivalence of Cassels-Swinnerton-Dyer conjecture with minimal set assertions and characterizes minimal sets for diagonal group actions in reductive groups.

Findings

Equivalence of conjecture and minimal set assertions.

A sufficient condition for minimal sets to be of a simple form.

The stabilizer of minimal sets contains no nontrivial connected unipotent subgroups.

Abstract

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.