Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics

TL;DR

This paper develops geometric results on linear actions of reductive Lie groups, providing tools to analyze limiting distributions in homogeneous dynamics, with applications to number theory problems.

Contribution

It offers general criteria to resolve linear dynamical questions in homogeneous dynamics using group theoretic and geometric conditions.

Findings

Resolved key linear dynamical questions using geometric conditions

Provided criteria for non-divergence of measures in homogeneous spaces

Connected linear actions to applications in number theory

Abstract

Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and linearization techniques, in view of Ratner's measure classification for unipotent flows, reduce such problems to dynamical questions about linear actions of reductive groups on finite dimensional vectors spaces. This article provides general results which resolve these linear dynamical questions in terms of natural group theoretic or geometric conditions.