Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces

TL;DR

This paper establishes a polynomial rate of effective equidistribution for large closed orbits of semisimple groups on homogeneous spaces, using spectral gaps and a closing lemma under specific conditions.

Contribution

It provides the first effective equidistribution results with explicit rates for these orbits, extending previous qualitative understandings.

Findings

Proves polynomial rate of equidistribution for large closed orbits.

Uses spectral gap techniques and a closing lemma in the proofs.

Requires the acting group to have a finite centralizer in the ambient group.

Abstract

We prove effective equidistribution, with polynomial rate, for large closed orbits of semisimple groups on homogeneous spaces, under certain technical restrictions (notably, the acting group should have finite centralizer in the ambient group). The proofs make extensive use of spectral gaps, and also of a closing lemma for such actions.