Limiting distribution of translates of the orbit of a maximal $\mathbb{Q}$-torus from identity on $SL(N,\mathbb{R})/SL(N,\mathbb{Z})$

TL;DR

This paper classifies all possible limit measures of translates of the orbit of a maximal $ ext{Q}$-torus in the space $SL(N, ext{R})/SL(N, ext{Z})$, extending previous work that focused only on $ ext{Q}$-split tori.

Contribution

It extends the classification of limit measures of torus orbits from $ ext{Q}$-split to all maximal $ ext{Q}$-tori in $SL(N, ext{R})/SL(N, ext{Z}).

Findings

Classified all possible limit measures for translated orbits.

Extended previous results from $ ext{Q}$-split to all maximal $ ext{Q}$-tori.

Provided a comprehensive framework for understanding orbit distributions.

Abstract

Given a maximal $\mathbb{Q}$-torus in $SL(N,\mathbb{Q})$, its orbit from identity coset in $SL(N,\mathbb{R})/SL(N,\mathbb{Z})$ naturally carries a possibly infinite Haar measure. We classify all possible limit measures of it when translated by a sequence of elements from $SL(N,\mathbb{R})$. This is a natural extension of Shapira and Zheng's work where only $\mathbb{Q}$-split tori are considered.