The space of homogeneous probability measures on $\overline{\Gamma\backslash X}^S_{\rm max}$ is compact

TL;DR

This paper proves the compactness of the space of homogeneous probability measures on the maximal Satake compactification of arithmetic locally symmetric spaces and explores implications for the distribution of weakly special subvarieties in Shimura varieties.

Contribution

It establishes the compactness of the measure space and connects this to the distribution of special subvarieties in Shimura varieties.

Findings

The space of homogeneous probability measures is compact.

Implications for the distribution of weakly special subvarieties.

Provides a new understanding of measure behavior on compactified spaces.

Abstract

In this paper we prove that the space of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space is compact. As an application, we explain some consequences for the distribution of weakly special subvarieties of Shimura varieties.