Convergence of measures on compactifications of locally symmetric spaces

TL;DR

This paper explores the behavior of homogeneous probability measures on compactifications of locally symmetric spaces, conjecturing their limits are also homogeneous and supported on boundary components, with proofs in specific cases.

Contribution

It introduces new tools to study measure convergence on compactifications and proves the conjecture for certain groups like SL_3(R).

Findings

Conjecture on compactness of measure sets is supported in specific cases.

Homogeneous measures' limits are supported on boundary components.

Tools developed aid in understanding measure convergence on compactified spaces.

Abstract

We conjecture that the set of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space $S=\Gamma\backslash G/K$ is compact. More precisely, given a sequence of homogeneous probability measures on $S$, we expect that any weak limit is homogeneous with support contained in precisely one of the boundary components (including $S$ itself). We introduce several tools to study this conjecture and we prove it in a number of cases, including when $G={\rm SL}_3(\mathbb{R})$ and $\Gamma={\rm SL}_3(\mathbb{Z})$.