Convergence of measures on compactifications of locally symmetric spaces

We conjecture that the set of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space $S=Γ\backslash G/K$ 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 $Γ={\rm SL}_3(\mathbb{Z})$.

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