Convergence of measures on compactifications of locally symmetric 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 $G={\mathrm{SL}}_3(\mathbb{R})$ and $\Gamma ={\mathrm{SL}}_3(\mathbb{Z})$.

arXiv comment45 pages

References in corpus (1)

• Inverses of Cartan Matrices of Lie Algebras and Lie Superalgebra