Rigidity of Higher-Rank Abelian Actions on Semisimple Lie Groups and Measure Classification

This monograph investigates the rigidity phenomena inherent in the actions of higher-rank abelian groups on homogeneous spaces of semisimple Lie groups. We focus specifically on the measure classification problem for actions of maximal split Cartan subgroups and their higher-rank sublattices. Building upon the foundational work of Katok, Spatzier, and the breakthrough methods of Einsiedler, Katok, and Lindenstrauss, we provide a rigorous exposition of the dichotomy between measure rigidity and algebraic structure. We analyze the entropy properties of invariant measures, establishing that under the assumption of positive entropy for at least one element of the action, the invariant measures are necessarily algebraic in nature. Furthermore, we discuss the implications of these results for the Zimmer program and the arithmetic quantum unique ergodicity conjecture. The technical core involves a detailed treatment of leafwise measures along stable foliations and the utilization of the pointwise entropy contribution of one-parameter subgroups to derive invariance under unipotent flows via the Marstrand-Matthews argument.