Invariant measures on G/Γ for split simple Lie groups G

We study the left action α of a Cartan subgroup on the space X = G /Γ, where Γ is a lattice in a simple split connected Lie group G of rank n > 1. Let μ be an α‐invariant measure on X . We give several conditions using entropy and conditional measures, each of which characterizes the Haar measure on X . Furthermore, we show that the conditional measure on the foliation of unstable manifolds has the structure of a product measure. The main new element compared to the previous work on this subject is the use of noncommutativity of root foliations to establish rigidity of invariant measures.