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.