Harmonic and Invariant Measures on Foliated Spaces

We consider the family of harmonic measures on a lamination L of a compact space X by locally symmetric spaces L of noncompact type, i.e. L ∼= ΓL\G/K. We establish a natural bijection between these measures and the measures on the associated lamination by G-orbits, Ò L , which are right invariant under a minimal parabolic (Borel) subgroup B < G. In the special case when G is split, these measures correspond to the measures that are invariant under the Weyl chamber flow and the stable horospherical action on the Weyl chamber lamination Ľ whose leaves are Ľ ∼= ΓL\G/M . We also show that the measures on Ò L right invariant under two fundamentally distinct minimal parabolics, and therefore all of G, are in bijective correspondence with the holonomy invariant ones.