Harmonic and invariant measures on foliated spaces

Abstract

We consider the family of harmonic measures on a lamination L \mathscr {L} of a compact space X X by locally symmetric spaces L L of noncompact type, i.e. L ≅ Γ L ∖ G / K L\cong \Gamma _L\backslash G/K . We establish a natural bijection between these measures and the measures on an associated lamination foliated by G G -orbits, L ^ \widehat {\mathscr {L}} , which are right invariant under a minimal parabolic (Borel) subgroup B > G B>G . In the special case when G G is split, these measures correspond to the measures that are invariant under both the Weyl chamber flow and the stable horospherical flows on a certain bundle over the associated Weyl chamber lamination. We also show that the measures on L ^ \widehat {\mathscr {L}} right invariant under two distinct minimal parabolics, and therefore all of G G , are in bijective correspondence with the holonomy invariant ones.