Harmonic and invariant measures on foliated spaces

Abstract:

We consider the family of harmonic measures on a lamination $\mathscr {L}$ of a compact space $X$ by locally symmetric spaces $L$ of noncompact type, i.e. $L\cong \Gamma _L\backslash G/K$. We establish a natural bijection between these measures and the measures on an associated lamination foliated by $G$-orbits, $\widehat {\mathscr {L}}$, 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 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 $\widehat {\mathscr {L}}$ right invariant under two distinct minimal parabolics, and therefore all of $G$, are in bijective correspondence with the holonomy invariant ones.