Harmonic measures on spaces foliated by locally symmetric spaces

Consider a lamination $\mathscr{L}$ of a compact topological space $X$ whose leaves are locally
symmetric spaces $L$ of noncompact type, i.e. $L\cong \Gamma_L\backslash G/K$. We
establish a natural bijection between the harmonic measures on $\mathscr{L}$ and the right
$B$-invariant measures on the associated lamination, $\hat\mathscr{L}$, foliated by $G$-orbits.
(Here $B$ is a minimal parabolic (Borel) subgroup $BG$.) In the cases 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. As a corollary, the measures on $\hat\mathscr{L}$ which are right invariant under
all of $G$, correspond to the holonomy invariant ones. This work is joint with Matilde Martinez.