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