Harmonic and invariant measures on foliated spaces

TL;DR

This paper explores harmonic measures on foliated spaces formed by locally symmetric spaces, establishing bijections with measures invariant under group actions and flows, revealing deep connections between geometry, dynamics, and measure theory.

Contribution

It introduces a natural correspondence between harmonic measures on laminations and invariant measures on associated foliations, extending understanding of measure invariance under group actions.

Findings

Bijection between harmonic measures and measures invariant under a minimal parabolic subgroup.

Correspondence between measures invariant under the entire group and holonomy-invariant measures.

Characterization of measures invariant under Weyl chamber and horospherical flows.

Abstract

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.