A Haar component for quantum limits on locally symmetric spaces

TL;DR

This paper establishes lower bounds on the entropy of limit measures for eigenfunctions on certain locally symmetric spaces, supporting the conjecture that these measures are absolutely continuous with a Lebesgue component.

Contribution

It provides new entropy bounds for eigenfunction limit measures on locally symmetric spaces, advancing understanding of their absolute continuity properties.

Findings

Lower bounds for entropy of limit measures proven.

Limit measures on specific quotients must have a Lebesgue component.

Supports conjecture of absolute continuity of limit measures.

Abstract

We prove lower bounds for the entropy of limit measures associated to non-degenerate sequences of eigenfunctions on locally symmetric spaces of non-positive curvature. In the case of certain compact quotients of the space of positive definite $n\times n$ matrices (any quotient for $n=3$, quotients associated to inner forms in general), measure classification results then show that the limit measures must have a Lebesgue component. This is consistent with the conjecture that the limit measures are absolutely continuous.