Quantum unique ergodicity on locally symmetric spaces: the degenerate lift

TL;DR

This paper constructs a measure on a homogeneous space that lifts a given measure on a locally symmetric space, demonstrating invariance under a subgroup and extending previous results to degenerate spectral parameters.

Contribution

It generalizes prior work by constructing invariant measures for eigenfunctions with degenerate spectral parameters on locally symmetric spaces.

Findings

Constructed a measure on $X$ lifting the measure on $Y$.

Proved invariance of the lifted measure under a subgroup $A_1$.

Extended previous results to cases with degenerate spectral parameters.

Abstract

Given a measure $\bar\mu$ on a locally symmetric space $Y=\Gamma\backslash G/K$, obtained as a weak-{*} limit of probability measures associated to eigenfunctions of the ring of invariant differential operators, we construct a measure $\mu$ on the homogeneous space $X=\Gamma\backslash G$ which lifts $\bar\mu$ and which is invariant by a connected subgroup $A_{1}\subset A$ of positive dimension, where $G=NAK$ is an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, then $\mu$ is also the limit of measures associated to Hecke eigenfunctions on $X$. This generalizes previous results of the author and A.\ Venkatesh to the case of "degenerate" limiting spectral parameters.