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# Quantum Unique Ergodicity on Locally Symmetric Spaces: the Degenerate Lift

Published:2015-04-21
Printed: Sep 2015
• Lior Silberman,
Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2, Canada
 Format: LaTeX MathJax PDF

## Abstract

Given a measure $\bar\mu_\infty$ 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 $\bar\mu_\infty$ on the homogeneous space $X=\Gamma\backslash G$ which lifts $\bar\mu_\infty$ 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 $\bar\mu_\infty$ is also the limit of measures associated to Hecke eigenfunctions on $X$. This generalizes results of the author with A. Venkatesh in the case where the spectral parameters stay away from the walls of the Weyl chamber.
 Keywords: quantum unique ergodicity, microlocal lift, spherical dual
 MSC Classifications: 22E50 - Representations of Lie and linear algebraic groups over local fields [See also 20G05] 43A85 - Analysis on homogeneous spaces

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