Quantum unique ergodicity on locally symmetric spaces: the degenerate lift

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 $X=\mathrm{\Gamma }\mathrm{\backslash }G$ 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.