  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.
