We establish sharp bounds for the second moment of symmetric-square L-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$, where the second moment is a sum over $t_j$ in a short interval. At the central point $s=1/2$ of the L-function, our interval is smaller than previous known results. More specifically, for $\left \lvert t_j\right \rvert $ of size T, our interval is of size $T^{1/5}$, whereas the previous best was $T^{1/3}$, from work of Lam. A little higher up on the critical line, our second moment yields a subconvexity bound for the symmetric-square L-function. More specifically, we get subconvexity at $s=1/2+it$ provided $\left \lvert t_j\right \rvert ^{6/7+\delta }\le \lvert t\rvert \le (2-\delta )\left \lvert t_j\right \rvert $ for any fixed $\delta>0$. Since $\lvert t\rvert $ can be taken significantly smaller than $\left \lvert t_j\right \rvert $, this may be viewed as an approximation to the notorious subconvexity problem for the symmetric-square L-function in the spectral aspect at $s=1/2$.

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 with eigenfunctions of the ring of invariant differential operators, we construct a measure ${{\bar{\mu }}_{\infty }}$ on the homogeneous space $X=\Gamma \backslash G$ that lifts ${{\bar{\mu }}_{\infty }}$ and 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 with 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.