  We prove lower bounds for the entropy of limit measures associated to
non-degenerate sequences of eigenfunctions on locally symmetric spaces of
non-positive curvature. In the case of certain compact quotients of the space
of positive definite $n\times n$ matrices (any quotient for $n=3$, quotients
associated to inner forms in general), measure classification results then show
that the limit measures must have a Lebesgue component. This is consistent with
the conjecture that the limit measures are absolutely continuous.
