A Haar component for quantum limits on locally symmetric spaces

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 × 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 Haar component. This is consistent with the conjecture that the limit measures are absolutely continuous.