| We report progress on the equidistribution problem of automorphic forms on locally symmetric spaces. First, generalizing work of Zelditch-Wolpert we construct a representation theoretic analog of the micro-local lift, showing that (under a technical condition of non-degeneracy) every weak-* limit of the generalized Wigner measures associated to a sequence of Maass forms with divergent spectral parameters on a locally symmetric space Gamma G/K can be lifted to a measure on the homogeneous space Gamma G which is invariant by a maximal split torus A in G. Secondly, we consider the case where G ≃ PGLd( R ) and Gamma < G is a lattice associated to a division algebra over Q of prime degree d. When the measures are associated to Hecke-Maass eigenforms, we generalize the work of Bourgain-Lindenstrauss to show that every non-trivial a ∈ A acts with positive entropy on each ergodic component of the lifted measure. Applying recent measure rigidity results of Einsiedler-Katok we find that the limit measure must be the Haar measure on GammaG. In particular we prove that a non-degenerate sequence of Hecke-Maass forms becomes equidistributed in GammaG/K in the semiclassical limit. |
