Abstract: The quantum unique ergodicity conjecture of Rudnick and Sarnak states roughly that for a negatively curved compact manifold the eigenfunctions of the Laplacian become more and more uniformly distributed as eigenvalue tends to infinity. Much research has been focused on the special case of arithmetic surfaces.

Surprisingly, for these special surfaces, the quantum unique ergodicity conjecture is intimately related with a seemingly unrelated topic with a long history of its own, the classification of measures on homogeneous spaces invariant under certain subgroups.

I will present new results regarding this classification, which in particular imply the quantum unique ergodicity conjecture for compact arithmetic surfaces (and something quite close also in the noncompact arithmetic case).