On quantum unique ergodicity for Γ ∖ ℍ × ℍ

The purpose of this note is to point out a connection between the Quantum Unique Ergodicity conjecture of Z. Rudnick and P. Sarnak (or, more precisely, natural higher rank generalizations of this conjecture) and conjectures of H. Furstenberg, A. Katok, and R. Spatzier, and G. Margulis regarding the scarcity of measures invariant under natural ℝ^d actions (d ≥ 2) in the ergodic theory of Lie groups. Our main tool is a new variant of the micro local lift of A. Schnirelman,Y. Colin de Verdière, and S. Zelditch with additional invariance properties. We also sharpen and generalize a related result of Rudnick and Sarnak on scarring of Hecke eigenforms.