Quantum unique ergodicity for Eisenstein series on P S L 2 ( ℤ ∖ P S L 2 ( ℝ )

In this paper we prove microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on P S L 2 ( ℤ ) ∖ P S L 2 ( ℝ ) . This generalizes a recent result of W. Luo and P. Sarnak who proves equidistribution for P S L 2 ( ℤ ) ∖ ℍ . The averaged versions of these results have been proven by Zelditch for an arbitrary finite-volume surface, but our proof depends essentially on the presence of Hecke operators and works only for congruence subgroups of S L 2 ( ℝ ) . In the proof the key estimates come from applying Meurman’s and Good’s results on L -functions associated to holomorphic and Maass cusp forms. One also has to use classical transformation formulas for generalized hypergeometric functions of a unit argument.