  The purpose of this note is to investigate the concentration properties of
spectral projectors on manifolds. This question has been intensively studied
(Logvinenko-Sereda, Nazarov, Jerison-Lebeau, Kovrizhkin,
Egidi-Seelmann-Veseli\'c, Burq-Moyano, etc) in connection with the uncertainty
principle. We provide the first high-frequency results in a geometric setting
which is neither Euclidean nor a perturbation of Euclidean. Namely, we prove
the natural (and optimal) uncertainty principle for the spectral projector on
the hyperbolic half-plane.
