Quantum unique ergodicity on locally symmetric spaces: the degenerate lift

Lior Silberman

Abstract: Given a measure μ¯ on a locally symmetric space Y=Γ∖G/K, obtained as a weak-{*} limit of probability measures associated to eigenfunctions of the ring of invariant differential operators, we construct a measure μ on the homogeneous space X=Γ∖G which lifts μ¯ and which is invariant by a connected subgroup A1⊂A of positive dimension, where G=NAK is an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, then μ is also the limit of measures associated to Hecke eigenfunctions on X. This generalizes previous results of the author and A.\ Venkatesh to the case of "degenerate" limiting spectral parameters.