2011/05/29 by Sönke Hansen, Hansen, Sönke, Joachim Hilgert +3
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #Spectral Theory (math.SP) #math.RT #math.SP
paper · pdf · doi:10.48550/arxiv.1105.5788
arxiv created 2011/05/29 · arxiv updated 2011/05/31
For a compact locally symmetric space \XG of non-positive curvature, we consider sequences of normalized joint eigenfunctions which belong to the principal spectrum of the algebra of invariant differential operators. Using an h-\psdiff calculus on \XG, we define and study lifted quantum limits as weak^*-limit points of Wigner distributions. The Helgason boundary values of the eigenfunctions allow us to construct Patterson--Sullivan distributions on the space of Weyl chambers. These distributions are asymptotic to lifted quantum limits and satisfy additional invariance properties, which makes them useful in the context of quantum ergodicity. Our results generalize results for compact hyperbolic surfaces obtained by Anantharaman and Zelditch.