The Paley-Wiener Theorem and the Local Huygens' Principle for Compact   Symmetric Spaces
  We prove a Paley-Wiener Theorem for a class of symmetric spaces of the compact type, in which all root multiplicities are even. This theorem characterizes functions of small support in terms of holomorphic extendability and exponential type of their (discrete) Fourier transforms. We also provide three independent new proofs of the strong Huygens' principle for a suitable constant shift of the wave equation on odd-dimensional spaces from our class. 
Thomas Branson, Gestur Olafsson, and Angela Pasquale