Abstract: In this talk I'll present a geometric relation between the Laplace-Beltrami spectra and eigenfunctions on compact Riemannian symmetric spaces and the Borel-Weil theory using ideas from symplectic geometry and geometric quantization. The examples of the classical simply-connected spaces of rank one and the space SU(3)/SO(3) will be considered in which oe can describe all eigenfunctions in terms of harmonic polynomials.