  We establish sharp-in-time pointwise kernel estimates for the Schr\"odinger
equation on noncompact symmetric spaces of general rank. A well-known
difficulty in higher rank analysis, namely the fact that the Plancherel density
is not a differential symbol in general, is overcome by using a spectral
decomposition introduced recently by two of the authors in the study of the
wave equation. We deduce the dispersive property of the Schr\"odinger
propagator and prove global-in-time Strichartz inequalities for a large family
of admissible pairs. As consequences, we extend the global well-posedness and
small data scattering results previously obtained on real hyperbolic spaces to
general Riemannian symmetric spaces of noncompact type.
