  We establish sharp-in-time kernel and dispersive estimates for the
Schr\"odinger equation on non-compact Riemannian symmetric spaces of any rank.
Due to the particular geometry at infinity and the Kunze-Stein phenomenon,
these properties are more pronounced in large time and enable us to prove the
global-in-time Strichartz inequality for a larger family of admissible couples
than in the Euclidean case. Consequently, we obtain the global well-posedness
for the corresponding semilinear equation with lower regularity data and some
scattering properties for small powers which are known to fail in the Euclidean
setting. The crucial kernel estimates are achieved by combining the stationary
phase method based on a subtle barycentric decomposition, a subordination
formula of the Schr\"odinger group to the wave propagator and an improved
Hadamard parametrix.
