Schr\"odinger equation on locally symmetric spaces

We prove dispersive and Strichartz estimates for Schr"o- dinger equations on a class of locally symmetric spaces {\Gamma}\X, where X = G/K is a symmetric space and {\Gamma} is a torsion free discrete sub- group of G. We deal with the cases when either X has rank one or G is complex. We present Strichartz estimates applications to the well-posedness and scattering for nonlinear Schr"odinger equations.