Category O for the Schr\"odinger algebra

We study category O for the (centrally extended) Schr\"odinger algebra. We determine the quivers for all blocks and relations for blocks of nonzero central charge. We also describe the quiver and relations for the finite dimensional part of O. We use this to determine the center of the universal enveloping algebra and annihilators of Verma modules. Finally, we classify primitive ideals with nonzero central charge.