  We establish scale-invariant Strichartz estimates for the Schr\"odinger flow
on any compact Lie group equipped with canonical rational metrics. In
particular, full Strichartz estimates without loss for some non-rectangular
tori are given. The highlights of this paper include estimates for some Weyl
type sums defined on rational lattices, different decompositions of the
Schr\"odinger kernel that accommodate different positions of the variable
inside the maximal torus relative to the cell walls, and an application of the
BGG-Demazure operators or Harish-Chandra's integral formula to the estimate of
the difference between characters.
