  Let G be a semisimple complex Lie group. In this article, we study Geometric
Invariant Theory on a flag variety G/B with respect to the action of a
principal 3-dimensional simple subgroup S of G. We determine explicitly the
GIT-equivalence classes of S-ample line bundles on G/B. We show that, under
mild assumptions, among the GIT-classes there are chambers, in the sense of
Dolgachev-Hu. The Hilbert quotients Y=X//S with respect to various chambers
form a family of Mori dream spaces, canonically associated with G. We are able
to determine three important cones in the Picard group of any of these
quotients: the pseudo-effective, the movable and the nef cones.
