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