Geometric invariant theory for principal three-dimensional subgroups acting on flag varieties

Abstract

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 $S\subset G$. 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 GIT-quotients with respect to various chambers form a family of Mori dream spaces, canonically associated with $G$. 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.