Open Orbits and Augmentations of Dynkin Diagrams
Abstract
Given any representation V of a complex linear reductive Lie group G0, we show that a larger semi-simple Lie group G with g=g0 + V + V* + ..., exists precisely when V has a finite number of G0-orbits. In particular, V admits an open G0-orbit. Furthermore, this corresponds to an augmentation of the Dynkin diagram of g0. The representation theory of g should be useful in describing the geometry of manifolds with stable forms as studied by Hitchin.
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