Open Orbits and Augmentations of Dynkin Diagrams

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Fan, Sin Tsun Edward
Leung, Naichung Conan
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Abstract
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Given any representation V of a complex linear reductive Lie group G_0, we show that a larger semi-simple Lie group G with g=g_0 + V + V* + ..., exists precisely when V has a finite number of G_0-orbits. In particular, V admits an open G_0-orbit. Furthermore, this corresponds to an augmentation of the Dynkin diagram of g_0. The representation theory of g should be useful in describing the geometry of manifolds with stable forms as studied by Hitchin.
Comment: After the first version was posted, we were informed that many of the results in it were obtained earlier by Kac, Rubenthaler and Vinberg. See Remark 1 for more details
Keywords
Mathematics - Representation Theory, Mathematics - Differential Geometry, 22E46(Primary), 17B10 (Secondary)
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