Open Orbits and Augmentations of Dynkin Diagrams

Details Open Orbits and Augmentations of Dynkin Diagrams Open Orbits and Augmentations of Dynkin Diagrams

2008-12-05

arxiv.org/abs/0812.1078v3

Mathematics

Representation Theory

After the first version was posted, we were informed that many of the results in it were obtained earlier by Kac, Rubenthaler

Scientific paper

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.

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