Open Orbits and Augmentations of Dynkin Diagrams
Representation Theory
2009-03-08 v3 Differential Geometry
Abstract
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.
Keywords
Cite
@article{arxiv.0812.1078,
title = {Open Orbits and Augmentations of Dynkin Diagrams},
author = {Sin Tsun Edward Fan and Naichung Conan Leung},
journal= {arXiv preprint arXiv:0812.1078},
year = {2009}
}
Comments
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