Open Orbits and Augmentations of Dynkin Diagrams

2008/12/05 by Sin Tsun Edward Fan, Naichung Conan Leung, Fan, Sin Tsun Edward +1
Mathematics · #17B10 (Secondary) #22E46(Primary) #Differential Geometry (math.DG) #FOS: Mathematics #Representation Theory (math.RT) #math.DG #math.RT #msc:17B10

paper · pdf · doi:10.48550/arxiv.0812.1078

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

arxiv created 2009/03/08 · arxiv updated 2009/12/01

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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