Visible actions on flag varieties of exceptional groups and a generalization of the Cartan decomposition
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We give a generalization of the Cartan decomposition for connected compact exceptional Lie groups motivated by the work on visible actions of T. Kobayashi [T. Kobayashi, J. Math. Soc. Japan 59 (2007) 669-691] for type A groups. This paper extends his results to the exceptional groups. First, we classify a pair of Levi subgroups of any compact exceptional simple Lie group G such that where 蟽 is a Chevalley-Weyl involution. This implies that the natural L-action on the generalized flag variety is strongly visible, and likewise the H-action on and the G-action on are strongly visible. Second, we find a generalized Cartan decomposition with B in by using the herringbone stitch method which was introduced by Kobayashi. Applications to multiplicity-free representations are also discussed.