Visible actions on flag varieties of exceptional groups and a generalization of the Cartan decomposition

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 (L,H) of any compact exceptional simple Lie group G such that G=LGσH where σ is a Chevalley-Weyl involution. This implies that the natural L-action on the generalized flag variety G/H is strongly visible, and likewise the H-action on G/L and the G-action on (G×G)/(L×H) are strongly visible. Second, we find a generalized Cartan decomposition G=LBH with B in Gσ by using the herringbone stitch method which was introduced by Kobayashi. Applications to multiplicity-free representations are also discussed.