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

作者：Tanaka Yuichiro^*

来源：Journal of Algebra, 2014, 399: 170-189.

DOI：10.1016/j.jalgebra.2013.10.002

• Cartan decomposition
• Multiplicity-free representation
• Exceptional Lie group
• Flag variety
• Visible action
• Herringbone stitch

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 = LG(sigma) H where sigma 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 x G)/(L x H) are strongly visible. Second, we find a generalized Cartan decomposition G = LBH with B in G(sigma) by using the herringbone stitch method which was introduced by Kobayashi. Applications to multiplicity-free representations are also discussed.

• 出版日期2014-2-1