Abstract
We give a generalization of the Cartan decomposition for connected compact Lie groups motivated by the work on visible actions of T. Kobayashi [J. Math. Soc. Japan, 2007] for type A group. This paper extends his results to type D group. First, we classify a pair of Levi subgroups (L,H) of a simple compact Lie group G of type D such that G = LGσH where σ is a Chevalley—Weyl involution. This gives the visibility of the L-action on the generalized flag variety G/H as well as that of the H-action on G/L and of the G-action on (G × G)/(L × H). Second, we find a generalized Cartan decomposition G = LBH with B in Gσ by using the herringbone stitch method which was introduced by Kobayashi in his 2007 paper. Applications to multiplicity-free theorems of representations are also discussed.
