Visible actions on flag varieties of type C and a generalization of the Cartan decomposition

Abstract

We give a generalization of the Cartan decomposition for connected compact Lie groups of type C motivated by the work on visible actions of T. Kobayashi [J. Math. Soc. Japan, 2007] for type A groups. Let $G$ be a compact simple Lie group of type C, $K$ a Chevalley--Weyl involution-fixed point subgroup and $L,H$ Levi subgroups. We firstly show that $G=LKH$ holds if and only if either Case I: $(G,H)$ and $(G,L)$ are both symmetric pairs or Case II: $L$ is a Levi subgroup of maximal dimension and $H$ is an arbitrary maximal Levi subgroup up to switch of $L,H$. This classification gives a visible action of $L$ on the generalized flag variety $G/H$, as well as that of the $H$-action on $G/L$ and of the $G$-action on the direct product of $G/L$ and $G/H$. Secondly, we find a generalized Cartan decomposition $G=LBH$ explicitly, where $B$ is a subset of $K$. An application to multiplicity-free theorems of representations is also discussed.