Visible actions on flag varieties of type B and a generalisation of the Cartan decomposition

We give a generalization of the Cartan decomposition for connected compact Lie groups of type B motivated by the work on visible actions of T. Kobayashi [J. Math. Soc. Japan, 2007] for type A groups. Suppose that $G$ is a connected compact Lie group of type B, $\sigma$ is a Chevalley--Weyl involution and $L$, $H$ are Levi subgroups. First, we prove that $G=LG^{\sigma}H$ holds if and only if either Case I: both $H$ and $L$ are maximal and of type A, or Case II: $(G,H)$ is symmetric and $L$ is the Levi subgroup of an arbitrary maximal parabolic subgroup up to switch of $H$ and $L$. 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 $(G\times G)/(L\times H)$. Second, we find an explicit `slice' $B$ with dim $B=$rank $G$ in Case I, and dim $B=2$ or $3$ in Case II such that a generalized Cartan decomposition $G=LBH$ holds. An application to multiplicity-free theorems of representations is also discussed. 10.1017/S000497271200061