Classification of visible actions on flag varieties

Abstract

We give a complete classification of the pairs $(L,H)$ of Levi subgroups of compact simple Lie groups $G$ such that the $L$-action on a generalized flag variety $G/H$ is strongly visible (or equivalently, the $H$-action on $G/L$ or the diagonal $G$-action on $(G\times G)/(L\times H)$). The notion of visible actions on complex manifolds was introduced by T. Kobayashi, and a classification was accomplished by himself for the type A groups [J. Math. Soc. Japan, 2007]. A key step is to classify the pairs $(L,H)$ for which the multiplication mapping $L\times G^{\sigma}\times H\to G$ is surjective, where $\sigma$ is a Chevalley–Weyl involution of $G$. We then see that strongly visible actions, multiplicity-free restrictions of representations (c.f. Littelmann, Stembridge), the decomposition $G=LG^{\sigma}H$ and spherical actions are all equivalent in our setting.