Visible Actions on Generalized Flag Varieties and a Generalization of the Cartan Decomposition

Abstract

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the theory of visible actions on complex manifolds. Our main results give a classification of triples $(G,H,L)$ for a compact Lie group $G$ and its Levi subgroups $H,L$, which satisfy $G=HBL$. Here, $B$ is a subset of a Chevalley--Weyl involution $\sigma$-fixed points subgroup $G^{\sigma}$ of $G$. The point here is that one decomposition $G=LBH$ produces three strongly visible actions on generalized flag varieties, and thus three finite-dimensional multiplicity-free representations (Kobayashi's triunity principle). Furthermore, we can also prove that the visibility of actions of compact Lie groups, the existence of a decomposition $G=LBH$ and the multiplicity-freeness property of finite-dimensional tensor product representations are all equivalent.