Abstract: |
With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible actions on complex manifolds. Our main results give a classification of a triple $(G,H,L)$ for a connected compact Lie group $G$ and its Levi subgroups $H,L$, which satisfies $G=HBL$. 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 $G/H$, $G/L$, $(G\times G)/(H\times L)$ of $L$, $H$, $G$, respectively, and three finite-dimensional multiplicity-free representations $\Ind(\chi {H})|_{L}$, $\Ind(\chi_{L})|_{H}$, $\Ind(\chi_{H})\otimes\Ind (\chi_{L})$. 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. |