Visible actions on flag varieties of exceptional groups and a generalization of the Cartan decomposition
Visible actions on flag varieties of exceptional groups and a generalization of the Cartan decomposition
复制标题
对特殊群体的标志变体的可见作用以及嘉当分解的推广
DOI：
10.1016/j.jalgebra.2013.10.002
复制
发表时间：
2014
影响因子：
0.9
通讯作者：
Yuichiro Tanaka
中科院分区：
文献类型：
--
作者：
瀧上舞;米田穣;田中雄一郎;Yuichiro Tanaka
在T. Kobayashi [T.]的可见作用研究的推动下，我们给出了连通紧例外李群的Cartan分解的推广。小林，J.数学。Soc。日本59 (2007)669-691]A类群体。本文将其结果推广到特殊群体。首先，我们对任意紧异常单李群G的一对Levi子群（L， H）进行分类，使得G= L G σ H，其中σ是Chevalley-Weyl对合。这意味着广义标志变体G/H上的自然L作用是强可见的，G/L上的H作用和(gx G)/(lx H)上的G作用也是强可见的。其次，利用Kobayashi引入的人字缝法，得到了G σ中有B的广义Cartan分解G= L B H。对无多重表示的应用也进行了讨论。
We give a generalization of the Cartan decomposition for connected compact exceptional Lie groups motivated by the work on visible actions of T. Kobayashi [T. Kobayashi, J. Math. Soc. Japan 59 (2007) 669–691] for type A groups. This paper extends his results to the exceptional groups. First, we classify a pair of Levi subgroups (L, H) of any compact exceptional simple Lie group G such that G= L G σ H where σ is a Chevalley–Weyl involution. This implies that the natural L-action on the generalized flag variety G/H is strongly visible, and likewise the H-action on G/L and the G-action on (G× G)/(L× H) are strongly visible. Second, we find a generalized Cartan decomposition G= L B H with B in G σ by using the herringbone stitch method which was introduced by Kobayashi. Applications to multiplicity-free representations are also discussed.