A generalized Cartan decomposition for the double coset space
A generalized Cartan decomposition for the double coset space
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双陪集空间的广义Cartan分解
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发表时间：
2006
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通讯作者：
Toshiyuki Kobayashi
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作者：
Toshiyuki Kobayashi
受复流形上可见作用的最新进展的启发，我们提出了一个问题：当G/H具有复结构且包含G′/G′ H作为全真实的子流形时，三个子群L，G′和H的乘积是否覆盖李群G.特别重要的情形是当G/L和G/H是广义旗簇时，我们对Levi子群对（L，H）进行分类，使得LG′H = G，或者等价地，在（G，G′）=（U（n），O（n））的情形下，真实的广义旗簇G′/H G′满足复广义旗簇G/H上的每个L-轨道.对于这样的对（L，H），我们引入了一种人字形缝合方法来寻找双陪集空间LG/H的广义Cartan分解，在非对称的情况下还没有一般的理论。我们的几何结果为一般线性群表示论中的各种无重数定理提供了一个统一的证明。部分由日本科学振兴会探索性研究补助金16654014资助
Motivated by recent developments on visible actions on complex manifolds, we raise a question whether or not the multiplication of three subgroups L, G′ and H surjects a Lie group G in the setting that G/H carries a complex structure and contains G′/G′ ∩ H as a totally real submanifold. Particularly important cases are when G/L and G/H are generalized flag varieties, and we classify pairs of Levi subgroups (L,H) such that LG′H = G, or equivalently, the real generalized flag variety G′/H ∩ G′ meets every L-orbit on the complex generalized flag variety G/H in the setting that (G,G′) = (U(n), O(n)). For such pairs (L,H), we introduce a herringbone stitch method to find a generalized Cartan decomposition for the double coset space LG/H, for which there has been no general theory in the non-symmetric case. Our geometric results provides a unified proof of various multiplicity-free theorems in representation theory of general linear groups. ∗Partly supported by Grant-in-Aid for Exploratory Research 16654014, Japan Society of the Promotion of Science