Mathematics – Representation Theory
Scientific paper
2006-06-30
Journal of Mathematical Society of Japan 59 (2007), pages 669-691,
Mathematics
Representation Theory
Scientific paper
10.2969/jmsj/05930669
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 $G'/G' \cap H$ 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 $L G' H = G$, or equivalently, the real generalized flag variety $G'/H \cap G'$ 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 \textit{herringbone stitch} method to find a generalized Cartan decomposition for the double coset space $L \backslash G/H$, 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.
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