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July, 2007 A generalized Cartan decomposition for the double coset space $(U(n_1) \times U(n_2) \times U(n_3)) \backslash U(n) / (U(p) \times U(q))$
Toshiyuki KOBAYASHI
J. Math. Soc. Japan 59(3): 669-691 (July, 2007). DOI: 10.2969/jmsj/05930669

Abstract

Motivated by recent developments on visible action 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.

Paticularly 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 $G^{\prime} / \cap G^{\prime}$ 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 $L \backslash G / H$, 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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Toshiyuki KOBAYASHI. "A generalized Cartan decomposition for the double coset space $(U(n_1) \times U(n_2) \times U(n_3)) \backslash U(n) / (U(p) \times U(q))$." J. Math. Soc. Japan 59 (3) 669 - 691, July, 2007. https://doi.org/10.2969/jmsj/05930669

Information

Published: July, 2007
First available in Project Euclid: 5 October 2007

zbMATH: 1124.22003
MathSciNet: MR2344822
Digital Object Identifier: 10.2969/jmsj/05930669

Subjects:
Primary: 22E4
Secondary: 11F67, 32A37, 43A85, 53C50, 53D20

Rights: Copyright © 2007 Mathematical Society of Japan

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Vol.59 • No. 3 • July, 2007
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