Journal of the Mathematical Society of Japan

A generalized Cartan decomposition for the double coset space $(U(n_1) \times U(n_2) \times U(n_3)) \backslash U(n) / (U(p) \times U(q))$


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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.

Article information

J. Math. Soc. Japan, Volume 59, Number 3 (2007), 669-691.

First available in Project Euclid: 5 October 2007

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Zentralblatt MATH identifier

Primary: 22E4
Secondary: 32A37: Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) [See also 46Exx] 43A85: Analysis on homogeneous spaces 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 53C50: Lorentz manifolds, manifolds with indefinite metrics 53D20: Momentum maps; symplectic reduction

Cartan decomposition double coset space multiplicity-free representation semisimple Lie group homogeneous space visible action flag variety


KOBAYASHI, Toshiyuki. 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 (2007), no. 3, 669--691. doi:10.2969/jmsj/05930669.

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