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July, 2013 Visible actions on flag varieties of type D and a generalization of the Cartan decomposition
Yuichiro TANAKA
J. Math. Soc. Japan 65(3): 931-965 (July, 2013). DOI: 10.2969/jmsj/06530931


We give a generalization of the Cartan decomposition for connected compact Lie groups motivated by the work on visible actions of T. Kobayashi [J. Math. Soc. Japan, 2007] for type A group. This paper extends his results to type D group. First, we classify a pair of Levi subgroups $(L,H)$ of a simple compact Lie group $G$ of type D such that $G=LG^{\sigma}H$ where $\sigma$ is a Chevalley–Weyl involution. This gives the visibility of the $L$-action on the generalized flag variety $G/H$ as well as that of the $H$-action on $G/L$ and of the $G$-action on $(G\times G)/(L\times H)$. Second, we find a generalized Cartan decomposition $G=LBH$ with $B$ in $G^{\sigma}$ by using the herringbone stitch method which was introduced by Kobayashi in his 2007 paper. Applications to multiplicity-free theorems of representations are also discussed.


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Yuichiro TANAKA. "Visible actions on flag varieties of type D and a generalization of the Cartan decomposition." J. Math. Soc. Japan 65 (3) 931 - 965, July, 2013.


Published: July, 2013
First available in Project Euclid: 23 July 2013

zbMATH: 1296.22015
MathSciNet: MR3079290
Digital Object Identifier: 10.2969/jmsj/06530931

Primary: 22E46
Secondary: 32A37 , 53C30

Keywords: Cartan decomposition , flag variety , herringbone stitch , multiplicity-free representation , semisimple Lie group , visible action

Rights: Copyright © 2013 Mathematical Society of Japan


Vol.65 • No. 3 • July, 2013
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