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2013 Visible actions on flag varieties of type C and a generalization of the Cartan decomposition
Yuichiro Tanaka
Tohoku Math. J. (2) 65(2): 281-295 (2013). DOI: 10.2748/tmj/1372182727


We give a generalization of the Cartan decomposition for connected compact Lie groups of type C motivated by the work on visible actions of T. Kobayashi [J. Math. Soc. Japan, 2007] for type A groups. Let $G$ be a compact simple Lie group of type C, $K$ a Chevalley--Weyl involution-fixed point subgroup and $L,H$ Levi subgroups. We firstly show that $G=LKH$ holds if and only if either Case I: $(G,H)$ and $(G,L)$ are both symmetric pairs or Case II: $L$ is a Levi subgroup of maximal dimension and $H$ is an arbitrary maximal Levi subgroup up to switch of $L,H$. This classification gives a visible action of $L$ on the generalized flag variety $G/H$, as well as that of the $H$-action on $G/L$ and of the $G$-action on the direct product of $G/L$ and $G/H$. Secondly, we find a generalized Cartan decomposition $G=LBH$ explicitly, where $B$ is a subset of $K$. An application to multiplicity-free theorems of representations is also discussed.


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Yuichiro Tanaka. "Visible actions on flag varieties of type C and a generalization of the Cartan decomposition." Tohoku Math. J. (2) 65 (2) 281 - 295, 2013.


Published: 2013
First available in Project Euclid: 25 June 2013

zbMATH: 1277.22014
MathSciNet: MR3079290
Digital Object Identifier: 10.2748/tmj/1372182727

Primary: 22E46
Secondary: 32A37, 53C30

Rights: Copyright © 2013 Tohoku University


Vol.65 • No. 2 • 2013
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