## Journal of the Mathematical Society of Japan

### Visible actions on flag varieties of type D and a generalization of the Cartan decomposition

Yuichiro TANAKA

#### Abstract

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.

#### Article information

Source
J. Math. Soc. Japan, Volume 65, Number 3 (2013), 931-965.

Dates
First available in Project Euclid: 23 July 2013

https://projecteuclid.org/euclid.jmsj/1374586630

Digital Object Identifier
doi:10.2969/jmsj/06530931

Mathematical Reviews number (MathSciNet)
MR3079290

Zentralblatt MATH identifier
1296.22015

#### Citation

TANAKA, Yuichiro. Visible actions on flag varieties of type D and a generalization of the Cartan decomposition. J. Math. Soc. Japan 65 (2013), no. 3, 931--965. doi:10.2969/jmsj/06530931. https://projecteuclid.org/euclid.jmsj/1374586630

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