Tohoku Mathematical Journal
- Tohoku Math. J. (2)
- Volume 65, Number 2 (2013), 281-295.
Visible actions on flag varieties of type C and a generalization of the Cartan decomposition
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.
Tohoku Math. J. (2), Volume 65, Number 2 (2013), 281-295.
First available in Project Euclid: 25 June 2013
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 22E46: Semisimple Lie groups and their representations
Secondary: 32A37: Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) [See also 46Exx] 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]
Tanaka, Yuichiro. Visible actions on flag varieties of type C and a generalization of the Cartan decomposition. Tohoku Math. J. (2) 65 (2013), no. 2, 281--295. doi:10.2748/tmj/1372182727. https://projecteuclid.org/euclid.tmj/1372182727