Open Access
June 2012 Classification of visible actions on flag varieties
Yuichiro Tanaka
Proc. Japan Acad. Ser. A Math. Sci. 88(6): 91-96 (June 2012). DOI: 10.3792/pjaa.88.91


We give a complete classification of the pairs $(L,H)$ of Levi subgroups of compact simple Lie groups $G$ such that the $L$-action on a generalized flag variety $G/H$ is strongly visible (or equivalently, the $H$-action on $G/L$ or the diagonal $G$-action on $(G\times G)/(L\times H)$). The notion of visible actions on complex manifolds was introduced by T. Kobayashi, and a classification was accomplished by himself for the type A groups [J. Math. Soc. Japan, 2007]. A key step is to classify the pairs $(L,H)$ for which the multiplication mapping $L\times G^{\sigma}\times H\to G$ is surjective, where $\sigma$ is a Chevalley–Weyl involution of $G$. We then see that strongly visible actions, multiplicity-free restrictions of representations (c.f. Littelmann, Stembridge), the decomposition $G=LG^{\sigma}H$ and spherical actions are all equivalent in our setting.


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Yuichiro Tanaka. "Classification of visible actions on flag varieties." Proc. Japan Acad. Ser. A Math. Sci. 88 (6) 91 - 96, June 2012.


Published: June 2012
First available in Project Euclid: 5 June 2012

zbMATH: 1250.32021
MathSciNet: MR2928897
Digital Object Identifier: 10.3792/pjaa.88.91

Primary: 22E46
Secondary: 32A37 , 53C30

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

Rights: Copyright © 2012 The Japan Academy

Vol.88 • No. 6 • June 2012
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