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VOL. 16 | 2015 Visible Actions on Generalized Flag Varieties and a Generalization of the Cartan Decomposition
Yuichiro Tanaka

Editor(s) Ivaïlo M. Mladenov, Andrei Ludu, Akira Yoshioka

Abstract

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the theory of visible actions on complex manifolds.

Our main results give a classification of triples $(G,H,L)$ for a compact Lie group $G$ and its Levi subgroups $H,L$, which satisfy $G=HBL$. Here, $B$ is a subset of a Chevalley--Weyl involution $\sigma$-fixed points subgroup $G^{\sigma}$ of $G$. The point here is that one decomposition $G=LBH$ produces three strongly visible actions on generalized flag varieties, and thus three finite-dimensional multiplicity-free representations (Kobayashi's triunity principle).

Furthermore, we can also prove that the visibility of actions of compact Lie groups, the existence of a decomposition $G=LBH$ and the multiplicity-freeness property of finite-dimensional tensor product representations are all equivalent.

Information

Published: 1 January 2015
First available in Project Euclid: 13 July 2015

MathSciNet: MR3363851

Digital Object Identifier: 10.7546/giq-16-2015-270-281

Rights: Copyright © 2015 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences

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