Open Access
July 2007 Visible actions on irreducible multiplicity-free spaces
Atsumu Sasaki
Proc. Japan Acad. Ser. A Math. Sci. 83(7): 109-113 (July 2007). DOI: 10.3792/pjaa.83.109


A holomorphic action of a Lie group $G$ on a complex manifold $D$ is called \textit{strongly visible} if there exist a totally real submanifold $S$ which meets every $G$-orbit in $D$ and an anti-holomorphic diffeomorphism $\sigma $ such that $\sigma |_{S}=\mathop{\mathrm{id}}_{S}$ and $\sigma$ preserves every $G$-orbit. In this paper, we prove that Kac’s multiplicity-free space is strongly visible, that is, if $(G_{\mathbf{C}},V)$ is an irreducible multiplicity-free space of a complex reductive Lie group $G_{\mathbf{C}}$, then a compact real form of $G_{\mathbf{C}}$ acts on $V$ in a strongly visible fashion. Furthermore, we give an explicit description of the choice of a totally real submanifold $S$ and an anti-holomorphic involution $\sigma$. This gives an evidence to Kobayashi’s conjecture , that is, $\dim_{\mathbf{R}}S$ coincides with the rank of the polynomial representation of $G_{\mathbf{C}}$ on $\mathbf{C}[V]$ in this setting.


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Atsumu Sasaki. "Visible actions on irreducible multiplicity-free spaces." Proc. Japan Acad. Ser. A Math. Sci. 83 (7) 109 - 113, July 2007.


Published: July 2007
First available in Project Euclid: 18 January 2008

zbMATH: 1144.32016
MathSciNet: MR2361421
Digital Object Identifier: 10.3792/pjaa.83.109

Primary: 32M05
Secondary: 20G05 , 22E46 , 32M15

Keywords: (strongly) visible action , complex manifold , multiplicity-free representation , multiplicity-free space , totally real submanifold

Rights: Copyright © 2007 The Japan Academy

Vol.83 • No. 7 • July 2007
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