Visible actions on irreducible multiplicity-free spaces

Abstract

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.