Visible actions of compact Lie groups on complex spherical varieties

Abstract

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible actions on complex manifolds. He proved that for a Lie group $U$ the space of global sections of a $U$-equivariant holomorphic vector bundle $\mathcal{W}$ is multiplicity-free if the $U$-action on the base space $X$ is strongly visible and if the isotropy representations on the fibers are multiplicity-free under some compatibility condition on an anti-holomorphic diffeomorphism of $\mathcal{W}$. Especially in the case of the trivial line bundle, his theorem says that if a Lie group acts on the base space strongly visibly then the space of holomorphic functions is multiplicity-free. In short, the visibility is a geometric condition that assures the multiplicity-freeness property. In this article we consider the converse direction when $U$ is a compact real form of a connected complex reductive algebraic group $G$ and $X$ is an irreducible complex algebraic $G$-variety. In this setting the multiplicity-freeness property of the space of algebraic sections of arbitrary $G$-line bundle is equivalent to the sphericity of the $G$-action on $X$. Then our main result says that the multiplicity-freeness property implies the visibility, namely, if $X$ is a $G$-spherical variety then the $U$-action on $X$ is strongly visible.