Visible actions on spherical nilpotent orbits in complex simple Lie algebras

TL;DR

This paper explores the conditions under which actions on complex nilpotent orbits are strongly visible, establishing a link with sphericity and providing explicit descriptions of slices in these actions.

Contribution

It proves that maximal compact group actions on nilpotent orbits are strongly visible if and only if the orbits are spherical, and describes slices explicitly.

Findings

Strongly visible actions correspond to spherical nilpotent orbits.

A concrete description of slices in strongly visible actions is provided.

Clarifies relationships among sphericity, multiplicity-free representations, and momentum maps.

Abstract

This paper studies nilpotent orbits in complex simple Lie algebras from the viewpoint of strongly visible actions in the sense of T. Kobayashi. We prove that the action of a maximal compact group consisting of inner automorphisms on a nilpotent orbit is strongly visible if and only if it is spherical, namely, admitting an open orbit of a Borel subgroup. Further, we find a concrete description of a slice in the strongly visible action. As a corollary, we clarify a relationship among different notions of complex nilpotent orbits: actions of Borel subgroups (sphericity); multiplicity-free representations in regular functions; momentum maps; and actions of compact subgroups (strongly visible actions).