Dimension Formula for Slice for Visible Actions on Spherical Nilpotent Orbits in Complex Simple Lie Algebras

Abstract

The recent paper [19] has studied spherical nilpotent orbits in complex simple Lie algebras from the viewpoint of the notion of strongly visible actions introduced by T. Kobayashi. The aim of this paper is to give a dimension formula for a slice for the strongly visible action on a spherical nilpotent orbit. This provides an unified explanation of the strong visibility for the action on a spherical nilpotent orbit. Further, we prove that our choice of slice for this action satisfies its dimension equals the rank of a nilpotent orbit, which suggests a deep relation between complex geometry and multiplicity-free representation in complex nilpotent orbits.