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This paper studies nilpotent orbits in complex simple Lie algebras from theviewpoint of strongly visible actions in the sense of T. Kobayashi. We provethat the action of a maximal compact group consisting of inner automorphisms ona 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 concretedescription of a slice in the strongly visible action. As a corollary, weclarify a relationship among different notions of complex nilpotent orbits:actions of Borel subgroups (sphericity); multiplicity-free representations inregular functions; momentum maps; and actions of compact subgroups (stronglyvisible actions).

Visible actions on spherical nilpotent orbits in complex simple Lie algebras