
Journal of Lie Theory 26 (2016), No. 3, 597649 Copyright Heldermann Verlag 2016 Visible Actions on Spherical Nilpotent Orbits in Complex Simple Lie Algebras Atsumu Sasaki Dept. of Mathematics, Tokai University, 411 Kitakaname, Hiratsuka 2591292, Japan atsumu@tokaiu.jp 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); multiplicityfree representations in regular functions; momentum maps; and actions of compact subgroups (strongly visible actions). Keywords: Visible action, multiplicityfree representation, nilpotent orbit, induction theorem. MSC: 22E46; 32M10, 32M05, 14M17 [ Fulltextpdf (468 KB)] for subscribers only. 