Journal of Lie Theory 26 (2016), No. 3, 597--649
Copyright Heldermann Verlag 2016
Visible Actions on Spherical Nilpotent Orbits in Complex Simple Lie Algebras
Atsumu Sasaki
Dept. of Mathematics, Tokai University, 4-1-1 Kitakaname, Hiratsuka 259-1292, Japan
atsumu@tokai-u.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); multiplicity-free representations in regular functions; momentum maps; and actions of compact subgroups (strongly visible actions).
Keywords: Visible action, multiplicity-free representation, nilpotent orbit, induction theorem.
MSC: 22E46; 32M10, 32M05, 14M17
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