Geodesic
Visible Actions on Spherical Nilpotent Orbits in Complex Simple Lie Algebras
Journal of Lie theory, Tome 26 (2016) no. 3, pp. 597-649
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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).
Classification : 22E46, 32M10, 32M05, 14M17
Mots-clés : Visible action, multiplicity-free representation, nilpotent orbit, induction theorem 
@article{JLT_2016_26_3_JLT_2016_26_3_a0,
     author = {A. Sasaki},
     title = {Visible {Actions} on {Spherical} {Nilpotent} {Orbits} in {Complex} {Simple} {Lie} {Algebras}},
     journal = {Journal of Lie theory},
     pages = {597--649},
     year = {2016},
     volume = {26},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JLT_2016_26_3_JLT_2016_26_3_a0/}
}
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A. Sasaki. Visible Actions on Spherical Nilpotent Orbits in Complex Simple Lie Algebras. Journal of Lie theory, Tome 26 (2016) no. 3, pp. 597-649. http://geodesic.mathdoc.fr/item/JLT_2016_26_3_JLT_2016_26_3_a0/