On spherical nilpotent orbits and beyond
Let \({\mathfrak g}\) be a semisimple Lie algebra over an algebraically closed field of characteristic zero. Write \(G\) for the corresponding adjoint group. This paper studies the complexity of nilpotent \(G\)-orbits \({\mathcal O}\) in \({\mathfrak g}\). Here the complexity is defined to be the minimal codimension of a Borel suborbit of \({\mathcal O}\). A nilpotent orbit, \({\mathcal O}\), of complexity zero is called spherical and the author has previously shown this notion is equivalent to the statement \(ht({\mathcal O}) \leq 3\). Here the height of \({\mathcal O}\) is defined as \(ht({\mathcal O})=\max\{n\in{\mathbb N} \mid(ad e)^n\not=0,\) \( e\in{\mathcal O}\}\). This paper gives an alternate characterization of spherical nilpotent orbits and provides a uniform proof. Namely, \({\mathcal O}\) is spherical if and only if it has a representative which is a sum of simple orthogonal root vectors. Additionally, the author shows the minimal non-spherical orbits in the simple Lie algebras are the ones of complexity 1 for \(SL_n\) and of complexity 2 for the remaining simple Lie algebras. The paper concludes with a study of the complexity nilpotent orbits of \(\theta\)-groups. \(\theta\)-groups are associated with finite order automorphisms, \(\theta\), of \({\mathfrak g}\).
scientific article; zbMATH DE number 1355988
scientific article