The nilpotent cone for classical Lie superalgebras

Abstract:

In this paper the authors introduce an analogue of the nilpotent cone, ${\mathcal N}$, for a classical Lie superalgebra, ${\mathfrak g}$, that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical simple Lie superalgebra, ${\mathfrak g}={\mathfrak g}_{\bar 0}\oplus {\mathfrak g}_{\bar 1}$ with $\text {Lie }G_{\bar 0}={\mathfrak g}_{\bar 0}$, it is shown that there are finitely many $G_{\bar 0}$-orbits on ${\mathcal N}$. Later the authors prove that the Duflo-Serganova commuting variety, ${\mathcal X}$, is contained in ${\mathcal N}$ for any classical simple Lie superalgebra. Consequently, our finiteness result generalizes and extends the work of Duflo-Serganova on the commuting variety. Further applications are given at the end of the paper.