Jet schemes of the closure of nilpotent orbits

TL;DR

This paper investigates the geometric properties of jet schemes of nilpotent orbit closures in complex reductive Lie algebras, revealing irreducibility for the nilpotent cone and reducibility for many other orbit closures.

Contribution

It demonstrates that while the nilpotent cone's jet schemes are irreducible, many other orbit closures have reducible jet schemes, using induction and restriction techniques.

Findings

Jet schemes of the nilpotent cone are irreducible.

Many nilpotent orbit closures have reducible jet schemes.

Results imply new geometric properties of orbit closures.

Abstract

We study in this paper the jet schemes of the closure of nilpotent orbits in a finite-dimensional complex reductive Lie algebra. For the nilpotent cone, which is the closure of the regular nilpotent orbit, all the jet schemes are irreducible. This was first observed by Eisenbud and Frenkel, and follows from a strong result of Musta\u{t}\c{a} (2001). Using induction and restriction of "little" nilpotent orbits in reductive Lie algebras, we show that for a large number of nilpotent orbits, the jet schemes of their closure are reducible. As a consequence, we obtain certain geometrical properties of these nilpotent orbit closures.