Nilpotent bicone and characteristic submodule of a reductive Lie algebra

TL;DR

This paper proves that the nilpotent bicone of a complex reductive Lie algebra is a complete intersection, confirming a conjecture, and explores related structures like the characteristic submodule and principal bicone using motivic integration.

Contribution

It establishes the nilpotent bicone as a complete intersection and introduces the principal bicone, advancing understanding of the commuting variety in Lie algebra theory.

Findings

Nilpotent bicone is a complete intersection.

Confirmed Kraft-Wallach conjecture on the nullcone.

Linked structures to jet schemes and motivic integration.

Abstract

The nilpotent bicone of a finite dimensional complex reductive Lie algebra g is the subset of elements in g x g whose subspace generated by the components is contained in the nilpotent cone of g. The main result of this note is that the nilpotent bicone is a complete intersection. This affirmatively answers a conjecture of Kraft-Wallach concerning the nullcone. In addition, we introduce and study the characteristic submodule of g. The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal of definition. In order to study the nilpotent bicone, we introduce another subvariety, the principal bicone. The nilpotent bicone, as well as the principal bicone, are linked to jet schemes. We study their dimensions using arguments from motivic integration. Namely, we follow methods developed in http://arxiv.org/abs/math/0008002v5 .