Nilpotent Bicone and Characteristic Submodule of a Reductive Lie Algebra

Abstract

Let $\mathfrak{g}$ be a finite-dimensional complex reductive Lie algebra and S($\mathfrak{g}$) its symmetric algebra. The nilpotent bicone of $\mathfrak{g}$ is the subset of elements (x, y) of $\mathfrak{g} \times \mathfrak{g}$ whose subspace generated by x and y is contained in the nilpotent cone. The nilpotent bicone is naturally endowed with a scheme structure, as nullvariety of the augmentation ideal of the subalgebra of ${\text{S}}{\left( \mathfrak{g} \right)} \otimes _{\mathbb{C}} {\text{S}}{\left( \mathfrak{g} \right)}$ generated by the 2-order polarizations of invariants of ${\text{S}}{\left( \mathfrak{g} \right)}$. The main result of this paper is that the nilpotent bicone is a complete intersection of dimension $3{\left( {{\text{b}}_{\mathfrak{g}} - {\text{rk}}\,\mathfrak{g}} \right)}$, where ${\text{b}}_{\mathfrak{g}}$ and ${\text{rk}}\,\mathfrak{g}$ are the dimensions of Borel subalgebras and the rank of $\mathfrak{g}$, respectively. This affirmatively answers a conjecture of Kraft and Wallach concerning the nullcone [KrW2]. In addition, we introduce and study in this paper the characteristic submodule of $\mathfrak{g}$. 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. The main difficulty encountered for this work is that the nilpotent bicone is not reduced. To deal with this problem, we introduce an auxiliary reduced variety, 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 by Mustaţǎ in [Mu]. Finally, we give applications of our results to invariant theory.