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Register a new userOn the generalized commuting varieties of a reductive Lie algebra
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Authors
Jean-Yves Charbonnel
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The generalized commuting and isospectral commuting varieties of a reductive Lie algebra have been introduced in a preceding article. In this note, it is proved that their normalizations are Gorenstein with rational singularities. Moreover, their canonical modules are free of rank 1. In particular, the usual commuting variety is Gorenstein with rational singularities and its canonical module is free of rank 1.
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The commuting variety of a reductive Lie algebra ${goth g}$ is the underlying variety of a well defined subscheme of $gg g{}$. In this note, it is proved that this scheme is normal. In particular, its ideal of definition is a prime ideal.
For a reductive Lie algbera over an algbraically closed field of charasteristic zero,we consider a borel subgroup $B$ of its adjoint group, a Cartan subalgebra contained inthe Lie algebra of $B$ and the closure $X$ of its orbit under $B$ in the Grassmannian.The variety $X$ plays an important role in the study of the commuting variety. In thisnote, we prove that $X$ is Gorenstein with rational singularities.
Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $mathbbm k$ of characteristic zero. We consider the commuting variety $mathcal C(mathfrak u)$ of the nilradical $mathfrak u$ of the Lie algebra $mathfrak b$ of a Borel subgroup $B$ of $G$. In case $B$ acts on $mathfrak u$ with only a finite number of orbits, we verify that $mathcal C(mathfrak u)$ is equidimensional and that the irreducible components are in correspondence with the {em distinguished} $B$-orbits in $mathfrak u$. We observe that in general $mathcal C(mathfrak u)$ is not equidimensional, and determine the irreducible components of $mathcal C(mathfrak u)$ in the minimal cases where there are infinitely many $B$-orbits in $mathfrak u$.
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 .
Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$, and assume that the characteristic of $k$ is zero or a pretty good prime for $G$. Let $P$ be a parabolic subgroup of $G$ and let $mathfrak p$ be the Lie algebra of $P$. We consider the commuting variety $mathcal C(mathfrak p) = {(X,Y) in mathfrak p times mathfrak p mid [X,Y] = 0}$. Our main theorem gives a necessary and sufficient condition for irreducibility of $mathcal C(mathfrak p)$ in terms of the modality of the adjoint action of $P$ on the nilpotent variety of $mathfrak p$. As a consequence, for the case $P = B$ a Borel subgroup of $G$, we give a classification of when $mathcal C(mathfrak b)$ is irreducible; this builds on a partial classification given by Keeton. Further, in cases where $mathcal C(mathfrak p)$ is irreducible, we consider whether $mathcal C(mathfrak p)$ is a normal variety. In particular, this leads to a classification of when $mathcal C(mathfrak b)$ is normal.
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