arXiv 0705.26852007-05math.RTmath.AG

Nilpotent bicone and characteristic submodule of a reductive Lie algebra

Jean-Yves Charbonnel, Anne Moreau

备注：: 48 pages. Remark 8 has been modified; one sentence was not correct. We thank Kari Vilonen for pointing out this error

英文摘要：

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 .

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