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Nilpotent bicone and characteristic submodule of a reductive Lie algebra
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 .Comment: 48 pages. Remark 8 has been modified; one sentence was not correct.
We thank Kari Vilonen for pointing out this erro
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