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Homology of the zero-set of a nilpotent vector field on a flag manifold. (English) Zbl 0646.14034

The authors study subvarieties of flag manifolds of connected complex reductive groups which are important in representation theory. Given a nilpotent element N of the Lie algebra of a connected complex reductive Lie group G they consider the variety consisting of all those Borel subalgebras of the Lie algebra of G which contain N. They show that the integral homology of this variety has no torsion, vanishes in odd degrees and all comes from algebraic cycles. Their method exploits the relationship between the homology of a variety and that of its set of fixed points under a torus action.
Reviewer: F.Kirwan

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
22E60 Lie algebras of Lie groups
14L30 Group actions on varieties or schemes (quotients)
20G15 Linear algebraic groups over arbitrary fields
14L24 Geometric invariant theory
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