On varieties in an orbital variety closure in semisimple Lie algebra

Details On varieties in an orbital variety closure in semisimple Lie algebra On varieties in an orbital variety closure in semisimple Lie algebra

2004-09-23

arxiv.org/abs/math/0409445v1

Journal of Algebra, 295, 2006, pp.44-50

Mathematics

Representation Theory

7 pages

Scientific paper

Let g be a semisimple complex Lie algebra. Let O be a nilpotent orbit in g. Fix a triangular decomposition g=n+h+n^-. An irreducible component of the intersection of O and n is called an orbital variety associated to O. It is a Lagrangian subvariety of O. In this note we discuss the closure of an orbital variety as a union of varieties. We show that if g contains factors not of type A_n then there are orbital varieties whose closure contains components which are not Lagrangian. We show that the argument does not work if all the factors are of type A_n and provide the facts supporting the conjecture claiming that if all the factors of g are of type A_n then the closure of an orbital variety is a union of orbital varieties.

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