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Dual pairs and Kostant-Sekiguchi correspondence. II: Classification of nilpotent elements. (English) Zbl 1107.22007

Summary: We classify the homogeneous nilpotent orbits in certain Lie color algebras and specialize the results to the setting of a real reductive dual pair.
For any member of a dual pair, we prove the bijectivity of the two Kostant-Sekiguchi maps by straightforward argument. For a dual pair we determine the correspondence of the real orbits, the correspondence of the complex orbits and explain how these two relations behave under the Kostant-Sekiguchi maps. In particular we prove that for a dual pair in the stable range there is a Kostant-Sekiguchi map such that the conjecture formulated in Part I [J. Algebra 250, No. 2, 408–426 (2002; Zbl 1003.22003)] of this paper holds. We also show that the conjecture cannot be true in general.

MSC:

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
20G05 Representation theory for linear algebraic groups
17B75 Color Lie (super)algebras

Citations:

Zbl 1003.22003
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References:

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