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Prehomogeneous spaces associated with nilpotent orbits in simple real Lie algebras \(E_{6(6)}\) and \(E_{6(-26)}\) and their relative invariants. (English) Zbl 1148.17004

Summary: We give an efficient and stable algorithm for computing highest weights in a large class of prehomogeneous spaces associated with the nilpotent orbits of the real Lie algebras \(E_{6(6)}\) and \(E_{6(-26)}\). This paper concludes our classification of such prehomogeneous spaces for all complex and real reductive Lie algebras. For classical algebras using the fact that the nilpotent orbits are parameterized by partitions of integers we have given general formulas in [J. Algebra 289, No. 2, 515–557 (2005; Zbl 1147.17300) and J. Algebra 305, No. 1, 194–269 (2006; Zbl 1147.17301)]. For complex or inner-type real exceptional algebras we have given general algorithms and tables in [Lect. Notes Comput. Sci. 3516, 611–618 (2005; Zbl 1120.68466) and Lect. Notes Comput. Sci. 3482, 512–521 (2005)]. The present paper considers the case of real exceptional algebras that are not of inner type.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20 Simple, semisimple, reductive (super)algebras
22E30 Analysis on real and complex Lie groups
11S90 Prehomogeneous vector spaces

Software:

LiE
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