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The closure diagram for nilpotent orbits of the split real form of \(E_8\). (English) Zbl 1050.17006
The closure diagrams for adjoint nilpotent orbits in noncompact real forms of the exceptional simple complex Lie algebras were determined by the author in a series of papers [J. Lie Theory 10, 213–220 and 491–510 (2000; Zbl 0945.22004 and Zbl 0974.17010), ibid. 11, 381–413 (2001; Zbl 1049.17006); Represent. Theory 5, 17–42 and 284–316 (2001; Zbl 1031.17004 and Zbl 1050.17007)], except for the split real form of \(E_8\).
This paper handles this last case, it can be viewed as a continuation of the author’s paper [Asian J. Math. 5, 561–584 (2001; Zbl 1033.17012)]. Concretely, let \({\mathfrak O}_1\) and \({\mathfrak O}_2\) be adjoint nilpotent orbits in a real semisimple Lie algebra. Write \({\mathfrak O}_1\geq {\mathfrak O}_2\) if \({\mathfrak O}_2\) is contained in the closure of \({\mathfrak O}_1\). This defines a partial order on the set of such orbits, known as the closure ordering. This paper determines this order for the split real form of the simple complex Lie algebra \(E_8\) and presents a comprehensive list of simple representatives of these orbits.

MSC:
17B25 Exceptional (super)algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
Software:
Maple; LiE
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