The closure diagram for nilpotent orbits of the split real form of 𝐸₇

Đoković, Dragomir

doi:10.1090/S1088-4165-01-00124-8

The closure diagram for nilpotent orbits of the split real form of ${E_7}$
HTML articles powered by AMS MathViewer

by Dragomir Ž. Đoković

Represent. Theory 5 (2001), 284-316

DOI: https://doi.org/10.1090/S1088-4165-01-00124-8

Published electronically: October 3, 2001

PDF | Request permission

Abstract:

Let $\mathcal {O}_1$ and $\mathcal {O}_2$ be adjoint nilpotent orbits in a real semisimple Lie algebra. Write $\mathcal {O}_1\geq \mathcal {O}_2$ if $\mathcal {O}_2$ is contained in the closure of $\mathcal {O}_1.$ This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form E V of $E_7.$

References

Dan Barbasch and Mark R. Sepanski, Closure ordering and the Kostant-Sekiguchi correspondence, Proc. Amer. Math. Soc. 126 (1998), no. 1, 311–317. MR 1422847, DOI 10.1090/S0002-9939-98-04090-8
W. M. Beynon and N. Spaltenstein, Green functions of finite Chevalley groups of type $E_{n}$ $(n=6,\,7,\,8)$, J. Algebra 88 (1984), no. 2, 584–614. MR 747534, DOI 10.1016/0021-8693(84)90084-X
N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1364, Hermann, Paris, 1975 (French). MR 0453824

Maple V Language reference Manual

David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060
Dragomir Ž. Đoković, Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers, J. Algebra 112 (1988), no. 2, 503–524. MR 926619, DOI 10.1016/0021-8693(88)90104-4
Dragomir Ž. Đoković, Explicit Cayley triples in real forms of $E_7$, Pacific J. Math. 191 (1999), no. 1, 1–23. MR 1725460, DOI 10.2140/pjm.1999.191.1
Dragomir Ž. Đoković, The closure diagrams for nilpotent orbits of real forms of $F_4$ and $G_2$, J. Lie Theory 10 (2000), no. 2, 491–510. MR 1774875

The closure diagrams for nilpotent orbits of real forms of $E_6$

11

The closure diagrams for nilpotent orbits of the real forms E VI and E VII of $E_7$

5

The closure diagram for nilpotent orbits of the real form E IX of $E_8$

Kenzo Mizuno, The conjugate classes of unipotent elements of the Chevalley groups $E_{7}$ and $E_{8}$, Tokyo J. Math. 3 (1980), no. 2, 391–461. MR 605099, DOI 10.3836/tjm/1270473003
M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155. MR 430336, DOI 10.1017/S0027763000017633
Mikio Sato, Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato’s lecture from Shintani’s note, Nagoya Math. J. 120 (1990), 1–34. Notes by Takuro Shintani; Translated from the Japanese by Masakazu Muro. MR 1086566, DOI 10.1017/S0027763000003214
J. Tits, Normalisateurs de tores. I. Groupes de Coxeter étendus, J. Algebra 4 (1966), 96–116 (French). MR 206117, DOI 10.1016/0021-8693(66)90053-6