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The closure diagrams for nilpotent orbits of the real forms EVI and EVII of $\mathbf {E_7}$


Author: Dragomir Ž. Đoković
Journal: Represent. Theory 5 (2001), 17-42
MSC (2000): Primary 05B15, 05B20; Secondary 05B05
DOI: https://doi.org/10.1090/S1088-4165-01-00112-1
Published electronically: February 2, 2001
Correction: Represent. Theory 5 (2001), 503-503.
MathSciNet review: 1826427
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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 two noncompact nonsplit real forms of the simple complex Lie algebra $E_7.$


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  • 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 https://doi.org/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 https://doi.org/10.1016/0021-8693%2884%2990084-X
  • N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
  • Ma B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan, and S.M. Watt, Maple V Language reference Manual, Springer–Verlag, New York, 1991, xv+267 pp.
  • 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 https://doi.org/10.1016/0021-8693%2888%2990104-4
  • DZ2 ---, Explicit Cayley triples in real forms of $E_7,$ Pacific J. Math. 191 (1999), 1–23. DZ3 ---, The closure diagrams for nilpotent orbits of real forms of $F_4$ and $G_2$, J. Lie Theory 10 (2000), 491–510. DZ4 ---, The closure diagrams for nilpotent orbits of real forms of $E_6$, J. Lie Theory (to appear). ED E.B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik 30 (1952), 349–462. (Amer. Math. Soc. Transl. Ser. 2 6 (1957), 111–245.)
  • Jun-ichi Igusa, A classification of spinors up to dimension twelve, Amer. J. Math. 92 (1970), 997–1028. MR 277558, DOI https://doi.org/10.2307/2373406
  • 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 https://doi.org/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
  • SSM M. Sato, T. Shintani, and M. Muro, Theory of prehomogeneous vector spaces (algebraic part), Nagoya Math. J. 120 (1990), 1–34. (The English translation of Sato’s lecture from Shintani’s notes.)
  • Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, vol. 946, Springer-Verlag, Berlin-New York, 1982 (French). MR 672610
  • LiE M.A.A. van Leeuwen, A.M. Cohen, and B. Lisser, "LiE”, a software package for Lie group theoretic computations, Computer Algebra Group of CWI, Amsterdam, The Netherlands.

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Additional Information

Dragomir Ž. Đoković
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Email: djokovic@uwaterloo.ca

Received by editor(s): August 15, 2000
Received by editor(s) in revised form: December 6, 2000
Published electronically: February 2, 2001
Additional Notes: Supported in part by the NSERC Grant A-5285.
Article copyright: © Copyright 2001 American Mathematical Society