The closure diagram for nilpotent orbits of the split real form of E8

• Autorzy: Dragomir Đoković
• Języki publikacji: EN

Abstrakty

EN

Let $$\mathcal{O}_1 $$ and $$\mathcal{O}_2 $$ be adjoint nilpotent orbits in a real semisimple Lie algebra. Write $$\mathcal{O}_1 $$ ≥ $$\mathcal{O}_2 $$ 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 of the simple complex Lie algebra, E 8. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible components of the boundaries $$\partial \mathcal{O}_1^i $$ and of the intersections $$\overline {\mathcal{O}_l^i } \cap \overline {\mathcal{O}_l^j } $$ .

Słowa kluczowe

EN

Exceptional Lie groups; adjoint action; closures of nilpotent orbits; normal triples; Kostant-Sekiguchi bijection; prehomogeneous vector spaces; 17B25; 17B45

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2003
• Tom: 1
• Numer: 4
• Strony: 573-643

Daty

wydano

2003-12-01

online

2003-12-01

Twórcy

autor

Dragomir Đoković

• University of Waterloo

Bibliografia

• [1] D. Barbasch and M.R. Sepanski: “Closure ordering and the Kostant-Sekiguchi correspondence”, Proc. Amer. Math. Soc., Vol. 126, (1998), pp. 311–317. http://dx.doi.org/10.1090/S0002-9939-98-04090-8
• [2] W.M. Beynon and N. Spaltenstein: “Green functions of finite Chevalley groups of type E n (n=6, 7, 8)”, J. Algebra, Vol. 88, (1984), pp. 584–614. http://dx.doi.org/10.1016/0021-8693(84)90084-X
• [3] N. Bourbaki: Groupes et algèbres de Lie, Chap. 7 et 8, Hermann, Paris, 1975.
• [4] R.W. Carter: “Conjugacy classes in the Weyl group”, Comp. Math., Vol. 25, (1972), pp. 1–59.
• [5] B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan, S.M. Watt: Maple V Language reference Manual, Springer-Verlag, New York, 1991, xv+267 pp.
• [6] D.H. Collingwood and W.M. McGovern: Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold, New York, 1993.
• [7] D.Ž. Đoković: “Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers”, J. Algebra, Vol. 112, (1988), pp. 503–524. http://dx.doi.org/10.1016/0021-8693(88)90104-4
• [8] D.Ž. Đoković: “Explicit Cayley triples in real forms of G 2, F 4, and E 6”, Pacific. J. Math., Vol. 184, (1998), pp. 231–255.
• [9] D.Ž. Đoković: “Explicit Cayley triples in real forms of E 8”, Pacific J. Math.”, Vol. 194, (2000), pp. 57–82. http://dx.doi.org/10.2140/pjm.2000.194.57
• [10] D.Ž. Đoković: “The closure diagrams for nilpotent orbits of real forms of F 4 and G 2”, J. Lie Theory, Vol. 10, (2000), pp. 491–510.
• [11] D.Ž. Đoković: “The closure diagrams for nilpotent orbits of real forms of E 6”, J. Lie Theory, 11, (2001), pp. 381–413.
• [12] D.Ž. Đoković: “The closure diagrams for nilpotent orbits of the real forms EVI and EVII of E 7”, Represent. Theory, Vol. 5, (2001), pp. 17–42. http://dx.doi.org/10.1090/S1088-4165-01-00112-1
• [13] D.Ž. Đoković: “The closure diagram for nilpotent orbits of the split real forms of E 7”, Represent. Theory, Vol. 5, (2001), pp. 284–316. http://dx.doi.org/10.1090/S1088-4165-01-00124-8
• [14] D.Ž. Đoković: “The closure diagram for nilpotent orbits of the real form EIX of E 8”, Asian J. Math., Vol. 5, (2001), pp. 561–584.
• [15] K. Mizuno: “The conjugate classes of unipotent elements of the Chevalley groups E 7 and E 8”, Tokyo J. Math., Vol. 3, (1980), pp. 391–461. http://dx.doi.org/10.3836/tjm/1270473003
• [16] A. Mortajine: Classification des espaces préhomogènes de type parabolique réguliers et de leurs invariants relatifs, Hermann, Paris, 1991.
• [17] M. Sato and T. Kimura: “A classification of irreducible prehomogeneous vector spaces and their relative invariants”, Nagoya Math. J., Vol. 65, (1977), pp. 1–155.
• [18] M. Sato, T. Shintani, M. Muro: “Theory of prehomogeneous vector spaces (algebraic part)”, Nagoya Math. J., Vol. 120, (1990), pp. 1–34.
• [19] M.A.A. van Leeuwen, A.M. Cohen, B. Lisser: LiE, version 2.1, a software package for Lie group theoretic computations, Computer Algebra Group of CWI, Amsterdam, The Netherlands.

Identyfikatory

DOI

10.2478/BF02475183