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On closedness and simple connectedness of adjoint and coadjoint orbits. (English) Zbl 0815.22004
Let $$\mathfrak g$$ be a finite dimensional real Lie algebra and $$\mathfrak h$$ a subalgebra of $$\mathfrak g$$. For a subset $$S$$ of a group we write $$\langle S\rangle$$ for the subgroup generated by $$S$$. We define $$\text{Inn}_{\mathfrak g}({\mathfrak h}) := \langle e^{\text{ad}{\mathfrak h}}\rangle$$ and $$\text{INN}_{\mathfrak g}({\mathfrak h}) := \overline{\text{Inn}_{\mathfrak g}({\mathfrak h})}$$. We also set $$\text{Inn}_{\mathfrak g} := \text{Inn}_{\mathfrak g}({\mathfrak g})$$ and $$\text{INN}_{\mathfrak g} := \text{INN}_{\mathfrak g}({\mathfrak g})$$. Then the adjoint action is the action of $$\text{Inn}_{\mathfrak g}$$ on the Lie algebra $$\mathfrak g$$ and the coadjoint action is the action of $$\text{Inn}_{\mathfrak g}$$ on $${\mathfrak g}^*$$ via $$g.v := (g^{-1})^*(v) = v \circ g^{-1}$$. The main results of this paper are: (1) Every coadjoint orbit of strict convexity type is closed and simply connected; (2) Every adjoint orbit meeting a Cartan subalgebra $$\mathfrak t$$ and every coadjoint orbit meeting $${\mathfrak t}^*$$ is closed and one component of its Zariski closure.
Reviewer: A.K.Guts (Omsk)

##### MSC:
 22E60 Lie algebras of Lie groups 17B05 Structure theory for Lie algebras and superalgebras
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##### References:
  [Bi71] Birkes, D.,Orbits of linear algebraic groups, Annals of Math.93(1971), 459–475 · Zbl 0212.36402  [Bo91] Borel, A., ”Linear Algebraic Groups”, Graduate Texts in Math.126, Springer Verlag, New York, Berlin, 1991  [BHC62] Borel, A., and Harish-Chandra,Arithmetic subgroups of algebraic groups, Annals of Math.75:3(1962), 485–535 · Zbl 0107.14804  [BT65] Borel, A., and J. Tits,Groupes réductifs, Publ. Math. I.H.E.S.27(1965), 55–150  [Bou71] Bourbaki, N.,Groupes et algèbres de Lie, Chapitres 1–3, Hermann, Paris, 1971 · Zbl 0213.04103  [Bou90]–,Groupes et algèbres de Lie, Chapitres 7 et 8, Masson, Paris, 1990  [BtD85] Bröcker, T., and T. tom Dieck, ”Representations of compact Lie groups”, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985 · Zbl 0581.22009  [Ch55] Chevalley, C., ”Théorie des groupes de Lie III”, Hermann, Paris, 1955  [Hel78] Helgason, S., ”Differential Geometry, Lie Groups, and Symmetric Spaces”, Acad. Press, London, 1978 · Zbl 0451.53038  [HHL89] Hilgert, J., K. H. Hofmann, and J. D. Lawson, ”Lie Groups, Convex Cones, and Semigroups”, Oxford University Press, 1989 · Zbl 0701.22001  [HiNe91] Hilgert, J., and K.-H. Neeb, ”Lie-Gruppen und Lie-Algebren”, Vieweg, Braunschweig, 1991 · Zbl 0760.22005  [HiNe93] Hilgert, J., ”Lie semigroups and their applications”, Lecture Notes in Mathematics1552, Springer, 1993  [Ho65] Hochschild, G. P., ”The Structure of Lie Groups”, Holden Day, San Francisco, 1965  [Ho81] Hochschild, G. P., ”Basic Theory of Algebraic Groups and Lie Algebras”, Springer, Graduate Texts in Mathematics75, 1981 · Zbl 0589.20025  [Kr84] Kraft, H., ”Geometrische Methoden in der Invariantentheorie”, Vieweg, Wiesbaden, 1984 · Zbl 0569.14003  [Ne93a] Neeb, K.-H.,Invariant subsemigroups of Lie groups, Memoirs of the AMS, to appear  [Ne93b] Neeb, K.-H., ”Holomorphic representation theory and coadjoint orbits of convexity type”, Habilitationsschrift, Technische Hochschule Darmstadt, Januar 1993  [Ne93c] Neeb, K.-H.,Kähler structures and convexity properties of coadjoint orbits, submitted  [Ol82] Ol’shanskiî, G. I.,Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series, Funct. Anal. and Appl.15 (1982), 275–285 · Zbl 0503.22011  [OV90] Onishchick, A. L. and E. B. Vinberg, ”Lie Groups and Algebraic Groups”, Springer, 1990  [Pa84] Paneitz, S.,Determination of invariant convec cones in simple Lie algebras, Arkiv för Mat.21 (1984), 217–228. · Zbl 0526.22016  [Se76] Segal, I. E., ”Mathematical Cosmology and Extragalactic Astronomy”, Acad. Press, New York, San Francisco, London, 1976  [Vi63] Vinberg, E. B.,The theory of convex homogeneous cones, Transactions of the Mosc. Math. Soc12 (1963), 303–358  [Wa72] Warner, G., ”Harmonic analysis on semisimple Lie groups I”, Springer, Berlin, Heidelberg, New York, 1972
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