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Real Group Orbits on Flag Manifolds

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Lie Groups: Structure, Actions, and Representations

Part of the book series: Progress in Mathematics ((PM,volume 306))

Abstract

We gather, partly with proofs, various results on the action of a real form of a complex semisimple group on its flag manifolds. In particular, we discuss the relationship between the cycle spaces of open orbits thereon and the crown of the symmetric space of non-compact type.

To Joseph Wolf on the occasion of his 75th birthday

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References

  1. D.N. Akhiezer, S.G. Gindikin, On Stein extensions of real symmetric spaces, Math. Ann. 286 (1990), 1–12.

    Google Scholar 

  2. A. Andreotti, H. Grauert, Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193–259.

    Google Scholar 

  3. L. Barchini, Stein extensions of real symmetric spaces and the geometry of the flag manifold, Math. Ann. 326 (2003), 331–346.

    Google Scholar 

  4. A. Borel, J. Tits, Éléments unipotents et sous-groupes paraboliques de groupes réductifs I, Invent. Math. 12, 2 (1971), 95–104.

    Google Scholar 

  5. R. Bremigan, J. Lorch, Orbit duality for flag manifolds, Manuscripta Math. 109, 2 (2002), 233–261.

    Google Scholar 

  6. M. Brion, Quelques propriétés des espaces homogènes sphériques, Manuscripta Math. 55, 2 (1986), 191–198.

    Google Scholar 

  7. D. Burns, S. Halverscheid, R. Hind, The geometry of Grauert tubes and complexification of symmetric spaces, Duke Math. J., 118, 3 (2003), 465–491.

    Article  MathSciNet  MATH  Google Scholar 

  8. C. Chevalley, Théorie des groupes de Lie: groupes algébriques, théorèmes généraux sur les algèbres de Lie, Paris, Hermann, 1951.

    Google Scholar 

  9. A. Dold, Lectures on algebraic topology, Springer, Berlin, Heidelberg, New York, second edition, 1980.

    MATH  Google Scholar 

  10. G. Fels, A.T. Huckleberry, Characterization of cycle domains via Kobayashi hyperbolicity, Bull. Soc. Math. France, 133, 1 (2005), 121–144.

    MathSciNet  MATH  Google Scholar 

  11. G. Fels, A.T. Huckleberry, J.A. Wolf, Cycle spaces of flag domains, a complex analytic viewpoint, Progress in Mathematics, V. 245, Birkhäuser, Boston, 2006.

    Google Scholar 

  12. S. Gindikin, B. Krötz, Complex crowns of Riemannian symmetric spaces and noncompactly causal symmetric spaces, Trans. Amer. Math. Soc. 354, 8 (2002), 3299–3327.

    Google Scholar 

  13. S. Gindikin, T. Matsuki, Stein extensions of Riemannian symmetric spaces and dualities of orbits on flag manifolds, Transform. Groups 8, 4 (2003), 333–376.

    Google Scholar 

  14. A. Huckleberry, On certain domains in cycle spaces of flag manifolds, Math. Ann. 323 (2002), 797–810.

    Google Scholar 

  15. A. Huckleberry, B. Ntatin, Cycle spaces of G-orbits in \({G}^{\mathbb{C}}\)-manifolds, Manuscripta Math. 112, 4 (2003), 433–440.

    Google Scholar 

  16. A. Huckleberry, J.A. Wolf, Schubert varieties and cycle spaces, Duke Math. J., 120, 2 (2003), 229–249.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Kobayashi, Hyperbolic complex spaces, Springer, Berlin, Heidelberg, 1998.

    Book  MATH  Google Scholar 

  18. B. Kostant, On the conjugacy of real Cartan subalgebras I, Proc. Nat. Acad. Sci U.S.A. 41, 11 (1955), 967–970.

    Google Scholar 

  19. B. Krötz, R. Stanton, Holomorphic extension of representations: (I) automorphic functions, Ann. of Math. 159, 2 (2004), 641–724.

    Google Scholar 

  20. B. Krötz, E. Opdam, Analysis on the crown domain, Geom. Funct. Anal. 18, 4 (2008), 1326–1421.

    Google Scholar 

  21. D. Luna, Slices étales, Bull. Soc. Math. France, Mémoire 33 (1973), 81–105.

    Google Scholar 

  22. F.M. Malyshev (Malys̆ev), Complex homogeneous spaces of semisimple Lie groups of the first category, Izv. Akad. Nauk SSSR, ser. mat. 39, 5 (1975), 992–1002, English transl.: Math. USSR Izv. 9, 5 (1975), 939–949.

    Google Scholar 

  23. F.M. Malyshev (Malys̆ev), Complex homogeneous spaces of semisimple Lie groups of type D n , Izv. Akad. Nauk SSSR, ser. mat. 41, 4 (1977), 829–852, English transl.: Math. USSR Izv. 11, 4 (1977), 783–805.

    Google Scholar 

  24. T. Matsuki, Orbits on affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math. J. 12, 2 (1982), 307–320.

    Google Scholar 

  25. T. Matsuki, Closure relations for orbits on affine symmetric spaces under the action of parabolic subgroups. Intersections of associated orbits, Hiroshima Math. J. 18, 1 (1988), 59–67.

    Google Scholar 

  26. T. Matsuki, Stein extensions of Riemann symmetric spaces and some generalization, J. Lie Theory 13, 2 (2003), 563–570.

    Google Scholar 

  27. T. Matsuki, Equivalence of domains arising from duality of orbits on flag manifolds I, II, III, Trans. Amer. Math. Soc. 358, 5 (2005), 2217–2245; Proc. Amer. Math. Soc. 134, 12 (2006), 3423–3428; Trans. Amer. Math. Soc. 359, 10 (2007), 4773–4786.

    Google Scholar 

  28. I. Mirković, T. Uzawa, K. Vilonen, Matsuki correspondence for sheaves, Invent. Math. 109, 2 (1992), 231–245.

    Google Scholar 

  29. D. Montgomery, Simply connected homogeneous spaces, Proc. Amer. Math. Soc. 1 (1950), 467–469.

    Google Scholar 

  30. A.L. Onishchik, Inclusion relations among transitive compact transformation groups, Trudy Mosk. Mat. Obshch. 11 (1962), 199–242, English transl.: Amer. Math. Soc. Transl. (2) 50 (1966), 5–58.

    Google Scholar 

  31. A.L. Onishchik, Decompositions of reductive Lie groups, Mat. Sbornik 80 (122) (1969), 553–599, English transl.: Math. USSR Sb. 9 (1969), 515–554.

    Google Scholar 

  32. A.L. Onishchik, Topology of transitive transformation groups, J.A.Barth, Leipzig, Berlin, Heidelberg, 1994.

    MATH  Google Scholar 

  33. W. Schmid, J. Wolf, A vanishing theorem for open orbits on complex flag manifolds, Proc. Amer. Math. Soc. 92, 3 (1984), 461–464.

    Google Scholar 

  34. M. Sugiura, Conjugate classes of Cartan subalgebras in real semi-simple Lie algebras, J. Math. Soc. Japan 11, 4 (1959), 374–434.

    Google Scholar 

  35. E.B. Vinberg, Complexity of actions of reductive groups, Functional Analysis and Appl. 20, 1 (1986), 1–11.

    Google Scholar 

  36. B. Weisfeiler, On one class of unipotent subgroups of semisimple algebraic groups, arXiv: math/0005149v1 [math. AG], translated from Russian: Uspekhi Mat. Nauk 21, 2 (128) (1966), 222–223.

    Google Scholar 

  37. R.O. Wells, Jr., J.A. Wolf, Poincaré series and automorphic cohomology on flag domains, Ann. Math. 105, 3 (1977), 397–448.

    Google Scholar 

  38. J.A. Wolf, The action of a real semisimple Lie group on a complex flag manifold. I: Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75, 6 (1969), 1121–1237.

    Google Scholar 

  39. J.A. Wolf, R. Zierau, Linear cycle spaces in flag domains, Math. Ann. 316 (2000), 529–545.

    Google Scholar 

  40. J.A. Wolf, R. Zierau, A note on the linear cycle space for groups of Hermitian type, J. Lie Theory 13, 1 (2003), 189–191.

    Google Scholar 

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Authors and Affiliations

  1. Institute for Information Transmission Problems, 19 B.Karetny per., 127994, Moscow, Russia

    Dmitri Akhiezer

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  1. Dmitri Akhiezer
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Correspondence to Dmitri Akhiezer .

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Editors and Affiliations

  1. Remagen, Germany

    Alan Huckleberry

  2. Jacobs University School of Engineering and Science, Bremen, Germany

    Ivan Penkov

  3. Yale University Dept. Mathematics, New Haven, Connecticut, USA

    Gregg Zuckerman

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Akhiezer, D. (2013). Real Group Orbits on Flag Manifolds. In: Huckleberry, A., Penkov, I., Zuckerman, G. (eds) Lie Groups: Structure, Actions, and Representations. Progress in Mathematics, vol 306. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-7193-6_1

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