Advertisement

Real Group Orbits on Flag Manifolds

  • Dmitri Akhiezer
Chapter
Part of the Progress in Mathematics book series (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.

Keywords

Reductive algebraic group Real form Flag manifold Flag domain Cycle space 

References

  1. [1]
    D.N. Akhiezer, S.G. Gindikin, On Stein extensions of real symmetric spaces, Math. Ann. 286 (1990), 1–12.Google Scholar
  2. [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. [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. [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. [5]
    R. Bremigan, J. Lorch, Orbit duality for flag manifolds, Manuscripta Math. 109, 2 (2002), 233–261.Google Scholar
  6. [6]
    M. Brion, Quelques propriétés des espaces homogènes sphériques, Manuscripta Math. 55, 2 (1986), 191–198.Google Scholar
  7. [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.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    C. Chevalley, Théorie des groupes de Lie: groupes algébriques, théorèmes généraux sur les algèbres de Lie, Paris, Hermann, 1951.Google Scholar
  9. [9]
    A. Dold, Lectures on algebraic topology, Springer, Berlin, Heidelberg, New York, second edition, 1980.MATHGoogle Scholar
  10. [10]
    G. Fels, A.T. Huckleberry, Characterization of cycle domains via Kobayashi hyperbolicity, Bull. Soc. Math. France, 133, 1 (2005), 121–144.MathSciNetMATHGoogle Scholar
  11. [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. [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. [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. [14]
    A. Huckleberry, On certain domains in cycle spaces of flag manifolds, Math. Ann. 323 (2002), 797–810.Google Scholar
  15. [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. [16]
    A. Huckleberry, J.A. Wolf, Schubert varieties and cycle spaces, Duke Math. J., 120, 2 (2003), 229–249.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    S. Kobayashi, Hyperbolic complex spaces, Springer, Berlin, Heidelberg, 1998.MATHCrossRefGoogle Scholar
  18. [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. [19]
    B. Krötz, R. Stanton, Holomorphic extension of representations: (I) automorphic functions, Ann. of Math. 159, 2 (2004), 641–724.Google Scholar
  20. [20]
    B. Krötz, E. Opdam, Analysis on the crown domain, Geom. Funct. Anal. 18, 4 (2008), 1326–1421.Google Scholar
  21. [21]
    D. Luna, Slices étales, Bull. Soc. Math. France, Mémoire 33 (1973), 81–105.Google Scholar
  22. [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. [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. [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. [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. [26]
    T. Matsuki, Stein extensions of Riemann symmetric spaces and some generalization, J. Lie Theory 13, 2 (2003), 563–570.Google Scholar
  27. [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. [28]
    I. Mirković, T. Uzawa, K. Vilonen, Matsuki correspondence for sheaves, Invent. Math. 109, 2 (1992), 231–245.Google Scholar
  29. [29]
    D. Montgomery, Simply connected homogeneous spaces, Proc. Amer. Math. Soc. 1 (1950), 467–469.Google Scholar
  30. [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. [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. [32]
    A.L. Onishchik, Topology of transitive transformation groups, J.A.Barth, Leipzig, Berlin, Heidelberg, 1994.MATHGoogle Scholar
  33. [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. [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. [35]
    E.B. Vinberg, Complexity of actions of reductive groups, Functional Analysis and Appl. 20, 1 (1986), 1–11.Google Scholar
  36. [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. [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. [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. [39]
    J.A. Wolf, R. Zierau, Linear cycle spaces in flag domains, Math. Ann. 316 (2000), 529–545.Google Scholar
  40. [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

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute for Information Transmission ProblemsMoscowRussia

Personalised recommendations