Transformation Groups

, Volume 1, Issue 4, pp 375–403 | Cite as

Wonderful varieties of rank two

  • B. Wasserman


LetG be a complex connected reductive group. Well known wonderfulG-varieties are those of rank zero, namely the generalized flag varietiesG/P, those of rank one, classified in [A], and certain complete symmetric varieties described in [DP] such as the famous space of complete conics. Recently, there is a renewed interest in wonderful varieties of rank two since they were shown to hold a keystone position in the theory of spherical varieties, see [L], [BP], and [K].

The purpose of this paper is to give a classification of wonderful varieties of rank two. These are nonsingular completeG-varieties containing four orbits, a dense orbit and two orbits of codimension one whose closuresD 1 andD 2 intersect transversally in the fourth orbit which is of codimension two. We have gathered our results in tables, including isotropy groups, explicit basis of Picard groups, and several combinatorial data in relation with the theory of spherical varieties.


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  1. [A]
    D. Akhiezer,Equivariant completion of homogeneous algebraic varieties by homogeneous divisors, Ann. Glob. Analysis and Geometry1 (1983), 49–78.CrossRefMathSciNetGoogle Scholar
  2. [Bi]
    A. Biaŀynicki-Birula,On induced actions of algebraic groups, Ann. Inst. Fourier Grenoble43, 2 (1993), 359–364.MathSciNetGoogle Scholar
  3. [BLV]
    M. Brion, D. Luna et Th. Vust,Espaces homogènes sphériques, Invent. Math.84 (1986), 617–632.CrossRefMathSciNetGoogle Scholar
  4. [BN]
    N. Bourbaki,Groupes et Algèbres de Lie, Chap. IV, V, VI, 2e édition, Masson, Paris, 1981.MATHGoogle Scholar
  5. [BP]
    M. Brion, et F. Pauer,Valuations des espaces homogènes sphériques, Comment. Math. Helv.62 (1987), 265–285.MathSciNetGoogle Scholar
  6. [B]
    M. Brion,Classification des espaces homogènes sphériques, Compositio Math.63 (1987), 189–208.MathSciNetGoogle Scholar
  7. [B1]
    M. Brion,On spherical varieties of rank one, CMS Conf. Proc.10 (1989), 31–41.MathSciNetGoogle Scholar
  8. [B2]
    M. Brion,Groupe de Picard et nombres caractéristiques des variétés sphériques, Duke Math.58, 2 (1989), 397–424.CrossRefMathSciNetGoogle Scholar
  9. [B3]
    M. Brion,Vers une généralisation des espaces symétriques, J. Algebra134 (1990), 115–143.CrossRefMathSciNetGoogle Scholar
  10. [BT]
    A. Borel, et J. Tits,Eléments unipotents et sous-groupes paraboliques de groupes réductifs I, Invent. Math.12 (1971), 95–104.CrossRefMathSciNetGoogle Scholar
  11. [DP]
    C. De Concini, and C. Procesi,Complete symmetric varieties, in Invariant theory, Proceedings, Montecatini (F. Gherardelli, ed.); Lect. Notes Math. 996, Springer-Verlag, 1983, 1–44.Google Scholar
  12. [F]
    W. Fulton,Introduction to Toric Varieties, Annals of Math. Studies 131, Princeton University Press, 1993.Google Scholar
  13. [KKLV]
    F. Knop, H. Kraft, D. Luna, and Th. Vust,Local properties of algebraic group actions, in: Algebraische Transformationsgruppen und Invariantentheorie (H. Kraft, P. Slodowy, and T. Springer, ed.); DMV Semin. 13, Birkhäuser, Basel-Boston-Berlin, 1989, 63–76.Google Scholar
  14. [K]
    F. Knop,Automorphisms, root systems, and compactifications of Homogeneous Varieties, J. Amer. Math. Soc., Vol.9, 1 (1996), 153–174.CrossRefMathSciNetGoogle Scholar
  15. [K1]
    F. Knop,The Luna-Vust theory of spherical embeddings, in: Proc. of the Hyderabad Conf. on Algebraic Groups, Manoj Prakashan, Madras, 1991, 225–249.Google Scholar
  16. [Kr]
    M. Krämer,Sphärische Untergrupen in kompakten zusammen hängenden Lie Gruppen, Compositio Math.38 (1979), 129–153.MathSciNetGoogle Scholar
  17. [L]
    D. Luna,Toute variété magnifique est sphérique, Transformation Groups1, 3 (1996), 249–258.MathSciNetGoogle Scholar
  18. [L1]
    D. Luna,Grosses cellules pour les variétés sphériques, in: Algebraic Groups and Related Subjects. Volume in honor of R. W. Richardson (G. I. Lehrer, et al., ed.); Cambridge University Press, 1996.Google Scholar
  19. [LV]
    D. Luna, et Th. Vust,Plongements d’espaces homogènes, Comment. Math. Helv.58 (1983), 186–245.MathSciNetGoogle Scholar
  20. [M]
    I. Mikityuk,On the integrability of invariant hamiltonian systems with homogeneous configuration spaces, Math. USSR, Sb.57 (1987), 527–546.CrossRefGoogle Scholar
  21. [Mo]
    G. Mostow,Fully reducible subgroups of algebraic groups, Amer. J. Math.78 (1956), 200–221.CrossRefMathSciNetGoogle Scholar
  22. [OV]
    A. L. Onishchik, and E. B. Vinberg,Seminar on Lie groups and Algebraic Groups, Berlin, Springer-Verlag, 1990.Google Scholar
  23. [P]
    F. Pauer,Caractérisation valuative d’une classe de sous-groupes d’un groupe algébrique, C.R. 109e Cong. Nat. Soc. Sav. Dijon Sci.3 (1984), 159–166.Google Scholar
  24. [S]
    H. Sumihiro,Equivariant completion, J. Math. Kyoto Univ.14 (1974), 1–28.MathSciNetGoogle Scholar

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© Birkhäuser 1996

Authors and Affiliations

  • B. Wasserman
    • 1
  1. 1.IFSaint Martin d’Hères CedexFrance

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