Transformation Groups

, Volume 16, Issue 1, pp 109–136 | Cite as

Spherical orbit closures in simple projective spaces and their normalizations



Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module. If G/HP(V) is a spherical orbit and if \( X = \overline {G/H} \) is its closure, then we describe the orbits of X and those of its normalization \( \tilde{X} \) . If, moreover, the wonderful completion of G/H is strict, then we give necessary and sufficient combinatorial conditions so that the normalization morphism \( \tilde{X} \to X \) is a homeomorphism. Such conditions are trivially fulfilled if G is simply laced or if H is a symmetric subgroup.


Simple Root Dynkin Diagram Distinguished Subset Borel Subgroup Spherical System 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ak]
    D. Akhiezer, Equivariant completions of homogeneous algebraic varieties by homogeneous divisors, Ann. Global Anal. Geom. 1 (1983), no. 1, 49–78.CrossRefMATHMathSciNetGoogle Scholar
  2. [Bra1]
    P. Bravi, Wonderful varieties of type E, Represent. Theory 11 (2007), 174–191.CrossRefMATHMathSciNetGoogle Scholar
  3. [Bra2]
    P. Bravi, Primitive spherical systems, preprint (2009), arXiv:math.AG/0909.3765v2, 70 pp.Google Scholar
  4. [BC]
    P. Bravi and S. Cupit-Foutou, Classification of strict wonderful varieties, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 2, 641–681.MATHMathSciNetGoogle Scholar
  5. [BGMR]
    P. Bravi, J. Gandini, A. Maffei, A. Ruzzi, Normality and nonnormality of group compactifications in simple projective spaces, preprint (2010), arXiv:math.AG/1005.2478v2, 16 pp. (to appear in Ann. Inst. Fourier (Grenoble)).Google Scholar
  6. [BL]
    P. Bravi, D. Luna, An introduction to wonderful varieties with many examples of type F4, preprint (2008), arXiv:math.AG/0812.2340v2, 65 pp. (to appear in J. Algebra).Google Scholar
  7. [BPe1]
    P. Bravi, G. Pezzini, Wonderful varieties of type D, Represent. Theory 9 (2005), 578–637.CrossRefMATHMathSciNetGoogle Scholar
  8. [BPe2]
    P. Bravi, G. Pezzini, Wonderful varieties of type B and C, preprint (2009), arXiv:mathAG/0909.3771v1, 24 pp.Google Scholar
  9. [Bri1]
    M. Brion, Groupe de Picard et nombres caractéristiques des variétés sphériques, Duke Math. J. 58 (1989), no. 2, 397–424.CrossRefMATHMathSciNetGoogle Scholar
  10. [Bri2]
    M. Brion, Variétés sphériques, Notes de la session de la S.M.F. Opérations hamiltoniennes et opérations de groupes algébriques, Grenoble (1997),, 59 pp.Google Scholar
  11. [Bri3]
    M. Brion, The total coordinate ring of a wonderful variety, J. Algebra 313 (2007), no. 1, 61–99.CrossRefMATHMathSciNetGoogle Scholar
  12. [BLV]
    M. Brion, D. Luna, Th. Vust, Espaces homogenès sphériques, Invent. Math. 84 (1986), no. 3, 617–632.CrossRefMATHMathSciNetGoogle Scholar
  13. [BP]
    M. Brion, F. Pauer, Valuations des espaces homogenès sphériques, Comment. Math. Helv. 62 (1987), no. 2, 265–285.CrossRefMATHMathSciNetGoogle Scholar
  14. [Cu]
    S. Cupit-Foutou, Wonderful varieties: A geometrical realization, preprint (2009), arXiv:math.AG/0907.2852v3, 37 pp.Google Scholar
  15. [CP]
    C. De Concini, C. Procesi, Complete symmetric varieties, in: Invariant Theory (Montecatini, 1982), Lecture Notes in Mathematics, Vol. 996, Springer-Verlag, Berlin, 1983, pp. 1–44.Google Scholar
  16. [Fo]
    A. Foschi, Variétés magnifiques et polytopes moment, PhD thesis, Institut Fourier, Université J. Fourier, Grenoble, 1998.Google Scholar
  17. [Kn1]
    F. Knop, The Luna–Vust Theory of spherical embeddings, in: Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Manoj Prakashan, Madras, 1991, 225–249.Google Scholar
  18. [Kn2]
    F. Knop, On the set of orbits for a Borel subgroup, Comment. Math. Helv. 70 (1995), no. 2, 285–309.CrossRefMATHMathSciNetGoogle Scholar
  19. [Kn3]
    F. Knop, Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc. 9 (1996), no. 1, 153–174.CrossRefMATHMathSciNetGoogle Scholar
  20. [Lo]
    I. Losev, Uniqueness property for spherical homogeneous spaces, Duke Math. J. 147 (2009), no. 2, 315–343.CrossRefMATHMathSciNetGoogle Scholar
  21. [Lu1]
    D. Luna, Toute variété magnifique est sphérique, Transform. Groups 1 (1996), no. 3, 249–258.CrossRefMATHMathSciNetGoogle Scholar
  22. [Lu2]
    D. Luna, Variétés sphériques de type A, Publ. Math. Inst. Hautes Études Sci. 94 (2001), 161–226.MATHMathSciNetGoogle Scholar
  23. [Lu3]
    D. Luna, La variété magnifique modèle, J. Algebra 313 (2007), no. 1, 292–319.CrossRefMATHMathSciNetGoogle Scholar
  24. [Ma]
    A. Maffei, Orbits in degenerate compactifications of symmetric varieties, Transform. Groups 14 (2009), no. 1, 183–194.CrossRefMATHMathSciNetGoogle Scholar
  25. [Pe1]
    G. Pezzini, Wonderful varieties of type C, PhD thesis, Università di Roma “La Sapienza”, Roma, 2003.Google Scholar
  26. [Pe2]
    G. Pezzini, Simple immersions of wonderful varieties, Math. Z. 255 (2007), no. 4, 793–812.CrossRefMATHMathSciNetGoogle Scholar
  27. [Ti]
    Д. А. Тимашев, Эквиварианmные компакmификации редукmивных групп, Мат. сб. 194 (2003), no. 4, 119–146. Engl. transl.: D. A. Timashev, Equivariant compactifications of reductive groups, Sb. Math. 194 (2003), no. 4, 589–616.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Dipartimento di Matematica, “Guido Castelnuovo”“Sapienza” Università di RomaRomaItaly

Personalised recommendations