Classification of strict wonderful varieties
Annales de l'Institut Fourier, Volume 60 (2010) no. 2, pp. 641-681.

In the setting of strict wonderful varieties we prove Luna’s conjecture, saying that wonderful varieties are classified by combinatorial objects, the so-called spherical systems. In particular, we prove that primitive strict wonderful varieties are mostly obtained from symmetric spaces, spherical nilpotent orbits and model spaces. To make the paper as self-contained as possible, we also gather some known results on these families and more generally on wonderful varieties.

D’après la conjecture de Luna, les variétés magnifiques peuvent être classifiées en termes d’objets combinatoires, les systèmes sphériques. Dans le présent article, nous prouvons cette conjecture dans le cas des variétés magnifiques dites strictes. Nous montrons, en particulier, que les variétés magnifiques strictes et primitives sont, pour la plupart, des variétés symétriques, des orbites nilpotentes sphériques ou des espaces modèles. Afin de faciliter la lecture de cet article, nous rappelons quelques faits connus sur ces variétés et, plus généralement, sur les variétés magnifiques.

DOI: 10.5802/aif.2535
Classification: 14M27, 14L30, 20G05
Keywords: Spherical varieties, wonderful varieties, symmetric varieties, spherical nilpotent orbits, model spaces
Mot clés : variétés sphériques, variétés magnifiques, variétés symétriques, orbites nilpotentes sphériques, espaces modèles
Bravi, Paolo 1; Cupit-Foutou, Stéphanie 2

1 Università di Roma La Sapienza Dipartimento di Matematica P. le Aldo Moro 5 00185 Roma (Italy)
2 Universität zu Köln Mathematisches Institut Weyertal Str. 86-90 50931 Köln (Germany)
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Bravi, Paolo; Cupit-Foutou, Stéphanie. Classification of strict wonderful varieties. Annales de l'Institut Fourier, Volume 60 (2010) no. 2, pp. 641-681. doi : 10.5802/aif.2535. http://www.numdam.org/articles/10.5802/aif.2535/

[1] Ahiezer, D. N. Equivariant completions of homogeneous algebraic varieties by homogeneous divisors, Ann. Global Anal. Geom., Volume 1 (1983), pp. 49-78 | DOI | MR | Zbl

[2] Bourbaki, N. Éléments de mathématique. Groupes et Algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines, Actualités Scientifiques et Industrielles, 1337, Hermann, Paris, 1968 | MR

[3] Bravi, P. Wonderful varieties of type E, Represent. Theory, Volume 11 (2007), pp. 174-191 | DOI | MR | Zbl

[4] Bravi, P.; Cupit-Foutou, S. Equivariant deformations of the affine multicone over a flag variety (arXiv:math.AG/0603690v2 )

[5] Bravi, P.; Cupit-Foutou, S. Equivariant deformations of the affine multicone over a flag variety, Adv. Math., Volume 217 (2008), pp. 2800-2821 | DOI | MR | Zbl

[6] Bravi, P.; Pezzini, G. Wonderful varieties of type D, Represent. Theory, Volume 9 (2005), pp. 578-637 | DOI | MR

[7] Brion, M. Classification des espaces homogènes sphériques, Compositio Math., Volume 63 (1987), pp. 189-208 | Numdam | MR | Zbl

[8] Brion, M. Variétés sphériques (1997) (Notes de la session de la S.M.F. “Opérations hamiltoniennes et opérations de groupes algébriques”, Grenoble)

[9] Brylinski, R. K.; Kostant, B.; et al., Donato P. The variety of all invariant symplectic structures on a homogeneous space and normalizers of isotropy subgroups, Symplectic Geometry and Mathematical Physics, Birkhauser, Basel, 1991, pp. 80-113 (Progr. Math. 99) | MR | Zbl

[10] De Concini, C.; Procesi, C. Complete symmetric varieties, Invariant theory (Montecatini, 1982), Springer, Berlin (1983), pp. 1-44 (Lecture Notes in Math. 996) | MR | Zbl

[11] Helgason, S. Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, 34, AMS, Providence, RI, 2001 | MR | Zbl

[12] Knop, F. The Luna-Vust theory of spherical embeddings, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Manoj Prakashan, Madras (1991), pp. 225-249 | MR | Zbl

[13] Knop, F. Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc., Volume 9 (1996), pp. 153-174 | DOI | MR | Zbl

[14] Krämer, M. Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compositio Math., Volume 38 (1979), pp. 129-153 | Numdam | MR | Zbl

[15] Losev, I. V. Uniqueness property for spherical homogeneous spaces, Duke Math. J., Volume 147 (2009), pp. 315-343 | DOI | MR | Zbl

[16] Luna, D. Toute variété magnifique est sphérique, Transform. Groups, Volume 1 (1996), pp. 249-258 | DOI | MR | Zbl

[17] Luna, D. Variétés sphériques de type A, Publ. Math. Inst. Hautes Études Sci., Volume 94 (2001), pp. 161-226 | Numdam | MR | Zbl

[18] Luna, D. La variété magnifique modèle, J. Algebra, Volume 313 (2007), pp. 292-319 | DOI | MR | Zbl

[19] Luna, D.; Vust, T. Plongements d’espaces homogènes, Comment. Math. Helv., Volume 58 (1983), pp. 186-245 | DOI | MR | Zbl

[20] Mikityuk, I. V. On the integrability of invariant hamiltonian systems with homogeneous configurations spaces (in Russian), Math. Sbornik, Volume 129 (1986), pp. 514-534 | MR | Zbl

[21] Panyushev, D. I. Complexity and nilpotent orbits, Manuscripta Math., Volume 83 (1994), pp. 223-237 | DOI | MR | Zbl

[22] Panyushev, D. I. Some amazing properties of spherical nilpotent orbits, Math. Z., Volume 245 (2003), pp. 557-580 | DOI | MR | Zbl

[23] Pezzini, G. Wonderful varieties of type C, Dipartimento di Matematica, Università La Sapienza, Rome (2003) (Ph. D. Thesis)

[24] Pezzini, G. Simple immersions of wonderful varieties, Math. Z., Volume 255 (2007), pp. 793-812 | DOI | MR | Zbl

[25] Steinberg, R. Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc., 80, AMS, Providence, RI, 1968 | MR | Zbl

[26] Timashev, D. Homogeneous spaces and equivariant embeddings (arXiv:math/0602228 )

[27] Vust, T. Plongements d’espaces symétriques algèbriques: une classification, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 17 (1990), pp. 165-195 | Numdam | MR | Zbl

[28] Wasserman, B. Wonderful varieties of rank two, Transform. Groups, Volume 1 (1996), pp. 375-403 | DOI | MR | Zbl

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