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Projective normality of model varieties and related results


Authors: Paolo Bravi, Jacopo Gandini and Andrea Maffei
Journal: Represent. Theory 20 (2016), 39-93
MSC (2010): Primary 14M27; Secondary 20G05
DOI: https://doi.org/10.1090/ert/477
Published electronically: February 12, 2016
MathSciNet review: 3458949
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Abstract: We prove that the multiplication of sections of globally generated line bundles on a model wonderful variety $ M$ of simply connected type is always surjective. This follows by a general argument which works for every wonderful variety and reduces the study of the surjectivity for every couple of globally generated line bundles to a finite number of cases. As a consequence, the cone defined by a complete linear system over $ M$ or over a closed $ G$-stable subvariety of $ M$ is normal. We apply these results to the study of the normality of the compactifications of model varieties in simple projective spaces and of the closures of the spherical nilpotent orbits. Then we focus on a particular case proving two specific conjectures of Adams, Huang and Vogan on an analogue of the model orbit of the group of type $ \mathsf E_8$.


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  • [1] Jeffrey Adams, Jing-Song Huang, and David A. Vogan Jr., Functions on the model orbit in $ E_8$, Represent. Theory 2 (1998), 224-263 (electronic). MR 1628031 (99g:20077), https://doi.org/10.1090/S1088-4165-98-00048-X
  • [2] Dmitry Ahiezer, Equivariant completions of homogeneous algebraic varieties by homogeneous divisors, Ann. Global Anal. Geom. 1 (1983), no. 1, 49-78. MR 739893 (85j:32052), https://doi.org/10.1007/BF02329739
  • [3] Dan Barbasch and David A. Vogan Jr., Unipotent representations of complex semisimple groups, Ann. of Math. (2) 121 (1985), no. 1, 41-110. MR 782556 (86i:22031), https://doi.org/10.2307/1971193
  • [4] I. N. Bernšteĭn, I. M. Gelfand, and S. I. Gelfand, Models of representations of compact Lie groups, Funkcional. Anal. i Priložen. 9 (1975), no. 4, 61-62 (Russian). MR 0414792 (54 #2884)
  • [5] P. Bravi, Primitive spherical systems, Trans. Amer. Math. Soc. 365 (2013), no. 1, 361-407. MR 2984062, https://doi.org/10.1090/S0002-9947-2012-05621-2
  • [6] Paolo Bravi and Stéphanie Cupit-Foutou, Classification of strict wonderful varieties, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 2, 641-681 (English, with English and French summaries). MR 2667789 (2011m:14085)
  • [7] Paolo Bravi, Jacopo Gandini, Andrea Maffei, and Alessandro Ruzzi, Normality and non-normality of group compactifications in simple projective spaces, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 6, 2435-2461 (2012) (English, with English and French summaries). MR 2976317, https://doi.org/10.5802/aif.2679
  • [8] P. Bravi and D. Luna, An introduction to wonderful varieties with many examples of type $ \rm F_4$, J. Algebra 329 (2011), 4-51. MR 2769314 (2012f:14102), https://doi.org/10.1016/j.jalgebra.2010.01.025
  • [9] Paolo Bravi and Guido Pezzini, Wonderful varieties of type $ D$, Represent. Theory 9 (2005), 578-637 (electronic). MR 2183057 (2006g:14078), https://doi.org/10.1090/S1088-4165-05-00260-8
  • [10] Michel Brion, Groupe de Picard et nombres caractéristiques des variétés sphériques, Duke Math. J. 58 (1989), no. 2, 397-424 (French). MR 1016427 (90i:14048), https://doi.org/10.1215/S0012-7094-89-05818-3
  • [11] Michel Brion, The total coordinate ring of a wonderful variety, J. Algebra 313 (2007), no. 1, 61-99. MR 2326138 (2008d:14067), https://doi.org/10.1016/j.jalgebra.2006.12.022
  • [12] Abraham Broer, Decomposition varieties in semisimple Lie algebras, Canad. J. Math. 50 (1998), no. 5, 929-971. MR 1650954 (99k:14077), https://doi.org/10.4153/CJM-1998-048-6
  • [13] Rocco Chirivì, Corrado de Concini, and Andrea Maffei, On normality of cones over symmetric varieties, Tohoku Math. J. (2) 58 (2006), no. 4, 599-616. MR 2297202 (2008g:14087)
  • [14] Rocco Chirivì, Peter Littelmann, and Andrea Maffei, Equations defining symmetric varieties and affine Grassmannians, Int. Math. Res. Not. IMRN 2 (2009), 291-347. MR 2482117 (2010b:14104), https://doi.org/10.1093/imrn/rnn132
  • [15] Rocco Chirivì and Andrea Maffei, Projective normality of complete symmetric varieties, Duke Math. J. 122 (2004), no. 1, 93-123. MR 2046808 (2005b:14085), https://doi.org/10.1215/S0012-7094-04-12213-4
  • [16] Rocco Chirivì and Andrea Maffei, Plücker relations and spherical varieties: application to model varieties, Transform. Groups 19 (2014), no. 4, 979-997. MR 3278858, https://doi.org/10.1007/s00031-014-9285-6
  • [17] David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060 (94j:17001)
  • [18] Mauro Costantini, On the coordinate ring of spherical conjugacy classes, Math. Z. 264 (2010), no. 2, 327-359. MR 2574980 (2010k:20071), https://doi.org/10.1007/s00209-008-0468-5
  • [19] Corrado De Concini, Normality and non normality of certain semigroups and orbit closures, Algebraic transformation groups and algebraic varieties, Encyclopaedia Math. Sci., vol. 132, Springer, Berlin, 2004, pp. 15-35. MR 2090668 (2005g:20068), https://doi.org/10.1007/978-3-662-05652-3_3
  • [20] C. De Concini and C. Procesi, Complete symmetric varieties, Invariant theory (Montecatini, 1982) Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, pp. 1-44. MR 718125 (85e:14070), https://doi.org/10.1007/BFb0063234
  • [21] W. A. de Graaf, Constructing representations of split semisimple Lie algebras, J. Pure Appl. Algebra 164 (2001), no. 1-2, 87-107. Effective methods in algebraic geometry (Bath, 2000). MR 1854331 (2002h:17007), https://doi.org/10.1016/S0022-4049(00)00150-X
  • [22] J. Gandini, Spherical orbit closures in simple projective spaces and their normalizations, Transform. Groups 16 (2011), no. 1, 109-136. MR 2785497 (2012f:14091), https://doi.org/10.1007/s00031-011-9120-2
  • [23] The GAP Group, Aachen, St Andrews,
    GAP-Groups, Algorithms, and Programming - a System for Computational Discrete Algebra,
    Version 4.5.4, 2012. http://www-gap.dcs.st-and.ac.uk/$ \sim $gap
  • [24] I. M. Gelfand and A. V. Zelevinskiĭ, Models of representations of classical groups and their hidden symmetries, Funktsional. Anal. i Prilozhen. 18 (1984), no. 3, 14-31 (Russian). MR 757246 (86i:22024)
  • [25] I. M. Gelfand and A. V. Zelevinsky, Representation models for classical groups and their higher symmetries, Astérisque Numero Hors Serie (1985), 117-128. The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837197 (88b:22022)
  • [26] Wim Hesselink, The normality of closures of orbits in a Lie algebra, Comment. Math. Helv. 54 (1979), no. 1, 105-110. MR 522033 (80g:14018), https://doi.org/10.1007/BF02566258
  • [27] S. Senthamarai Kannan, Projective normality of the wonderful compactification of semisimple adjoint groups, Math. Z. 239 (2002), no. 4, 673-682. MR 1902056 (2003f:20076), https://doi.org/10.1007/s002090100319
  • [28] Misha Kapovich and John J. Millson, Structure of the tensor product semigroup, Asian J. Math. 10 (2006), no. 3, 493-539. MR 2253157 (2007h:22009), https://doi.org/10.4310/AJM.2006.v10.n3.a2
  • [29] Friedrich Knop, Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc. 9 (1996), no. 1, 153-174. MR 1311823 (96c:14037), https://doi.org/10.1090/S0894-0347-96-00179-8
  • [30] Friedrich Knop, Hanspeter Kraft, and Thierry Vust, The Picard group of a $ G$-variety, Algebraische Transformationsgruppen und Invariantentheorie, DMV Sem., vol. 13, Birkhäuser, Basel, 1989, pp. 77-87. MR 1044586
  • [31] Allen Knutson and Terence Tao, The honeycomb model of $ {\rm GL}_n({\bf C})$ tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), no. 4, 1055-1090. MR 1671451 (2000c:20066), https://doi.org/10.1090/S0894-0347-99-00299-4
  • [32] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753-809. MR 0311837 (47 #399)
  • [33] Shrawan Kumar, Tensor product decomposition, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1226-1261. MR 2827839
  • [34] D. Luna, Toute variété magnifique est sphérique, Transform. Groups 1 (1996), no. 3, 249-258 (French, with English summary). MR 1417712 (97h:14066), https://doi.org/10.1007/BF02549208
  • [35] D. Luna, Variétés sphériques de type $ A$, Publ. Math. Inst. Hautes Études Sci. 94 (2001), 161-226 (French). MR 1896179 (2003f:14056), https://doi.org/10.1007/s10240-001-8194-0
  • [36] D. Luna, La variété magnifique modèle, J. Algebra 313 (2007), no. 1, 292-319 (French, with English and French summaries). MR 2326148 (2008e:14070), https://doi.org/10.1016/j.jalgebra.2006.10.042
  • [37] Andrea Maffei, Orbits in degenerate compactifications of symmetric varieties, Transform. Groups 14 (2009), no. 1, 183-194. MR 2480858 (2010f:14055), https://doi.org/10.1007/s00031-008-9040-y
  • [38] Dmitri I. Panyushev, Some amazing properties of spherical nilpotent orbits, Math. Z. 245 (2003), no. 3, 557-580. MR 2021571 (2004j:14051), https://doi.org/10.1007/s00209-003-0555-6
  • [39] Guido Pezzini, Simple immersions of wonderful varieties, Math. Z. 255 (2007), no. 4, 793-812. MR 2274535 (2007j:14069), https://doi.org/10.1007/s00209-006-0050-y
  • [40] R. W. Richardson, On orbits of algebraic groups and Lie groups, Bull. Austral. Math. Soc. 25 (1982), no. 1, 1-28. MR 651417 (83i:14041), https://doi.org/10.1017/S0004972700005013
  • [41] John R. Stembridge, The partial order of dominant weights, Adv. Math. 136 (1998), no. 2, 340-364. MR 1626860 (2000a:06038), https://doi.org/10.1006/aima.1998.1736
  • [42] Dmitry A. Timashev, Homogeneous spaces and equivariant embeddings, Encyclopaedia of Mathematical Sciences, vol. 138, Springer, Heidelberg, 2011. Invariant Theory and Algebraic Transformation Groups, 8. MR 2797018 (2012e:14100)
  • [43] David A. Vogan Jr., The method of coadjoint orbits for real reductive groups, Representation theory of Lie groups (Park City, UT, 1998) IAS/Park City Math. Ser., vol. 8, Amer. Math. Soc., Providence, RI, 2000, pp. 179-238. MR 1737729 (2001k:22027)
  • [44] B. Wasserman, Wonderful varieties of rank two, Transform. Groups 1 (1996), no. 4, 375-403. MR 1424449 (97k:14051), https://doi.org/10.1007/BF02549213

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Additional Information

Paolo Bravi
Affiliation: Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy
Email: bravi@mat.uniroma1.it

Jacopo Gandini
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
Email: jacopo.gandini@sns.it

Andrea Maffei
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
Email: maffei@dm.unipi.it

DOI: https://doi.org/10.1090/ert/477
Received by editor(s): February 19, 2015
Received by editor(s) in revised form: December 29, 2015
Published electronically: February 12, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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