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Transactions of the American Mathematical Society

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The $ G$-stable pieces of the wonderful compactification

Author: Xuhua He
Journal: Trans. Amer. Math. Soc. 359 (2007), 3005-3024
MSC (2000): Primary 20G15, 14L30
Published electronically: February 21, 2007
MathSciNet review: 2299444
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Abstract: Let $ G$ be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification $ \bar {G}$ of $ G$ into finite many $ G$-stable pieces, which was introduced by Lusztig. In this paper, we will investigate the closure of any $ G$-stable piece in $ \bar {G}$. We will show that the closure is a disjoint union of some $ G$-stable pieces, which was first conjectured by Lusztig. We will also prove the existence of cellular decomposition if the closure contains finitely many $ G$-orbits.

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  • [DP] 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, 10.1007/BFb0063234
  • [H] Xuhua He, Unipotent variety in the group compactification, Adv. Math. 203 (2006), no. 1, 109–131. MR 2231043, 10.1016/j.aim.2005.04.004
  • [L1] G. Lusztig, Total positivity in reductive groups, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 531–568. MR 1327548
  • [L2] G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR 1974442
  • [L3] George Lusztig, Parabolic character sheaves. I, Mosc. Math. J. 4 (2004), no. 1, 153–179, 311 (English, with English and Russian summaries). MR 2074987
  • [L4] G. Lusztig, Parabolic character sheaves. II, Mosc. Math. J. 4 (2004), no. 4, 869–896, 981 (English, with English and Russian summaries). MR 2124170
  • [Q] Daniel Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167–171. MR 0427303
  • [S] T. A. Springer, Intersection cohomology of 𝐵×𝐵-orbit closures in group compactifications, J. Algebra 258 (2002), no. 1, 71–111. With an appendix by Wilberd van der Kallen; Special issue in celebration of Claudio Procesi’s 60th birthday. MR 1958898, 10.1016/S0021-8693(02)00515-X
  • [SL] Peter Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815, Springer, Berlin, 1980. MR 584445
  • [Su] A. A. Suslin, Projective modules over polynomial rings are free, Dokl. Akad. Nauk SSSR 229 (1976), no. 5, 1063–1066 (Russian). MR 0469905
  • [VS] L. N. Vaseršteĭn and A. A. Suslin, Serre’s problem on projective modules over polynomial rings, and algebraic 𝐾-theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 5, 993–1054, 1199 (Russian). MR 0447245

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Additional Information

Xuhua He
Affiliation: Department of Mathematics, M.I.T., Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794

Received by editor(s): March 4, 2005
Published electronically: February 21, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.