Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [DP] C. De Concini and C.Procesi, Complete symmetric varieties, Invariant theory (Montecatini 1982), Lect. Notes Math., vol. 996, Springer, 1983, pp. 1-44. MR 0718125 (85e:14070)
  • [H] X. He, Unipotent variety in the group compactification, Adv. in Math., 203 (2006), 109-131. MR 2231043
  • [L1] G. Lusztig, Total positivity in reductive groups, Lie Theory and Geometry: in honor of Bertram Kostant, Progress in Math. 123 (1994), 531-568. MR 1327548 (96m:20071)
  • [L2] G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series, 18, American Mathematical Society, Providence, RI, 2003. MR 1974442 (2004k:20011)
  • [L3] G. Lusztig, Parabolic character sheaves I, Moscow Math.J 4 (2004), 153-179. MR 2074987 (2006d:20091a)
  • [L4] G. Lusztig, Parabolic character sheaves II, Moscow Math.J 4 (2004), 869-896. MR 2124170 (2006d:20091b)
  • [Q] D. Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167-171. MR 0427303 (55:337)
  • [S] T. A. Springer, Intersection cohomology of $ B \times B$-orbits closures in group compactifications, J. Alg. 258 (2002), 71-111. MR 1958898 (2004a:14025)
  • [SL] P. Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, 815, Springer, Berlin, 1980. MR 0584445 (82g:14037)
  • [Su] A. A. Suslin, Projective modules over polynomial rings are free (Russian), Dokl. Akad. Nauk SSSR 229 (1976), no. 5, 1063-1066. MR 0469905 (57:9685)
  • [VS] L. N. Vaserstein and A. A. Suslin, Serre's problem on projective modules over polynomial rings, and algebraic $ K$-theory (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 5, 993-1054, 1199. MR 0447245 (56:5560)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20G15, 14L30

Retrieve articles in all journals with MSC (2000): 20G15, 14L30

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.

American Mathematical Society