Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



Cohomology of Deligne-Lusztig varieties for unipotent blocks of $ \mathrm{GL}_n(q)$

Author: Olivier Dudas
Journal: Represent. Theory 17 (2013), 647-662
MSC (2010): Primary 20C33
Published electronically: December 10, 2013
MathSciNet review: 3139556
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the cohomology of parabolic Deligne-Lusztig varieties associated to unipotent blocks of $ \mathrm {GL}_n(q)$. We show that the geometric version of Broué's conjecture over $ \overline {\mathbb{Q}}_\ell $, together with Craven's formula, holds for any unipotent block whenever it holds for the principal $ \Phi _1$-block.

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

  • [1] Cédric Bonnafé and Raphaël Rouquier, Coxeter orbits and modular representations, Nagoya Math. J. 183 (2006), 1–34. MR 2253885
  • [2] Michel Broué and Gunter Malle, Zyklotomische Heckealgebren, Astérisque 212 (1993), 119–189 (German). Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235834
  • [3] Michel Broué, Gunter Malle, and Jean Michel, Generic blocks of finite reductive groups, Astérisque 212 (1993), 7–92. Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235832
  • [4] Michel Broué and Jean Michel, Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées, Finite reductive groups (Luminy, 1994) Progr. Math., vol. 141, Birkhäuser Boston, Boston, MA, 1997, pp. 73–139 (French). MR 1429870
  • [5] Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication. MR 1266626
  • [6] J. Chuang and R. Rouquier, Calabi-Yau algebras and perverse Morita equivalences, In preparation.
  • [7] David Craven, On the cohomology of Deligne-Lusztig varieties, arXiv:math/1107.1871, Preprint (2011).
  • [8] F. Digne and J. Michel, Endomorphisms of Deligne-Lusztig varieties, Nagoya Math. J. 183 (2006), 35–103. MR 2253886
  • [9] F. Digne and J. Michel, Parabolic Deligne-Lusztig varieties, arXiv:math/1110.4863, preprint (2011).
  • [10] François Digne, Jean Michel, and Raphaël Rouquier, Cohomologie des variétés de Deligne-Lusztig, Adv. Math. 209 (2007), no. 2, 749–822 (French, with English summary). MR 2296313, 10.1016/j.aim.2006.06.001
  • [11] Olivier Dudas, Quotient of Deligne-Lusztig varieties, J. Algebra 381 (2013), 1–20. MR 3030506, 10.1016/j.jalgebra.2013.01.032
  • [12] G. Lusztig, Coxeter orbits and eigenspaces of Frobenius, Invent. Math. 38 (1976/77), no. 2, 101–159. MR 0453885
  • [13] George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 20C33

Retrieve articles in all journals with MSC (2010): 20C33

Additional Information

Olivier Dudas
Affiliation: Université Denis Diderot - Paris 7, UFR de Mathématiques, Institut de Mathématiques de Jussieu, Case 7012, 75205 Paris Cedex 13, France

Received by editor(s): January 16, 2013
Received by editor(s) in revised form: July 10, 2013
Published electronically: December 10, 2013
Additional Notes: The author was supported by the EPSRC, Project No EP/H026568/1 and by Magdalen College, Oxford.
Article copyright: © Copyright 2013 American Mathematical Society