Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165

 
 

 

Relative Kazhdan-Lusztig cells


Author: Meinolf Geck
Journal: Represent. Theory 10 (2006), 481-524
MSC (2000): Primary 20C08; Secondary 20G40
DOI: https://doi.org/10.1090/S1088-4165-06-00287-1
Published electronically: November 14, 2006
MathSciNet review: 2266700
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study the Kazhdan-Lusztig cells of a Coxeter group $ W$ in a ``relative'' setting, with respect to a parabolic subgroup $ W_I \subseteq W$. This relies on a factorization of the Kazhdan-Lusztig basis $ \{\mathbf{C}_w\}$ of the corresponding (multi-parameter) Iwahori-Hecke algebra with respect to $ W_I$. We obtain two applications to the ``asymptotic case'' in type $ B_n$, as introduced by Bonnafé and Iancu: we show that $ \{\mathbf{C}_w\}$ is a ``cellular basis'' in the sense of Graham and Lehrer, and we construct an analogue of Lusztig's canonical isomorphism from the Iwahori-Hecke algebra to the group algebra of the underlying Weyl group of type $ B_n$.


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

  • 1. S. ARIKI, Robinson-Schensted correspondence and left cells. Combinatorial methods in representation theory (Kyoto, 1998), 1-20, Adv. Stud. Pure Math., 28, Kinokuniya, Tokyo, 2000. MR 1855588 (2002i:20003)
  • 2. D. BARBASCH AND D. VOGAN, Primitive ideals and orbital integrals in complex exceptional groups, J. Algebra 80 (1983), 350-382. MR 0691809 (84h:22038)
  • 3. C. BONNAFÉ AND L. IANCU, Left cells in type $ B_n$ with unequal parameters, Represent. Theory 7 (2003), 587-609. MR 2017068 (2004j:20007)
  • 4. C. BONNAFÉ, Two-sided cells in type $ B$ in the asymptotic case, Journal of Algebra 304 (2006), 216-236.
  • 5. R. DIPPER AND G. D. JAMES, Representations of Hecke algebras of type $ B_n$, J. Algebra 146 (1992), 454-481. MR 1152915 (93c:20019)
  • 6. R. DIPPER, G. D. JAMES AND G. E. MURPHY, Hecke algebras of type $ {B}_n$ at roots of unity, Proc. London Math. Soc. (3) 70 (1995), 505-528. MR 1317512 (96b:20004)
  • 7. M. GECK, Kazhdan-Lusztig cells and decomposition numbers, Represent. Theory 2 (1998), 264-277 (electronic). MR 1628035 (99e:20015)
  • 8. M. GECK, On the induction of Kazhdan-Lusztig cells, Bull. London Math. Soc. 35 (2003), 608-614. MR 1989489 (2004d:20003)
  • 9. M. GECK, Computing Kazhdan-Lusztig cells for unequal parameters, J. Algebra 281 (2004), 342-365; section "Computational Algebra". MR 2091976 (2005g:20007)
  • 10. M. GECK AND L. IANCU, Lusztig's $ a$-function in type $ B_n$ in the asymptotic case, Nagoya Math. J. 182 (2006), 199-240. MR 2235342
  • 11. M. GECK AND G. PFEIFFER, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Math. Soc. Monographs, New Series 21 (Oxford University Press, 2000). MR 1778802 (2002k:20017)
  • 12. J. J. GRAHAM AND G. I. LEHRER, Cellular algebras, Invent. Math. 123 (1996), 1-34. MR 1376244 (97h:20016)
  • 13. D. A. KAZHDAN AND G. LUSZTIG, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 0560412 (81j:20066)
  • 14. G. LUSZTIG, On a theorem of Benson and Curtis, J. Algebra 71 (1981), 490-498. MR 0630610 (83a:20053)
  • 15. G. LUSZTIG, Left cells in Weyl groups, Lie Group Representations, I (eds R. L. R. Herb and J. Rosenberg), Lecture Notes in Mathematics 1024 (Springer, Berlin, 1983), pp. 99-111. MR 0727851 (85f:20035)
  • 16. G. LUSZTIG, Characters of reductive groups over a finite field, Annals Math. Studies 107 (Princeton University Press, 1984). MR 0742472 (86j:20038)
  • 17. G. LUSZTIG, Cells in affine Weyl groups, Advanced Studies in Pure Math. 6, Algebraic groups and related topics, Kinokuniya and North-Holland, 1985, 255-287. MR 0803338 (87h:20074)
  • 18. G. LUSZTIG, Hecke algebras with unequal parameters, CRM Monographs Ser. 18, Amer. Math. Soc., Providence, RI, 2003. MR 1974442 (2004k:20011)
  • 19. M. NEUNHÖFFER, Untersuchungen zu James' Vermutung über Iwahori-Hecke-Algebren vom Typ $ A$, Ph.D. Thesis, RWTH Aachen, 2003.
  • 20. T. A. SPRINGER, Quelques applications de la cohomologie d'intersection, Séminaire Bourbaki (1981/82), exp. 589, Astérisque 92-93 (1982). MR 0689533 (85i:32016b)

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 20C08, 20G40

Retrieve articles in all journals with MSC (2000): 20C08, 20G40


Additional Information

Meinolf Geck
Affiliation: Institut Camille Jordan, bat. Jean Braconnier, Université Lyon 1, 21 av Claude Bernard, F–69622 Villeurbanne Cedex, France
Address at time of publication: Department of Mathematical Sciences, King’s College, Aberdeen University, Aberdeen AB24 3UE, UK
Email: geck@maths.abdn.ac.uk

DOI: https://doi.org/10.1090/S1088-4165-06-00287-1
Received by editor(s): May 30, 2005
Published electronically: November 14, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society