Left cells and constructible representations
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- by Meinolf Geck
- Represent. Theory 9 (2005), 385-416
- DOI: https://doi.org/10.1090/S1088-4165-05-00245-1
- Published electronically: May 2, 2005
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Erratum: Represent. Theory 11 (2007), 172-173.
Abstract:
We consider the partition of a finite Coxeter group $W$ into left cells with respect to a weight function $L$. In the equal parameter case, Lusztig has shown that the representations carried by the left cells are precisely the so-called constructible ones. We show that this holds for general $L$, assuming that the conjectural properties (P1)–(P15) in Lusztig’s book on Hecke algebras with unequal parameters hold for $W,L$. Our proofs use the idea (Gyoja, Rouquier) that left cell representations are projective in the sense of modular representation theory. This also gives partly new proofs for Lusztig’s result in the equal parameter case.References
- Dean Alvis, The left cells of the Coxeter group of type $H_4$, J. Algebra 107 (1987), no. 1, 160–168. MR 883878, DOI 10.1016/0021-8693(87)90082-2
- Dean Alvis and George Lusztig, The representations and generic degrees of the Hecke algebra of type $H_{4}$, J. Reine Angew. Math. 336 (1982), 201–212. MR 671329, DOI 10.1515/crll.1982.336.201
- Susumu Ariki, On the decomposition numbers of the Hecke algebra of $G(m,1,n)$, J. Math. Kyoto Univ. 36 (1996), no. 4, 789–808. MR 1443748, DOI 10.1215/kjm/1250518452
- Susumu Ariki, Robinson-Schensted correspondence and left cells, Combinatorial methods in representation theory (Kyoto, 1998) Adv. Stud. Pure Math., vol. 28, Kinokuniya, Tokyo, 2000, pp. 1–20. MR 1855588, DOI 10.2969/aspm/02810001
- Dan Barbasch and David Vogan, Primitive ideals and orbital integrals in complex exceptional groups, J. Algebra 80 (1983), no. 2, 350–382. MR 691809, DOI 10.1016/0021-8693(83)90006-6
- Cédric Bonnafé and Lacrimioara Iancu, Left cells in type $B_n$ with unequal parameters, Represent. Theory 7 (2003), 587–609. MR 2017068, DOI 10.1090/S1088-4165-03-00188-2
- Michel Broué and SungSoon Kim, Familles de caractères des algèbres de Hecke cyclotomiques, Adv. Math. 172 (2002), no. 1, 53–136 (French, with French summary). MR 1943901, DOI 10.1006/aima.2002.2078
- Richard Dipper, Gordon James, and Eugene Murphy, Hecke algebras of type $B_n$ at roots of unity, Proc. London Math. Soc. (3) 70 (1995), no. 3, 505–528. MR 1317512, DOI 10.1112/plms/s3-70.3.505
- Meinolf Geck, On the representation theory of Iwahori-Hecke algebras of extended finite Weyl groups, Represent. Theory 4 (2000), 370–397. MR 1780716, DOI 10.1090/S1088-4165-00-00093-5
- Meinolf Geck, Constructible characters, leading coefficients and left cells for finite Coxeter groups with unequal parameters, Represent. Theory 6 (2002), 1–30. MR 1915085, DOI 10.1090/S1088-4165-02-00128-0
- Meinolf Geck, On the induction of Kazhdan-Lusztig cells, Bull. London Math. Soc. 35 (2003), no. 5, 608–614. MR 1989489, DOI 10.1112/S0024609303002236
- Meinolf Geck, Computing Kazhdan-Lusztig cells for unequal parameters, J. Algebra 281 (2004), no. 1, 342–365. MR 2091976, DOI 10.1016/j.jalgebra.2004.07.029
- Meinolf Geck, Gerhard Hiss, Frank Lübeck, Gunter Malle, and Götz Pfeiffer, CHEVIE—a system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput. 7 (1996), no. 3, 175–210. Computational methods in Lie theory (Essen, 1994). MR 1486215, DOI 10.1007/BF01190329
- Meinolf Geck and Götz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000. MR 1778802
- Meinolf Geck and Raphaël Rouquier, Centers and simple modules for Iwahori-Hecke algebras, Finite reductive groups (Luminy, 1994) Progr. Math., vol. 141, Birkhäuser Boston, Boston, MA, 1997, pp. 251–272. MR 1429875, DOI 10.1007/978-1-4612-4124-9_{9}
- Meinolf Geck and Raphaël Rouquier, Filtrations on projective modules for Iwahori-Hecke algebras, Modular representation theory of finite groups (Charlottesville, VA, 1998) de Gruyter, Berlin, 2001, pp. 211–221. MR 1889346, DOI 10.1515/9783110889161.211
- Akihiko Gyoja, Cells and modular representations of Hecke algebras, Osaka J. Math. 33 (1996), no. 2, 307–341. MR 1416051
- Jens Carsten Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1359532, DOI 10.1090/gsm/006
- David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
- Bernard Leclerc and Hyohe Miyachi, Constructible characters and canonical bases, J. Algebra 277 (2004), no. 1, 298–317. MR 2059632, DOI 10.1016/j.jalgebra.2004.02.023
- George Lusztig, Representations of finite Chevalley groups, CBMS Regional Conference Series in Mathematics, vol. 39, American Mathematical Society, Providence, R.I., 1978. Expository lectures from the CBMS Regional Conference held at Madison, Wis., August 8–12, 1977. MR 518617, DOI 10.1090/cbms/039
- George Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981), no. 2, 490–498. MR 630610, DOI 10.1016/0021-8693(81)90188-5
- George Lusztig, A class of irreducible representations of a Weyl group. II, Nederl. Akad. Wetensch. Indag. Math. 44 (1982), no. 2, 219–226. MR 662657, DOI 10.1016/S1385-7258(82)80013-9
- George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472, DOI 10.1515/9781400881772
- George Lusztig, Sur les cellules gauches des groupes de Weyl, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 1, 5–8 (French, with English summary). MR 827096
- G. Lusztig, Leading coefficients of character values of Hecke algebras, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 235–262. MR 933415, DOI 10.1090/pspum/047.2/933415
- G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR 1974442, DOI 10.1090/crmm/018
- Gunter Malle and Raphaël Rouquier, Familles de caractères de groupes de réflexions complexes, Represent. Theory 7 (2003), 610–640 (French, with French summary). MR 2017069, DOI 10.1090/S1088-4165-03-00193-6
- Yuval Roichman, Induction and restriction of Kazhdan-Lusztig cells, Adv. Math. 134 (1998), no. 2, 384–398. MR 1617785, DOI 10.1006/aima.1997.1712
- Raphaël Rouquier, Familles et blocs d’algèbres de Hecke, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 12, 1037–1042 (French, with English and French summaries). MR 1735880, DOI 10.1016/S0764-4442(00)88470-0
Bibliographic Information
- Meinolf Geck
- Affiliation: Institut Girard Desargues, bat. Jean Braconnier, Université Lyon 1, 21 av Claude Bernard, F–69622 Villeurbanne Cedex, France
- MR Author ID: 272405
- Email: geck@igd.univ-lyon1.fr
- Received by editor(s): May 10, 2004
- Received by editor(s) in revised form: March 26, 2005
- Published electronically: May 2, 2005
- © Copyright 2005 American Mathematical Society
- Journal: Represent. Theory 9 (2005), 385-416
- MSC (2000): Primary 20C08
- DOI: https://doi.org/10.1090/S1088-4165-05-00245-1
- MathSciNet review: 2133765