Conjugacy classes of involutions and Kazhdan–Lusztig cells

Bonnafé, Cédric; Geck, Meinolf

doi:10.1090/S1088-4165-2014-00456-4

Conjugacy classes of involutions and Kazhdan–Lusztig cells
HTML articles powered by AMS MathViewer

by Cédric Bonnafé and Meinolf Geck PDF

Represent. Theory 18 (2014), 155-182 Request permission

Abstract:

According to an old result of Schützenberger, the involutions in a given two-sided cell of the symmetric group $\mathfrak {S}_n$ are all conjugate. In this paper, we study possible generalizations of this property to other types of Coxeter groups. We show that Schützenberger’s result is a special case of a general result on “smooth” two-sided cells. Furthermore, we consider Kottwitz’s conjecture concerning the intersections of conjugacy classes of involutions with the left cells in a finite Coxeter group. Our methods lead to a proof of this conjecture for classical types which, combined with further recent work, settles this conjecture in general.

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
Cédric Bonnafé, Two-sided cells in type $B$ (asymptotic case), J. Algebra 304 (2006), no. 1, 216–236. MR 2255826, DOI 10.1016/j.jalgebra.2006.05.035
Cédric Bonnafé, Meinolf Geck, Lacrimioara Iancu, and Thomas Lam, On domino insertion and Kazhdan-Lusztig cells in type $B_n$, Representation theory of algebraic groups and quantum groups, Progr. Math., vol. 284, Birkhäuser/Springer, New York, 2010, pp. 33–54. MR 2761947, DOI 10.1007/978-0-8176-4697-4_{3}
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
Cédric Bonnafé and Raphaël Rouquier, Calogero-Moser versus Kazhdan-Lusztig cells, Pacific J. Math. 261 (2013), no. 1, 45–51. MR 3037558, DOI 10.2140/pjm.2013.261.45
Cédrick Bonnafé and Raphaël Rouquier, Cellules de Calogero-Moser; preprint arXiv:1302.2720.
Bill Casselman, Verifying Kottwitz’ conjecture by computer, Represent. Theory 4 (2000), 32–45. MR 1740179, DOI 10.1090/S1088-4165-00-00052-2
Fokko du Cloux, Positivity results for the Hecke algebras of noncrystallographic finite Coxeter groups, J. Algebra 303 (2006), no. 2, 731–741. MR 2255133, DOI 10.1016/j.jalgebra.2005.10.004
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, Left cells and constructible representations, Represent. Theory 9 (2005), 385–416. MR 2133765, DOI 10.1090/S1088-4165-05-00245-1
Meinolf Geck, Kazhdan-Lusztig cells and the Murphy basis, Proc. London Math. Soc. (3) 93 (2006), no. 3, 635–665. MR 2266962, DOI 10.1017/S0024611506015930
Meinolf Geck, On Iwahori-Hecke algebras with unequal parameters and Lusztig’s isomorphism theorem, Pure Appl. Math. Q. 7 (2011), no. 3, Special Issue: In honor of Jacques Tits, 587–620. MR 2848588, DOI 10.4310/PAMQ.2011.v7.n3.a5
Meinolf Geck, Kazhdan-Lusztig cells and the Frobenius-Schur indicator, J. Algebra 398 (2014), 329–342. MR 3123768, DOI 10.1016/j.jalgebra.2013.01.019
Meinolf Geck, $\mathsf {PyCox}$: computing with (finite) Coxeter groups and Iwahori-Hecke algebras, LMS J. Comput. Math. 15 (2012), 231–256. MR 2988815, DOI 10.1112/S1461157012001064
Meinolf Geck, On Kottwitz’ conjecture for twisted involutions, see arXiv:1206.0443.
Meinolf Geck and Abbie Halls, On the Kazhdan-Lusztig cells in type $E_8$, Mathematics of Computation, to appear.
Meinolf Geck and Nicolas Jacon, Representations of Hecke algebras at roots of unity, Algebra and Applications, vol. 15, Springer-Verlag London, Ltd., London, 2011. MR 2799052, DOI 10.1007/978-0-85729-716-7
Meinolf Geck and Gunter Malle, Frobenius-Schur indicators of unipotent characters and the twisted involution module, Represent. Theory 17 (2013), 180–198. MR 3037782, DOI 10.1090/S1088-4165-2013-00430-2
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
Jérémie Guilhot, On the lowest two-sided cell in affine Weyl groups, Represent. Theory 12 (2008), 327–345. MR 2448287, DOI 10.1090/S1088-4165-08-00334-8
Christophe Hohlweg, A generalization of plactic-coplactic equivalences and Kazhdan-Lusztig cells, J. Algebra 283 (2005), no. 2, 671–689. MR 2111216, DOI 10.1016/j.jalgebra.2004.09.018
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
Robert E. Kottwitz, Involutions in Weyl groups, Represent. Theory 4 (2000), 1–15. MR 1740177, DOI 10.1090/S1088-4165-00-00050-9
G. Lusztig, Left cells in Weyl groups, Lie group representations, I (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1024, Springer, Berlin, 1983, pp. 99–111. MR 727851, DOI 10.1007/BFb0071433
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, Cells in affine Weyl groups, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 255–287. MR 803338, DOI 10.2969/aspm/00610255
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 10.1007/bf01389157
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
G. Lusztig, A bar operator for involutions in a Coxeter group, Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), no. 3, 355–404. MR 3051318
George Lusztig and David A. Vogan Jr., Hecke algebras and involutions in Weyl groups, Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), no. 3, 323–354. MR 3051317
Eric Marberg, How to compute the Frobenius-Schur indicator of a unipotent character of a finite Coxeter system, Adv. Math. 240 (2013), 484–519. MR 3046317, DOI 10.1016/j.aim.2013.02.023
M.-P. Schützenberger, La correspondance de Robinson, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) Lecture Notes in Math., Vol. 579, Springer, Berlin, 1977, pp. 59–113 (French). MR 0498826