×

Lifting of elements of Weyl groups. (English) Zbl 1395.20024

Let \(G\) be a reductive algebraic group, let \(T\leq G\) be a Cartan subgroup, let \(N\) be the nomalizer of \(T\) in \(G\) and let \(W=N/T\) be its corresponding Weyl group. In this paper, the authors discuss the problem of determining the orders of the elements of \(N\) that are lifts of elements of \(W\). Whilst it is straightforward to see that an element of \(W\) of order \(d\) will lift to an element of order either \(d\) or \(2d\), it is harder to determine precisely which holds and it is this question that is addressed here.
The first main theorem of this paper is concerned with the case in which elements lift to elements of the same order, the case of the underlying field having characteristic 2 being very different from the rest. The other two main results of this paper are more technical and focus specifically on the case in which the characteristic of the underlying field is not 2.

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
20G99 Linear algebraic groups and related topics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adams, Jeffrey; Vogan, David A., Contragredient representations and characterizing the local Langlands correspondence, Amer. J. Math., 138, 3, 657-682 (2016) · Zbl 1354.11039
[2] Adams, Jeffrey, Atlas of Lie groups and representations (2014), Website
[3] Bourbaki, Nicolas, Lie Groups and Lie Algebras, Elements of Mathematics (Berlin) (2005), Springer-Verlag: Springer-Verlag Berlin, Chaps. 7-9, Translated from the 1975 and 1982 French originals by Andrew Pressley · Zbl 1139.17002
[4] Broué, Michel; Michel, Jean, Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées, (Finite Reductive Groups. Finite Reductive Groups, Luminy, 1994. Finite Reductive Groups. Finite Reductive Groups, Luminy, 1994, Progr. Math., vol. 141 (1997), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 73-139 · Zbl 1029.20500
[5] Curtis, Morton; Wiederhold, Alan; Williams, Bruce, Normalizers of maximal tori, (Localization in Group Theory and Homotopy Theory, and Related Topics. Localization in Group Theory and Homotopy Theory, and Related Topics, Sympos., Battelle Seattle Res. Center, Seattle, WA, 1974. Localization in Group Theory and Homotopy Theory, and Related Topics. Localization in Group Theory and Homotopy Theory, and Related Topics, Sympos., Battelle Seattle Res. Center, Seattle, WA, 1974, Lecture Notes in Math., vol. 418 (1974), Springer: Springer Berlin), 31-47 · Zbl 0301.22007
[6] Geck, Meinolf; Kim, Sungsoon; Pfeiffer, Götz, Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. Algebra, 229, 2, 570-600 (2000) · Zbl 1042.20026
[7] Geck, Meinolf; Michel, Jean, “Good” elements of finite Coxeter groups and representations of Iwahori-Hecke algebras, Proc. Lond. Math. Soc. (3), 74, 2, 275-305 (1997) · Zbl 0877.20027
[8] Geck, Meinolf; Pfeiffer, Götz, Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras, London Mathematical Society Monographs. New Series, vol. 21 (2000), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York · Zbl 0996.20004
[9] He, Xuhua, Minimal length elements in some double cosets of Coxeter groups, Adv. Math., 215, 2, 469-503 (2007) · Zbl 1149.20035
[10] He, Xuhua; Nie, Sian, Minimal length elements of finite Coxeter groups, Duke Math. J., 161, 15, 2945-2967 (2012) · Zbl 1272.20042
[11] Ihara, Shin-ichiro; Takeo, Yokonuma, On the second cohomology groups (Schur-multipliers) of finite reflection groups, J. Fac. Sci. Univ. Tokyo Sect. I, 11, 15, 155-171 (1965) · Zbl 0136.28802
[12] Onishchik, A. L.; Vinberg, È. B., Lie Groups and Algebraic Groups, Springer Series in Soviet Mathematics (1990), Springer-Verlag: Springer-Verlag Berlin, Translated from the Russian and with a preface by D.A. Leites · Zbl 0722.22004
[13] Reeder, Mark; Levy, Paul; Yu, Jiu-Kang; Gross, Benedict H., Gradings of positive rank on simple Lie algebras, Transform. Groups, 17, 4, 1123-1190 (2012) · Zbl 1310.17017
[14] Rostami, S., On the canonical representatives of a finite Weyl group (2016)
[15] Springer, T. A., Regular elements of finite reflection groups, Invent. Math., 25, 159-198 (1974) · Zbl 0287.20043
[16] Springer, T. A., Linear Algebraic Groups, Modern Birkhäuser Classics (2009), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 1202.20048
[17] Tits, J., Normalisateurs de tores. I. Groupes de Coxeter étendus, J. Algebra, 4, 96-116 (1966) · Zbl 0145.24703
[18] Zaremsky, Matthew C. B., Representatives of elliptic Weyl group elements in algebraic groups, J. Group Theory, 17, 1, 49-71 (2014) · Zbl 1323.20040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.