Abstract
Let G be a connected reductive algebraic group defined over a finite field with q elements. In the 1980’s, Kawanaka introduced generalised Gelfand–Graev representations of the finite group \( G\left({\mathbbm{F}}_q\right) \), assuming that q is a power of a good prime for G. These representations have turned out to be extremely useful in various contexts. Here we investigate to what extent Kawanaka’s construction can be carried out when we drop the assumptions on q. As a curious by-product, we obtain a new, conjectural characterisation of Lusztig’s concept of special unipotent classes of G in terms of weighted Dynkin diagrams.
Change history
29 May 2021
A Correction to this paper has been published: https://doi.org/10.1007/s00031-021-09654-z
References
R. W. Carter, Centralizers of semisimple elements in the finite classical groups, Proc. London Math. Soc. 42 (1981), 1–41.
R. W. Carter, Finite Groups of Lie type: Conjugacy Classes and Complex Characters, Wiley, New York, 1985.
M. C. Clarke, A. Premet, The Hesselink stratification of nullcones and base change, Invent. Math. 191 (2013), 631–669.
J. Dong, G. Yang, Geck’s conjecture and the generalized Gelfand–Graev representations in bad characteristic, arXiv:1910.03764 (2019).
A. W. M. Dress, W. Wenzel, A simple proof of an identity concerning Pfaffians of skew symmetric matrices, Advances in Math. 112 (1995), 120–134.
O. Dudas, G. Malle, Modular irreducibility of cuspidal unipotent characters, Invent. Math. 211 (2018), 579–589.
H. Enomoto, The characters of the finite symplectic group Sp(4, q), q = 2f, Osaka J. Math. 9 (1972), 75–94.
H. Enomoto, The characters of the finite Chevalley group G2(q), q = 3f, Japan. J. Math. 2 (1976), 191–248.
H. Enomoto, H. Yamada, The characters of G2(2n), Japan. J. Math. 12 (1986), 325–377.
The GAP Group, GAP–Groups, Algorithms, and Programming, Version 4.8.10, 2018 (see http://www.gap-system.org).
M. Geck, On the construction of semisimple Lie algebras and Chevalley groups, Proc. Amer. Math. Soc. 145 (2017), 3233–3247.
M. Geck, D. Hézard, On the unipotent support of character sheaves, Osaka J. Math. 45 (2008), 819–831.
M. Geck, G. Hiss, Modular representations of finite groups of Lie type in non-defining characteristic, in: Finite Reductive Groups (Luminy, 1994; ed. M. Cabanes), Progress in Math., Vol. 141, Birkhäuser, Boston, MA, 1997, pp. 195–249.
M. Geck, G. Malle, On the existence of a unipotent support for the irreducible characters of finite groups of Lie type, Trans. Amer. Math. Soc. 352 (2000), 429–456.
G.-M. Greuel, G. Pfister, A Singular introduction to commutative algebra, Springer–Verlag, Berlin, 2002 (see also http://www.singular.uni-kl.de).
N. Kawanaka, Generalized Gelfand–Graev representations and Ennola duality, in: Algebraic Groups and Related Topics, Advanced Studies in Pure Math., Vol. 6, Kinokuniya, Tokyo, and North-Holland, Amsterdam, 1985, pp. 175–206.
N. Kawanaka, Generalized Gelfand–Graev representations of exceptional algebraic groups I, Invent. Math. 84 (1986), 575–616.
N. Kawanaka, Shintani lifting and Gelfand–Graev representations, in: The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math., Vol. 47, Part 1, Amer. Math. Soc., Providence, R.I., 1987, pp. 147–163.
D. E. Knuth, Overlapping pfaffians, Electron. J. Combin. 3 (1996), R5.
F. Lübeck, Centralizers and numbers of semisimple classes in exceptional groups of Lie type, online data at http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/CentSSClasses.
G. Lusztig, A class of irreducible representations of a Weyl group, Proc. Kon. Nederl. Akad. (A) 82 (1979), 323–335.
G. Lusztig, Characters of Reductive Groups Over a Finite Field, Ann. Math. Studies 107, Princeton U. Press, Princeton, New Jersey, 1984.
G. Lusztig, A unipotent support for irreducible representations, Advances in Math. 94 (1992), 139–179.
G. Lusztig, Notes on unipotent classes, Asian J. Math. 1 (1997), 194–207.
G. Lusztig, Unipotent elements in small characteristic, Transform. Groups 10 (2005), 449–487.
G. Lusztig, Unipotent elements in small characteristic II, Transform. Groups 13 (2008), 773–797.
G. Lusztig, Unipotent classes and special Weyl group representations, J. Algebra 321 (2009), 3418–3449.
G. Lusztig, Unipotent elements in small characteristic III, J. Algebra 329 (2011), 163–189.
G. Lusztig, Unipotent elements in small characteristic IV, Transform. Groups 15 (2010), 921–936.
G. Lusztig, N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. 19 (1979), 41–52.
A. Premet, Nilpotent orbits in good characteristic and the Kempf–Rousseau theory (Special issue celebrating the 80th birthday of Robert Steinberg), J. Algebra 260 (2003), 338–366.
A. Premet, A modular analogue of Morozov’s theorem on maximal subalgebras of simple Lie algebras, Advances in Math. 311 (2017), 833–884.
N. Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Math., Vol. 946, Springer, Berlin, 1982.
T. A. Springer, Linear Algebraic Groups, 2nd ed., Birkhäuser, Boston, 1998.
R. Steinberg, Lectures on Chevalley Groups, mimeographed notes, Department of Math., Yale University, 1967/68; now available as Vol. 66 of the University Lecture Series, Amer. Math. Soc., Providence, RI, 2016.
J. Taylor, Generalized Gelfand–Graev representations in small characteristics, Nagoya Math. J. 224 (2016), 93–167.
T. Xue, Nilpotent elements in the dual of odd orthogonal algebras, Transform. Groups 17 (2012), 571–592.
Author information
Authors and Affiliations
Corresponding author
Additional information
Funding Information
Open Access funding enabled and organized by Projekt DEAL.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
GECK, M. GENERALISED GELFAND–GRAEV REPRESENTATIONS IN BAD CHARACTERISTIC ?. Transformation Groups 26, 305–326 (2021). https://doi.org/10.1007/s00031-020-09575-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-020-09575-3