Skip to main content

Advertisement

SpringerLink
Go to cart
  1. Home
  2. Transformation Groups
  3. Article
GENERALISED GELFAND–GRAEV REPRESENTATIONS IN BAD CHARACTERISTIC ?
Download PDF
Your article has downloaded

Similar articles being viewed by others

Slider with three articles shown per slide. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide.

On Refined Bruhat Decompositions and Endomorphism Algebras of Gelfand-Graev Representations

10 April 2019

Alessandro Paolini & Iulian I. Simion

GENERICALLY FREE REPRESENTATIONS II: IRREDUCIBLE REPRESENTATIONS

27 July 2020

SKIP GARIBALDI & ROBERT M. GURALNICK

On the irreducibility of the extensions of Burau and Gassner representations

18 September 2021

Mohamad N. Nasser & Mohammad N. Abdulrahim

Representations of a central extension of the simple Lie superalgebra $$\mathfrak p(3)$$ p ( 3 )

31 July 2018

Vera Serganova

Jacobi-Trudi Type Formula for Character of Irreducible Representations of 𝖌 𝖑 ( m | 1 ) $\frak {gl}(m|1)$

23 July 2018

Nguyên Luong Thái Bình, Nguyên Thi Phuong Dung & Phùng Hô Hai

PSEUDOCHARACTERS OF HOMOMORPHISMS INTO CLASSICAL GROUPS

06 August 2020

M. WEIDNER

The Navarro–Tiep Galois conjecture for $$p=2$$ p = 2

13 February 2019

Gunter Malle

Study of multiplicities in induced representations of $$GL_n$$ G L n through a symmetric reduction

09 February 2022

Taiwang Deng

GENERICALLY FREE REPRESENTATIONS III: EXTREMELY BAD CHARACTERISTIC

31 July 2020

SKIP GARIBALDI & ROBERT M. GURALNICK

Download PDF
  • Open Access
  • Published: 22 May 2020

GENERALISED GELFAND–GRAEV REPRESENTATIONS IN BAD CHARACTERISTIC ?

  • MEINOLF GECK1 

Transformation Groups volume 26, pages 305–326 (2021)Cite this article

  • 207 Accesses

  • 3 Citations

  • Metrics details

A Correction to this article was published on 29 May 2021

This article has been updated

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.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

Change history

  • 29 May 2021

    A Correction to this paper has been published: https://doi.org/10.1007/s00031-021-09654-z

References

  1. R. W. Carter, Centralizers of semisimple elements in the finite classical groups, Proc. London Math. Soc. 42 (1981), 1–41.

    Article  MathSciNet  Google Scholar 

  2. R. W. Carter, Finite Groups of Lie type: Conjugacy Classes and Complex Characters, Wiley, New York, 1985.

    MATH  Google Scholar 

  3. M. C. Clarke, A. Premet, The Hesselink stratification of nullcones and base change, Invent. Math. 191 (2013), 631–669.

    Article  MathSciNet  Google Scholar 

  4. J. Dong, G. Yang, Geck’s conjecture and the generalized Gelfand–Graev representations in bad characteristic, arXiv:1910.03764 (2019).

  5. 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.

    Article  MathSciNet  Google Scholar 

  6. O. Dudas, G. Malle, Modular irreducibility of cuspidal unipotent characters, Invent. Math. 211 (2018), 579–589.

    Article  MathSciNet  Google Scholar 

  7. H. Enomoto, The characters of the finite symplectic group Sp(4, q), q = 2f, Osaka J. Math. 9 (1972), 75–94.

    MathSciNet  MATH  Google Scholar 

  8. H. Enomoto, The characters of the finite Chevalley group G2(q), q = 3f, Japan. J. Math. 2 (1976), 191–248.

    Article  MathSciNet  Google Scholar 

  9. H. Enomoto, H. Yamada, The characters of G2(2n), Japan. J. Math. 12 (1986), 325–377.

    Article  MathSciNet  Google Scholar 

  10. The GAP Group, GAP–Groups, Algorithms, and Programming, Version 4.8.10, 2018 (see http://www.gap-system.org).

  11. M. Geck, On the construction of semisimple Lie algebras and Chevalley groups, Proc. Amer. Math. Soc. 145 (2017), 3233–3247.

    Article  MathSciNet  Google Scholar 

  12. M. Geck, D. Hézard, On the unipotent support of character sheaves, Osaka J. Math. 45 (2008), 819–831.

    MathSciNet  MATH  Google Scholar 

  13. 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.

  14. 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.

    Article  MathSciNet  Google Scholar 

  15. G.-M. Greuel, G. Pfister, A Singular introduction to commutative algebra, Springer–Verlag, Berlin, 2002 (see also http://www.singular.uni-kl.de).

  16. 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.

  17. N. Kawanaka, Generalized Gelfand–Graev representations of exceptional algebraic groups I, Invent. Math. 84 (1986), 575–616.

    Article  MathSciNet  Google Scholar 

  18. 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.

  19. D. E. Knuth, Overlapping pfaffians, Electron. J. Combin. 3 (1996), R5.

    Article  MathSciNet  Google Scholar 

  20. 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.

  21. G. Lusztig, A class of irreducible representations of a Weyl group, Proc. Kon. Nederl. Akad. (A) 82 (1979), 323–335.

  22. G. Lusztig, Characters of Reductive Groups Over a Finite Field, Ann. Math. Studies 107, Princeton U. Press, Princeton, New Jersey, 1984.

  23. G. Lusztig, A unipotent support for irreducible representations, Advances in Math. 94 (1992), 139–179.

    Article  MathSciNet  Google Scholar 

  24. G. Lusztig, Notes on unipotent classes, Asian J. Math. 1 (1997), 194–207.

    Article  MathSciNet  Google Scholar 

  25. G. Lusztig, Unipotent elements in small characteristic, Transform. Groups 10 (2005), 449–487.

    Article  MathSciNet  Google Scholar 

  26. G. Lusztig, Unipotent elements in small characteristic II, Transform. Groups 13 (2008), 773–797.

    Article  MathSciNet  Google Scholar 

  27. G. Lusztig, Unipotent classes and special Weyl group representations, J. Algebra 321 (2009), 3418–3449.

    Article  MathSciNet  Google Scholar 

  28. G. Lusztig, Unipotent elements in small characteristic III, J. Algebra 329 (2011), 163–189.

    Article  MathSciNet  Google Scholar 

  29. G. Lusztig, Unipotent elements in small characteristic IV, Transform. Groups 15 (2010), 921–936.

    Article  MathSciNet  Google Scholar 

  30. G. Lusztig, N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. 19 (1979), 41–52.

    Article  MathSciNet  Google Scholar 

  31. 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.

  32. A. Premet, A modular analogue of Morozov’s theorem on maximal subalgebras of simple Lie algebras, Advances in Math. 311 (2017), 833–884.

    Article  MathSciNet  Google Scholar 

  33. N. Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Math., Vol. 946, Springer, Berlin, 1982.

  34. T. A. Springer, Linear Algebraic Groups, 2nd ed., Birkhäuser, Boston, 1998.

  35. 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.

  36. J. Taylor, Generalized Gelfand–Graev representations in small characteristics, Nagoya Math. J. 224 (2016), 93–167.

    Article  MathSciNet  Google Scholar 

  37. T. Xue, Nilpotent elements in the dual of odd orthogonal algebras, Transform. Groups 17 (2012), 571–592.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IAZ - Lehrstuhl für Algebra, Universität Stuttgart, Pfaffenwaldring 57, D-70569, Stuttgart, Germany

    MEINOLF GECK

Authors
  1. MEINOLF GECK
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to MEINOLF GECK.

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/.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Published: 22 May 2020

  • Issue Date: March 2021

  • DOI: https://doi.org/10.1007/s00031-020-09575-3

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Over 10 million scientific documents at your fingertips

Switch Edition
  • Academic Edition
  • Corporate Edition
  • Home
  • Impressum
  • Legal information
  • Privacy statement
  • Your US state privacy rights
  • How we use cookies
  • Your privacy choices/Manage cookies
  • Accessibility
  • FAQ
  • Contact us
  • Affiliate program

Not affiliated

Springer Nature

© 2023 Springer Nature Switzerland AG. Part of Springer Nature.