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3 results

  • On quasi and weak Steinberg characters of general linear groups

  • Part of
  • Surinder Kaur
  • Journal:
    Proceedings of the Edinburgh Mathematical Society , First View
    Published online by Cambridge University Press:
    19 December 2024, pp. 1-18
    • Article
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  • Let G be a finite group and r be a prime divisor of the order of G. An irreducible character of G is said to be quasi r-Steinberg if it is non-zero on every r-regular element of G. A quasi r-Steinberg character of degree $\displaystyle |Syl_r(G)|$ is said to be weak r-Steinberg if it vanishes on the r-singular elements of $G.$ In this article, we classify the quasi r-Steinberg cuspidal characters of the general linear group $GL(n,q).$ Then we characterize the quasi r-Steinberg characters of $GL(2,q)$ and $GL(3,q).$ Finally, we obtain a classification of the weak r-Steinberg characters of $GL(n,q).$

  • 7 - The Duality Functor and the Steinberg Character

  • François Digne, Université de Picardie Jules Verne, Amiens, Jean Michel, Centre National de la Recherche Scientifique (CNRS), Paris
  • Book:
    Representations of Finite Groups of Lie Type
    Published online:
    14 February 2020
    Print publication:
    05 March 2020, pp 113-129

  • Summary

    We associate a sign to F-stable reductive subgroups of a reductive group with a Frobenius root F, and define Curtis–Alvis duality. We prove its properties, in particular commutation with Harish–Chandra induction and to restriction to centralisers of semi-simple elements. We define the Steinberg character as the dual of the identity and use it to compute the number of unipotent elements.

  • 3 - Harish-Chandra Theories

  • Meinolf Geck, Universität Stuttgart, Gunter Malle, Technische Universität Kaiserslautern, Germany
  • Book:
    The Character Theory of Finite Groups of Lie Type
    Published online:
    20 February 2020
    Print publication:
    27 February 2020, pp 211-270

  • Summary

    We present the ordinary Harish-Chandra theory for finite groups with a BN-pair in arbitrary non-defining characteristic and the relation to Hecke algebras. We then introduceLusztig induction for finite reductive groups, explain its basic properties, use it to define and investigate the duality operation on the character ring and the Steinberg character. In the final section we explain the d-Harish-Chandra theories for finite reductive groups which play a fundamenhtal role in modular representation theory of finite reductive groups.