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3 results

  • ON THE DISTINCTION OF IWAHORI-SPHERICAL DISCRETE SERIES REPRESENTATIONS

  • Part of
  • Paul Broussous
  • Journal:
    Journal of the Institute of Mathematics of Jussieu / Volume 24 / Issue 1 / January 2025
    Published online by Cambridge University Press:
    17 April 2024, pp. 1-15
    Print publication:
    January 2025
    • Article
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  • Let $E/F$ be a quadratic unramified extension of non-archimedean local fields and $\mathbb H$ a simply connected semisimple algebraic group defined and split over F. We establish general results (multiplicities, test vectors) on ${\mathbb H} (F)$-distinguished Iwahori-spherical representations of ${\mathbb H} (E)$. For discrete series Iwahori-spherical representations of ${\mathbb H} (E)$, we prove a numerical criterion of ${\mathbb H} (F)$-distinction. As an application, we classify the ${\mathbb H} (F)$-distinguished discrete series representations of ${\mathbb H} (E)$ corresponding to degree $1$ characters of the Iwahori-Hecke algebra.

  • 6 - Iwahori–Hecke Algebras

  • 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 91-112

  • Summary

    This chapter studies endomorphism algebras of the Harish–Chandra induced representation from a cuspidal pair, at first on an arbitrary field, then in characteristic 0. This algebra is viewed as an Iwahori–Hecke algebra, allowing definition of Schur elements and generic degrees. The chapter ends with the character table of an Iwahori–Hecke algebra of type G2 in the generic case.

  • 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.