Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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The values of unipotent characters at unipotent elements for groups of type $\mathsf {E}_8$ and ${}^{2}{\mathsf {E}}_6$
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by Jonas Hetz;
Trans. Amer. Math. Soc. 378 (2025), 4865-4902
DOI: https://doi.org/10.1090/tran/9397
Published electronically: March 5, 2025

Abstract:

In order to tackle the problem of generically determining the character tables of the finite groups of Lie type $\mathbf G(q)$ associated to a connected reductive group $\mathbf G$ over $\overline {\mathbb {F}}_{p}$, Lusztig developed the theory of character sheaves in the 1980s. The subsequent work of Lusztig and Shoji in principle reduces this problem to specifying certain roots of unity. The situation is particularly well understood as far as character values at unipotent elements are concerned. We complete the computation of the values of unipotent characters at unipotent elements for the groups $\mathbf G(q)$ where $\mathbf G$ is the simple group of type $\mathsf E_8$, by specifying the aforementioned roots of unity for all prime powers $q$. We also resolve this task for the groups ${}^{2}{\mathsf E}_6(q)$ when $q$ is a power of $p=2$. Our results thus conclude the project of computing the values of unipotent characters at unipotent elements for the simple exceptional groups of Lie type.
References
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Bibliographic Information
  • Jonas Hetz
  • Affiliation: Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen, Pontdriesch 14/16, D–52062 Aachen, Germany
  • MR Author ID: 1334861
  • ORCID: 0009-0009-5215-4849
  • Email: jonas.hetz@rwth-aachen.de
  • Received by editor(s): September 12, 2024
  • Received by editor(s) in revised form: December 9, 2024, December 11, 2024, and December 18, 2024
  • Published electronically: March 5, 2025
  • Additional Notes: The author was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) — Project-ID 286237555 – TRR 195.
  • © Copyright 2025 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 378 (2025), 4865-4902
  • MSC (2020): Primary 20C33; Secondary 20G40, 20G41
  • DOI: https://doi.org/10.1090/tran/9397