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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Subfield symmetric spaces for finite special linear groups
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by Toshiaki Shoji and Karine Sorlin PDF
Represent. Theory 8 (2004), 487-521 Request permission

Abstract:

Let $G$ be a connected algebraic group defined over a finite field ${\mathbf F}_q$. For each irreducible character $\rho$ of $G(\mathbf F_{q^r})$, we denote by $m_r(\rho )$ the multiplicity of $1_{G({\mathbf F}_q)}$ in the restriction of $\rho$ to $G({\mathbf F}_q)$. In the case where $G$ is reductive with connected center and is simple modulo center, Kawanaka determined $m_2(\rho )$ for almost all cases, and then Lusztig gave a general formula for $m_2(\rho )$. In the case where the center of $G$ is not connected, such a result is not known. In this paper we determine $m_2(\rho )$, up to some minor ambiguity, in the case where $G$ is the special linear group.

We also discuss, for any $r \ge 2$, the relationship between $m_r(\rho )$ with the theory of Shintani descent in the case where $G$ is a connected algebraic group.

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Additional Information
  • Toshiaki Shoji
  • Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
  • Karine Sorlin
  • Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
  • Address at time of publication: LAMFA, Université de Picardie-Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex, France
  • Received by editor(s): March 2, 2004
  • Received by editor(s) in revised form: September 13, 2004
  • Published electronically: November 15, 2004
  • Additional Notes: The second author would like to thank the JSPS for support which made this collaboration possible
  • © Copyright 2004 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 8 (2004), 487-521
  • MSC (2000): Primary 20G40; Secondary 20G05
  • DOI: https://doi.org/10.1090/S1088-4165-04-00233-X
  • MathSciNet review: 2110358