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Representation Theory

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Subfield symmetric spaces for finite special linear groups

Authors: Toshiaki Shoji and Karine Sorlin
Journal: Represent. Theory 8 (2004), 487-521
MSC (2000): Primary 20G40; Secondary 20G05
Published electronically: November 15, 2004
MathSciNet review: 2110358
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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
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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