## Subfield symmetric spaces for finite special linear groups

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- by Toshiaki Shoji and Karine Sorlin
- Represent. Theory
**8**(2004), 487-521 - DOI: https://doi.org/10.1090/S1088-4165-04-00233-X
- Published electronically: November 15, 2004
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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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## Bibliographic 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