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

DOI:
https://doi.org/10.1090/S1088-4165-04-00233-X

Published electronically:
November 15, 2004

MathSciNet review:
2110358

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Abstract | References | Similar Articles | Additional Information

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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[S4] S4 T. Shoji; Lusztig’s conjecture for finite special linear groups, in preparation.

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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.