Subfield symmetric spaces for finite special linear groups

Abstract

Let G G be a connected algebraic group defined over a finite field F q {\mathbf F}_q . For each irreducible character ρ \rho of G ( F q r ) G(\mathbf F_{q^r}) , we denote by m r ( ρ ) m_r(\rho ) the multiplicity of 1 G ( F q ) 1_{G({\mathbf F}_q)} in the restriction of ρ \rho to G ( F q ) G({\mathbf F}_q) . In the case where G G is reductive with connected center and is simple modulo center, Kawanaka determined m 2 ( ρ ) m_2(\rho ) for almost all cases, and then Lusztig gave a general formula for m 2 ( ρ ) m_2(\rho ) . In the case where the center of G G is not connected, such a result is not known. In this paper we determine m 2 ( ρ ) m_2(\rho ) , up to some minor ambiguity, in the case where G G is the special linear group. We also discuss, for any r ≥ 2 r \ge 2 , the relationship between m r ( ρ ) m_r(\rho ) with the theory of Shintani descent in the case where G G is a connected algebraic group.