Subfield symmetric spaces for finite special linear groups

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.