Subfield symmetric spaces for finite special linear groups

Let $G$ be a finite reductive group defined over a finite field $F_q$. In the case where $G$ is a special linear group, we compute the multiplicities of irreducible characters of $G(F_{q^2})$ with the character of $G(F_{q^2})$ induced from the trivial character of $G(F_q)$. We discuss the relationship between these multiplicities with the theory of Shintani descent for finite reductive groups in general. We also give some formula concerning the decomposition of the character of $G(F_{q^r})$ induced from the trivial character of $G(F_q)$ for reductive groups $G$ with any positive integer $r$.