  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 $r$.
