  Let ${\mathbf G}^F$ be a finite group of Lie type, where ${\mathbf G}$ is a
reductive group defined over ${\overline{\mathbb F}_q}$ and $F$ is a Frobenius
root. Lusztig's Jordan decomposition parametrises the irreducible characters in
a rational series${\mathcal E}({{\mathbf G}^F},(s)_{{\mathbf G}^{*F^*}})$ where
$s\in{{\mathbf G}^{*F^*}}$ by the series ${\mathcal E}(C_{{\mathbf
G}^*}(s)^{F^*},1)$.We conjecture that the Shintani twisting preserves the space
of class functions generated by the union of the ${\mathcal E}({{\mathbf
G}^F},(s')_{{\mathbf G}^{*F^*}})$ where$(s')_{{\mathbf G}^{*F^*}}$ runs over
the semi-simple classes of ${{\mathbf G}^{*F^*}}$ geometrically conjugate to
$s$;further, extending the Jordan decomposition by linearity to this space, we
conjecture that there is a way to fix Jordan decomposition such that it maps
the Shintani twisting to the Shintani twisting on disconnected groups defined
by Deshpande, which acts on the linear span of $\coprod_{s'}{\mathcal
E}(C_{{\mathbf G}^*}(s')^{F^*},1)$. We show a non-trivial case of this
conjecture, the case where ${\mathbf G}$ is of type $A_{n-1}$with $n$ prime.
