Commutation of Shintani descent and Jordan decomposition (2007.08819v3)

Abstract: 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.