Commutation of Shintani descent and Jordan decomposition
Résumé
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.
Origine : Fichiers produits par l'(les) auteur(s)
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