Localisation de faisceaux caract\`eres

TL;DR

This paper derives a formula for the characteristic function of a character sheaf using representation theory of finite groups linked to the Weyl group, generalizing prior results and explicitly identifying relevant subgroups for quasi-simple groups.

Contribution

It generalizes previous formulas for character sheaves by incorporating reductive subgroups with cuspidal character sheaves and explicitly classifies these subgroups for quasi-simple groups.

Findings

Derived a new formula for character sheaf functions.

Connected the formula to representation theory of finite groups.

Explicitly classified relevant subgroups for quasi-simple groups.

Abstract

Nous obtenons une formule pour les valeurs de la fonction caract\'eristique d'un faisceau caract\`ere en fonction de la th\'eorie des repr\'esentations de certains groupes finis, li\'es au groupe de Weyl. Cette formule, qui g\'en\'eralise des r\'esultats ant\'erieurs de M{\oe}glin et de Waldspurger, d\'epend de la connaissance de certains sous-groupes r\'eductifs admettant un faisceau caract\`ere cuspidal. Dans un second temps, afin de rendre la formule plus explicite dans le cas d'un groupe quasi-simple, nous d\'eterminons ces sous-groupes \`a conjugaison pr\`es. We obtain a formula for the values of the characteristic function of a character sheaf, in terms of the representation theory of certain finite groups related to the Weyl group. This formula, a generalization of previous results due to M{\oe}glin and Waldspurger, depends on knowledge of certain reductive subgroups that admit cuspidal character sheaves. For quasi-simple groups, we make the formula truly explicit by determining all such subgroups upto conjugation.