On semisimple classes and semisimple characters in finite reductive groups

Brunat, Olivier

Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1671-1716.

Dans cet article, on étudie les éléments de centralisateur non connexe du complexe de Brauer associé à un groupe algébrique simple $\mathbf{G}$ défini sur un corps fini. On déduit alors, lorsque le groupe fondamental est d’ordre premier, le nombre de classes de conjugaison semi-simples rationnelles de $\mathbf{G}$ dont les représentants ont un centralisateur non connexe. On étudie également l’extensibilité des caractères semisimples du groupe des points fixes ${\mathbf{G}}^F$ à leur groupe d’inertie dans le groupe des automorphismes de ${\mathbf{G}}^F$, où $F$ est l’endomorphisme de Frobenius de $\mathbf{G}$ relatif à la structure rationnelle. Comme conséquence, on montre qu’un groupe fini simple de type $E_6$ vérifie la condition inductive de McKay en caractéristique naturelle. Ce travail s’inscrit dans le programme général initialisé par Isaacs, Malle et Navarro pour prouver la conjecture de McKay en théorie des représentations des groupes finis.

In this article, we study the elements with disconnected centralizer in the Brauer complex associated to a simple algebraic group $\mathbf{G}$ defined over a finite field with corresponding Frobenius map $F$ and derive the number of $F$-stable semisimple classes of $\mathbf{G}$ with disconnected centralizer when the order of the fundamental group has prime order. We also discuss extendibility of semisimple characters of the fixed point subgroup ${\mathbf{G}}^F$ to their inertia group in the full automorphism group. As a consequence, we prove that “twisted” and “untwisted” simple groups of type $E_6$ are “good” in defining characteristic, which is a contribution to the general program initialized by Isaacs, Malle and Navarro to prove the McKay Conjecture in representation theory of finite groups.

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