On The Mackey Formula for Connected Centre Groups

Abstract

Let $\mathbf{G}$ be a connected reductive algebraic group over ${\overline{\mathbb{F}}}_p$ and let $F:\mathbf{G}\to \mathbf{G}$ be a Frobenius endomorphism endowing $\mathbf{G}$ with an ${\mathbb{F}}_q$-rational structure. Bonnafé--Michel have shown that the Mackey formula for Deligne--Lusztig induction and restriction holds for the pair $(\mathbf{G},F)$ except in the case where $q=2$ and $\mathbf{G}$ has a quasi-simple component of type ${\mathsf{E}}_6$, ${\mathsf{E}}_7$, or ${\mathsf{E}}_8$. Using their techniques we show that if $q=2$ and $Z(\mathbf{G})$ is connected then the Mackey formula holds unless $\mathbf{G}$ has a quasi-simple component of type ${\mathsf{E}}_8$. This establishes the Mackey formula, for instance, in the case where $(\mathbf{G},F)$ is of type ${\mathsf{E}}_7(2)$. Using this, together with work of Bonnafé--Michel, we can conclude that the Mackey formula holds on the space of unipotently supported class functions if $Z(\mathbf{G})$ is connected.

7 pages; v2., minor changes, added Lemma 3.4 for clarity

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