On The Mackey Formula for Connected Centre Groups

Let $\mathbf{G}$ be a connected reductive algebraic group over ${\bar{\mathbb{F}}}_p$ and let $F:\mathbf{G}\to \mathbf{G}$ be a Frobenius endomorphism endowing $\mathbf{G}$ with an ${\mathbb{F}}_q$-rational structure. Bonnaf'e–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 ${\mathsf{E}}_{\mathsf{6}}$, ${\mathsf{E}}_{\mathsf{7}}$, or ${\mathsf{E}}_{\mathsf{8}}$. 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 ${\mathsf{E}}_{\mathsf{8}}$. This establishes the Mackey formula, for instance, in the case where $(\mathbf{G},F)$ is of type ${\mathsf{E}}_{\mathsf{7}}(\mathsf{2})$. Using this, together with work of Bonnaf'e–Michel, we can conclude that the Mackey formula holds on the space of unipotently supported class functions if $Z(\mathbf{G})$ is connected.