On The Mackey Formula for Connected Centre Groups

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. 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 $\sf{E}_6$, $\sf{E}_7$, or $\sf{E}_8$. 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 $\sf{E}_8$. This establishes the Mackey formula, for instance, in the case where $(\mathbf{G},F)$ is of type $\sf{E}_7(2)$. 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.