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.