On the Mackey formula for connected centre groups

Abstract

Abstract Let 𝐆 {\mathbf{G}} be a connected reductive algebraic group over 𝔽 ¯ p {\overline{\mathbb{F}}_{p}} and let F : 𝐆 → 𝐆 {F:\mathbf{G}\to\mathbf{G}} be a Frobenius endomorphism endowing 𝐆 {\mathbf{G}} with an 𝔽 q {\mathbb{F}_{q}} -rational structure. Bonnafé–Michel have shown that the Mackey formula for Deligne–Lusztig induction and restriction holds for the pair ( 𝐆 , F ) {(\mathbf{G},F)} except in the case where q = 2 {q=2} and 𝐆 {\mathbf{G}} has a quasi-simple component of type 𝖤 6 {\mathsf{E}_{6}} , 𝖤 7 {\mathsf{E}_{7}} , or 𝖤 8 {\mathsf{E}_{8}} . Using their techniques, we show that if q = 2 {q=2} and Z ⁢ ( 𝐆 ) {Z(\mathbf{G})} is connected then the Mackey formula holds unless 𝐆 {\mathbf{G}} has a quasi-simple component of type 𝖤 8 {\mathsf{E}_{8}} . This establishes the Mackey formula, for instance, in the case where ( 𝐆 , F ) {(\mathbf{G},F)} is of type 𝖤 7 ⁢ ( 2 ) {\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 ⁢ ( 𝐆 ) {Z(\mathbf{G})} is connected.