On The Mackey Formula for Connected Centre Groups

Let G be a connected reductive algebraic group over Fp and let F : G → G be a Frobenius endomorphism endowing G with an Fq-rational structure. Bonnafé–Michel have shown that the Mackey formula for Deligne–Lusztig induction and restriction holds for the pair (G, F) except in the case where q = 2 and G has a quasi-simple component of type E6, E7, or E8. Using their techniques we show that if q = 2 and Z(G) is connected then the Mackey formula holds unless G has a quasi-simple component of type E8. This establishes the Mackey formula, for instance, in the case where (G, F) is of type E7(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) is connected.