关于连通中心群的Mackey公式
On The Mackey Formula for Connected Centre Groups
摘要：
令$\mathbf{G}$是$\overline{\mathbb{F}}_p$上的连通约化代数群，令$F : \mathbf{G} \to \mathbf{G}$是赋予$\mathbf{G}$ $\mathbb{F}_q$ -有理结构的Frobenius自同态。Bonnafé-Michel证明了Deligne-Lusztig归纳和限制的Mackey公式对$(\mathbf{G},F)$是成立的，除非$q = 2$和$\mathbf{G}$有一个$\sf{E}_6$、$\sf{E}_7$或$\sf{E}_8$型的准单分量。使用他们的技术，我们表明，如果$q = 2$和$Z(\mathbf{G})$是连接的，那么Mackey公式成立，除非$\mathbf{G}$有一个准简单的组件类型$\sf{E}_8$。例如，在$(\mathbf{G},F)$是$\sf{E}_7(2)$类型的情况下，这建立了Mackey公式。利用这一点，再加上Bonnafé-Michel的工作，我们可以得出结论，如果$Z(\mathbf{G})$是连通的，则Mackey公式在幂等支持的类函数空间上成立。
英文摘要：
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 $\sf{E}_6$, $\sf{E}_7$, 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 Bonnafé--Michel, we can conclude that the Mackey formula holds on the space of unipotently supported class functions if $Z(\mathbf{G})$ is connected.
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