On The Mackey Formula for Connected Centre Groups

TL;DR

This paper extends the validity of the Mackey formula for Deligne--Lusztig induction in connected reductive groups over finite fields, particularly addressing cases where the field size is two and the group has specific types.

Contribution

It proves the Mackey formula holds for groups with connected center when q=2, except for type E8, refining previous results and covering new cases such as E7(2).

Findings

Mackey formula holds for groups with connected center when q=2, excluding E8.

Established validity of the Mackey formula for E7(2) groups.

Confirmed the formula on unipotently supported class functions for connected center 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. 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.