On $e$-local structures for ${\mathbb{Z}}_{\ell }$-spetses

Abstract

Let $q$ be a prime power, $\ell$ a prime not dividing $q$, and $e$ the order of $q$ modulo $\ell$. We show that the geometric realisation of the nerve of the transporter category of $e$-split Levi subgroups of a finite reductive group $G$ over ${\mathbb{F}}_q$ is homotopy equivalent to the classifying space $BG$ up to $\ell$-completion. We suggest a generalisation of this equivalence to the setting of ${\mathbb{Z}}_{\ell }$-reflection cosets and establish a related fact involving the associated orbit spaces. We also establish a Dade-like formula for unipotent characters of ${\mathbb{Z}}_{\ell }$-spetses inspired by a question of Broué.

18 pages