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

TL;DR

This paper explores the homotopy equivalence between certain geometric realizations related to finite reductive groups and classifying spaces, extending to $bZ_ell$-reflection cosets and unipotent characters.

Contribution

It generalizes the homotopy equivalence to $bZ_ell$-reflection cosets and introduces a Dade-like formula for unipotent characters of $bZ_ell$-spetses.

Findings

Homotopy equivalence between geometric realization of transporter categories and classifying spaces.

Extension of equivalence to $bZ_ell$-reflection cosets and orbit spaces.

A Dade-like formula for unipotent characters of $bZ_ell$-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 Brou\'e.