The Brown complex in non-defining characteristic and applications

Abstract

We study the Brown complex associated to the poset of $\ell$-subgroups in the case of a finite reductive group defined over a field $\mathbb{F}_q$ of characteristic prime to $\ell$. First, under suitable hypotheses, we show that its homotopy type is determined by the generic Sylow theory developed by Brou\'e and Malle and, in particular, only depends on the multiplicative order of $q$ modulo $\ell$. This result leads to several interesting applications to generic Sylow theory, mod $\ell$ homology decompositions, and $\ell$-modular representation theory. Then, we conduct a more detailed study of the Brown complex in order to establish an explicit connection between the local-global conjectures in representation theory of finite groups and the generic Sylow theory. This is done by isolating a family of $\ell$-subgroups of finite reductive groups that corresponds bijectively to the structures controlled by the generic Sylow theory.