  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.
