  We prove a long-standing conjecture of Geck which predicts that cuspidal
unipotent characters remain irreducible after $\ell$-reduction. To this end, we
construct a progenerator for the category of representations of a finite
reductive group coming from generalised Gelfand--Graev representations. This is
achieved by showing that cuspidal representations appear in the head of
generalised Gelfand--Graev representations attached to cuspidal unipotent
classes, as defined and studied in \cite{GM96}.
