Modular irreducibility of cuspidal unipotent characters

We prove a long-standing conjecture of Geck which predicts that cuspidal unipotent characters remain irreducible after $\mathrm{\ell }$-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}.