Modular irreducibility of cuspidal unipotent characters
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 Geck and Malle (J Lond Math Soc 2(53):63–78, 1996).
Mathematics Subject ClassificationPrimary 20C33 Secondary 20C08
We thank Meinolf Geck and Jay Taylor for valuable comments on an earlier version.