Sheaf homology and complete reducibility

AbstractIn the study of finite simple groups by means of the geometries provided by their local subgroups, problems of structure often reduce to questions about modular representations in finite characteristic. Of particular interest are modules spanned by fixed points of Sylow groups (including “high weight” modules for Chevalley groups) for which the homology methods of Ronan and Smith [6]can be very useful. This note presents an elementary sufficient condition for splitting of some reducible modules of this type. Among the applications, we generalize a number of results appearing in recent work of Timmesfeld [11], which provided the original inspiration for this analysis. Some of the ideas had appeared, in a special case, in [10]