Simulating Perverse Sheaves in Modular Representation Theory

For some time we have studied the representation theory of semisimple algebraic groups in characteristic p > 0. Of special interest is a celebrated conjecture of Lusztig [Ll] describing the characters of the simple modules when p is not too small relative to the root system. A similar conjecture, by Kazhdan and Lusztig [KLl], for the composition factor multiplicities of Verma modules for semisimple complex Lie algebras, has already been proved [BB] and [BK]. The method of proof was to establish a correspondence-actually an equivalence of categories-between a category containing the relevant Lie algebra modules and a category of “perverse sheaves”, where powerful geometric methods had already decided the issue (see Kazhdan and Lusztig [KL], and later treatments by Lusztig and Vogan [LV] and MacPherson [Sp]). Our recent research has centered on constructing a framework putting the salient features of the characteristic p modules as well as the characteristic 0 Lie algebra modules and perverse sheaves under one algebraic roof. The relevant notion is that of an abstract highest weight category, and the related concept of a quasi-hereditary algebra, as defined and developed by us in [CPSl], [CPS2], [CPS3]. In [PSI the second two authors succeeded in showing that the relevant perverse sheaves formed such an abstract highest weight category, and that their derived category coincided with the relative derived category of constructible sheaves appearing in the arguments of MacPherson cited above. Of course, our framework also encompasses the characteristic p modules we wish to study, so now it is possible and interesting to take a different viewpoint: What properties of perverse sheaves can be proved in the