Modular representations and cohomology

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Abstract for award DMS-0400966 of Parshall and ScottThere are three great classes of representations of the finite groups of Lie type. First, in the defining characteristic theory, the underlying module is taken over a field with the same characteristic as the defining characteristic of the group. Second, in the cross-characteristic theory, modules are taken over fields of characteristic different from the defining characteristic. Cross-characteristic studies are unavoidable if one wants to understand all ways that one finite Lie group might embed in another. The third class involves the representation theory in all characteristics of the underlying Weyl group. In all cases, representations of continuous Lie groups provide starting point for representations. This is most apparent in the defining characterstic case, where an ambient algebraic group is available. In cross characteristic, quantum groups plays a similar role, but, so far, only for the finite general linear groups. Arguably, there is fourth class of "finite group of Lie type," viz., the Hecke algebras associated to Weyl---or even Coxeter---groups. These algebras often connect the three classes; e.g., in the cross-characteristic theory, they provide a link between the finite general linear groups and quantum groups. Though an analogous role for quantum groups in other types has yet to appear, the Hecke algebra "link" remains, often with interpretations in continuous geometry. This project will advance research in these areas, both in current investigations and in focusing anew on finding a unified approach to the representation theory of the finite Lie groups. For example, can everything be reduced to Hecke algebras in a suitably rich sense (allowing homological or geometric structure)? Recently, the authors "reduced" the defining characteristic Lusztig conjecture for finite general linear groups to proposed symmetric group cohomology properties. Is there a similar reduction for other types or analogous cross-characteristic conjectures? Progress on these questions will involve much further development of the authors' recent work on Hecke algebra cohomology, as well as continued pursuit of Parshall's nullcone research. Other continuing projects, including Scott's computer work with undergraduates, fit in well with these themes.The groups and representations studied here comprise the most important basic ingredients for creating a general theory of all finite group representations. Over the past century, similar theories for continuous groups played a large role in quantum theory and the theory of elementary particles. Their finite analogs have already proved valuable in the design of communications and data storage devices, though this finite theory remains very incomplete. In the future, one expects that the finite discrete worlds of computers and communications will become even more important. The task of creating a viable general theory of finite group representations-the investigators' long-term goal-is, thus, a central problem for the future.