This research concerns the modular representation theory and cohomology of finite groups of Lie type. A major thrust involves the defining characteristic theory, in which the underlying module is taken over a field having the "same" characteristic as the group. The cross-characteristic theory is another theme; here, the characteristic of the module differs from that of the group. In both cases, the (already known) representations of continuous Lie groups provide a 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. The major unsolved problems involve classifying the irreducible modules, determining their dimensions and characters, and understanding their homological properties. Issues involving Weyl groups (and Hecke algebras), finite dimensional algebras (e.g., endomorphism and quasi-hereditary algebras), the geometric theory of perverse sheaves, etc. arise naturally in many attacks on these problems. Parshall and Scott have developed and contributed to many of the various approaches, and the current project will continue in that direction, but also apply new (and deep results), such as the recent solution of the Broue conjecture for symmetric groups (by Chuang and Rouquier). For example, the authors will combine this work with their own recent work relating Kazhdan-Lusztig polynomial character formulas for representations of the special linear groups in the defining characteristic to the representations of symmetric groups.
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. Finally, graduate and undergraduate students will be fully involved in the project.
