Cohomology and Modular Representation Theory of Finite and Algebraic Groups

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CLINE.TEXT The principal investigator of this grant will work in collaboration with Brian Parshall and Leonard Scott of the University of Virginia for a significant portion of the research activity on this grant. The group will study the modular representation theory of algebraic and finite groups of Lie type. In the case where the (prime) characteristic of the base field is the natural characteristic of the ambient algebraic group, the main problem centers upon a famous conjecture by Lusztig for the characters of certain irreducible representations. Recent progress shows that this conjecture is true for all but a finite number of primes. However, at present, there is no effective bound on the size of the prime. One goal of this project is to continue work toward a complete proof of this conjecture. In the case where the characteristic is not the natural characteristic, the theory is much less well developed. Recent work of Cline, in collaboration with Parshall and Scott, yields techniques which apply, generally, to the modular representation theory of the finite groups of Lie type in all characteristics. A second goal of this project therefore is to develop these new techniques into an effective tool for the study of these representations in the non-natural characteristic cases. In particular, one hopes eventually to obtain significant insight into the nature of the irreducible modules in these cases. The theory of groups and their representations is a central area of research in mathematics because its methods apply generally to the study of any object, in particular to many mathematical structures, with a high degree of symmetry. For a given group, the determination of its irreducible representations is a central problem. The research of the principal investigator in this grant will have as one focus the determination of these irreducible representations for the finite groups of Lie type, and their associated algebraic groups. The problem divid es into two major cases. The first case, in which major progress has been made on the Lusztig conjecture, is much more well developed than in the second where little is known in general. Therefore the work of the principal investigator on this grant will be of a more preliminary nature in the second case than in the first. Progress in either case would have applications in many other areas of mathematics including cohomology theory, Lie theory, the theory of finite dimensional algebras and the study of the subgroup structure of the finite and algebraic groups of Lie type.