Modular representations and cohomology for algebraic, finite and quantum groups

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The PIs will build on their recent work on bounds on cohomology groups for semi-simple algebraic groups. For large primes, this both directly and indirectly involves quantum groups, in methods pioneered by the PIs. There are immediate consequences, from a classical "generic cohomology" theory, obtained by the PIs years ago in collaboration with other authors, to asymptotic estimates for bounds for finite groups of Lie type, with more modern methods sometimes allowing these to be improved to actual bounds. This program has been carried out for degree 1 cohomology, and to the point of generic cohomology for all higher degrees. These results lead to important new questions involving the rates of growth of the cohomology spaces. Again the issues in the structures are intertwined, and the study of algebraic groups is a decided advantage for analyzing the quantum case, with both structures contributing to estimates for the growth of sizes of Kazhdan-Lusztig polynomials. For algebraic groups, there are many open questions, especially that of a polynomial rate of growth. Such growth rate issues occur broadly in mathematics, especially in algorithmic issues, and are prominent in theoretical computer science. The PIs will also continue to study Koszul properties for the finite dimensional algebras which come up in the representations of semi-simple groups and quantum groups. Koszul structures arise, or may be conjectured, from geometric considerations (perverse sheaves and their filtrations), but the authors have been pushing entirely algebraic methods into areas where geometry may not directly apply. Applications to the graded and filtered structures of standard (Weyl) modules have already been found, with additional results expected, as well as applications to filtrations of resolutions and cohomology groups of these modules. Often this work uses a conjecture due to Lusztig, which has been proved true for large primes. These studies exhibit deeper consequences of the conjecture (and could even provide insight for establishing it in more cases). Finally, the PIs will continue their work in small characteristic and the calculation of support varieties.This proposal concerns the representation and cohomology theory of important algebraic structures, including semisimple algebraic groups and their finite and infinitesimal subgroups, quantum groups, and Kazhdan-Lusztig polynomials. These structures are interrelated, so can be profitably studied together. A central aspect includes representations of important classes of finite groups. 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, it will surely be even more important in the future. This project also points to the future in its manifold involvement of graduate and undergraduate students.