Sheaves on Affine Flags, Springer Fibers, and Representation Theory

Northwestern University, Evanston IL

Investigators

Abstract

Previous results of the investigator and collaborators establish a relation between coherent sheaves on a Steinberg variety of a complex simple algebraic group, and perverse sheaves on the affine flag variety of the Langlands dual group. Since the latter are known, by the work of Kazhdan and Lusztig, to be related to representations of quantum groups at a root of unity, and hence also to representations of algebraic groups in prime characteristic, our results have implications for that theories. They also indicate that there should be a relation between the geometry of perverse sheaves on affine flag varieties, and non-restricted representations of quantum groups or Lie algebras in prime characteristics. The principal goal of the present project is to develop this kind of relationship, thus providing a new tool for the theory of non-restricted representations. It is expected, in particular, to yield a proof and a conceptual explanation for recent numerical conjectures by Lusztig. The methods and ideology of the present work are partly based on the geometric approach to the Langlands program, due to Drinfeld. Another goal of the project is to investigate consequences of the above mentioned results to that theory. A key to new developments, and a source of inspiration in mathematics often lie in discovery of parallelism (equivalence) between seemingly unrelated objects or theories. Representation theory is a rich source of examples of that kind: though formally being a branch of algebra, the modern theory relies heavily on geometric disciplines, such as topology and algebraic geometry. The principle goal of the present project is to develop geometric language for a branch of representation theory where it has been lacking so far, namely the so called theory of non-restricted modular representations. In the corresponding algebraic constructions divisibility properties of integral numbers by a particular prime number are relevant. The geometry apparently related to this deals with particular infinite dimensional objects, encountered also in mathematical physics (conformal field theory), and in number theory (Langlands duality theory). This construction is expected to provide new tools for the above mentioned branch of representation theory; in particular, it is expected to yield a proof of certain conjectures by leading experts in the field.