The aim of the project is to continue development of algebro-geometric and categorical methods in representation theory. In particular, the PI plans to extend his approach to character sheaves (worked out jointly with Finkelberg and Ostrik) to loop groups. This is expected to provide a local algebro-geometric view of endoscopy, complementing the results of B.-C. Ngo obtained by global methods. The PI also plans to work on an algebraic version of the local trace formula, continue to study quantization of algebraic symplectic varieties in positive characteristic and its relation to non-commutative geometry and develop a relation between non-restricted representations of quantum groups at roots of unity and affine Lie algebras.
Representation theory is a branch of algebra studying the algebraic structure of symmetries. The basic question of representation theory is: given an abstract structure of symmetry (the idea rigorously expressed in such concepts as a group, a Lie algebra etc.) to describe all possible ways to realize it as symmetries of a concrete algebraic object. During some hundred years of its existence representation theory found tremendously powerful applications in quantum physics, number theory, algebraic geometry and its original "home territory" of abstract algebra. In recent decades a lot of progress in topology, algebra and some branches of quantum physics has been obtained by applying a general concept of "categorification". The latter is based on the idea that some known algebraic objects should be realized as "shadows" of something more sophisticated but enjoying a much richer structure. An earlier work by the PI is devoted to a particular example of this strategy arising in the study of representations of finite Chevalley groups (such as the group of invertible matrices whose entries are residues modulo a given prime number). One of the goals of the current project is to apply this strategy to much more complicated, infinite dimensional, groups, known as loop groups. Quantum structures whose coefficients are residues modulo a prime are the subject of the other sections of the proposal.
