The principal investigators conduct research in geometric representation theory for unipotent groups over fields of positive characteristic and in the theory of modular tensor categories. They define and study the notion of a character sheaf on a unipotent algebraic group. Using this notion they introduce certain subcategories, called "blocks", of the equivariant derived category of bounded constructible l-adic complexes on this group. The principal investigators formulate certain conjectures about the structure of the blocks and the relation between character sheaves on a unipotent group and the irreducible characters of its group of points over a finite field; their goal is to prove the conjectures. One of the conjectures says that each block is equivalent to the derived category of some modular category, another conjecture describes all possible modular categories that can arise in this way.
The subject of the research lies at the intersection of several domains of modern mathematics and mathematical physics -- geometric representation theory, algebraic geometry, and conformal field theory. The research will deepen our understanding of geometric representation theory by developing it in the new context of unipotent groups in positive characteristic, where this theory has not been studied before. It will also strengthen the connection between geometric representation theory and the theory of modular tensor categories.
