Character Sheaves for Unipotent Groups

主要研究人员在正特征域上的幂幺群的几何表示理论和模张量范畴理论方面进行研究。他们定义和研究的概念，一个字符层上的一个幂幺代数群。利用这一概念，他们介绍了某些子类别，称为“块”，等变派生类别的有界建设的L-进复杂的这个群体。主要研究人员制定了一些关于块的结构和幂幺群上的特征层与有限域上点群的不可约特征之间的关系的命题;他们的目标是证明这些命题。其中一个猜想说每个块都等价于某个模范畴的导出范畴，另一个猜想描述了所有可能的模范畴都可以通过这种方式产生。研究的主题位于现代数学和数学物理的几个领域的交叉点--几何表示论，代数几何和共形场论。这一研究将加深我们对几何表示理论的理解，使其在正特征幂幺群的新背景下得到发展，而这一理论以前从未被研究过。它也将加强几何表示理论和模张量范畴理论之间的联系。

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.