Character Sheaves for Unipotent Groups

主要研究方向为正特征域上幂偶群的几何表示理论和模张量范畴理论。他们定义并研究了单幂代数群上的特征簇的概念。利用这一概念，他们引入了群上有界可构一元复形的等变派生范畴的某些子范畴，称为“块”。主要研究人员对有限域上的单能群上的块的结构和它的点群上的不可约群的特征束之间的关系作了若干推测；他们的目标是证明这些猜想。其中一个猜想说，每个块都等价于某个模范畴的派生范畴，另一个猜想描述了所有可能以这种方式出现的模范畴。该研究的主题是现代数学和数学物理的几个领域的交叉点——几何表示理论、代数几何和共形场论。本研究将几何表征理论置于正特征单能群的新背景下，加深我们对几何表征理论的理解。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.