arXiv 1311.05512013-11-04math.RT

幂幺群上的模范畴、轨道方法和character层

Modular categories, orbit method and character sheaves on unipotent groups

Swarnendu Datta

摘要：

设$G$是特征为$p > 0$的域上的幂幺群. $G$上的特征层理论是由V. Drinfeld提出的，并与D. Boyarchenko他们还介绍了$\mathbb{L}$的概念-包的字符层。每个$\mathbb{L}$分组可以按照模块类别来描述。现在假设$G$的零阶类小于$p$。然后$\mathbb{L}$ -包与协伴轨道集$\mathfrak{g}^*/G$双射，其中$\mathfrak{g}$是使用Lazard对应从$G$获得的李环方案，而$\mathfrak{g}^*$是$G$的Serre对偶。如果$Ω$是余伴随轨道，那么相应的模范畴可以用$Ω$上的$G$ -等变局部系统范畴来标识。这反过来又等价于有限群的有限维表示范畴。然而，关联性、编织和带状约束是不平凡的。Drinfeld在2006年对这些限制进行了理论上的描述。本文证明了当$\dim(Ω)$为偶数时描述带状结构的公式。

英文摘要：

Let $G$ be a unipotent group over a field of characteristic $p > 0$. The theory of character sheaves on $G$ was initiated by V. Drinfeld and developed jointly with D. Boyarchenko. They also introduced the notion of $\mathbb{L}$-packets of character sheaves. Each $\mathbb{L}$-packet can be described in terms of a modular category. Now suppose that the nilpotence class of $G$ is less than $p$. Then the $\mathbb{L}$-packets are in bijection with the set $\mathfrak{g}^*/G$ of coadjoint orbits, where $\mathfrak{g}$ is the Lie ring scheme obtained from $G$ using the Lazard correspondence and $\mathfrak{g}^*$ is the Serre dual of $G$. If $Ω$ is a coadjoint orbit, then the corresponding modular category can be identified with the category of $G$-equivariant local systems on $Ω$. This in turn is equivalent to the category of finite dimensional representations of a finite group. However, the associativity, braiding and ribbon constraints are nontrivial. Drinfeld gave a conjectural description of these constraints in 2006. In this article, we prove the formula describing the ribbon structure when $\dim(Ω)$ is even.

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