论文标题
Offine类别O，Koszul二元性和Zuckerman函数
Affine category O, Koszul duality and Zuckerman functors
论文作者
论文摘要
抛物线类别$ \ MATHCAL {O} $ for Aggine $ {\ Mathfrak {\ Mathfrak {gl}} _ n $在级别$ -N-E $ at Level $ -n-e $ penters of $ \ wideTilde {\ m athfrak {sl}} _ e $的类别表示结构。此类别包含一个较小的类别$ \ mathbf {a} $，该类别对更高级别的Fock空间进行分类。我们证明，类别$ \ mathbf {a} $中的函数$ e $和$ f $是Zuckerman fuctors的Koszul Dual。 证明的关键点是证明类别$ \ mathbf {a} $在级别$ -n-e $上的函数$ f $可以根据级别$ \ mathbf {a a}的函数$ f $的组件来分解级别$ -n-e-n-e-e-e-1 $。为了证明这一点，我们使用以下事实：具有$ \ widetilde {\ mathfrak sl} _ {e+1} $的操作的类别包含一个（典型定义的）子类别，其动作$ \ didetilde {\ mathfrak sl}} _ {e} $。 我们还证明了一个一般性陈述，该声明说，在某些一般情况下，满足公理列表的函子自动对某种Zuckerman函数是双重的。
The parabolic category $\mathcal{O}$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of $\widetilde{\mathfrak{sl}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a smaller category $\mathbf{A}$ that categorifies the higher level Fock space. We prove that the functors $E$ and $F$ in the category $\mathbf{A}$ are Koszul dual to Zuckerman functors. The key point of the proof is to show that the functor $F$ for the category $\mathbf{A}$ at level $-N-e$ can be decomposed in terms of the components of the functor $F$ for the category $\mathbf{A}$ at level $-N-e-1$. To prove this, we use the following fact: a category with an action of $\widetilde{\mathfrak sl}_{e+1}$ contains a (canonically defined) subcategory with an action of $\widetilde{\mathfrak sl}_{e}$. We also prove a general statement that says that in some general situation a functor that satisfies a list of axioms is automatically Koszul dual to some sort of Zuckerman functor.
