标题： 从最小幂零有限$W$-代数的有限维模中得到的不可约尖点$\mathfrak{sl}_{n+1}$-模 显示英文标题

标题： Irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules from finite-dimensional modules over the minimal nilpotent finite $W$-algebra

摘要： 一个权重量子群$\mathfrak{sl}_{n+1}$-模如果它的权重量空间都是有限维的，并且$\mathfrak{sl}_{n+1}$的每个根向量在此模上都作用为单射，则称其为尖点模。在\cite{LL}中已经证明，任意带有尖点$\mathfrak{sl}_{n+1}$-模范畴广义中心特征标的块，均等价于$\mathfrak{sl}_{n+1}$对应的最小幂零有限$W$-代数$W(e)$上有限维模范畴中的一个块。 本文利用了$W(e)$的中心化子实现和一个显式的嵌入$W(e)\rightarrow U(\mathfrak{gl}_n)$，证明了每个有限维不可约$W(e)$-模都同构于某个有限维不可约$\mathfrak{gl}_n$-模的不可约$W(e)$-商模。 作为一个应用，我们可以利用有限维不可约的 $\mathfrak{gl}_n$-模给出了所有不可约尖点 $\mathfrak{sl}_{n+1}$-模的非常明确的实现，避免使用在 \cite{M}中引入的扭曲局部化方法和协调族。 显示英文摘要

摘要： A weight $\mathfrak{sl}_{n+1}$-module with finite-dimensional weight spaces is called a cuspidal module, if every root vector of $\mathfrak{sl}_{n+1}$ acts injectively on it. In \cite{LL}, it has been shown that any block with a generalized central character of the cuspidal $\mathfrak{sl}_{n+1}$-module category is equivalent to a block of the category of finite-dimensional modules over the minimal nilpotent finite $W$-algebra $W(e)$ for $\mathfrak{sl}_{n+1}$. In this paper, using a centralizer realization of $W(e)$ and an explicit embedding $W(e)\rightarrow U(\mathfrak{gl}_n)$, we show that every finite-dimensional irreducible $W(e)$-module is isomorphic to an irreducible $W(e)$-quotient module of some finite-dimensional irreducible $\mathfrak{gl}_n$-module. As an application, we can give very explicit realizations of all irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules using finite-dimensional irreducible $\mathfrak{gl}_n$-modules, avoiding using the twisted localization method and the coherent family introduced in \cite{M}.