CCNU 代数学系列报告 (十二)：Whittaker category for the Lie algebra of polynomial vector fields

发布时间：2022-12-08 作者：浏览次数：

Speaker:	刘根强	DateTime:	2022年12月9日（周五）14:00-15：00
Brief Introduction to Speaker: 刘根强，河南大学教授，研究领域为仿射李代数，Virasoro代数，Witt代数及量子环面李代数的结构与表示理论，主持多项国家自然科学基金项目，在国际顶尖数学杂志上发表论文20余篇。		Place:	腾讯会议 : 318 1530 4736
Abstract:For any positive integer $n$, let $A_n=\mathbb{C}[t_1,\dots,t_n]$, $W_n=\text{Der}(A_n)$ and $\Delta_n=\text{Span}\{\frac{\partial}{\partial{t_1}},\dots,\frac{\partial}{\partial{t_n}}\}$. Then $(W_n, \Delta_n)$ is a Whittaker pair. A $W_n$-module $M$ on which $\Delta_n$ operates locally finite is called a Whittaker module. We show that each block $\Omega_{\mathbf{a}}^{\widetilde{W}}$ of the category of $(A_n,W_n)$-Whittaker modules with finite dimensional Whittaker vector spaces is equivalent to the category of finite dimensional modules over $L_n$, where $L_n$ is the Lie subalgebra of $W_n$ consisting of vector fields vanishing at the origin. As a corollary, we classify all simple non-singular Whittaker $W_n$-modules with finite dimensional Whittaker vector spaces using $\mathfrak{gl}_n$-modules. We also obtain an analogue of Skryabin's equivalence for the non-singular block $\Omega_{\mathbf{a}}^W$.