江苏师大70周年校庆系列学术讲座（二十七）

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 $\Omega_{\mathbf{a}}^W$.

刘根强，男，河南大学副教授、博士生导师，主要研究领域为李代数和结合代数的表示理论，在Transformation Groups，Bull. Lond. Math. Soc.,Israel J. Math.,J. Algebra等著名SCI杂志上发表学术论文20余篇。主持国家自然科学基金面上项目、青年项目等项目。

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