The category of Whittaker modules over the Cartan Type Lie algebra ${\bar{S}}_2$

Abstract

The Lie algebra ${\bar{S}}_2$ of polynomial vector fields on ${\mathbb{C}}^2$ with constant divergence is an important Cartan type Lie algebra. In this paper, we study Whittaker ${\bar{S}}_2$-modules that are locally finite over $\text{span}\{\frac{\partial }{\partial t_1},\frac{\partial }{\partial t_2}\}$. We first show that each block ${\Omega }_{\mathbf{a}}^{{\tilde{S}}_2}$ of the category of $(A_2,{\bar{S}}_2)$-Whittaker modules with finite-dimensional Whittaker vector spaces is equivalent to the category of finite-dimensional modules over the parabolic subalgebra ${\bar{S}}_2^{\ge 0}$. Then we classify all simple Whittaker ${\bar{S}}_2$-modules in every block ${\Omega }_{\mathbf{a}}^{{\bar{S}}_2}$ . Finally, we establish an equivalence between ${\Omega }_1^{{\bar{S}}_2}$ and the category $H_1$-fmod of finite-dimensional modules over an associative algebra $H_1$, whose generators are also determined.