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 $\Omega^{\widetilde{S}_2}_{\mathbf{a}}$ 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 $\bar{S}_2^{\geq 0}$. Then we classify all simple Whittaker $\bar{S}_2$-modules in every block $\Omega^{\bar{S}_2}_{\mathbf{a}}$ . Finally, we establish an equivalence between $\Omega^{\bar{S}_2}_{\mathbf{1}}$ and the category $H_{\mathbf{1}}$-fmod of finite-dimensional modules over an associative algebra $H_{\mathbf{1}}$, whose generators are also determined.
From: Genqiang Liu [view email]
[v1]
Tue, 28 Apr 2026 03:43:53 UTC (16 KB)
[v2]
Thu, 18 Jun 2026 03:00:25 UTC (16 KB)
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