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

TL;DR

This paper classifies and establishes categorical equivalences for Whittaker modules over the Lie algebra of polynomial vector fields with constant divergence, linking them to modules over certain subalgebras and associative algebras.

Contribution

It provides a classification of simple Whittaker modules and proves categorical equivalences involving blocks of these modules over the Cartan type Lie algebra.

Findings

Each block of Whittaker modules is equivalent to modules over a parabolic subalgebra.

All simple Whittaker modules with finite-dimensional vectors are classified via gl_2-modules.

An equivalence between a category of Whittaker modules and modules over an associative algebra is established.

Abstract

Let $\bar{S}_2$ be the Lie algebra of polynomial vector fields on $A_2=\mathbb{C}[t_1,t_2]$ with constant divergence.In this paper, 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 finite-dimensional module category over the parabolic subalgebra $\bar{S}_2^{\geq 0}$. Then we classify all simple Whittaker $\bar{S}_2$-modules with finite-dimensional Whittaker vector spaces using $\mathfrak{gl}_2$-modules. 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 $H_{\mathbf{1}}$.