Whittaker modules for the derivation Lie algebra of torus with two variables

TL;DR

This paper classifies Whittaker modules for the derivation Lie algebra of a two-variable Laurent polynomial ring, providing explicit descriptions of vectors, submodules, and simple modules.

Contribution

It introduces a universal Whittaker module for this algebra and characterizes all its Whittaker vectors, submodules, and simple modules, advancing the understanding of its representation theory.

Findings

Explicit description of all Whittaker vectors.

Complete classification of submodules of the universal Whittaker module.

Identification of all simple Whittaker modules of the given type.

Abstract

Let $\mathcal{L}$ be the derivation Lie algebra of ${\mathbb C}[t_1^{\pm 1},t_2^{\pm 1}]$. Given a triangle decomposition $\mathcal{L} =\mathcal{L}^{+}\oplus\mathfrak{h}\oplus\mathcal{L}^{-}$, we define a nonsingular Lie algebra homomorphism $\psi:\mathcal{L}^{+}\rightarrow\mathbb{C}$ and the universal Whittaker $\mathcal{L}$-module $W_{\psi}$ of type $\psi$. We obtain all Whittaker vectors and submodules of $W_{\psi}$, and all simple Whittaker $\mathcal{L}$-modules of type $\psi$.