On Harish-Chandra modules of the Lie algebra arising from the $2$-Dimensional Torus

Abstract: Let $A=\mathbb{C}[t_1^{\pm1},t_2^{\pm1}]$ be the algebra of Laurent polynomials in two variables and $B$ be the set of skew derivations of $A$. Let $L$ be the universal central extension of the derived Lie subalgebra of the Lie algebra $A\rtimes B$. Set $\widetilde{L}=L\oplus\mathbb{C} d_1\oplus\mathbb{C} d_2$, where $d_1$, $d_2$ are two degree derivations. A Harish-Chandra module is defined as an irreducible weight module with finite dimensional weight spaces. In this paper, we prove that a Harish-Chandra module of the Lie algebra $\widetilde{L}$ is a uniformly bounded module or a generalized highest weight (GHW for short) module. Furthermore, we prove that the nonzero level Harish-Chandra modules of $\widetilde{L}$ are GHW modules. Finally, we classify all the GHW Harish-Chandra modules of $\widetilde{L}$.