Classification of simple Harish-Chandra modules of the Cartan type Lie algebra S¯2

Let ${\overline{S}}_2$ be the Lie algebra of vector fields on $C^2$ with constant divergence. In this paper, we classify all simple Harish-Chandra modules of ${\overline{S}}_2$ including bounded and non-bounded simple Harish-Chandra modules. In simple terms, any simple Harish-Chandra module of ${\overline{S}}_2$ is isomorphic to a tensor module $F(P,M)$ or its simple sub-quotient where P is a simple weight $D_2(:=C[t_1,t_2,\frac{\partial }{\partial t_1},\frac{\partial }{\partial t_2}\left]\right)$ module and M is a simple weight ${\mathrm{gl}}_2$ module.