Jet modules for the vector field Lie algebras

For a commutative algebra $A$ over $\mathbb{C}$, let $\mathfrak{g}=\text{Der}(A)$. A module over  the smash product $A\# U(\mathfrak{g})$ is called a jet $\mathfrak{g}$-module, where $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$. In this talk, we talk about jet modules when $A=\mathbb{C}[t_1^{\pm 1},t_2]$. We show that  $A\#U(\mathfrak{g})\cong\mathcal{D}\otimes U(L)$, where $\mathcal{D}$ is the Weyl   algebra $\mathbb{C}[t_1^{\pm 1},t_2, \frac{\partial}{\partial t_1},\frac{\partial}{\partial t_2}]$, and $L$ is a Lie subalgebra of  $A\# U(\mathfrak{g})$ called the jet Lie algebra corresponding to $\mathfrak{g}$. Using a Lie algebra isomorphism $\theta:L \rightarrow \mathfrak{m}_{1,0}\Delta$,  where $\mathfrak{m}_{1,0}\Delta$ is the subalgebra of vector fields vanishing at the point $(1,0)$, we show that any irreducible finite dimensional  $L$-module is isomorphic to an irreducible $\gl_2$-module. As an application, we  give  tensor product realizations of  irreducible   jet modules over $\mathfrak{g}$ with uniformly bounded weight spaces。