Submodule structures of $\mathbb{C}[s,t]$ over $W(0,b)$ and a new class of irreducible modules over the Virasoro algebra

For any $a,b\in \mathbb{C}$, $W(a,b)$ is the Lie algebra with basis $\{L_m,M_m\mathrm{\mid }m\in \mathbb{Z}\}$ and relations $[L_m,L_n]=(n-m)L_{m+n},$ $[L_m,W_n]=(a+n+bm)W_{m+n}$, $[W_m,W_n]=0$ for $m,n\in \mathbb{Z}$. For any $\lambda \in {\mathbb{C}}^{\ast },$ $\alpha \in \mathbb{C}$, $h:=h(t)\in \mathbb{C}[t]$, there exists a non-weight module over $W(0,b)$ (resp., $W(0,1)$), denoted by $\mathrm{\Phi }(\lambda ,\alpha ,h)$ (resp. $\mathrm{\Theta }(\lambda ,h)$), which is defined on the space $\mathbb{C}[s,t]$ of polynomials on variables $s,t$ and is free of rank one over the enveloping algebra $U(\mathbb{C}{L}_0\oplus \mathbb{C}{W}_0)$ of $\mathbb{C}{L}_0\oplus \mathbb{C}{W}_0$. In the present paper, by introducing two sequences of useful operators on $\mathbb{C}[s,t]$, we determine all submodules of $\mathbb{C}[s,t]$. We also study submodules of $\mathbb{C}[s,t]$ regarded as modules over the Virasoro algebra $\mathcal{V}\text{\hspace{0.17em}⁣}$ (with the trivial action of the center), and prove that these submodules are finitely generated if and only if $\mathrm{d}\mathrm{e}\mathrm{g}h(t)\ge 1$. In addition, it is proven that $\Phi(\lambda, \alpha,h)$isanirreducible$\mathscr V!$-moduleifandonlyif$b=-1$,${\rm deg}, h(t)=1$,$\alpha\neq0$. Finally, we obtain a large family of new irreducible modules over the Virasoro algebra $\mathcal{V}\text{\hspace{0.17em}⁣}$, by taking various tensor products of a finite number of irreducible modules $\mathrm{\Phi }({\lambda }_i,{\alpha }_i,h_i)$ for ${\lambda }_i,{\alpha }_i\in {\mathbb{C}}^{\ast },$ $h_i\in \mathbb{C}[t]$ with an irreducible $\mathcal{V}\text{\hspace{0.17em}⁣}$-module $V$, where $V$ satisfies that there exists a nonnegative integer $R_V$ such that $L_m$ acts locally finitely on $V$ for $m\ge R_V$.