Whittaker modules over the loop Virasoro algebra

In this paper, we first study two classes of Whittaker modules over the loop Witt algebra ${\mathfrak g}:=\mathcal{W}\otimes\mathcal{A}$, where $\mathcal{W}=\text{Der}({\mathbb{C}}[t])$, $\mathcal{A}={\mathbb{C}}[t,t^{-1}]$. The necessary and sufficient conditions for these Whittaker modules being simple are determined. Furthermore, we study a family of Whittaker modules over the loop Virasoro algebra $\mathfrak{L}:=Vir\otimes\mathcal{A}$, where $Vir$ is the Virasoro algebra. The irreducibility criterion for these Whittaker modules are obtained. As an application, we give the irreducibility criterion for universal Whittaker modules of the affine Lie algebra $\widehat{\mathfrak{sl}_{2}}$.