Whittaker modules over the loop-Virasoro algebra

In this paper, we study Whittaker modules over the loop-Virasoro algebra, which is the tensor product of the Virasoro algebra and the Laurent polynomial algebra, in terms of the natural triangular decomposition. We obtain the necessary and sufficient conditions for the Whittaker module over the loop-Virasoro algebra to be irreducible, as well as the necessary and sufficient conditions for the irreducible Whittaker module to be quasi-finite.