Restricted modules over some ℤ-graded Lie algebras

In this paper, we answer a question asked by Kaiming Zhao at the mini course “Simple representations of the Virasoro algebra” at Xiamen mathematical center in the fall term of 2021. More precisely, we prove that if one special element in the Lie algebra g (including the Virasoro algebra, the Heisenberg-Virasoro algebra and affine Virasoro algebras) acts locally nilpotently on an irreducible module V, then V is a restricted g-module.