Supports of weight modules over Witt algebras

As the first step towards a classification of simple weight modules with finite dimensional weight spaces over Witt algebras Wn, we explicitly describe the supports of such modules. We also obtain some descriptions of the support of an arbitrary simple weight module over a ℤn-graded Lie algebra $\mathfrak{g}$ having a root space decomposition $\smash{\bigoplus_{\alpha\in\mathbb{Z}^n}\mathfrak{g}_\alpha}$ with respect to the abelian subalgebra $\mathfrak{g}_0$, with the property $\smash{[\mathfrak{g}_\alpha,\mathfrak{g}_\beta] = \mathfrak{g}_{\alpha+\beta}}$ for all α, β ∈ ℤn, α ≠ β (this class contains the algebra Wn).