  We classify all simple $W_n$-modules with finite-dimensional weight spaces.
Every such module is either of a highest weight type or is a quotient of a
module of tensor fields on a torus, which was conjectured by Eswara Rao. This
generalizes the classical result of Mathieu on simple weight modules for the
Virasoro algebra. In our proof of the classification we construct a functor
from the category of cuspidal $W_n$-modules to the category of $W_n$-modules
with a compatible action of the algebra of functions on a torus. We also
present a new identity for certain quadratic elements in the universal
enveloping algebra of $W_1$, which provides important information about
cuspidal $W_1$-modules.

Representation Theory