You have $U_3$ acting on $W$ (presumably a vector space ove the complex numbers). If $w\in W$ is arbitrary, we can write
w=\frac13\left([w+\tau w+\tau^2w]+[w+\omega^2\tau w+\omega\tau^2w]+[w+\omega \tau w+\omega^2\tau^2w]\right),
because $1+\omega+\omega^2=0$. Here the vectors surrounded by square brackets are eigenvectors of $\tau$. The first belongs to eigenvalue $\lambda=1$, the second to $\lambda=\omega$ and the last to $\lambda=\omega^2$.
In other words $W$ decomposes into a direct sum of representations of $U_3$. These are all known to be 1-dimensional, and thus $\tau$ acts on any irreducible $U_3$-module by scalar multiplication. After you have learned a bit more representation theory, you will observe that you can replace $U_3$ with any finite abelian group. Some people would view this decomposition as discrete Fourier analysis. The representation theory of finite cyclic (actually all finite abelian) groups is surprisingly ubiquitous.