Let $G$ be a compact, connected and simply connected Lie group. Let $T\subset G$ be a maximal torus and let $W$ be the corresponding Weyl group.

Then we have the diagonal action of $W$ on $T^{n}$ for $n\ge 0$. I would like to know if the cohomology groups $H^{*}(T^{n}/W;\mathbb{Z})$ have been computed or if anything is known about them.

The case $n=1$ is particularly simple as $T/W$ is contractible when $G$ simply connected. However, if $n>1$, one gets a complicated space. Any ideas?