Cohomology of $T^n/W$ for compact Lie groups 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? 
 A: In the case $G=\mathrm{U}(m)$, this is the character variety $\mathrm{Hom}(\mathbb{Z}^n, \mathrm{U}(m))/\mathrm{U}(m)$. 
See my paper here for general results on its topology (and similar results for other compact and complex reductive $G$).
In particular, the rational cohomology ring is worked out at the end.  So from this you can at least get the free part of the integral cohomology, as discussed in this answer.
Also see this paper for the computation of its fundamental group.
Examples:


*

*For $n=1$ and $G$ simply connected you always get closed ball.  See Remark 6.2 here.

*For $n=2$ and $G=\mathrm{SU}(2)$ you get a 2-sphere.  Here is a 3D print of it.  It is the boundary of $\mathrm{Hom}(F_2,\mathrm{SU}(2))/\mathrm{SU}(2)$.  See Example 3.14 here.

*For $n=3$ and $G=\mathrm{SU}(2)$ you get 3-dimensional
orbifold with 8 isolated singularities.  See the discussion on page 23 here for a more detailed discussion.  Also, there is this video giving a visualization of this orbifold.
A: The case n=2 is well studied and has beautiful anwser.  This quotient space is a weighted projective space (and for $SU(n)$ it is $CP^{n-1}$). See E. Looijenga, Root systems and elliptic curves, Invent. Math. 38 (1976), 17–32. And I.N. Bernshtein and O.V. Shvartsman, Chevalley’s theorem for complex crystallographic Cox-
eter groups, Funct. Anal. Appl. 12 (1978), 308–310.  As pointed out in Sean's answer these quotient spaces are moduli spaces of flat $G$ connections in the trivial $G$ bundle over the torus.  It is interesting to extend this story to non-trivial $G$ bundles. A wonderful book about some of what goes on "Almost commuting elements in compact Lie groups" by Borel, Friedman and Morgan.
