TITLE: Cohomology of $T^n/W$ for compact Lie groups
QUESTION [11 upvotes]: 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?

REPLY [4 votes]: 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.