The cochains on G/G can be calculated as the Hochschild cochains of cochains on BG (this uses compactness of G - we'd get a kind of dual picture with Hochschild chains if we looked at free loops in a FINITE CW complex rather than BG). Now let me do something maybe evil and ignore gradings. Then we have a polynomial algebra, which is functions on the Chevalley vector space h/W. Its Hochschild cochains can be calculated as we would for any smooth affine variety as polyvector fields. Since the tangent bundle is trivial we get the tensor product of polynomials on h/W with the exterior algebra on the tangent space h/W. Now remembering the gradings we see precisely the cochains on BG tensor the cochains on G (this latter exterior algebra). (I'm assuming G is connected and simply connected just to be safe). I think this is all kosher algebraically - the crutch of using the HKR theorem for smooth affine varieties is just a way to avoid actually writing down the cyclic bar complex and calculating HH, but the result is the same..

The cochains on G/G can be calculated as the Hochschild cochains of cochains on BG (this uses compactness of G - we'd get a kind of dual picture with Hochschild chains if we looked at free loops in a FINITE CW complex rather than BG). Now let me do something maybe evil and ignore gradings. Then we have a polynomial algebra, which is functions on the Chevalley vector space h/W. Its Hochschild cochains can be calculated as we would for any smooth affine variety as polyvector fields. Since the tangent bundle is trivial we get the tensor product of polynomials on h/W with the exterior algebra on the tangent space h/W. Now remembering the gradings we see precisely the cochains on BG tensor the cochains on G (this latter exterior algebra). (I'm assuming G is connected and simply connected just to be safe). I think this is all kosher algebraically - the crutch of using the HKR theorem for smooth affine varieties is just a way to avoid actually writing down the cyclic bar complex and calculating HH, but the result is the same..

EDIT: This discussion (in particular the appeal to HKR) is of rational cochains.. not sure what happens with torsion. Also I'm ignoring completion issues (related to the ignored grading), don't know how fatal they are. It's probably better to think of cochains on G/G as Hochschild cochains of the algebra of chains on G under convolution, but I think the answer comes out the same.