Just want to make sure I am not imagining things - are the inner products you describe are elements of the Gram matrix for the Shapovalov form? I think there are some elegant formulas for them in the affine case (there was some paper by David Hill a couple of years ago). Also, you might want to look at what they do when proving Kac formulas for Z(g)=S(h)^W where the Shapovalov form is used.

I guess your assumption about at least half a brain is wrong for the team that was responsible for that routine! :-/

@Igor: yes, that totally makes sense. What does not, is the case of $S_7$ - that gap still surprises me a lot, even though it can be thought of as a weird "small values effect".

@Igor: That's very interesting. Thanks a lot! I hope you don't mind me not accepting this answer for a while hoping that someone will also explain what's so special in numbers 5,6,8 for $S_n$...

@Kevin: while what you say is one of the reasons, it does not really explain the gap ($S_6$ and $S_8$ are not quotients, while $S_7$ is) ;-)

@Henrik: just a remark on your "of course one knows precisely" - I think it is true that it's possible to present these elements explicitly, but the most straightforward explanation is in a sense an existence theorem only (as it uses Bertrand's postulate), see the update.

Good to know :-)

Jason, you must be kidding. Out=Aut/Inn is the automorphism group of the Dynkin diagram.

Good, that looks convincing now. Thanks!

Pasha, I am actually confused by what you say after thinking it over. If we take the orthogonal complement of I^2, would not it contain all polynomials of degree 1 (not all of which are invariant)? I miss something obvious, I am sure, but I need someone to enlighten me.

I just have a brief remark of what you say in brackets about the Duflo isomorphism. The actual Duflo map is a bit different from your guess; see, e.g. the introduction to arxiv.org/abs/math/0506499 for a precise formulation. Also, remarkably it turns out that for metric Lie algebras the Duflo theorem is in a sense trivial - see arxiv.org/abs/0909.3743 (where the term "quadratic" is a synonym for "metric"). Sorry, these were random thoughts, nothing to say about the main question.

There is a slightly more elegant exposition (I think) of Zolotarev's proof written down by Prasolov (tinyurl.com/y9ltg5x - unfortunately, it might only be available in Russian...)