TITLE: Reference request: Coxeter length and irreducible characters
QUESTION [8 upvotes]: Let $S_n$ be the symmetric group on $\{1,2,\ldots, n\}$ and $\ell$ the Coxeter length on $S_n$. There is a well-known formula to compute this length, namely for a $\pi \in S_n$ we have
$$\ell(\pi) = |\{1 \leq i < j \leq n \mid \pi(i) > \pi(j)\}|.$$
Furthermore, we know that
$$\sum_{\pi \in S_n} sign(\pi) \ell(\pi) = 0$$
for all $n \geq 4$. So I was wondering: What about other characters or class functions? Given a class function $f \in Cl_{\mathbb{Q}}(S_n)$, what can we say about
$$\sum_{\pi \in S_n} f(\pi)\ell(\pi)\,\,\, ?$$
What about the case where $f$ is an irreducible character?
Given that the length is already known for a long time, I was rather sure that this problem has already been solved. However, I couldn't even find 
$$\sum_{\pi \in S_n} \ell(\pi) fix(\pi) = \frac{n!n(n-1)}{4} - \frac{(n+1)!}{6}$$
in the OEIS, where $fix(\pi)$ denotes the number of fixed points of $\pi$ (and yes, the two summands there are exactly the sum for the two irreducible characters that make up $fix$).
So, after this long introduction, my question is: Did someone already consider this question and am I just not able to find their work? Does anyone here know of a paper mentioning or even solving this problem, is there an application of such a solution in the study of the Coxeter length or other combinatorial functions? Can you suggest any paper, author, book,... where I might find something about this topic, is there maybe a standard textbook on combinatorial representation theory?

REPLY [4 votes]: A big thanks to Axel Hultman, who sent me an email to tell me that he basically answered my question in his 2014 paper Permutation statistics of products of random permutations (theorem 6.2).
Once again I'm really amazed by the fast and good feedback you get here on MathOverflow.
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