Before you jump on the "duplicate" buttom, let me say that I do not want to hear about Weingarten calculus and I do not want to see a character of the symmetric group.


What I would like is a formula for the (normalized) Haar measure integral
$$
\int\_{U(N)} g\_{i\_1 j\_1}\cdots g\_{i\_n j\_n} {\bar{g}}\_{k\_1 l\_1}\cdots {\bar{g}}\_{k\_n l\_n}\ d\mu(g)
$$
of the form
$$
\left.\mathcal{D}\ g\_{i\_1 j\_1}\cdots g\_{i\_n j\_n} {\bar{g}}\_{k\_1 l\_1}\cdots {\bar{g}}\_{k\_n l\_n}\right|\_{g=\bar{g}=0}
$$
where $\mathcal{D}$ is an explicit constant coefficient differential operator of infinite order.
Of course, in this formula the $g$'s and $\bar{g}$'s are treated are $2N^2$ completely unrelated formal variables.


As an example of what I would like, in the case of $SU(N)$ and $\bar{g}$-free monomials
$$
\mathcal{D}=\sum\_{n=0}^{\infty} \frac{0!1!\cdots (N-1)!}{n!(n+1)!\cdots(n+N-1)!}\ ({\rm det}(\partial g))^n
$$
works. 


As per the "Additional remark" in my second answer to [this MO question](https://mathoverflow.net/questions/255492/how-to-constructively-combinatorially-prove-schur-weyl-duality), I had a vague recollection of seeing a math-physics paper with
such a formula, but maybe my memory is faulty. So I think it's better to ask the experts.



Formula for $U(N)$ integration wanted
To expand on my comments, this paper <https://arxiv.org/pdf/hep-th/9209083v2.pdf> by Shatashvili deals
with ``correlation functions'' of Haar unitary matrices of the form
$$
\int\_{U(N)}^{} d\mu(U) e\_{}^{tr(UAU\_{}^{-1}B)}
U\_{i\_1j\_1}^{}\bar U\_{k\_1\ell\_1}^{}\ldots 
U\_{i\_mj\_m}^{}\bar U\_{k\_m\ell\_m},
$$
and provides a certain combinatorial formula for these. Then setting $A=0$ would probably
recover what you're asking about. 


The same correlation functions (and an alternative formula for them) are also discussed in 
<https://arxiv.org/pdf/hep-th/0502041.pdf>


