TITLE: Formula for $U(N)$ integration wanted
QUESTION [11 upvotes]: 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, 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.

REPLY [6 votes]: 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