Integration over special unitary group It is known that for $SU(N)$ 
$$
\int \chi_{\mu_1}(UV_1)\chi_{\mu_2}(U^{-1}V_2)\, dU = \delta_{\mu_1\mu_2}\frac{\chi_{\mu_1}(V_1V_2)}{\dim(\mu_1)}
$$
where $dU$ is Haar measure on $SU(N)$ normalized such that with respect to it  $\operatorname{Vol}(SU(N))=1$; and $\chi_{\mu}(U)$ means the trace of $U$ in the irreducible representation $\mu$.
I came across this integral in "On quantum gauge theories in two dimensions", Edward Witten, Commun. Math. Phys. 141, 153-209 (1991).
My question is simple.
What else is known about integrals on $SU(N)$ of the form
$$ \int \chi_{\mu_1}(U^{\pm 1}V_1)\dots\chi_{\mu_n}(U^{\pm 1}V_n)\, dU \ ? $$
 A: The first formula go under the name the Schur orthogonality relations. There is a reason for this formula to be of such a nice form. It actually holds for any compact group. Could you explain why you want to consider it? 
$SU(1)$ is easy. For $SU(2)$, this involves integrals of the Chebyschev polynomials. If you can't find anything useful in the usual integral tables, there is no hope to find something for any other $n$.
Edit: As a second thought, the character $\chi_{\mu_1}\cdot \chi_{\mu_2}$ equals the character $\chi_{\mu_1 \otimes \mu_2}$, where $\mu_1 \otimes \mu_2$ is the tensor product of $SU(n)$-reps. I think that Fulton-Harris' book about representation theory explains you what you get for tensoring $SU(n)$-reps, i.e., how they decompose into irreducible things. So you have to clue together those $\chi$'s with $U$ and those with $U^{-1}$, and you can use the Schur orthogonality relations.
A: Do the Weingarten formulas help at all? See Theorem 2.5, and the surrounding discussions, in
http://arxiv.org/abs/0903.5143
