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
$$ \int \chi\_{\mu\_1}(U^{\pm 1}V\_1)\dots\chi\_{\mu\_n}(U^{\pm 1}V\_n)\, dU \ ? $$



Integration over special unitary group
Do the Weingarten formulas help at all? See Theorem 2.5, and the surrounding discussions, in
<http://arxiv.org/abs/0903.5143>


