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I am looking for families of invariant integrals $\int_G dg f(g)$ (where $dg$ is a Haar measure) over a semisimple Lie group that can be evaluated in closed form, together with references where I can find proofs.

I don't know yet what I'll need, but $f$ can be quite involved, so I want to study the techniques that are available for deriving such formulas. My primary interest is in $SU(n)$ for general $n$, but results on other groups are likely to be instructive, too.

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Integration with respect to the Haar measure on unitary, orthogonal and symplectic group will tell you pretty much all we know for integrals of polynomial functions. (For a more recent paper, see Elementary derivation of Weingarten functions of classical Lie groups.)

There is not much more. Regrettably, even the integral over U(N) of a rational function has no closed-form expression.

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