I've been learning about some representation theory of compact Lie groups, and one of the $G$-averaging tricks has thrown me for a bit of a loop. Given a finite group $G$, I'm familiar with the trick of summing over the group to obtain a Hermitian inner product on the class functions: $$\langle \phi, \psi\rangle := \frac{1}{|G|} \sum_{g\in G} \phi(g)\overline{\psi(g)}$$ I'm trying to understand how to boost this idea to the compact Lie group setting. Let $|dg|$ be a Haar measure on a compact Lie group $G$. It makes sense that $$\langle \phi,\psi\rangle:= \frac{1}{\text{vol}(G)} \int_G \phi(g)\overline{\psi(g)} \ |dg|$$ is the desired generalization, since restricting to the case $\dim G=0$ recovers the above summation. However, I'm having trouble understanding how to make sense of the integral. The way I learned how to integrate on (compact) Lie groups is via densities: one simply pulls a density back to $\mathbb{R}^n$ via a coordinate chart and integrates it there (using the usual un-oriented integral). But the integrand here is complex-valued. Am I meant to interpret this integral by splitting the integrand into real and imaginary parts and integrating each separately? Or is there some sort of notion of a "complex density bundle" that I need to use?
Any help would be greatly appreciated!
Yes, the integral of a complex-value function is just given by integrating its real and imaginary parts separately. This is not special to Lie groups in any way, and is a standard convention whenever one speaks of any sort of integral in any context. (For instance, surely you have also seen integrals of complex-valued functions on $\mathbb{R}$!)
In your case, you could also define it using a "complex density bundle", which would be nothing but the complexification of the ordinary real density bundle. This isn't anything fancy, though: as a real bundle, it's just a direct sum of two copies of the real density bundle (one for the real part and one for the imaginary part), and then when you pull back to $\mathbb{R}^n$ and integrate, you are again just integrating the real and imaginary parts separately.