$\DeclareMathOperator\End{End}$Over the past few months, I have taught myself the classification of reductive groups, and continued to non-abelian (as well as a small venture to non-compact) Harmonic Analysis.

I am now trying to put everything that I learn together into something coherent. Let's, therefore, take the case of compact real Lie groups. By Chevalley the category of compact real Lie groups is equivalent to the category of $\mathbb{R}$-anisotropic reductive linear algebraic groups whose connected components have $\mathbb{R}$-points. In particular, all of the representations of a compact real Lie group are algebraic.

Therefore, by the Theorem of the Highest Weight, we have a classification of the unitary dual (the set of irreducible unitary representations) of a real compact Lie group $G$.

From the Harmonic Analysis perspective: $$L^2(G)\cong\bigoplus_{\pi\in\hat{G}} \End(\pi)(\cong \bigoplus_{\pi\in\hat{G}} \pi\otimes\pi^*).$$ It remains, therefore, to choose a basis of each $\End(\pi)$, for each $\pi$ guaranteed by the Theorem of the Highest Weight.

I looked at the examples in Folland's book on Harmonic Analysis, and did not see any mention of the Theorem of the Highest Weight. This seemed to have been done completely ad hoc.

I was also told in an old question of mine to take a look at Zonal Spherical Functions, in the particular case that $K=1$. I must confess, I find Zonal Spherical Functions to be quite confusing. I suspect that the point is that these are the method by which Harish Chandra proved the Plancherel Theorem (the non-compact variant of Peter-Weyl), but it's not clear to me how to use this in practice.

But either way, I end up wondering: what is the point of the Theorem of the Highest Weight? Does it provide a basis for each $\End(\pi)$? If it does, then what is it? If it doesn't, and we end up using some other method for finding the basis for each $\End(\pi)$ (a method that appears to ad hoc show that we have exhausted all of the irreducible unitary representations), then what good is the Theorem of the Highest Weight?

To put it succinctly: what utility does the Theorem of the Highest Weight provide, and how, if at all, does it fit into the picture of finding a basis for each $\End(\pi)$?

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